Fracture Lecture of Abaqus

Basic Concepts of Fracture Mechanics

Lecture 1

L1.2

Modeling Fracture and Failure with Abaqus

Overview

• Introduction

• Fracture Mechanisms

• Linear Elastic Fracture Mechanics

• Small Scale Yielding

• Energy Considerations

• The J-integral

• Nonlinear Fracture Mechanics

• Mixed-Mode Fracture

• Interfacial Fracture

• Creep Fracture

• Fatigue

L1.3


Overview

• This lecture is optional.

• It aims to introduce the necessary fracture mechanics concepts and

quantities that are relevant to the Abaqus functionality that is presented

in the subsequent lectures.

• If you are already familiar with these concepts, this lecture may be

omitted.

Introduction

L1.5


Introduction

• Fracture mechanics is the field of solid mechanics that deals with the

behavior of cracked bodies subjected to stresses and strains.

• These can arise from primary applied loads or secondary self-

equilibrating stress fields (e.g., residual stresses).

L1.6


Introduction

• Objective of fracture mechanics

• The objective of fracture mechanics is to characterize the local

deformation around a crack tip in terms of the asymptotic field around

the crack tip scaled by parameters that are a function of the loading and

global geometry.

Fracture Mechanisms

L1.8


Fracture Mechanisms

• For engineering materials, such as metals, there are

two primary modes of fracture: brittle and ductile.

• Brittle fracture

• Cracks spread very rapidly with little or no

plastic deformation.

• Cracks that initiate in a brittle material tend to

continue to grow and increase in size provided

the loading will cause crack growth.

• Ductile fracture

• Three stages: void nucleation, growth, and

coalescence.

• The crack moves slowly and is accompanied by

a large amount of plastic deformation.

• The crack typically will not grow unless the

applied load is increased.

L1.9


Fracture Mechanisms

• Brittle fracture in polycrystalline materials displays either cleavage

(transgranular) or intergranular fracture.

• This depends upon whether the grain boundaries are stronger or

weaker than the grains .

Cleavage fracture

L1.10


Fracture Mechanisms

• Ductile fracture has a dimpled, cup-and-cone fracture appearance .

• Ductile fracture surfaces have larger necking regions and an

overall rougher appearance than a brittle fracture surface.

L1.11


Fracture Mechanisms

• Fracture process zone

• The fracture process zone is the region around the crack tip where

dislocation motions, material damage, etc. occur.

• It is a region of nonlinear deformation.

• The fracture process zone size is characterized by

• a number of grain sizes for brittle fracture or

• either inclusion or second phase particle spacings for ductile

fracture.

• Different theories have been advanced to describe the fracture process

in order to develop predictive capabilities

• LEFM

• Cohesive zone models

• EPFM

• Etc.

Linear Elastic Fracture Mechanics

L1.13



• Fracture modes

• Linear Elastic Fracture Mechanics (LEFM)

considers three distinct fracture modes: Modes

I, II, and III

• These encompass all possible ways a crack

tip can deform.

• Mode I:

• The forces are perpendicular to the crack,

pulling the crack open.

• This is referred to as the opening mode.

L1.14



• Mode II:

• The forces are parallel to the crack.

• One force pushes the top half of the

crack back and the other pulls the

bottom half of the crack forward, both

along the same line.

• This creates a shear crack: the

crack slides along itself.

• This is referred to as the in-plane shear

mode.

• The forces do not cause out-of-

plane deformation.

L1.15



• Mode III:

• The forces are transverse to the crack.

• This causes the material to separate

and slide along itself, moving out of

its original plane

• This is referred to as the out-of-plane

shear mode.

• The objective of LEFM is to predict the critical

loads that will cause a crack to grow in a brittle

material.

L1.16



• Stress intensity factor

• For isotropic, linear elastic materials, LEFM characterizes the local

crack-tip stress field in the linear elastic (i.e., brittle) material using a

single parameter called the stress intensity factor K.

• K depends upon the applied stress, the size and placement of the

crack, as well as the geometry of the specimen.

• K is defined from the elastic stresses near the tip of a sharp crack

under remote loading (or residual stresses).

• K is used to predict the stress state ("stress intensity") near the tip

of a crack.

• When this stress state (i.e., K) becomes critical, a small crack

grows ("extends") and the material fails.

• This critical value is denoted KC and is known as the fracture

toughness (it is a material property; discussed further later).

L1.17



• Asymptotic crack tip solutions

• The stress and strain fields in the vicinity of the crack tip are expressed

in terms of asymptotic series of solutions around the crack tip.

• They are valid only is a small region near the crack tip.

• This size of this region is quantified by small scale yielding

assumptions (discussed later).

• The stress intensity factor is the parameter that relates the local

crack-tip fields with the global aspects of the problem.

L1.18


• The leading-order terms of the asymptotic solution are:

where

r is the distance from the crack tip,

= atan(x2/x1),

KI is the Mode I (opening) stress intensity factor,

KII is the Mode II (in-plane shear) stress intensity factor,

KIII is the Mode III (transverse shear) stress intensity factor, and the

fija define the angular variation of the stress for mode a.


( , ) ( ) ( ) ( )2 2 2

I II IIII II IIIij ij ij ij

K K Kr f f f

r r r

,

x1

x2 r

L1.19



• Crack-tip singularity

• The predicted stress state at the crack tip in a linear elastic (brittle)

material possesses a square-root singularity:

• In reality, the crack tip is surrounded by the fracture process zone

where plastic deformation and material damage occur.

• Inside this zone, the LEFM solution is not valid.

• Outside of this zone (i.e., sufficiently "far" from the fracture

process zone), the LEFM is accurate provided the

plastic/damage zone is “small enough.”

• This is called small-scale yielding (discussed further later).

1

r .

L1.20



• Some comments on fracture toughness

• Fracture toughness is strongly dependent on temperature.

• The brittle-ductile transition temperature range depends on the material.

• For many common metals it may lie within the reasonable operating

temperature range for the design, so the temperature dependence

of the fracture toughness must be considered.

Fra

ctu

re t

ou

gh

ness

Temperature

L1.21



• Experimentally, the fracture toughness KC is a function of specimen

thickness.

• Since plane strain gives the practical minimum value of KC …

• The plane strain value is usually the value that is determined

experimentally.

• However, if the application is fracture of thin sheets of material, KC

values somewhere between the plane stress and plane strain values

may be appropriate.

Fra

ctu

re t

ou

gh

ne

ss

Thickness →

KC

L1.22


• Aside from temperature and thickness, the fracture toughness is also a

function of the crack extension.

• The fracture toughness as a function of crack extension is called the

resistance curve (shown below).

• The resistance curve is used to predict crack growth stability.


Variation in fracture toughness

with crack growth is Kr(Da):

Kr(0)= KC

brittle

ductile

L1.23



• Crack growth and stability

• The condition for continued crack growth for a crack length a + Da is

• The condition for stable continued crack growth is

( )applied RK K a D .

applied R

load

K dK

a d a

D.

Small-Scale Yielding

L1.25



• Small-scale yielding (SSY) means the region of inelastic deformation at

the crack tip is contained well within the zone dominated by the LEFM

asymptotic solution.

• For LEFM to be valid, there must be an annular region around the

crack tip in which the asymptotic solution to the linear elasticity

problem gives a good approximation to the complete stress field.

K-dominated zone

Transition zone

Plastic zone

L1.26



• The size of the process zone and the plastic region must be

sufficiently small so that this is true. Typical shapes of plastic zones

follow:

plane strain plane stress

(diffuse)

plane stress

(Dugdale)

L1.27



• We can estimate the plastic zone size, rp, by setting 22 = 0 in the LEFM

asymptotic solution, where 0 is the yield stress. This gives (for Mode I)

• Since the tractions across the boundary of the plastic zone have no net

force or moments (St. Venant’s principle), the effect on the elastic field

surrounding the plastic zone decays rapidly with distance from the

boundary, becoming negligible at ~3rp.

• LEFM predicts infinite stress at the crack tip—obviously this is unrealistic.

• But we can use LEFM results if the region of inelastic deformation near

the crack tip is small enough that there is a finite zone outside this

region where the LEFM asymptotic solution is accurate.

2 2

0 0

1 1

2 6

I Ip

K Kr

.

L1.28


• If a is a characteristic dimension in the problem, such as remaining ligament

size or thickness or crack length, then, to have a finite zone rK in which the

K-field dominates, we need

or

• This is the limit on specimen size in ASTM Standard E-399 for a valid

KIC test.

• KIC is KC (the fracture toughness) in Mode I.

• The fracture toughness represents the critical value of K required

to initiate crack growth.

2

0

1/5 3

2

ICK p

Ka r r

2

0

2.5 ICKa

.

ASTM Standard for

validity of LEFM


L1.29



• For some typical metal materials rp is calculated by matching the yield

stress to the Mises stress of the K field and the minimum characteristic

length is calculated using the ASTM standard limit.

• For materials with high fracture toughness the size of the specimen

for a valid fracture test is very large.

MaterialT

(ºC)

0

(MPa)

KIC

(MN/m3/2)

rp

(mm)

Characteristic

dimension

(mm)

A061-T651 (Al) 20 269 33 5 38

A075-T651 (Al) 20 620 36 0.35 8.4

AISI 4340 (Steel) 0 1500 33 0.05 1.2

A533-B (Steel) 93 620 200 11 260

Energy Considerations

L1.31



• Energy principles play an important role in studying crack problems.

• This is motivated by the fact that crack propagation always involves

dissipation of energy. Sources of energy dissipation include:

• Surface energy, plastic dissipation, etc.

• By considering fracture from an energetic point of view, crack

growth criteria can be postulated in terms of energy release rates.

• This approach offers an alternative to the K-based fracture

criteria discussed earlier and reinforces the connection

between global and local fields in fracture problems.

• The energy release rate is a global parameter while the stress

intensity factor is a local crack-tip parameter.

L1.32



• The energy available to grow a crack

is defined as

where PE is the potential energy and

G is the Energy Release Rate.

• We consider the difference in the

energy for two essentially identical

specimens, one with crack length a,

the other with crack length a + Da.

• The area under the load-

displacement curve gives -PE for

elastic materials.

( )

Loads

PE

a

-

,G

L1.33



• For isotropic linear elastic materials, one can show that

and

• In a three-dimensional body under general loading that contains a crack

with a smoothly changing crack-tip line, the energy release rate

(assuming linear elasticity) per unit crack front length is

• Thus, we see the stress intensity factors are directly related to the

energy release rate associated with infinitesimal crack growth in an

isotropic linear elastic material.

221 v

KE

- for plane strainG

2K

E for plane stress.G

22 2 21 1

( )2

I II III

vK K K

E G

- .G

L1.34



• Initiation of crack growth in SSY

• The necessary condition for crack growth expressed in terms of the

energy release rate is G GC.

• GC is a material property and represents the energy per unit crack

advance going into:

• the formation of new surfaces,

• the fracture process, and

• plastic deformation.

• As noted earlier, for linear elastic materials, G and K are related.

• This leads to an alternative condition for K KC.

• Recall KC is the fracture toughness of the material.

The J-integral

L1.36


The J-integral

• The J-integral is used in rate-independent quasi-static fracture analysis

to characterize the energy release associated with crack growth.

• It can be related to the stress intensity factor if the material

response is linear.

• As will become apparent in the next section, it also has the

advantage that it provides a method for analyzing fracture in

nonlinear materials.

L1.37


• J is defined as follows:

• It is path independent when contours are taken around a crack tip.

• The definition of J assumes:

• The material is homogeneous in the crack direction.

• The material is elastic.

• For linear elastic materials, the value of J is equal to the energy

release rate associated with crack advance:

x1

x2

The J-integral

11

iij j

uJ Wn n ds

x

-

J G

L1.38


The J-integral

• J in small-scale yielding

• Choose , the contour for J, to fall entirely within the annular region in

which the K fields dominate.

• The integrand for J can be evaluated directly in terms of the (known) Kfields. Direct calculation for Mode I in a linear elastic material gives

22

2

1

1

I

I

vJ K

E

J KE

-

for plane strain and

for plane stress.

G

G

3rp

Nonlinear Fracture Mechanics

L1.40



• LEFM applies when the nonlinear deformation of the material is confined

to a small region near the crack tip.

• For brittle materials, it accurately establishes the criteria for failure.

• However, severe limitations arise when the region of the material

subject to plastic deformation before a crack propagates is not

negligible.

• Nonlinear fracture mechanics attempts to extend LEFM to consider

inelastic effects.

• The theory is sometimes called Elastic-Plastic Fracture Mechanics

(EPFM).

• However, the theory is not based on an elastic-plastic material

model, but rather a nonlinear elastic material.

L1.41



• Consider a material that has a power-law hardening form,

where 0 is the effective yield stress, e0 = 0 / E is the associated yield

strain, E is Young's modulus, and a and n are chosen to fit the stress-

strain data for the material.

0 0

ne

ae

,

n

L1.42



• For such a material, Hutchinson, Rice, and Rosengren (extended to mixed

mode loading by Shih) showed that the near-tip fields have the form

• Here is the displacement relative to the displacement of the crack

tip, . These fields are commonly referred to as the HRR crack-tip fields.

ˆi iu u-ˆiu

1

1

00 0

1

00 0

1

00 0

( )

( )

ˆ ( )

n

ij ijn

n

n

ij ijn

n

n

i i in

J

I r

J

I r

Ju u r u

I r

a e

e e e a e

ae a e

-

,

,

.

Loading parameter is J

L1.43



• Why not elastic-plastic?

• The HRR field assumes a nonlinear

elastic power law material:

• Under monotonic loading, this

nonlinear elastic material can be

matched to the behavior of an

elastic-plastic material whose

hardening behavior is accurately

modeled by a power law.

• Thus, evaluating J allows us to

characterize the strength of the

singularity in the crack-tip region in

an elastic-plastic material subjected

to monotonic loading.

0 0

ne

ae

L1.44



• In unloading situations, the HRR fields do not describe the state around

the crack tip, and hence J does not characterize the strength of the

stress state ahead of a crack tip for plastic materials. Use caution when:

• The loading is not monotonic and an incremental plasticity material

is used

• Crack growth occurs under monotonic loading (individual material

particles may unload even when the overall structure is being

loaded).

• The HRR solution:

• Gives the leading term in an asymptotic expansion of the

deformation around the crack tip for a power law material; and

• Does not take into account finite-strain effects.

L1.45



• Some comments on the HRR fields

• The HRR fields, thus, describe the near-tip crack fields in terms of J.

• J gives the strength of the near-tip singularity in any power-law material

(nonlinear elastic or plastic) solid

• Recall that in LEFM K plays this role in linear elastic materials.

• J-based fracture mechanics is applied in much the same way as LEFM.

• Crack growth initiates when J reaches a critical value: J JC .

• To apply the theory, must ensure conditions for J-dominance are

satisfied (discussed next).

L1.46



• J-dominance

• J-dominance refers to situations when J can be used as a method of

predicting fracture.

• In general, J is an adequate characterization when there exists a state of

high triaxial tension (high triaxiality) ahead of the crack tip.

• High triaxiality ahead of the crack tip leads to low fracture

toughness.

• Examples: states of small-scale and well-contained yielding (where

the plastic zone is surrounded by an elastic zone):

• Deeply notched bend specimen

c «dd

c

L1.47



• In some situations the crack-tip stress field does not exhibit high triaxiality.

• Example: large-scale yielding (the plastic zone extends to the free

boundaries of the body):

• Fully plastic flow of single-edge cracked specimens under tension

loading

• Shallow cracks under bending

• Center-cracked panel

• A two-parameter approach can be used to extend the fracture

characterization to such cases (discussed next).

L1.48



• Two-parameter fracture mechanics

• The Williams’ expansion of the Mode I stress field about a sharp crack in

a linear elastic body with respect to r, the distance from the crack tip, is

• The T-stress thus represents a stress parallel to the crack faces.

• The magnitude of the T-stress affects the size and shape of the

plastic zone and the region of tensile triaxiality ahead of the crack

tip.

• For positive T-stress, J-dominance exists and a single parameter Jcan be used for a fracture criterion.

• For negative T-stress, a two-parameter approach (J, T) is required

to characterize the stress fields.

1/21 1( , ) ( ) ( )

2

Iij ij i j

Kr f T O r

r

.

Mixed-Mode Fracture

L1.50


Mixed-Mode Fracture

• Under general loading almost all theories for the direction of crack growth

assume or predict that the continued crack growth will be with KII = 0.

• Can assume that macroscopic cracks growing with continuously

turning tangents will advance straight ahead, presumably under Mode

I conditions.

• The crack curvature will evolve in such a way as to maintain this in

response to the loading.

• If the loading changes such that the local crack-tip stress field

experiences a large change in local stress intensities, mixed-mode

fracture will occur.

L1.51


Mixed-Mode Fracture

• Different criteria for homogeneous,

isotropic linear elastic materials have

been proposed, including:

• The maximum tangential

stress criterion.

• The maximum energy release

rate criterion.

• The KII = 0 criterion.

• Although all three imply that

KII = 0 as the crack extends, they

predict slightly different angles for

crack initiation.

Comparison of predictions of crack

propagation direction for different

ratios of KII / KI

Interfacial Fracture

L1.53



• Many engineering applications involve bonded materials.

• Examples:

• adhesive joints;

• protective coatings;

• composite materials;

• etc.

• Engineers must be able to predict the strength of the bond.

• Interfacial fracture mechanics provides a method by which to do this.

• It extends LEFM to predict the behavior of cracks between two

linear elastic materials.

L1.54



• Once a crack has started to grow in an

isotropic, homogeneous material, it

generally does so in an opening mode;

that is, in Mode I.

• A crack lying on an interface can

kink off the interface and grow

under Mode I conditions, or it can

grow along the interface under

mixed mode conditions.

• Whether the crack kinks off the

interface or propagates along it is

frequently determined through energy

considerations.

L1.55



• If the crack kinks off the interface, the fact that there is an interface is

important only in how it influences the stress and strain fields.

• If the crack grows along the interface, it grows under mixed mode

conditions due to material asymmetry and possibly (though not

necessarily) under mixed remote loading conditions.

• In such situations the conditions for crack growth depend on the

interface properties. It is not sufficient to define crack initiation and

growth criterion based on the conventional fracture toughness, KC.

• Specifically KC = KC ().

• Toughness depends strongly on the mode mixity .

L1.56



• Asymptotic fields

• The asymptotic stress field for an interfacial crack between linear elastic

materials is given by

where K* = K1 iK2 is the complex stress intensity factor (i.e., it has real

and imaginary parts) and is a complex function of the angle

and material mismatch parameter e :

*

Re ( , )2

iij ij

Kr

r

e e

1 2 2 1

1 2 2 1

( 1) ( 1)1 1log ,

2 1 ( 1) ( 1)

3

1

3 4

e

- - --

-

-

where , and

for plane stress

for plane strain, axi, 3D

,ij e

L1.57



• The complex exponent rie indicates that the stresses will oscillate near

the crack tip:

• Both the stresses and crack opening displacements will oscillate wildly

as the crack tip is approached.

• At some distance ahead of the crack tip, the fields settle down.

• The fracture criterion should be measured at this point. Provided the

location of this point is the same in different specimens, a fracture

criterion is valid.

Creep Fracture

L1.59


Creep Fracture

• High-temperature fracture

• For temperatures above 0.3M (where M is the melting temperature on

an absolute scale), metals will typically creep.

• In plastics creep can occur even at room temperature.

• There are typically two mechanisms that are active in creep fracture:

• Blunting of the crack tip due to a relaxing stress field.

• This tends to retard crack growth.

• Accumulation of creep damage (microcracks, void growth, and

coalescence).

• This enhances crack growth.

• Steady-state creep crack growth occurs when the two effects balance

one another.

L1.60


Creep Fracture

• The stress state around a crack tip in a material that can creep is more

complicated than for the corresponding plasticity problem.

• Because of the time-dependent effects there is no one parameter that

can characterize the stress state around the crack tip for all

possibilities.

• This makes measuring the relevant parameters more difficult.

• Hence, creep fracture is not as well established as elastic-plastic

fracture.

Initially, the crack-tip field is the elastic field.

Stationary crack: around the

crack tip (RR field); around this field

(K field).

Growing crack: region develops where

(HR field), which is in turn surrounded by the RR

field. Eventually the HR field envelops the RR

field (which ultimately disappears).

( ) ( )cr elO O e e( ) ( )el crO O e e

( ) ( )el crO O e e

L1.61


Creep Fracture

• Contour integrals

• The contour integral for creep fracture is called the C(t)-integral.

• It plays an analogous role to the J-integral in the context of time-

dependent creep fracture.

• Its development assumes a power law creep material:

• The C(t)-integral is proportional to the rate of growth of the crack-tip

creep zone for a stationary crack under small-scale creep conditions:

• Under steady-state creep conditions, when creep dominates throughout

the specimen, C(t) becomes path independent and is known as C*.

0

11

( )1r

jij ij i ij

unC t n n ds

n x e

-

.

00

n

el cr

E

e e e e

L1.62


Creep Fracture

• Asymptotic fields for stationary crack

• The near tip stress and strain fields were obtained by Riedel and Rice in

terms of C(t). They are known as the RR fields and are analogous to the

HRR fields in power law hardening plasticity.

Here In is a function of n and the magnitude of is approximately

1.

1

1

00 0

1

00 0

( )( , )

( )( , )

n

ij ijn

n

ncrij ij

n

C tn

I r

C tn

I r

e

e e e e

( , )ij n

C(t) acts like a time-dependent

loading parameter

Crack tip fields are

similar to those for

an elastic-plastic

material

L1.63


Creep Fracture

• Small-scale vs. extensive creep

• For the case of no crack growth the

loading parameters that characterize the

crack-tip fields are reasonably well

understood.

• Under small-scale creep conditions

with secondary creep, K is the loading

parameter characterizing the crack-tip

field.

• For extensive secondary creep C* is

a loading parameter characterizing

the crack-tip field upon which a

fracture criterion may be based.

• Suitable criteria for crack extension that

will predict an initiation time for crack

growth for general cases are not yet

available.

( )K

r

creep

zone

Small-scale creep

Extensive creep

Fatigue

L1.65


Fatigue

• Fatigue is a special kind of failure in which cracks gradually grow under

a prolonged period of subcritical loading.

• It is the single most common cause of failure in metallic structures.

• The Paris Law can be used to predict crack growth as a function of

cycles (or time):

max min

( ) ,ndaC K

dN

K K K

D

D -

where

Damage at the ball grid array

(BGA) in a solder joint after

2700 thermal loading cycles

L1.66


Fatigue

• Abaqus offers a direct cyclic low-cycle fatigue capability based on the

Paris Law.

• Models progressive damage and failure both in bulk materials and

at material interfaces for a structure subjected to a sub-critical cyclic

loading.

• For more advanced fatigue analysis capabilities, consult www.safetechnology.com.

• fe-safe is a suite of fatigue analysis software that has a direct

interface to Abaqus.

Modeling Cracks

Lecture 2

L2.2


Overview

• Crack Modeling Overview

• Modeling Sharp Cracks in Two Dimensions

• Modeling Sharp Cracks in Three Dimensions

• Finite-Strain Analysis of Crack Tips

• Limitations Of 3D Swept Meshing For Fracture

• Modeling Cracks with Keyword Options

Crack Modeling Overview

L2.4



• A crack can be modeled as either

• Sharp

• Small-strain analysis

• Singular behavior at the crack tip

• Requires special attention

• In Abaqus, a sharp crack is modeled

using seam geometry

• Blunted

• Finite-strain analysis

• Non-singular behavior at crack tip

• In Abaqus, a blunted crack is modeled

using open geometry

• For example, a notch

L2.5



• Mesh refinement

• Crack tips cause stress concentrations.

• Stress and strain gradients are large as a crack tip is approached.

• The finite element mesh must be refined in the vicinity of the crack

tip to get accurate stresses and strains.

• The J-integral is an energy measure; for LEFM, accurate J values can

generally be obtained with surprisingly coarse meshes, even though the

local stress and strain fields are not very accurate.

• For plasticity or rubber elasticity, the crack-tip region has to be

modeled carefully to give accurate results.

L2.6



• The crack-tip singularity in small-strain analysis

• For mesh convergence in a small-strain analysis, the singularity at the

crack tip must be considered.

• J values are more accurate if some singularity is included in the

mesh at the crack tip than if no singularity is included.

• The stress and strain fields local to the crack tip will be modeled

more accurately if singularities are considered.

• In small-strain analysis, the strain singularity is:

• Linear elasticity r -½

• Perfect plasticity r -1

• Power-law hardening r -n/(n+1)

Modeling Sharp Cracks in Two

Dimensions

L2.8


Modeling Sharp Cracks in Two Dimensions

• In two dimensions…

• The crack is modeled as an internal edge

partition embedded (partially or wholly) inside

a face.

• This is called a seam crack

• The edge along the seam will have

duplicate nodes such that the elements

on the opposite sides of the edge will not

share nodes.

• Typically, the entire 2D part is filled with a

quad or quad-dominated mesh.

• At the crack tip, a ring of triangles are

inserted along with concentric layers of

structured quads.

• All triangles in the contour domains must

be represented as degenerated quads.

L2.9



• Example: Slanted crack in a plate

• In Abaqus/CAE a seam is defined by

through the Crack option underneath the

Special menu of the Interaction module.

• The seam will generate duplicate

nodes along the edge.

Seam

Create face partition to represent

the seam; assign a seam to the

partition.

L2.10



• To define the crack, you must specify

• Crack front and the crack-tip

• Normal to the crack plane or the

direction of crack advance

• The crack advance direction is

called the q vector.

The crack extension direction (q vector)

defines the direction in which the crack

would extend if it were growing.

It is used for contour integral

calculations.

Crack tip

same as

crack

front in

this case

Select the vertex at either

end as the crack front.

(Repeat for the other end.)

L2.11



• Other options for defining the crack front and crack tip

Crack front may be:

Vertex/Node

Edges/Element edges

Faces/Elements

Crack tip may be:

Vertex/Node

Geometric

Instances

Orphan

Mesh

Geometric

Instances

Orphan

Mesh

Crack tip for an

orphan mesh

Crack front for a

geometric instance

L2.12



• Example: crack on a symmetry plane

• If the crack is on a symmetry plane, you

do not need to define a seam.

• This feature can be used only for

Mode I fracture.

Crack tip

Crack normal

L2.13



• Modeling the crack-tip singularity with second-order quad elements

• To capture the singularity in an 8-node isoparametric element:

• Collapse one side (e.g., the side made up by nodes a, b, and c) so

that all three nodes have the same geometric location at the crack

tip.

• Move the midside nodes on the sides connected to the crack tip to

the ¼ point nearest the crack tip.

L2.14



• If nodes a, b, and c are free to move independently, then

everywhere in the collapsed element.

• If nodes a, b, and c are constrained to move together, A = 0:

• The strains and stresses are square-root singular (suitable for

linear elasticity).

• If nodes a, b, and c are free to move independently and the midside

nodes remain at the midsides, B = 0 :

• The singularity in strain is correct for the perfectly plastic case.

• For materials in between linear elastic and perfectly plastic (most metals),

it is better to have a stronger singularity than necessary.

• The numerics will force the coefficient of this singularity to be small.

0A B

rr r

as

L2.15



• Usage:

Quarter-point midside

nodes on the sides

connected to the crack tip

The crack tip nodes are

constrained: r -½ singularity

The crack tip nodes are

independent: r -1 singularity

1,1,2,3

2

13

1,2,3,4

3

1, 24

L2.16



• Aside: Controlling the position of midside nodes for orphan meshes

• Singularity controls cannot be applied to orphan meshes.

• Use the Mesh Edit tools to adjust their position.

L2.17



• If the side of the element is not collapsed but the midside nodes on the

sides of the element connected to the crack tip are moved to the ¼point:

• The strain is square root singular along the element edges but not in

the interior of the element.

• This is better than no singularity but not as good as the collapsed

element.

nodes moved to ¼ points

L2.18



• Angular resolution

• We need enough elements to resolve the angular dependence of the

strain field around the crack tip.

• Reasonable results are obtained for LEFM if typical elements

around the crack tip subtend angles in the range of 10 (accurate) to

22.5 (moderately accurate).

• Nonlinear material response usually requires finer meshes.

L2.19



• Modeling the crack-tip singularity with first-order quad elements

• Collapsing the side of a first-order quadrilateral element with

independent nodes on the collapsed side gives

0A

rr

as .

L2.20



• Example: Slanted crack in a plate

• To enable the creation of degenerate quads, you must create swept

meshable regions around the crack tips (using partitions) and specify a

quad-dominated mesh.

Quad-dominated mesh + swept

technique for the circular regions

surrounding the crack tips

CPE8R elements; typical nodal

connectivity shows repeated node at crack tip:

8, 8, 583, 588, 8, 1969, 1799, 1970

All crack-tip elements repeat node 8 in

this example (nodes are constrained).

Quadratic element type

assigned to part

Quarter-

point

nodes

24 elements around

crack tip: 15 angles

L2.21



• Example (cont’d):

Alternate meshes

• No degeneracy:

• Degenerate with

duplicate nodes:

With swept meshable region:

CPE6M elements at crack tip —

cannot be used for fracture

studies in Abaqus.

CPE8R elements at crack tip but no

repeated nodes:

1993, 1992, 583, 588, 2016, ...

Coincident nodes

located at crack tip

With arbitrary mesh,

singularity only along edges

connected to crack tip.

L2.22



• Example (cont’d): Deformed shape

Focused mesh; deformation

scale factor = 100

Arbitrary mesh;

deformation scale

factor = 100

Modeling Sharp Cracks in Three

Dimensions

L2.24


Modeling Sharp Cracks in Three Dimensions

• In three dimensions…

• The seam crack is modeled as a

face partition that is either partially

or totally embedded into a solid

body.

• This can be done by

partitioning or using a cut

(Boolean) operation.

• The face along the seam will have

duplicate nodes such that the

elements on the opposite sides of

the face will not share nodes.

• Wedge elements must be created

along the crack front.

• Generally, this will require

partitioning.

Penny-shaped seam

crack: Full modelQuarter model

Meshed modelWedge elements

L2.25



• Options for defining the crack front and crack line

Crack front may be:

Edges/Element edges

Faces/Element faces

Cells/Elements

Geometric

Instances

Crack line may be:

Edges/Element edges

Orphan

Mesh

Geometric

Instances

Orphan

Mesh

Crack line for an

orphan mesh

Crack front for a

geometric instance

L2.26



• Specifying the crack growth direction in three dimensions

• In 3D you can specify either the

• normal to the crack plane (only when the crack is planar)

or the

• virtual crack extension direction (the q vector).

• Only a single q vector can be defined for geometric instances.

• The implications of this will be discussed shortly.

L2.27



• Modeling the crack-tip singularity in three dimensions

• 20-node and 27-node bricks can be used with a collapsed face to create

singular fields.

C3D20(RH)

2 nodes collapsed to

the same location

crack line


the same location

midplane

edge plane

midside nodes

moved to ¼ points

L2.28



• On an edge plane (orthogonal to the

crack line):

Crack line

Double-edge notch specimen

(symmetry model)

0A B

rr r

as0

Ar

r as

Edge plane nodes

displace together

Edge plane nodes

displace independently

0B

rr

as

L2.29


• On a midplane for 20-node bricks:

• If the two nodes on the collapsed face at the midplane can displace

independently, r -1 at the midplane (i.e., element interior).

• If on each plane there is only one node along the crack line, no

singularity is represented within the element.

• In either case the interpolation is not the same on the midplane as

on an edge plane.

• This generally causes local oscillations in the J-integral values

along the crack line.


L2.30



• On a midplane for 27-node bricks with all the extra nodes on the

element faces:

C3D27(RH)


same location

midplane

edge plane

3 nodes collapsed to same

location

centroid

crack line

L2.31



• If all midface nodes and the centroid node are included and moved with

the midside nodes to the ¼ points, the singularity can be made the same

on the edge planes and midplane.

• Abaqus does not allow the centroid node to be moved from the

geometric centroid of the element.

• Therefore, the behavior at the midplane will never be the same as at

the edge planes.

• This usually causes some small oscillation of the crack fields along

the crack line.

• The midface node marked “A” is frequently omitted.

• This creates differences in interpolation between the midplane and

the edge planes and, hence, causes further oscillation in the crack-

tip fields.

• These oscillations are minor in most cases.

L2.32



• Example: Conical crack in a half-

space

• A conical crack in an infinite half-

space is considered.

• Only the aspects related to the

geometric modeling are

considered here.

• The results of this analysis

(J-integral values, etc) will

be considered in the next

lecture.

• The modeling procedure is

outlined next.

L2.33



Example (cont’d): Create the basic geometry

• Because of symmetry, only a quarter model is created

Large solid block (300 × 300 ×300)

used to represent the half-space. Conical shell of revolution (revolved 90º);

this will be used to cut the block.

1

a = 15

q = 45º

r = 10

L2.34



Example (cont’d): Merge the block and cone

• This will create the edges and surface

necessary to define the seam and the crack.

Instance and merge the

two parts to create a

new part. The instance

must be independent.

2

L2.35



Example (cont’d): Define the seam and the crack front/line

Only one q vector can be defined

for geometry. The q vectors will

be adjusted at the end of the

modeling process by editing an

orphan mesh.

3

L2.36



Example (cont’d): Partition the block for meshing

A small curved tube is centered

at the crack tip; this region is

meshed with a single layer of

wedge elements. This mesh is

swept along the length of the

tube.

The regions surrounding the

crack front are partitioned to

permit structured meshing.

4

L2.37



• Aside: Why is the small curved tube needed?

The swept meshing technique sweeps a

mesh through a cross section.

For the curved tube, this implies the

sweep direction is along its length. In

order for Abaqus to automatically create

a focused mesh at the crack tip,

however, it would need to sweep around

the circumference.

To overcome this, two concentric tubes

are used; the smaller one is meshed

with a single layer of wedge elements

(which is then swept along the length of

the tube).

If only a single curved tube was created

(shown at right), the mesh around the

crack tip would be arbitrary—not

focused (wedge elements not created).

L2.38



• Aside: What about the seam?

• After all the partitions are created for meshing purposes, the definition of

the seam remains intact.

Mesh seam

L2.39



Example (cont’d): Mesh the part

• Specify appropriate edge seeds to create

a focused mesh around the crack front

with minimal mesh distortion.

5

L2.40



Example (cont’d): Adjust the q vectors

• As noted earlier, only a single q vector

can be defined for geometry. As seen in

the figure, the vector that was defined is

only accurate at the left end of the crack

line.

• Individual q vectors can be defined on

an orphan mesh, however. Thus,

either…

• Create a mesh part (Mesh module)

or

• Write an input file and import the

model

• This approach has the

advantage that it preserves

attributes (sets, loads, etc).

To take advantage of the input file

approach, define a set that

contains the conical region before

writing the input file. Then you will

be able to easily create a display

group based on this set when

manipulating the orphan mesh.

6

L2.41



• For the orphan mesh, adjust each

vector individually

To redefine

this particular

vector, select

these nodes

as the start

and end points

of the vector.

L2.42



• For all elements, the singularities are modeled best if the element edges

are straight.

• In three dimensions the planes of the element perpendicular to the crack

line should be flat.

• If they are not, when the midside nodes are moved to the ¼ points,

the Jacobian of the element at some integration points may be

negative.

• One way to correct this is to move the midside nodes slightly away

from the ¼ points toward the midpoint.

L2.43



• Example: Conical crack model

Finite-Strain Analysis of Crack Tips

L2.45



• Finite-strain analyses:

• Singular elements should not be used (normally).

• The mesh must be sufficiently refined to model the very high strain

gradients around the crack tip if details in this region are required.

• Even if only the J-integral is required, the deformation around the

crack tip may dominate the solution and the crack-tip region will

have to be modeled with sufficient detail to avoid numerical

problems.

L2.46



• Physically, the crack tip is not perfectly sharp, and such modeling makes it

difficult to obtain results.

• Instead, we model the tip as a blunted notch, with a suggested radius

10-3rp.

• Here, rp is the size of the plastic zone (discussed in Lecture 1).

• The notch must be small enough that under the applied loads, the

deformed shape of the notch no longer depends on the original

geometry.

• Typically, the notch must blunt out to more than four times its

original radius for this to be true.

L2.47



• Geometric modeling of blunt cracks

• In 2D, the geometry of a blunted (or

open) crack is modeled as a cut

having a significant thickness.

• Meshing is done in the usual way.

• A very fine mesh is required at

the crack tip.

• This can be achieved by simply

assigning small element sizes to

the notch.

L2.48



• 3D open cracks can be created in

Abaqus/CAE in one of two ways:

• Adding a Cut feature in the

Part module.

• Subtracting a flaw from the

original part with a Boolean

operation in the Assembly

module.

• Hex meshing more difficult

due to irregular geometry.

• Creating a fine mesh at the

crack front generally requires

many partitions.

Quarter model Meshed model

Partitions to control mesh Refined mesh

Penny shaped open

crack: Full model

L2.49



• The size of the elements around the notch must be about 1/10th the

notch-tip radius.

SEN specimen

crack-tip mesh

rnotch

10% of rnotch

Biased edge seeds can

reduce the size of the mesh

by focusing small elements

towards the crack tip.

L2.50


• For J-integral evaluation, the region on the surface of the blunted notch

should be used to define the crack front.

• For the J- and Ct-integrals to be path independent, the crack surfaces

must be parallel to one another (or parallel to the symmetry plane).

• If this is not the case, Abaqus automatically generates normals on

the crack surface.

• If the notch radius shrinks to zero, all nodes that would be at the crack

tip should be included in the crack-tip node set.


Crack surface

is detected

automatically

The blunted notch

surface is the crack

front region Symmetry plane

Crack tip

region

q vector

L2.51



• If the mesh is so coarse that the integration points nearest the crack tip

are far from the tip, most of the details (accurate stresses and strains) of

the finite-strain region around the crack tip will be lost.

• However, accurate J values may still be obtained if cracks are

modeled as sharp.

L2.52



• Example: SEN specimen

Deformed shape

Undeformed

shape

Moderate blunting

Severe blunting

Contours of PEEQDeformed vs Undeformed Shapes

L2.53



• In situations involving finite rotations but small strains, such as the bending of slender structures, a small keyhole around the crack tip should be modeled.

• The region defining the crack front for the contour integral consists of the region on the keyhole.

• The elements should not be singular.

crack-front

region

Limitations Of 3D Swept Meshing For

Fracture

L2.55


Limitations Of 3D Swept Meshing For Fracture

• For curved regions cannot generate wedges at the center using a hex-

dominated approach and then sweep along the length of the region.

• This was discussed earlier in the context of the conical crack problem.

• To create a focused mesh in this case, embed a small tube within a

larger concentric tube. Mesh the smaller tube with a single layer of

wedge elements; the surrounding regions are meshed with hex

elements.

Sweep direction

L2.56



• Partition for a penny-shaped crack

• Illustrates the limitation that the path for the partition must be

perpendicular to its bounding surfaces; thus, cannot properly partition

along the arc of a circle as shown in this example:

Partition by sweeping

circular edge along arc

Cross-sectional

view of block

Tangent direction of arc

arc (not a semi-circle as

in previous example)

L2.57



• The workaround is to partition the face with circular arcs, and then

partition the cell using the n-sided patch technique.

n-sided patchFace partition

Note that the cross-sectional area of the swept

region is not constant along its length because

the tangents at the ends are not perpendicular

to the block (generalized sweep meshing)Resulting mesh around

the crack front using

wedge elements

Modeling Cracks with Keyword

Options

L2.59


Modeling Cracks with Keyword Options

• Defining a crack with keyword options:

• The *CONTOUR INTEGRAL option is used to define both, the crack itself and the fracture output, in an Abaqus input (.inp) file.

• In this section, we focus solely on the crack-specific parameters of this

option.

• These include:

*CONTOUR INTEGRAL, SYMM, NORMAL

• In the next lecture, we discuss the output-specific parameters of this

option.

• As noted earlier, the main requirements in defining a crack are:

• Defining the crack front

• Defining the crack extension direction

L2.60



• Crack symmetry

*CONTOUR INTEGRAL, SYMM

• The crack lies on a plane of

symmetry and only half the

structure is being modeled

• This feature should only be

used for Mode I problems.

L2.61



• Crack extension

*CONTOUR INTEGRAL, NORMAL

• The NORMAL parameter is used to

define the normal to the crack plane

when the crack is planar.

• Usage:

*contour integral, normal

nx, ny, nz

nodeSet1, nodeSet2, ...

• In this case, give a list of the node

set names defining the crack front

from one end to the other end, in

sequential order, without missing

any points on the crack line.

• In two-dimensional cases,

only one node set is needed.

These sets define the crack front;

the first node in each set defines

the crack tip node for that set.

(An optional CRACK TIP NODES

parameter is available to specify

the crack tip nodes directly).

L2.62



• Example: Penny-shaped crack in an infinite space

*Contour integral, symm, normal, ...

0.0, 1.0, 0.0

Crack-Front-1, Crack-Front-2, Crack-Front-3, ...

Crack-Front-1

L2.63



• If the NORMAL parameter is omitted, we must give the crack-tip node

set name, and the crack propagation direction q, at each node set

defining the crack front.

• Usage:

*contour integral, ...

nodeSet1, (qx)1, (qy)1, (qz)1nodeSet2, (qx)2, (qy)2, (qz)2

:

• Data must start with the node set at one end and be given for each

node set defining the crack line sequentially until the other end of

the crack is reached.

• The first node in each set is the crack tip node for that set

unless the CRACK TIP NODES parameter is used.

• This format allows nonplanar cracks to be analyzed.

L2.64



• Example: conical crack in an infinite

half-space

*Contour integral, ...

Crack-Front-1, 0.707107, -0.707107, 0.

Crack-Front-2, 0.705994, -0.707107, 0.0396478

Crack-Front-3, 0.702661, -0.707107, 0.0791708

Crack-Front-1

L2.65



• Generating a focused mesh with keyword options

• Example: DEN specimen

• The focused mesh shown in the figure will be generated with the

use of keyword options.

• The options include

*NODE

*NGEN

*NFILL

*ELEMENT

*ELGEN

L2.66



• Node definitions

*node

1, 0.0125, 0.0000

16001, 0.0125, 0.0000

101, 0.0250, 0.0000

4101, 0.0250, 0.0125

12101, 0.0000, 0.0125

16101, 0.0000, 0.0000

*ngen, nset=tip

1, 16001, 1000

*ngen, nset=outer

101, 4101, 1000

4101, 12101, 1000

12101, 16101, 1000

101

410112101

16101

tip

2101

8101

14101

Increment in

node number

Start

nodeEnd node

*NGEN generates nodes

incrementally between any two

previously defined nodes.

In this example, 17 crack-tip nodes are created (contained in the set tip);

the 17 nodes on the outer boundary are contained in set outer.

L2.67



• Quarter-point nodes

*nfill, singular=1

tip, outer, 10, 10

Start set:

first boundEnd set:

second bound

11 21

4021

31

8021

40112021

1021

*NFILL generate nodes for a region of a

mesh by filling in nodes between two

bounds.

In this example, 10 rows of nodes are generated between each tip node and its

corresponding outer node.

Number of

intervals between

bounding nodes

Node

number

increment

This parameter generates quarter-point nodes; the 1 indicates the first

bound represents the crack tip

L2.68



• Element definitions

*element, type=cps8r

1, 1, 21, 2021, 2001, 11, 1021, 2011, 1001

*elgen, elset=plate

1, 5, 20, 10, 8, 2000, 1000

1

11

21

1021

2021

*ELGEN generates elements

incrementally.

In this example, 5 elements form the

first row (extending radially outward

from the tip); a total of 8 rows of

elements (based on the first row) are

created around the crack tip.

1

Nodes 1, 1001,

and 2001 are

coincident

First row of

elementsTotal number of

rows

L2.69



• Crack-tip nodes

• If the crack-tip nodes are permitted to behave independently, the

strength of the strain-field singularity is r -1.

• The crack-tip nodes can be constrained using equations, multi-point

constraints, using repeated nodes in the element definition, etc. For

example, to constrain the crack-tip nodes with a multi-point

constraint:

*nset, nset=constrain, generate

1, 15001, 1000

*mpc

tie, constrain, 16001

• Only node 16001 is independent in this case.

• The strain-field singularity is r -½.

Fracture Analysis

Lecture 3

L3.2


Overview

• Calculation of Contour Integrals

• Examples

• Nodal Normals in Contour Integral Calculations

• J-Integrals at Multiple Crack Tips

• Through Cracks in Shells

• Mixed-Mode Fracture

• Material Discontinuities

• Numerical Calculations with Elastic-Plastic Materials

• Workshop 1

• Workshop 2

Calculation of Contour Integrals

L3.4



• Abaqus offers the evaluation of J-integral values, as well as several

other parameters for fracture mechanics studies. These include:

• The KI, KII, and KIII stress intensity factors, which are used mainly

in linear elastic fracture mechanics to measure the strength of local

crack tip fields;

• The T-stress in linear elastic calculations;

• The crack propagation direction: an angle at which a preexisting

crack will propagate; and

• The Ct-integral, which is used with time-dependent creep behavior.

• Output can be written to the output database (.odb), data (.dat), and

results (.fil) files.

L3.5



• Domain representation of J

• For reasons of accuracy, J is evaluated

using a domain integral.

• The domain integral is evaluated over

an area/volume contained within a

contour surrounding the crack tip/line.

• In two dimensions, Abaqus defines the

domain in terms of rings of elements

surrounding the crack tip.

• In three dimensions, Abaqus defines a

tubular surface around the crack line.

L3.6



• Different contours (domains) are

created automatically by Abaqus.

• The first contour consists of the

crack front and one layer of

elements surrounding it.

• Ring of elements from one

crack surface to the other (or

the symmetry plane).

• The next contour consists of the

ring of elements in contact with the

first contour as well as the

elements in the first contour.

• Each subsequent contour is

defined by adding the next ring of

elements in contact with the

previous contour.

Contour 1 Contour 2

Contour 3 Contour 4

L3.7



• The J-integral and the Ct-integral at steady-state creep should be

path (domain) independent.

• The value for the first contour is generally ignored.

• Examples of contour domains:

2nd

contour

1st

contour

Crack-tip node crack-front nodes

1st contour2nd contour

Crack-tip node

L3.8



• Usage:

*CONTOUR INTEGRAL, CONTOURS= n,

TYPE={J, C, T STRESS, K FACTORS},

DIRECTION = {MTS, MERR, KII0}

Note: In this lecture, we focus on the output-specific parameters of the *CONTOUR INTEGRAL

option. The crack-specific parameters SYMM and NORMAL were discussed in the previous lecture.

Specifies the number of contours (domains)

on which the contour integral will be

calculated

This is the output

frequency in

increments

L3.9



• Usage (cont’d):




• J for J-integral output,

• C for Ct-integral output.

• T STRESS to output T-stress

calculations

• K FACTORS for stress intensity

factor output

L3.10



• Usage (cont’d):




• Use with TYPE=K FACTORS to specify the criterion to be

used for estimating the crack propagation direction in

homogenous, isotropic, linear elastic materials:

• Maximum tangential stress criterion (MTS)

• Maximum energy release rate criterion (MERR)

• KII = 0 criterion (KII0)

Three criteria to calculate the crack

propagation direction at initiation

L3.11



• Output files

*CONTOUR INTEGRAL, OUTPUT

• Set OUTPUT=FILE to store the

contour integral values in the results (.fil) file.

• Set OUTPUT=BOTH to print

the values in the data and

results files.

• If the parameter is omitted, the

contour integral values will be printed in the data (.dat) file

but not stored in the results (.fil) file.

L3.12



• Loads

• Loads included in contour integral calculations:

• Thermal loads.

• Crack-face pressure and traction loads on continuum elements as well as those applied using user subroutines DLOAD and UTRACLOAD.

• Surface traction and crack-face edge loads on shell elements as well as those applied using user subroutine UTRACLOAD.

• Uniform and nonuniform body forces.

• Centrifugal loads on continuum and shell elements.

• Not all types of distributed loads (e.g., hydrostatic pressure and gravity

loads) are included in the contour integral calculations.

• The presence of these loads will result in a warning message.

L3.13



• Other loads not included in contour integral calculations:

• Contributions due to concentrated loads are not included.

• If needed, modify the mesh to include a small element and

apply a distributed load to the element.

• Contributions due to contact forces are not included.

• Initial stresses are not considered in the definition of contour

integrals.

Examples

L3.15


Examples

• Penny-shaped crack in an infinite space

• Model characteristics

• The mesh is extended far enough

from the crack tip so that the finite

boundaries will not influence the

crack-tip solution.

• The radius of the penny-shaped

crack is 1.

• Two types of loading are

considered:

• Uniform far-field loading

• Nonuniform loading on the

crack face: p = Arn.

L3.16


Examples

• Different mesh characteristics:

• Axisymmetric or three-dimensional

• Fine or coarse focused meshes

• With or without ¼ point elements

• Various element types used:

• First- and second-order

• With and without reduced integration

Axisymmetric model

20

20

Focused mesh around

crack tip

Crack tip

L3.17


Examples

• Fine mesh vs. coarse mesh (axisymmetric and 3D models)

~0.08

0.080.0004

The fine mesh is shown to the left;

the coarse mesh above. The length

perpendicular to crack line of the

crack-tip elements are indicated.

L3.18


Examples

• Axisymmetric model: geometry

Model geometry

Close up of crack tip region for

coarse mesh model (identical for

fine mesh model—only the inner

semicircular region is smaller)

Symmetry planes

L3.19


Examples

• Axisymmetric model: crack definition

Crack tip with extension direction

Set to 0.5 to use mid-

point rather than ¼ point

elements

L3.20


Examples

• 3D model: geometry and mesh

• A 90 sector is modeled because

of symmetry.

Additional partition

required for swept

mesh

On planes perpendicular to the crack

front, the mesh is very similar to the

axisymmetric mesh

In the circumferential direction around

the crack line, 12 elements are used.

Partitions used for coarse mesh model

(identical for fine mesh model—only

the inner semicircular region is smaller)

Fine 3D mesh

Symmetry planes

L3.21


Examples

• Why is the additional partition required?

• Without the additional partition, the region shown below would require

irregular elements at the vertex located on the axis of symmetry.

• This is not supported by Abaqus.

A 7-node element

is an example of an

irregular element.

Irregular elements

required here

because revolving

about a point

L3.22


Examples

• 3D model: crack definition

• Orphan mesh created to edit q

vectors.

L3.23


Examples

• Contour integral output requests (axisymmetric and 3D)

Separate output

requests are required

for J, K-factors, and the

T-stress.

L3.24


Examples

• Loads (axisymmetric and 3D)

The far-field load is suppressed.

L3.25


Examples

• Results

• MISES stress shown below for

the axisymmetric fine mesh.

Analytical Contour 1 Contour 2 Contour 3 Contour 4 Contour 5

5.796E-02 5.8169E-02 5.8095E-02 5.8121E-02 5.8104E-02 5.8084E-02

Contour 6 Contour 7 Contour 8 Contour 9 Contour 10

5.8064E-02 5.8044E-02 5.8024E-02 5.8005E-02 5.7985E-02

Deformation scale factor = 250

100%analytical numerical

analytical

J J

J

L3.26


Examples

J values from meshes with ¼ point elements (reduced integration)

• Abaqus values are based on the average of contours 3−5 in each mesh.

LoadingAnalytical

result

3-D Axisymmetric

C3D20R CAX8R

Coarse Fine Coarse Fine

Uniform

far field.0580 .0578 .0580 .0579 .0581

Uniform

crack face.0580 .0578 .0580 .0579 .0581

Nonuniform

crack face (n = 1).0358 .0356 .0357 .0356 .0358

Nonuniform

crack face (n = 2).0258 .0256 .0260 .0256 .0258

Nonuniform

crack face (n = 3).0201 .0199 .0206 .0200 .0202

L3.27


Examples

J values from meshes with ¼ point elements (full integration)


LoadingAnalytical

result

3-D Axisymmetric

C3D20 CAX8

Coarse Fine Coarse Fine

Uniform

far field.0580 .0577 .0572 .0578 .0580

Uniform

crack face.0580 .0577 .0572 .0578 .0580

Nonuniform

crack face (n = 1).0358 .0355 .0352 .0356 .0358

Nonuniform

crack face (n = 2).0258 .0255 .0253 .0255 .0258

Nonuniform

crack face (n = 3).0201 .0198 .0197 .0199 .0201

L3.28


Examples

J values from meshes without ¼ point elements (reduced integration)


LoadingAnalytical

result

3-D Axisymmetric

C3D20R C3D8R CAX8R CAX4R

Coarse Fine Coarse Coarse Fine Coarse

Uniform

far field.0580 .0574 .0580 .0563 .0574 .0581 .0562

Uniform

crack face.0580 .0574 .0580 .0563 .0574 .0581 .0562

Nonuniform

crack face (n = 1).0358 .0350 .0357 .0336 .0350 .0358 .0337

Nonuniform

crack face (n = 2).0258 .0250 .0260 .0234 .0250 .0258 .0236

Nonuniform

crack face (n = 3).0201 .0193 .0206 .0177 .0193 .0202 .0179

L3.29


Examples

J values from meshes without ¼ point elements (full integration)


LoadingAnalytical

result

3-D Axisymmetric

C3D20 C3D8 CAX8 CAX4

Coarse Fine Coarse Coarse Fine Coarse

Uniform

far field.0580 .0573 .0572 .0552 .0574 .0580 .0557

Uniform

crack face.0580 .0573 .0572 .0552 .0574 .0580 .0557

Nonuniform

crack face (n = 1).0358 .0350 .0352 .0329 .0350 .0358 .0333

Nonuniform

crack face (n = 2).0258 .0249 .0253 .0229 .0250 .0258 .0232

Nonuniform

crack face (n = 3).0201 .0193 .0197 .0172 .0193 .0201 .0175

L3.30


Examples

• Conclusions

• 3D fine meshes with second-order elements are more sensitive to the

choice of integration rule when determining J.

• The results are still very accurate (within 2% of analytical value).

• The inclusion of the singularity helps most in the coarser meshes.

• For mesh convergence in small strain, the singularity must be

included.

L3.31


Examples

• Conical crack in a half-space

• At each node set along the crack front, the crack propagation direction is

different.

L3.32


Examples

• Three-dimensional model

• Displaced shape and Mises stress distribution of full three-

dimensional model.

Deformation scale factor = 1.e6

L3.33


Examples

• J values of three-dimensional mesh

• There is some oscillation between J values evaluated at corner

nodes compared to J values evaluated at midside nodes.

Variation of J with angular position

1.328E-07

1.330E-07

1.332E-07

1.334E-07

1.336E-07

1.338E-07

0 45 90

Angle (degrees)

J-i

nte

gra

l 3D contour 5

3D contour 4

3D contour 3

3D contour 2

L3.34


Contours 3-5 have

converged

Examples

• Axisymmetric model and results

Axisymmetric results are

used as reference results.

L3.35


Examples

• Comparison of axisymmetric and 3D results


Contour 1

1.300E-07

1.320E-07

1.340E-07

1.360E-07

1.380E-07

0 45 90

Angle (degrees)

J-i

nte

gra

l

3D

AXI


Contour 2

1.329E-07

1.330E-07

1.331E-07

1.332E-07

1.333E-07

1.334E-07

0 45 90

Angle (degrees)

J-i

nte

gra

l

3D

AXI


Contour 3

1.328E-07

1.330E-07

1.332E-07

1.334E-07

1.336E-07

0 45 90

Angle (degrees)

J-i

nte

gra

l

3D

AXI


Contour 5

1.328E-07

1.330E-07

1.332E-07

1.334E-07

1.336E-07

1.338E-07

0 45 90

Angle (degrees)

J-i

nte

gra

l

3D

AXI

L3.36


Examples

• Since the three-dimensional mesh is quite coarse around the axis of

symmetry, these results are considered to be good—the error is less

than 0.5% for all but the first contour.

% difference in J between AXI and 3D results

0.00.51.01.5

2.02.53.03.5

0 45 90

Angle (degrees)

% d

iffe

ren

ce Contour 1

Contour 2

Contour 3

Contour 4

Contour 5

L3.37


Examples

• Submodeling

• We can use submodeling to create

two meshes that are significantly

smaller than the full three-

dimensional model.

• The top-right figure is the

coarse mesh global model in

the vicinity of the crack.

• The bottom-right figure shows

the refined submodel mesh

overlaid on the global model

mesh.

L3.38


Examples

• J values of submodel:

• Inaccuracies are introduced

by the coarser mesh used in

the global model.

• Errors in J are less than 1%.

• CPU time was reduced by a

factor of 3.


1.318E-07

1.320E-07

1.322E-07

1.324E-07

1.326E-07

0 45 90

Angle (degrees)

J-i

nte

gra

l 3D contour 5

3D contour 4

3D contour 3

3D contour 2


Contour 5

1.315E-07

1.320E-07

1.325E-07

1.330E-07

1.335E-07

0 45 90

Angle (degrees)

J-i

nte

gra

l

3D

AXI

% difference in J between AXI and 3D results

0.00.51.01.52.02.53.03.54.04.5

0 45 90

Angle (degrees)

% d

iffe

ren

ce Contour 1

Contour 2

Contour 3

Contour 4

Contour 5

L3.39


Examples

• Compact Tension Specimen

• This is one of five standardized specimens defined by the ASTM for the

characterization of fracture initiation and crack growth.

• The ASTM standardized testing apparatus uses a clevis and a pin to

hold the specimen and apply a controlled displacement.

L3.40


Examples

• Model details

• Plane strain conditions assumed.

• The initial crack length is 5 mm.

• Elastic-plastic material

• Low alloy ferritic steel

Crack seam

q-vector

1/√r singularity modeled in

the crack-tip elements

Prescribed load line displacement

L3.41


Examples

• Results

Small strain analysis Finite strain analysis

L3.42


Examples

At small to moderate strain levels,

the small and finite strain models

yield similar results.

Finite strain effects must be

considered to represent this level of

deformation and strain accurately.

Nodal Normals in Contour Integral

Calculations

L3.44


Nodal Normals in Contour Integral Calculations

• Sharp curved cracks

• For sharp cracks, if the crack faces

are curved, Abaqus automatically

determines the normal directions of

the nodes on the portions of the crack

faces that lie within the contour

integral domains.

• This improves the accuracy of the

contour integral estimation.

• The normal is not used at the

crack-tip node, however.

Normals to top crack

surface nodes

n (normal to

crack plane)

Normals to bottom

crack surface nodes

q

L3.45



• Example: sharp curved crack

Contour # 1 2 3 4 5

J without normals 3.363 2.980 2.475 1.888 1.283

J with normals 3.600 3.602 3.605 3.605 3.605

L3.46



• Blunt cracks and notches

• All nodes on the notch should be included in the crack-tip node set.

• The J-integral results are more accurate since the q vector is

parallel to the crack surface in this case, as illustrated below.

Crack surfaceCrack surface

Single node in crack-tip node set;

normals calculated on nodes of

blunted surface; q not parallel to

crack surface.

All nodes on blunted surface in

crack-tip node set; q parallel to

crack surface.

Paths for contour

integrals

n

q q

J-Integrals at Multiple Crack Tips

L3.48


J-Integrals at Multiple Crack Tips

• Abaqus can calculate J (or Ct) at multiple crack tips

• Abaqus/CAE: multiple crack tips and history

output requests

• Input file: repeated use of the *CONTOUR

INTEGRAL option.

• If the domain for one crack tip envelopes the other

crack tip, the J value will go to zero (as it should).

Through Cracks in Shells

L3.50



• Second-order quadrilateral shell elements must be used if contour

integral output is requested.

• Sides of S8R elements should not be collapsed. If a focused mesh is

used, the crack tip must be modeled as a keyhole whose radius is small

compared to the other dimensions measured in the plane of the shell.

Crack-tip mesh for S8R elementsShell mesh

L3.51


• S8R5 elements can be collapsed and midside nodes moved to the 1/4 points.

• The q vector must lie in the shell surface.

• It should be tangent to the surface.


Crack-tip mesh for S8R5 elementsShell mesh

L3.52



• Example: Circumferential through crack under axial load

• Mean radius R = 10.5 in

• Wall thickness t = 0.525 in

• Crack half-angle q = p / 4

• Longitudinal membrane stress = 100 psi

L3.53



• Model details

• Axial load is applied using

a shell edge load

• Symmetry used to reduce

mode size

Edge loads

symmetry

L3.54



• Modeling a crack with a keyhole

Crack tip

Crack front

q vector

L3.55



• Results

Deformed shape—axial loading

J values—axial loading

L3.56



• In shell element meshes, mechanical loads which act normal to the shell

surface and are applied within the contour integral domain are not taken

into account in the calculation of the contour integral.

• For example, pressure loads are not considered because they act

normal to the shell surface

• Conversely, axial edge loads are considered because they act in

the shell surface.

• Two workarounds exist:

• Run successive shell models with differing crack lengths and

numerically differentiate the potential energy

• Use solid elements (if the response is membrane dominated)

L3.57



• Using numerical differentiation to obtain J:

• The PE values should be obtained from two separate analyses, with

crack lengths differing by Da.

• The values of PE in the Abaqus data (.dat) file are generally not

printed to a sufficient number of figures to be useful for this calculation and must be read from the results (.fil) file.

• A similar technique can be used to get Ct at long times.

Constant Load

Constant Load

( )

a a a

PEJ

a

PE PE

a

D

=

=

D.

Potential energy:

PE = ALLSE ALLWK

L3.58



• Using solid elements:

• If membrane deformation is dominant, the shell can be modeled

with a single layer of 20-node bricks since these solid elements

include loading contributions to contour integrals.

L3.59



• To obtain accurate values of J through the shell thickness with solid

elements, more than one element should be used in the thickness

direction.

J values will show significant path dependence unless

averaged.

• If only one element is used through the thickness, the values can be

averaged by thinking of J as a force per unit length:

• The average is calculated as if the J values were equivalent

nodal forces:

4

6

A B C

shell

J J JJ

= .

CB

A

L3.60



• Aside: Generating a solid element mesh from a shell mesh.

• A shell mesh can easily be converted to a solid one using the ―Offset

Mesh‖ tool.

• Creates solid layers from a shell mesh.

L3.61



• Example: Circumferential through crack in

an internally pressurized, closed-end pipe

• The same pipe discussed earlier, now

subjected to 10 psi internal pressure +

axial load (which simulates the closed

end).

• Comparison of J values using one layer

of C3D20R elements through the

thickness :

CONTOURJ values 100

1 2 3 4 5

At Node A 2.0965 2.1317 2.1505 2.1557 2.1697

At Node B 3.7396 3.6992 3.7004 3.6968 3.6904

At Node C 5.0226 5.0501 5.0813 5.1471 5.2373

Averaged 3.6796 3.6631 3.6722 3.6817 3.6948

CB

A

L3.62



• Example: Circumferential through crack under axial load revisited

• Now we revisit the problem in which the pipe is subjected to an axial

load.

• Comparison of J values using one layer of C3D20R elements through

the thickness:

CONTOURJ values 100

1 2 3 4 5

At Node A 2.2122 2.2524 2.2700 2.2740 2.2850

At Node B 3.7629 3.7202 3.7212 3.7184 3.7136

At Node C 4.9560 4.9893 5.0175 5.0737 5.1492

Averaged 3.7033 3.6871 3.6954 3.7036 3.7148

Analytical 3.7181

L3.63



• Comparing these results with the

shell element results presented

earlier:

• Errors with respect to the

analytical solution for the 3D

model are less than 1%.

• Much closer agreement because

transverse shear effects are

considered in the 3D model.

• Only in-plane stress and strain

terms are included in the Abaqus

J calculations for shells.

• Transverse shear terms are

neglected.

Mixed-Mode Fracture

L3.65


Mixed-Mode Fracture

• Abaqus uses interaction integrals to

compute the stress intensity factors.

• This approach accounts for

mixed-mode loading effects.

• Note that the J- or Ct-integrals

do not distinguish between

modes of loading.

• Usage:

*CONTOUR INTEGRAL,

TYPE=K FACTORS

• Stress intensity factors can

only be calculated for linear

elastic materials.

L3.66


Mixed-Mode Fracture

0K a p=

Element

type

22.5º CPE8 0.185 (2.9%) 0.403 (0.2%)

22.5º CPE8R 0.185 (2.9%) 0.403 (0.2%)

67.5º CPE8 1.052 (3.6%) 0.373 (1.0%)

67.5º CPE8R 1.053 (3.8%) 0.374 (1.3%)

22.5 = 67.5 =

• Example: Center slant cracked plate under tension

*Values enclosed in parentheses are

percentage differences with respect to

the reference solution. See Abaqus

Benchmark Problem 4.7.4 for more

information.

*

Material Discontinuities

L3.68



• The J-integral will be path independent if the material is homogeneous in

the direction of crack propagation in the domain used for the contour

integral calculation.

• If there is material discontinuity ahead of the crack in this region, the

*NORMAL option can be used to correct the calculation of J so that

it will still be path independent.

• The normal to the material discontinuity line must

be specified for all nodes on the material

discontinuity that will lie in a contour integral domain.

n

L3.69



• Example: J-integral analysis of a two material plate

• As an example, the figure shows a single-edge

notch specimen made from two materials in

which the material interface runs at an angle to

the sides of the specimen.

• The material containing the crack (left) has a

Young’s modulus of 2 105 MPa and a

Poisson’s ratio of 0.3.

• The uncracked material (right) has Young’s

modulus of 2 104 MPa and a Poisson’s ratio

of 0.1.

• The specimen is stretched by uniform

displacement at its ends.

L3.70



• J-integral analysis of a two material plate (cont’d)

• Along the material discontinuity, the normal to

the discontinuity is given using the *NORMAL

option.

• The normal needs to be defined on both

sides of the discontinuity.

*NORMAL

LEFT, NORM, 1.0, 0.125, 0.0

RIGHT, NORM, -1.0, -0.125, 0.0

L3.71



• The calculated J-integral values for 10 contours are as follows:

• The need for the normals on the interface (contours 5–10) is clear.

ContourJ (N/mm)

Without normals With normals

1 55681 55681

2 57085 57085

3 57052 57052

4 57058 57058

5 35188 57116

6 31380 57114

7 27536 57114

8 23512 57113

9 19172 57116

10 14181 57094

Numerical Calculations with

Elastic-Plastic Materials

L3.73


Numerical Calculations with Elastic-Plastic Materials

• For Mises plasticity the plastic deformation is incompressible.

• The rate of total deformation becomes incompressible (constant

volume) as the plastic deformation starts to dominate the response.

• All Abaqus quadrilateral and brick elements suitable for use in J-integral

calculations can handle this rate incompressibility condition except for

the ―fully‖ integrated quadrilaterals and brick elements without the

―hybrid‖ formulation.

• Do not use CPE8, CAX8, C3D20 elements with these materials.

They will ―lock‖ (become overconstrained) as the material becomes

more incompressible.

L3.74



• Second-order elements with reduced integration (CPE8R,

C3D20R, etc.) work best for stress concentration problems in

general and for crack tips in particular.

• If the displaced shape plot shows a regular pattern of deformation,

this state is an indication of mesh locking.

• Locking can be seen in quilt contour plots of hydrostatic

pressure for first-order elements—the pressure shows a

checkerboard pattern.

• Change to reduced integration elements if you are using fully

integrated elements.

• Increase the mesh density if you already using reduced

integration elements.

• If these steps do not help, use hybrid elements.

• Hybrid elements must be used for fully incompressible materials (such

as hyperelasticity, linear elasticity with n = 0.5).

L3.75



• Results with elastic-plastic materials (and nonlinear materials in general)

are more sensitive to meshing than for small-strain linear elasticity.

• Meshes adequate for linear elasticity may have to be refined.

• The more complex the solution, the more J values tend to be path

dependent.

• A lack of path dependence can be an indication of a lack of mesh

convergence; however, path independence of J does not prove

mesh convergence.

Workshop 1

L3.77


Workshop 1

• Crack in a three-point bend specimen

• Two-dimensional geometry

• Mesh sensitivity study

• Focus vs. unfocused mesh

• Quarter-point vs. mid-side nodes

Workshop 2

L3.79


Workshop 2

• Crack in a helicopter airframe component

• Three-dimensional geometry

• Create mesh and evaluate response for cracks at different locations

Material Failure and Wear

Lecture 4

L4.2


Overview

• Progressive Damage and Failure

• Damage Initiation for Ductile Metals

• Damage Evolution

• Element Removal

• Damage in Fiber-Reinforced Composite Materials

• Failure in Fasteners

• Material Wear and Ablation

Progressive Damage and Failure

L4.4



• Abaqus offers a general capability for modeling progressive damage

and failure in engineering structures

• Material failure refers to the complete loss of load carrying capacity that

results from progressive degradation of the material stiffness.

• Stiffness degradation is modeled using damage mechanics.

• Progressive damage and failure can be modeled for:

• Ductile materials

• Continuum constitutive behavior

• Fiber-reinforced composites

• Interface materials

• Cohesive elements with a traction-separation law

• Damage and failure of cohesive elements are discussed in the next

lecture.

L4.5



• Two distinct types of ductile material

failure can be modeled with Abaqus

• Ductile fracture of metals

• Void nucleation, coalescence, and

growth

• Shear band localization

• Necking instability in sheet-metal

forming

• Forming Limit Diagrams

• Marciniak-Kuczynski (M-K) criterion

• Damage in sheet metals is not

discussed further in this seminar.

L4.6


Multiple damage definitions are allowed

Keywords

*MATERIAL

*ELASTIC

*PLASTIC

*DAMAGE INITIATION,CRITERION=criterion

*DAMAGE EVOLUTION

*SECTION CONTROLS, ELEMENT DELETION=YES


• Components of material definition

• Undamaged constitutive

behavior (e.g., elastic-plastic

with hardening)

• Damage initiation (point A)

• Damage evolution (path A–B)

• Choice of element removal

(point B)

A

B

Undamaged response

Damaged

response

Typical material response showing

progressive damage

Damage Initiation Criteria for

Ductile Metals

L4.8


Damage Initiation Criteria for Ductile Metals

• Damage initiation defines the point of initiation of degradation of stiffness

• It is based on user-specified criteria

• Ductile or shear

• It does not actually lead to damage unless damage evolution is also specified

• Output variables associated with each criterion

• Useful for evaluating the severity of current deformation state

• Output

DMICRT

Ductile Shear

Different damage initiation criteria on

an aluminum double-chamber profile

DMICRT > 1 indicates

damage has initiated

L4.9


• Ductile criterion:

• Appropriate for triggering damage

due to nucleation, growth, and

coalescence of voids

• The model assumes that the

equivalent plastic strain at the onset

of damage is a function of stress

triaxiality and strain rate.

• Stress triaxiality h = - p / q

• The ductile criterion can be used with

the Mises, Johnson-Cook, Hill, and

Drucker-Prager plasticity models,

including equation of state.Ductile criterion for Aluminum Alloy AA7108.50-T6

(Courtesy of BMW)

Pressure stress

Mises stress


L4.10



• Usage:

• Specify the equivalent plastic strain at the onset of damage as a

tabular function of

• Stress triaxiality

• Strain rate

*DAMAGE INITIATION,

CRITERION=DUCTILE

• Output:

DUCTCRT (wD)

, , , , pl pliT f h

Temperature and field

variable dependence

optional

Equivalent fracture strain

at damage initiation

The criterion for damage initiation is met when wD = 1.

L4.11


• Shear criterion:

• Appropriate for triggering damage

due to shear band localization

• The model assumes that the

equivalent plastic strain at the onset

of damage is a function of the shear

stress ratio and strain rate.

• Shear stress ratio defined as:

• The shear criterion can be used with

the Mises, Johnson-Cook, Hill, and

Drucker-Prager plasticity models,

including equation of state.Shear criterion for Aluminum Alloy AA7108.50-T6

(Courtesy of BMW)

ks = 0.3

qs = (q + ks p) /tmax


L4.12



• Usage:

• Specify the equivalent plastic strain at the onset of damage as a

tabular function of

• Shear stress ratio

• Strain rate

*DAMAGE INITIATION,

CRITERION=SHEAR, KS=ks

• Output:

SHRCRT (wS)

Temperature and field

variable dependence

optional

, , , , pl pls iT f q

The criterion for damage initiation is met when wS = 1.

Equivalent fracture strain

at damage initiation

ks is a material parameter

L4.13



• Example: Axial crushing of an aluminum

double-chamber profile

Cross

section

Quasi-static buckling mode

L4.14



• Model details

• Steel base:

• C3D8R elements

• Enhanced hourglass control

• Elastic-plastic material

• Aluminum chamber:

• S4R elements

• Stiffness hourglass control

• Rate-dependent plasticity

• Damage initiation

• General contact

• Variable mass scaling Steel base: bottom

is encastred.

Rigid plate

with initial

downward

velocity

Aluminum

chamber

L4.15


*MATERIAL, NAME=ALUMINUM

*DENSITY

2.70E-09

*ELASTIC

7.00E+04, 0.33

*PLASTIC,HARDENING=ISOTROPIC,RATE=0

:

*DAMAGE INITIATION, CRITERION=DUCTILE

5.7268, 0.000, 0.001

4.0303, 0.067, 0.001

2.8377, 0.133, 0.001

:

4.4098, 0.000, 250

2.5717, 0.067, 250

1.5018, 0.133, 250

:


• Material definition : Keywords interface

0

1

2

3

4

5

6

7

0 0.2 0.4 0.6

stress triaxiality

str

ain

at

da

ma

ge

in

itia

tio

n

strain rate=0.001/s

strain rate=250/s

Ductile criteria for Aluminum Alloy AA7108.50-

T6 (Courtesy of BMW)

Equivalent fracture strain at

damage initiation, pl

Strain rate, pl

Stress triaxiality, h

L4.16


*MATERIAL, NAME=ALUMINUM

:


5.7268, 0.000, 0.001

4.0303, 0.067, 0.001

:

*DAMAGE INITIATION, CRITERION=SHEAR, KS=0.3

0.2761, 1.424, 0.001

0.2613, 1.463, 0.001

0.2530, 1.501, 0.001

:

0.2731, 1.424, 250

0.3025, 1.463, 250

0.3323, 1.501, 250

:


Equivalent fracture strain at

damage initiation, pl

Strain rate, pl

Shear stress ratio,sq

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1.6 1.7 1.8 1.9 2

shear stress ratio

str

ain

at

da

ma

ge

in

itia

tio

n

strain rate=0.001/s

strain rate=250/s

Shear criteria for Aluminum Alloy

AA7108.50-T6 (Courtesy of BMW)

• Material definition : Keywords interface (cont'd)

L4.17



• Material definition :

Abaqus/CAE interface

:


5.7268, 0.000, 0.001

4.0303, 0.067, 0.001

2.8377, 0.133, 0.001

:

4.4098, 0.000, 250

2.5717, 0.067, 250

1.5018, 0.133, 250

:

L4.18



• Material definition :

Abaqus/CAE interface (cont'd)

:

*DAMAGE INITIATION,

CRITERION=SHEAR, KS=0.3

0.2761, 1.424, 0.001

0.2613, 1.463, 0.001

0.2530, 1.501, 0.001

:

0.2731, 1.424, 250

0.3025, 1.463, 250

0.3323, 1.501, 250

:

L4.19



• Results (without damage evolution)

DuctileShear

Quasi-static response

Damage Evolution

L4.21


Damage Evolution

• Damage evolution defines the post damage-initiation material behavior.

• That is, it describes the rate of degradation of the material stiffness

once the initiation criterion is satisfied.

• The formulation is based on scalar damage approach:

• The overall damage variable d captures the combined effect of all

active damage mechanisms.

• When damage variable d = 1, material point has completely failed.

• In other words, fracture occurs when d = 1.

(1 )d= - Stress due to undamaged response

L4.22


Damage Evolution

• Elastic-plastic materials

• For a elastic-plastic material,

damage manifests in two forms

• Softening of the yield stress

• Degradation of the elasticity

• The strain softening part of the

curve cannot represent a

material property.

• The above argument is

based on

• Fracture mechanics

considerations

• Mesh sensitivitySchematic representation of elastic-plastic

material with progressive damage.

EEd )1( -

d-

0

E

pl

fpl

0

0y)0( =d

softeningDegradation of

elasticity

Undamaged

response

L4.23


Damage Evolution

• To address the strain softening issue, Hillerborg’s (1976) proposal is

adopted.

• The fracture energy to open a unit area of crack, Gf , is assumed to be a

material property.

• The softening response after damage initiation is characterized by a

stress-displacement response (rather than a stress-strain response)

• This requires the introduction of a characteristic length L associated

with a material point.

L4.24


Damage Evolution

• The fracture energy is written as

where is the equivalent plastic displacement.

• The characteristic length L is computed automatically by Abaqus based

on element geometry.

• Elements with large aspect ratios should be avoided to minimize mesh

sensitivity.

• The damage evolution law can be specified either in terms of fracture

energy (per unit area) or in terms of the equivalent plastic

displacement.

• Both approaches take into account the characteristic length of the

element.

• The formulation ensures that mesh-sensitivity is minimized.

plu

0

0

pl plf f

pl

upl pl

f y yG L u

= =

L4.25


Damage Evolution

• Displacement-based damage evolution

*DAMAGE EVOLUTION,TYPE=DISPLACEMENT,

SOFTENING={TABULAR,LINEAR,EXPONENTIAL}

d

plu

1

0

d

plu

1

0pl

fu

d

plu

1

0pl

fu(a) Tabular (b) Linear (c) Exponential

L4.26


• Procedure for generating d vs

table from tensile test data

Schematic representation of tensile test data

in stress – displacement space for

elastic-plastic materials

0

0y

1. Plot true stress, vs. total

displacement u measured over

the gauge length L

2. For stress values in the

softening branch (i.e. beyond

damage initiation), compute

damage parameter d from the

expression

3. Compute the corresponding

plastic displacement as

shown in the schematic.

4. In the absence of intermediate

data, choose linear softening

and provide value of

(1 )d= -

plu

L

E

L

Ed)1( -

d-

u

L

E

softening

Undamaged

response

pl

fu

plupl

fu

0;0 == upl

d

uupl

f

pl

d == ;1

plu

Damage Evolution

L4.27


Damage Evolution

• Energy-based damaged evolution

*DAMAGE EVOLUTION,TYPE=ENERGY,

SOFTENING={LINEAR,EXPONENTIAL}

pl

fu

y

0y

plu

fG

y

0y

plu

fG

(a) Linear (b) Exponential

0

2

y

fpl

f

Gu

= NOTE: The response is linear or

exponential only if the undamaged

response is perfectly plastic

L4.28


Damage Evolution

• Example: Tearing of an X-shaped cross section

Pull and twist this

this end

Fix this endTie constraints

*damage initiation, criterion=fld

0.20,

*damage evolution, type=displacement, softening=tabular

0.0, 0.0

1.0, 0.003damage-plastic displacement data pairs

Failure modeled with different mesh

densities

L4.29


Damage Evolution

• Comparison of reaction forces and moments confirms mesh insensitivity

of the results.

L4.30


• Example: Axial crushing of an aluminum double-chamber profile

• Dynamic response with damage evolution

*Material, name=Aluminum

:

*Damage initiation, criterion=Ductile

:

*Damage evolution, type=displacement

0.1,

*Damage initiation, criterion=Shear, ks=0.3

:

*Damage evolution, type=displacement

0.1,

Damage Evolution

L4.31


Damage Evolution

• With damage evolution, the simulation response is a good approximation

of the physical response.

Simulation without

damage evolutionSimulation with

damage evolutionAluminum double-chamber

after dynamic impact

Element Removal

L4.33


Element Removal

• Abaqus offers the choice to

remove the element from the

mesh once the material stiffness

is fully degraded (i.e., once the

element has failed).

• An element is said to have

failed when all section

points at any one

integration point have lost

their load carrying capacity.

• By default, failed elements

are deleted from the mesh.

L4.34


Element Removal

• Removing failed elements before complete degradation

• The material point is assumed to fail when the overall damage variable

D reaches the critical value Dmax.

• You can specify the value for the maximum degradation Dmax.

• The default value of Dmax is 1 if the element is to be removed from

the mesh upon failure.

L4.35


Element Removal

• Usage:

*SECTION CONTROLS, NAME=Ec-1, ELEMENT DELETION=YES, MAX DEGRADATION=0.9

:

** Refer to the section controls by name on the element section definition.

*SOLID SECTION, ELSET=Elset_1, CONTROLS=Ec-1, MATERIAL=Material_1

:

L4.36


Element Removal

• Retaining failed elements

• You may choose not to remove failed elements

from the mesh.

*SECTION CONTROLS, ELEMENT DELETION = NO

• In this case the default value of Dmax is

0.99, which ensures that elements will

remain active in the simulation with a

residual stiffness of at least 1% of the

original stiffness.

• Here Dmax represents

• the maximum degradation of the shear stiffness (three-dimensional),

• the total stiffness (plane stress), or

• the uniaxial stiffness (one-dimensional).

• Failed elements that have not been removed from the mesh can

sustain hydrostatic compressive stresses.

L4.37


Failed elements removed

by default when STAUS

output is available

Element Removal

• Output

• The output variable SDEG

contains the value of D.

• The output variable STATUS

indicates whether or not an

element has failed.

• STATUS=0 for failed elements

• STATUS=1 for active elements

• Abaqus/Viewer will

automatically remove failed

elements when the output database (.odb) file includes

STATUS.

Deactivate status variable to view failed elements

failed

elements

Damage in Fiber-Reinforced Composite

Materials

L4.39


Common damage types in

composite laminates

Damage in Fiber-Reinforced Composite Materials

• Abaqus offers a general capability for modeling progressive damage

and failure in fiber-reinforced composites.

• Material failure refers to the complete loss of load carrying capacity that

results from progressive degradation of the material stiffness.

• Stiffness degradation is modeled using damage mechanics.

• The model must be used with elements with a plane stress formulation

(plane stress, shell, continuum shell, and membrane elements)

• Four different modes of failure are considered:

• fiber rupture in tension;

• fiber buckling and kinking in

compression;

• matrix cracking under transverse

tension and shearing; and

• matrix crushing under transverse

compression and shearing

L4.40



• User interface

• Damage Initiation

*DAMAGE INITIATION, CRITERION=HASHIN, ALPHA=<alpha>

XT,XC,YT,YC,SL,ST

L4.41




*DAMAGE EVOLUTION,

TYPE=ENERGY,

SOFTENING=LINEAR

Gft,Gfc ,Gmt,Gmc

• Viscous Regularization

*DAMAGE STABILIZATION

ηft, ηfc, ηmt, ηmc

L4.42



• Output

• Initiation Criteria Variables

• HSNFTCRT – tensile fiber Hashin’s criterion

• HSNFCCRT – compressive fiber Hashin’s criterion

• HSNMTCRT – tensile matrix Hashin’s criterion

• HSNMCCRT – compressive matrix Hashin’s criterion

• Damage Variables

• DAMAGEFT – tensile fiber damage

• DAMAGEFC – compressive fiber damage

• DAMAGEMT – tensile matrix damage

• DAMAGEMC – compressive matrix damage

L4.43



• Output (cont'd)

• Status

• STATUS – element status (1 – present, 0 – removed)

• Energies

• Damage energy (ALLDMD,DMENER,ELDMD,EDMDDEN)

• Viscous regularization (ALLCD, CENER, ELCD, ECDDEN)

L4.44



• Example: Analysis of blunt notched fiber metal laminate

• Fiber metal laminates (FMLs) are composed of:

• laminated thin aluminum layers

• Intermediate glass fiber-reinforced epoxy layers

L4.45



• Geometry of blunt notched fiber metal laminate (Glare 3 3/2–0.3)

• Through-thickness view of the laminate:

Example Problem 1.4.6, "Failure of

blunt notched fiber metal laminates”

1/8 part model Aluminum core

and exterior

glass fiber-reinforced

epoxy layers

a through-thickness hole

L4.46



• Results

damage in matrix and damage in fibers

for one of glass fiber-reinforced epoxy layers Net blunt notch strength (MPa)

Test (De Vries, 2001) 446

Abaqus 453

L4.47



• Abaqus allows the import of the damage model

for fiber-reinforced composites from

Abaqus/Explicit to Abaqus/Standard.

• Details of the import capability will not be

covered in this lecture (please refer to

―Importing and transferring results,‖ Section

9.2 of the Abaqus Analysis User’s Manual).

• One typical application is the analysis of Barely

Visible Impact Damage (BVID) in composite

structures used in aerospace applications.

• Non-visible damage to composite structures is

a significant concern in the aerospace

industry.

from McGowan, D.M., and Ambur, D.R., NASA TM-110303

Damage-Tolerance Characteristics of Composite Fuselage Sandwich Structures With Thick Facesheets

Damage in Fasteners

L4.49


Damage in Fasteners

• Connection methodologies—point fasteners

• Fastener (spot weld) compliance and failure are available in Abaqus.

multiple layers

radius of influence

attachment

points

L4.50


Damage in Fasteners

• Fastener failure

• Model combines plasticity and progressive damage

Plasticity

0

45

90

Schematic representation of the

predicted numerical response

F

plu

– Stages:

• Rigid plasticity with

variable hardening

damage

initiation

boundary


Plasticity + Damage

• Progressive damage

evolution using fracture

energy

90N

0S

– Response depends on loading angle (normal/shear)

Spot weld

L4.51


Damage in Fasteners

• Example

• Spot-welded hat section of three layers of sheet metals subjected to

severe compressive loading

Rigid spot welds Compliant spot welds with damage

Failed fasteners

Deformable fastener

still holding

Material Wear and Ablation

L4.53



• Material wear/erosion in Abaqus/Standard

• Many applications require the modeling

of wear/erosion of material at one or

more surfaces

• Capability enables modeling of material

wear/erosion on the surface of the body

• Idea is to erode material while receding

mesh away from surface (with same

number and topology of elements)

• Involves remeshing, state

mapping—handled through an

Arbitrary Lagrangian-Eulerian (ALE)

technique

• User interface takes advantage of

existing adaptive meshing

framework to define mesh motion

Adaptive mesh domain for modeling

material wear. Wear extent/velocity

applied as mesh constraints

L4.54



• Applications

• Geotechnical

• Well bore sand production

• Plastic strain, fluid velocity

• Aerospace

• Rocket motor ablation

• Pyrolysis, char formation

• Solid propellants

• Automotive

• Tire wear

• Disk brake wear

• Manufacturing

• Machining

Fluid velocity dependent

wear of a well bore

L4.55



• User interface

*Adaptive mesh, elset=...

*Adaptive mesh constraint, type=[velocity|displacement],

User

*Adaptive mesh controls

• Adaptive mesh constraints define mesh motion (wear extent or velocity)

• Wear criterion

• General descriptions possible through user subroutine UMESHMOTION

• User access to solution variables

• Nodal

• Material

• Contact

• A local surface coordinate system is provided

L4.56



• Example of wear criterion

• Tire wear

• Use of CSLIP, CSHEAR, CPRESS

h E=

Rate of recession

of tread

Rate of frictional

energy dissipation

Proportionality

constant

h

L4.57



• Example: erosion of material

from oil bore hole perforation

tunnel

• Setup consists of bore hole

with perforations, loaded by

weight of material above

• Pore pressure gradient leads

to flow into perforation

• Material wear rate controlled

by fluid flux, transport

concentration, porosity, sand

production coefficient, and the

local plastic deformation

• Optimum design to minimize

wear rate

• Example Problem 1.1.22

Perforation tunnel

Bore hole

Geometry of oil well

Courtesy of Exxon

L4.58



• Analysis steps

• Geostatic

• Model change removal of well bore and casing (drilling operation)

• Apply pore pressure; establish steady state conditions

• Transient soils consolidation (during which the erosion occurs)

• Ablation relation:

V = 10 × (PEEQ - 0.028)

Erosion

velocity

L4.59



• Adaptive mesh constraints

*Adaptive mesh, elset=Adaptive-Zone, Freq=1, Mesh=40

*Adaptive mesh constraint, constraint type=Lagrangian

Lag

*Adaptive mesh constraint, type=velocity, user

Rock-Perf, 1, 1, 1.0

Adaptive-Zone Rock-Perf

Lag: Nodes on back face of

adaptive domain

Cut section of the adaptive mesh

domain showing the perforation tunnel

L4.60



• User subroutinesubroutine umeshmotion(uref,ulocal,node,nndof,lnodetype,alocal,

$ ndim,time,dtime,pnewdt,kstep,kinc,kmeshsweep,jmatyp,jgvblock,lsmooth)

c

include 'aba_param.inc'

c

parameter (zero=0.d0, ten=10.d0, peeqCrit=0.028d0)

parameter (nelemmax=100)

dimension array(1000)

dimension ulocal(*)

dimension jgvblock(*),jmatyp(*)

dimension alocal(ndim,*)

dimension jelemlist(nelemmax),jelemtype(nelemmax)

locnum = 0

jtyp = 1

peeq = zero

nelems = nelemmax

call getNodeToElemConn(node,nelems,jelemlist,

$ jelemtype,jrcd,jgvblock)

call getVrmAvgAtNode(node, jtyp, 'PE', array, jrcd,

$ jelemlist, nelems, jmatyp, jgvblock)

peeq = array(7)

if (peeq .gt. peeqCrit) then

ulocal(ndim) = ulocal(ndim)- ten*(peeq - peeqCrit)

end if

return

end

When NDIM=3 the 3-direction

is normal to the surface

ulocal passed in as the value determined by

the mesh smoothing algorithm

L4.61



• Results

Material wear at bore hole/perforation junction Total volume lost due to erosion is available

with history output variable VOLC

L4.62



• Mesh smoothing

• Two options

• Original configuration projection

method

• Smoothing performed according

to the original configuration

• Volume-based smoothing

• Either method can include a

geometric-based enhancement

Original-configuration

smoothing

Volumetric

smoothing

L4.63



• Smoothing permitted in conjunction with UMESHMOTION constraints

• Enables UMESHMOTION to describe normal mesh motions, while the

smoothing algorithm handles the tangential mesh motions.

L4.64



• Limitations

• Available for a subset of continuum elements

• Available only for following procedures using geometric nonlinearity

• Static

• Soils

• Coupled Temperature-Displacement

• Tracer particles not supported

Element-based Cohesive Behavior

Lecture 5

L5.2


Overview

• Introduction

• Element Technology

• Constitutive Response

• Viscous Regularization

• Modeling Techniques

• Examples

• Workshop 3 (Part 1)

• Workshop 4

L5.3


Overview

• Historical perspective

• The concept of a cohesive zone has been around for some time:

• Dugdale (1960) and Barenblatt (1962) were the first to apply the

concept of a cohesive stress zone to fracture modeling.

• Many extensions since then.

• For example, Needleman (1987) recognized that cohesive

elements are particularly attractive when interface strengths are

relatively weak compared to the adjoining materials.

• Examples: composite laminates and parts bonded with adhesives

Introduction

L5.5


Introduction

• Cohesive behavior is useful in modeling adhesives, bonded interfaces, and gaskets.

• Models separation between two initially bonded surfaces

• Progressive failure of adhesives

• Delamination in composites

• Idealize complex fracture mechanisms with a macroscopic “cohesive law,” which relates the traction across the interface to the separation.

• The cohesive behavior can be:

• Element-based

• Modeled with cohesive elements

• Surface-based

• Modeled with contact pairs in Abaqus/Standard and general contact in Abaqus/Explicit

Rail crush: Cohesive surfaces

Failed adhesive is red (CSDMG = 1)

T-peel analysis: Cohesive elements are

used for modeling adhesive patches

L5.6


Introduction

• Element-based cohesive behavior—cohesive elements

• Cohesive elements allow very detailed modeling of adhesive

connections, including

• specification of detailed adhesive material properties, direct control

of the connection mesh, modeling of adhesives of finite thickness,

etc.

• Cohesive elements in Abaqus primarily address two classes of

problems:

• Adhesive joints

• Adhesive layer with finite thickness

• Typically the bulk material properties are known

• Delamination

• Adhesive layer of “zero” thickness

• Typically the bulk material properties are not known

L5.7


Introduction

• The constitutive modeling depends on the class of problem:

• Based on macroscopic properties (stiffness, strength) for adhesive

joints

• Continuum description: any Abaqus material model can be used

• Modeling technique is relatively straightforward: cohesive layer

has finite thickness; standard material models (including damage).

• The continuum description is not discussed further in this lecture.

• Based on a traction-separation description for delamination

• Linear elasticity with damage

• Modeling technique is less straightforward: typical applications use

zero-thickness cohesive elements; non-standard constitutive law

• This application is the primary focus of this lecture

L5.8


Introduction

• In addition, the uniaxial response of a laterally unconstrained adhesive

patch can also be modeled

• This represents the behavior of a gasket.

• Limited capability for modeling gaskets with cohesive elements:

• The complexity of the response in the thickness direction is not

as rich as with gasket elements available in Abaqus/Standard.

• Compared to gasket elements, however, cohesive elements:

• are fully nonlinear (can be used with finite strains and

rotations);

• can have mass in a dynamic analysis; and

• are available in both Abaqus/Standard and Abaqus/Explicit.

• The use of cohesive elements for modeling gaskets is not discussed

further in this lecture.

L5.9


Introduction

• Surface-based cohesive behavior—cohesive surfaces

• This is a simplified and easy way to model cohesive connections, using

the traction-separation interface behavior.

• It offers capabilities that are very similar to cohesive elements

modeled with the traction-separation constitutive response.

• However, it does not require element definitions.

• In addition, cohesive surfaces can bond anytime contact is

established (“sticky” contact)

• It is primarily intended for situations in which interface thickness is

negligibly small.

• It must be defined as a surface interaction property.

• Damage for cohesive surfaces is an interaction property, not a

material property.

• The kinematics of cohesive surfaces is different from that of cohesive

elements.

• By default, the initial stiffness of the interface is computed

automatically.

L5.10


Introduction

• Cohesive elements are the focus of this lecture.

• Cohesive surfaces are discussed in the next lecture.

• A workshop exercise will allow you to compare and contrast the two

cohesive modeling techniques in the context of a simple problem.

Element Technology

L5.12


Element Technology

• Element types*

• 3D elements

• COH3D8

• COH3D6

• 2D element

• COH2D4

• Axisymmetric element

• COHAX4

• These elements can be embedded

in a model via

• shared nodes or

• tie constraints.

Bottom face

Top face

*Cohesive pore pressure elements are also available.

L5.13


Element Technology

• Element and section definition

*ELEMENT, TYPE = COH3D8

*COHESIVE SECTION, ELSET =...,

RESPONSE = {TRACTION SEPARATION, CONTINUUM,

GASKET },

THICKNESS = { SPECIFIED, GEOMETRY},

MATERIAL = ...

Specify thickness in dataline (default is 1.0)

L5.14


Element Technology

• Default thickness of cohesive elements

• Traction-separation response:

• Unit thickness

• Continuum and gasket response

• Geometric thickness based on nodal coordinates

L5.15


Element Technology

• Output variables

• Scalar damage (i.e., degradation) variable

• SDEG

• Variables indicating whether damage initiation criteria met or exceeded

• Discussed shortly

• Element status flag

• STATUS

L5.16


Element Technology

• Import of cohesive elements

• The combination of Abaqus/Standard and Abaqus/Explicit expands the

range of applications for cohesive elements.

• For example, you can simulate the damage in a structure due to an

impact event then study the effect of the damage on the structure's load

carrying capacity.

Constitutive Response

L5.18



• Delamination applications

• Traction separation law

• Typically characterized by peak strength (N) and fracture energy (GTC)

• Mode dependent


• Available in both Abaqus/Standard and Abaqus/Explicit

• Modeling of damage under the general framework introduced earlier


• Traction or separation-based criterion

• Damage evolution

• Removal of elements0

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8 1

Mode Mix

GT

C

Normal mode

Shear mode

Dependence of fracture toughness

on mode mix

Typical traction-separation response

T

T CG

N

L5.19




• Linear elasticity

• Defines behavior before the

initiation of damage

• Relates nominal stress to nominal

strain

• Nominal traction to separation

with default choice of unit

thickness

• Uncoupled traction behavior:

nominal stress depends only on

corresponding nominal strain

• Coupled traction behavior is more

general*ELASTIC, TYPE = { TRACTION,

COUPLED TRACTION }

L5.20



• The elastic modulus for the traction

separation law should be interpreted as a

penalty stiffness.

• For example, for the opening mode:

Kn = Nmax / ninit

• In Abaqus, nominal stress and strain

quantities are used for the traction

separation law.

• If unit thickness is specified for the

element, then the nominal strain

corresponds to the separation value.

• Elastic response governed by Kn.

• If you specify a non-unit thickness for

the cohesive element, you must scale

your data to obtain the correct

stiffness Kn. Example on next slide.

N

n

maxN

initn fail

n

nK

1

Displacement at damage

initiation in normal

(opening) mode

L5.21



• Example: Peel test model

A

En=Knheff

/ /

n n

n n

n n eff n n eff

N E

K

h K E h

=

=

= =

Abaqus evaluates this…

…which is equivalent to this

Geometric thickness (based

on nodal coordinates) of the

adhesive hgeom = 1e-3

Assume separation at initiation = 1e-3

and Nmax = 6.9e9.

For model A: use geometric thickness

heff = hgeom =1e-3 = /heff = 1;

Nmax = En = 6.9e9 Kn = 6.9e12

For model B: specify unit thickness

heff = 1 = / heff =1e-3;

Nmax = 6.9e9 En = Kn = 6.9e12

initn

initn

initn

initninit

n

B

L5.22



• Mixed mode conditions

• Maximum stress

(or strain) criterion:

• Output:

• MAXSCRT

• MAXECRT


max max max

, , 1n t sMAX

N T S

=

0

0 0

n nn

n

=

for

for

* DAMAGE INITIATION, CRITERION = { MAXS, MAXE }

L5.23



• For example, for Mode I (opening mode) the MAXS condition implies

damage initiates when n = Nmax.

N

n

maxN

*Damage initiation,criterion=MAXS

290.0E6, 200.0E6, 200.0E6

Damage initiation point

Nmax Tmax Smax

L5.24



• Quadratic stress (or strain) interaction criterion:

• No damage initiation under

pure compression

• Output:

• QUADSCRT

• QUADECRT

2 2 2

max max max

1n t s

N T S

=

* DAMAGE INITIATION,

CRITERION = { QUADS, QUADE }

L5.25



• Summary of damage initiation criteria

max max max, , 1

n s t

n s t

MAX

=

Maximum nominal strain criterion

*DAMAGE INITIATION, CRITERION=MAXEmax max max, ,n s t

2 2 2

max max max1

n s t

n s t

=

Quadratic nominal stress criterion

*DAMAGE INITIATION, CRITERION=QUADEmax max max, ,n s t

2 2 2

max max max

1n s t

N S T

=

Quadratic nominal stress criterion

*DAMAGE INITIATION, CRITERION=QUADS

max max max, ,N S T

n: nominal stress in the pure normal mode

s: nominal stress in the first shear direction

t: nominal stress in the second shear direction

n: nominal strain in the pure normal mode

s: nominal strain in the first shear direction

t: nominal strain in the second shear direction

*DAMAGE INITIATION, CRITERION=MAXS

max max max, ,N S T

max max max

, , 1n s tMAX

N S T

=

Maximum nominal stress criterion

, ,n s tn s t

o o oT T T

= = =Note : where n, s, and t are components of relative displacement

between the top and bottom of the cohesive element; and To

is the original thickness of the cohesive element.

L5.26




• Post damage-initiation response defined by:

• d is the scalar damage variable

d = 0: undamaged

d = 1: fully damaged

d monotonically increases

1 d= -

Typical damaged response

0K

d-

(1 )d-

0K

0(1 )d Κ-

L5.27



• Damage evolution is based on

energy or displacement

• Specify either the total

fracture energy or the post

damage-initiation effective

displacement at failure

• May depend on mode mix

• Mode mix may be defined in

terms of energy or traction

N

n

T CG

Area under the curve

is the fracture energy

maxN

Displacement at failure

in normal (opening) mode

failn

L5.28



• Displacement-based damage evolution

• Damage is a function of an effective

displacement:

• The post damage-initiation softening

response can be either

• Linear

• Exponential

• Tabular

Traction

Linear post-

initiation response

init fail

2 2 2n s t =

L5.29



• Keywords interface for displacement-based damage evolution

• For LINEAR and EXPONENTIAL softening:

• Specify the effective displacement at complete failure fail relative to

the effective displacement at initiation init.

• For TABULAR softening:

• Specify the scalar damage variable d directly as a function of

–init.

• Optionally specify the effective displacement as function of mode mix in tabular form.

• Abaqus assumes that the damage evolution is mode independent otherwise.

*DAMAGE EVOLUTION, TYPE = DISPLACEMENT,

SOFTENING = { LINEAR | EXPONENTIAL | TABULAR },

MIXED MODE BEHAVIOR = TABULAR

L5.30



• Abaqus/CAE interface for displacement-based damage evolution

L5.31



• Energy-based damage evolution

• The fracture energy can be defined as a function of mode mix using

either a tabular form or one of two analytical forms:

• Power law

• BK (Benzeggagh-Kenane)

1I II III

IC IIC IIIC

G G G

G G G

=

For isotropic failure

(GIC = GIIC), the

response is insensitive to

the value of .

shearIC IIC IC TC

T

shear II III

T I shear

GG G G G

G

G G G

G G G

- =

=

=

where

L5.32



• Keywords interface for energy-based damage evolution

• Specify fracture energy as function of mode mix in tabular form, or

• Specify the fracture energy in pure normal and shear deformation modes

and choose either the POWER LAW or the BK mixed mode behavior

*DAMAGE EVOLUTION, TYPE = ENERGY,

SOFTENING = { LINEAR | EXPONENTIAL},

MIXED MODE BEHAVIOR = { TABULAR | POWER LAW | BK },

POWER = value

L5.33



• Abaqus/CAE interface for energy-based damage evolution

L5.34



• Example

• The preceding discussion was very

general in the sense that the full

range of options for modeling the

constitutive response of cohesive

elements was presented.

• In the simplest case, Abaqus requires

that you input the adhesive thickness

heff and 10 material parameters:

*Elastic, type=traction

En, Et, Es

*Damage initiation, criterion =

maxs

Nmax, Tmax, Smax

*Damage evolution, type=energy,

mixed mode behavior=bk, power=

GIC, GIIC , GIIIC

What do you do when you only

have 1 property and the adhesive

thickness is essentially zero?

Diehl, T., "Modeling Surface-Bonded Structures with

ABAQUS Cohesive Elements: Beam-Type Solutions,"

ABAQUS Users' Conference, Stockholm, 2005.

Normal (opening) mode:

maxN

initn fail

n

nK

1Tra

ctio

n

(no

min

al

stre

ss)

Separation

(area under

entire curve)

GIC

Cohesive material law:

Traction, Damage Evolution

nn

eff

EK

h=

L5.35



• Example (cont’d)

• Common case: you know GTC for the surface bond.

• Assume isotropic behavior

GIC = GIIC = GIIIC = GTC

• For MIXED MODE BEHAVIOR = BK, this makes the response

independent of term, so set = any valid input value (e.g.,

1.0)

• Bond thickness is essentially zero

• Specify the cohesive section property thickness heff = 1.0

Nominal strains = separation; elastic moduli = stiffness

• Isotropy also implies the following:

En = Et = Es = Eeff (=Keff since we chose heff = 1.0)

Nmax = Tmax = Smax = Tult

L5.36




• Introduce concept of damage initiation ratio:

ratio= init /fail, where 0 ratio 1.

• Use GC and equation of a triangle to relate back to Keff and Tult :

• The problem now reduces to two penalty terms: fail and ratio.

• Assume ratio = ½.

• Choose fail as a fraction of the typical cohesive element mesh size.

• For example, use fail = 0.050 typical cohesive element size

as a starting point.

2

2 TCeff

ratio fail

GK

=

2 TCult

fail

GT

=

L5.37




• Thus, after choosing the two penalty terms, a single (effective)

traction-separation law applies to all modes (normal + shear):

*Cohesive section, thickness=SPECIFIED, ...

1.0,

:

:

*Elastic, type=TRACTION

Keff, Keff, Keff*Damage initiation, criterion = MAXS

Tult, Tult, Tult*Damage evolution, type=ENERGY,

mixed mode behavior=BK, power=1

GTC, GTC , GTC

Effective properties:

ultT

init fail

effK

1Tra

ctio

n

(no

min

al

stre

ss)

Separation

(area under

entire curve)

GTC

Cohesive material law:

Traction, Damage Evolution

effeff

eff

EK

h=

L5.38




• What if the response is dynamic? What about the density?

• The density of the cohesive layer should also be considered a

penalty quantity.

• For Abaqus/Explicit, the effective density should not adversely affect

the stable time increment. Diehl suggests the following rule:

• The Abaqus Analysis User’s Manual provides additional guidelines

for determining a cohesive element density that minimizes the effect

on the stable time increment in Abaqus/Explicit.

• Dtstable = stable time increment

without cohesive elements in the model

• ft2D = 0.32213 (for cohesive elements

whose original nodal coordinates relate

to zero element thickness)

2

2

stableeff eff

t D eff

tE

f h

D=

Viscous Regularization

L5.40



• Cohesive elements have the potential to cause numerical difficulties in

the following cases

• Stiff cohesive behavior may lead to reduced maximum stable time

increment in Abaqus/Explicit

• Potentially addressed through selective mass scaling

• Unstable crack propagation may lead to convergence difficulties in

Abaqus/Standard

• Potentially addressed through built-in viscous regularization option

specific to cohesive elements

L5.41



• Viscous regularization

• Material models with damage often lead to severe convergence

difficulties in Abaqus/Standard

• Viscous regularization helps in such cases

• Helps make the consistent tangent stiffness of softening material

positive for sufficiently small time increments

• Similar approach used in the concrete damaged plasticity model in

Abaqus/Standard

1 vd= -

1

v vd d d

= -

L5.42



• Consistent material tangent stiffness

K0 is the undamaged elastic stiffness

f is a factor that depends on the details of the damage model

• Viscous regularization ensures that when ,

• “Offending” second term is eliminated when the analysis cuts back

drastically

0t

D

01d

d f

= - -

D K

0(1 )d= -D K

L5.43



• User interface for viscous regularization

*COHESIVE SECTION, CONTROLS = control1

*SECTION CONTROLS, NAME = control1,

VISCOSITY = factor

• Add-on transverse shear stiffness may

provide additional stability

*COHESIVE SECTION

*TRANSVERSE SHEAR STIFFNESS

• Output

• Energy associated with viscous regularization: ALLCD

L5.44



• Example: Multiple delamination

problem (Alfano & Crisfield, 2001)

– Industry standard Alfano-

Crisfield nonsymmetric

delamination examples

• Plies are initially bonded with

predefined cracks, then peeled

apart in a complex sequence

• Example done in

Abaqus/Standard and

Abaqus/Explicit

• Effect of viscous regularization

is investigated

10 layers

12 layers2 layers

Interface elementsInitial cracks

a1

a2

a2

L

L5.45



.e= -1 4

.e= -1 3

= 0

. e= -2 5 4

.e= -5 4

L5.46



• Effect of viscous regularization on convergence of multiple delamination

problem:

• Significant improvements with small regularization factor

Viscous

regularization

factor

Total number of

increments

0. 375

1.0e-4 171

2.5e-4 153

1.0e-3 164

Modeling Techniques

L5.48


Modeling Techniques

• Model problem: double-cantilever beam

• Alfano and Crisfield (2001)

• Pure Mode I

• Displacement control

• Analyzed using

• 1D (B21),

• 2D (CPE4I), and

• 3D (C3D8I) elements

• Delamination assumed to occur along a straight line

• Beams: Orthotropic material

• Cohesive layer: Traction-separation with damage

Initial crack

u

- u

L5.49


Modeling Techniques

• One-dimensional model

• Use tie constraints between the cohesive layer and the beams

• Require distinct parts for the beam and cohesive zone geometry

• Geometry

L5.50


Modeling Techniques

• One-dimensional model (cont’d)

• Assembly

Create 2 instances of the beam;

one of the cohesive zone

Position the parts to leave gaps

between them; this will later

facilitate picking surfaces

L5.51


Modeling Techniques


• Tie constraints

Define tie constraints between

mating surfaces.

The cohesive side should be the

slave surface (because it is a

softer material)

This approach is required when

quadratic displacement elements

are used.

beam-top

beam-bot

coh-top

coh-bot

L5.52


Modeling Techniques


• Properties: beam

L5.53


Modeling Techniques


• Properties: adhesive

L5.54


Modeling Techniques


• MeshingCohesive elements can only

be assigned to sweep

meshable regions

Sweep path must be aligned

with thickness direction

Assign seeds and mesh

Only one element

through the thickness

1

3

Assign cohesive element

type to the swept region2

L5.55


Modeling Techniques


• Meshing (cont’d)

Edit the nodal coordinates of each part instance

so that they all have the same 2-coordinate

Toggle this off; otherwise, nodes will

project back to their original positions

Final mesh

4

L5.56


Modeling Techniques

• Two-dimensional model

• All geometry is 2D and planar

• Properties, attributes, etc. treated in a

similar manner to the 1D case presented

earlier

• Modeling options include:

• Shared nodes

• Tie constraints

• Similar to the 1D model

L5.57


Modeling Techniques

• Two-dimensional model (cont’d)

• Shared nodes

Define a finite thickness slit in the beam as shown below

• Use the actual overall thickness of the DCB

• The center region represents the cohesive layer

Mesh the part:

1

2

L5.58


Modeling Techniques


• Shared nodes (cont’d)

Edit the coordinates of the nodes along the interface3

L5.59


Modeling Techniques


• Tie constraints

Create two instances of the beams and position them as shown

below.

• Suppress the visibility of the instances to facilitate picking

surfaces, etc.

Create a finite thickness cohesive layer, position it appropriately in

the horizontal direction, define surfaces, etc.

• After meshing, adjust the coordinates of all the nodes in the

cohesive layer so that they lie along the interface between the

two beams.

1

2

L5.60


Modeling Techniques

• Three-dimensional model

• All geometry is 3D

• Solid geometry for beams

• Solid or shell geometry for cohesive layer

• Modeling options include

• Shared nodes

• Tie constraints

L5.61


Modeling Techniques

• Three-dimensional model (cont’d)

• Shared nodes

Partition the geometry and

define a mesh seam

between these two faces

1

L5.62


Modeling Techniques



Mesh the part with solid

(continuum elements).

Create a orphan mesh

Mesh→Create Mesh Part

2

3

L5.63


Modeling Techniques

Create a single zero-thickness

solid layer by offsetting from the

midplane (selected by angle) of

the orphan mesh created in the

previous step

Tip 2: Use the selection

options tools to facilitate

picking. In particular, select

from interior entities.

Create a set for the new layer so you

can easily assign element type and

section properties.

Tip 1: Remove elements from

top region with display groups

(select by angle)4

L5.64


Modeling Techniques



Assign section properties and

the element type to the set

created in the previous step

5

L5.65


Modeling Techniques


• Tie constraints

• The cohesive region can be defined

as

• Solid (with finite thickness)

• Edit nodal coordinates of

cohesive elements as in

previous examples

• Shell geometry

• Mesh geometry then create

orphan mesh

• Offset a zero-thickness layer of

solid elements from the orphan

mesh

Define surfaces automatically to

facilitate tie constraints

L5.66


Modeling Techniques


• Tie constraints (cont’d)

When defining the tie constraints,

query the mesh stack direction to

determine when the “top” and

“bottom” surfaces should be used

Brown = top Purple = bottom

L5.67


Modeling Techniques

• What if I don't use Abaqus/CAE?

• In this case do the following in the preprocessor of your choice:

1. Generate the mesh for the structure and cohesive layer

(temporarily assigning an arbitrary element type to the cohesive

layer)

2. Position the layer of cohesive elements over the interface

3. Define surfaces on the structure and cohesive layer

4. Write the input file

Surface top-beam

Surface bot-beam

Surface top-coh

Surface bot-coh

L5.68


Modeling Techniques

• Edit the input file:

5. Change the element type assigned to the cohesive layer

6. Assign cohesive section properties

*element, elset=coh, type=coh2d4

*cohesive section, elset=coh, material=cohesive,

response=traction separation, stack direction=2, controls=visco

1.0, 0.02

:

*material, name=cohesive

*elastic, type=traction

5.7e+14, 5.7e+14, 5.7e+14

*damage initiation, criterion=quads

5.7e7, 5.7e7, 5.7e7

*damage evolution, type=energy, mixed mode behavior=bk, power=2.284

280.0, 280.0, 280.0

L5.69


Modeling Techniques

• The stack direction defines the thickness direction based on the

element isoparametric directions.

• Set STACK DIRECTION = { 1 | 2 | 3 } to define the element

thickness direction along an isoparametric direction.

• 2D example (extends to 3D):

Thickness

direction

Element connectivity: 101, 102, 202, 201

Stack direction = 2

Element connectivity: 102, 202, 201, 101

Stack direction = 1

101

201 202

102

1

2

101

201 202

102

1

2

L5.70


Modeling Techniques

• Edit the input file (cont'd):

7. Define tie constraints between the surfaces

*tie, name=top, adjust=yes, position tolerance=0.002

top-coh, top-beam

*tie, name=bot, adjust=yes, position tolerance=0.002

bot-coh, bot-beam

Setting adjust=yes will force Abaqus to

move the slave (cohesive element) nodes

onto the master surface. By adjusting both

the top and bottom cohesive surfaces in this

way, a zero-thickness cohesive layer is

produced.

The position tolerance should be large

enough to contain the slave nodes when

measured from the master surface. In this

case the overclosure is equal to 0.001 on

either side of the interface so a position

tolerance of 0.002 is sufficient to capture all

slave nodes.

0.001

Cohesive surface

is the slave

L5.71


Modeling Techniques

• Results

L5.72


Modeling Techniques

• Effect of viscous regularization

Viscous

regularization

factor

Total number

of increments

1.e-5 636

2.5e-5 163

5.0e-5 129

1.0e-4 90

L5.73


Modeling Techniques

• Effect of mesh refinement

• Typically, you will need to use

a much finer mesh (for both

the stress/displacement and

cohesive elements) than may

be necessary for a problem

without cohesive elements.

L5.74


Modeling Techniques

• Non-planar geometry

• The technique for embedding a layer of solid elements into an orphan

mesh is not restricted to planar geometry.

• As an example, consider the following fiber-matrix pullout model

Orphan mesh

fiber

matrix

L5.75


Modeling Techniques

• Failure driven by mismatch in CTEs

View cut of the matrix-fiber interface at

100% of the applied load (magnified 5×) Failure levels at 38% of the applied load

L5.76


Modeling Techniques

• Cohesive elements on a symmetry plane

• The traction-separation law is based on

the separation between the top and

bottom faces of the cohesive element.

• On a symmetry plane, however, the

separation that is computed is ½ the

actual value.

• To account for this, specify:

• 2 the cohesive stiffness that would

be used in a full model.

• ½ the fracture toughness that would

be used in a full model.

• Linear equations between the

nodes on the top and bottom faces

in the lateral directions.

N

n

maxN

2

initn

2

failn

2 nK

1

2

CGarea =

22

/ 2

n nn

eff eff

E EK

h h= =

L5.77


Modeling Techniques

• Symmetry example

Symmetric model

Full model

Symmetric model (top)

overlaid on full model

Constitutive thickness is

same as for the full model so

double the elastic modulus to

double the cohesive stiffness

Constraint on lateral

displacements

Examples

L5.79


Examples

• Composite components in

aerospace structures

(Courtesy: NASA)

• Stress concentrations

around stiffener

terminations and flanges

• Residual thermal strains at

the interface at room

temperature

• Analysis of the effects of

residual strains on

skin/stiffener debonding

• Delamination initiation and

propagation

Beginning of separation After separation

Abaqus/Standard simulation of skin/stiffener debonding

Example Problem 1.4.5

L5.80


Examples

Abaqus/Standard simulation of

skin/stiffener debonding

L5.81


Examples

• Electronic packaging (Courtesy: INTEL)

• Solder to motherboard fracture due to static overload

• Experiments to assess integrity of solder joints under various loading conditions (e.g., board bending)

• Strain in motherboard at which solder joint fails

Ball grid array

L5.82


Examples

Damage severity in cohesive layer between

motherboard and solder balls

Debonded solder balls

L5.83


Cohesive layers

Examples

• Delamination of a composite

• This model is a representative of composite

delamination.

• It comprises 3 layers of composite with

adhesive layers applied between

composite layers.

• The composite delaminates under the

impact of a heavy mass displayed in

light greenish shade in the animation.

L5.84


Examples

• Impact of moving mass with a stationary wall

• Brick wall modeled with adhesives applied

to each face of each brick.

• Simulating damage of the (stationary) wall

from high velocity impact with a heavy

mass

• Analysis performed in Abaqus/Explicit.

• This model is a representative of several

problems that can be modeled using

cohesive elements

• Hydroplaning

• Machining

• Oil Drilling

• Excavation

• Effect of explosion on a building.

Section of the model illustrating

the application of cohesive

layers around the bricks.

L5.85


Examples

• Deformation sequence

Workshop 3 (Part 1)

L5.87


Workshop 3 (Part 1)

• Crack growth in a three-point bend specimen using element-based

cohesive behavior

• Generate cohesive element mesh

• Define/assign traction-separation behavior and damage properties

Layer of

cohesive

elements

Workshop 4 (Optional)

L5.89


Workshop 4 (Optional)

• Crack growth in a helicopter airframe

• Use the mesh offset tool to create a layer of cohesive elements

• Impose symmetry conditions on the cohesive elements using linear

equations

Cohesive element

thickness shrunk to zero

Surface-based Cohesive Behavior

Lecture 6

L6.2


Overview

• Surface-based Cohesive Behavior

• Element- vs. Surface-based Cohesive Behavior

• Workshop 3 (Part 2)


L6.4



• Surface-based cohesive behavior provides a simplified way to model

cohesive connections with negligibly small interface thicknesses using the

traction-separation constitutive model.

• It can also model “sticky” contact (surfaces can bond after coming into

contact).

• The cohesive surface behavior can be defined for general contact in

Abaqus/Explicit and contact pairs in Abaqus/Standard (with the

exception of the finite-sliding, surface-to-surface formulation).

• Cohesive surface behavior is defined as a surface interaction property.

• To prevent overconstraints in Abaqus/Explicit, a pure master-slave

formulation is enforced for surfaces with cohesive behavior.

L6.5



• User interface

*SURFACE INTERACTION, NAME=cohesive

*COHESIVE BEHAVIOR

...

*CONTACT PAIR, INTERACTION=cohesive

surface1, surface2

Abaqus/Standard


*COHESIVE BEHAVIOR

...

*CONTACT

*CONTACT PROPERTY ASSIGNMENT

surface1, surface2, cohesive

Abaqus/Explicit

Abaqus/CAE

L6.6



• The formulae and laws that govern surface-based cohesive behavior are

very similar to those used for cohesive elements with traction-separation

behavior:

• linear elastic traction-separation,

• damage initiation criteria, and

• damage evolution laws.

• However, it is important to recognize that damage in surface-based

cohesive behavior is an interaction property, not a material property.

• Traction and separation are interpreted differently for cohesive elements

and cohesive surfaces:

Relative displacement ()between the top and bottom

of the cohesive layer

Cohesive elements Cohesive surfaces

Nominal strain () = Contact separation ()Initial thickness (To)

Nominal stress () Contact force (F)Contact stress (t) =

Current area (A) at each contact point

separation

traction

traction

separation

GC

L6.7



• Linear elastic traction-separation behavior

• Relates normal and shear stresses to the normal and shear separations

across the interface before the initiation of damage.

• By default, elastic properties are based on underlying element stiffness.

• Can optionally specify the properties.

• Recall this specification is required for cohesive elements.

• The traction-separation behavior can be uncoupled (default) or coupled.

*COHESIVE BEHAVIOR, TYPE= { UNCOUPLED, COUPLED}

Optional data line to specify Knn, Kss, Ktt

L6.8



• Controlling the cohered nodes

• The slave nodes to which cohesive behavior is applied can be controlled

to define a wider range of cohesive interactions: Can include:

• All slave nodes

• Only slave nodes initially in contact

• Initially bonded node set

• Applying cohesive behavior to all slave nodes (default)

• Cohesive constraint forces potentially act on all nodes of the

slave surface.

• Slave nodes that are not initially contacting the master surface

can also experience cohesive forces if they contact the master

surface during the analysis.

*COHESIVE BEHAVIOR,

ELIGIBILITY = CURRENT CONTACTS

1

L6.9



Applying cohesive behavior only to slave nodes initially in contact

• Restrict cohesive behavior to only those slave nodes that are in

contact with the master surface at the start of a step.

• Any new contact that occurs during the step will not experience

cohesive constraint forces.

• Only compressive contact is modeled for new contact.

*COHESIVE BEHAVIOR,

ELIGIBILITY = ORIGINAL CONTACTS

2

L6.10



Applying cohesive behavior only to an initially bonded node set

(Abaqus/Standard only)

• Restrict cohesive behavior to a subset of slave nodes defined

using *INITIAL CONDITIONS, TYPE=CONTACT.

• All slave nodes outside of this set will experience only

compressive contact forces during the analysis.

• This method is particularly useful for modeling crack

propagation along an existing fault line.

3

*COHESIVE BEHAVIOR,

ELIGIBILITY = SPECIFIED CONTACTS

L6.11



• Example: Double cantilever beam (DCB)

• Analyze debonding of the DCB model using the surface-based cohesive behavior in Abaqus/Standard.

• To model debonding using surface-based cohesive behavior,

• you must define:

• contact pairs and initially bonded crack surfaces;

• the traction-separation behavior;

• the damage initiation criterion; and

• the damage evolution.

• You may also

• specify viscous regularization to facilitate solution convergence in Abaqus/Standard.

• Note: Steps 3, 4, and 5, will be covered later in this lecture.

Initial crack

u

- uCohesive interface

1

2

3

4

5

Note: Only the Keywords interface is illustrated in the example;

the Abaqus/CAE interface is illustrated in the workshop exercise.

L6.12



• Define contact pairs and initially bonded crack surfaces

• The initially bonded portion of the slave surface (i.e., node set bond)

is identified with the *INITIAL CONDITIONS, TYPE=CONTACT

option.

1

Note: Frictionless contact is assumed.

*NSET, NSET=bond, GENERATE

1, 121, 1

*SURFACE, NAME=TopSurf

_TopBeam_S1, S1

*SURFACE, NAME=BotSurf

_BotBeam_S1, S1

*CONTACT PAIR, INTER=cohesive

TopSurf, BotSurf

*INITIAL CONDITIONS, TYPE=CONTACT

TopSurf, BotSurf, bond

slave surface master surface a list of slave nodes

that are initially bonded

BotSurfTopSurf

bond

L6.13



• Define traction-separation behavior

• In this model, the cohesive behavior is only enforced for the node set bond.

• Use the ELIGIBILITY=SPECIFIED CONTACTS

parameter to enforce this behavior.

• Recall the default elastic properties are based

on underlying element stiffness. Here we

specify the properties.

...


TopSurf, BotSurf




*COHESIVE BEHAVIOR,

ELIGIBILITY=SPECIFIED CONTACTS

5.7e14, 5.7e14, 5.7e14

2

Kn Ks Kt

t

1

Kn (Ks , Kt)

Kn, Ks, and Kt: normal and

tangential stiffness components

Optional

BotSurfTopSurf

bond

L6.14



separations at failure

, ,f f fn s t and :

peak values of the contact stress

max max max, ,n s tt t tand :

peak values of the contact separation

max max max, ,n s t and :

max max max,n s tt t t

,f f fn s t

t

max max max,n s t

• Damage modeling for cohesive

surfaces

• Damage of the traction-separation

response for cohesive surfaces is

defined within the same general

framework used for cohesive

elements.

• The difference between the two

approaches is that for cohesive

surfaces damage is specified as

part of the contact interaction

properties.

L6.15



• User interface


*COHESIVE BEHAVIOR

*DAMAGE INITIATION

*DAMAGE EVOLUTION

*CONTACT PAIR, INTERACTION=cohesive

surface1, surface2

Abaqus/Standard


*COHESIVE BEHAVIOR

*DAMAGE INITIATION

*DAMAGE EVOLUTION

*CONTACT

*CONTACT PROPERTY ASSIGNMENT

surface1, surface2, cohesive

Abaqus/Explicit

Abaqus/CAE

L6.16



• Damage initiation criteria

max max max, , 1

n s t

n s t

MAX

Maximum separation criterion

*DAMAGE INITIATION, CRITERION=MAXUmax max max, ,n s t

2 2 2

max max max1

n s t

n s t

Quadratic separation criterion

*DAMAGE INITIATION, CRITERION=QUADUmax max max, ,n s t

2 2 2

max max max1

n s t

n s t

t t t

t t t

Quadratic stress criterion

*DAMAGE INITIATION, CRITERION=QUADSmax max max, ,n s tt t t

tn: normal contact stress in the pure normal mode

ts: shear contact stress along the first shear direction

tt: shear contact stress along the second shear direction

n: separation in the pure normal mode

s: separation in the first shear direction

t: separation in the second shear direction

*DAMAGE INITIATION, CRITERION=MAXSmax max max, ,n s tt t t

max max max, , 1

n s t

n s t

t t tMAX

t t t

Maximum stress criterion

Note: Recall the damage initiation criteria for the cohesive elements: if the initial constitutive thickness To = 1,

then = /To = . In this case, the separation measures for both approaches are exactly the same.

L6.17



• Example: Double cantilever beam

• Define the damage initiation criterion

• The quadratic stress criterion is specified for this problem.

3

...


TopSurf, BotSurf




*COHESIVE BEHAVIOR,


5.7e14, 5.7e14, 5.7e14


5.7e7, 5.7e7, 5.7e7

max max maxn s tt t t

BotSurfTopSurf

bond

L6.18




• For surface-based cohesive behavior, damage evolution describes the

degradation of the cohesive stiffness.

• In contrast, for cohesive elements damage evolution describes the

degradation of the material stiffness.

• Damage evolution can be based on energy or separation (same as for

cohesive elements).

• Specify either the total fracture energy (a property of the cohesive

interaction) or the post damage-initiation effective separation at

failure.

• May depend on mode mix

• Mode mix may be defined

in terms of energy or traction


,f f fn s t

t

max max max,n s t

GTC

L6.19


• Separation-based damage evolution

• Damage is a function of an effective

separation:

• As with cohesive elements, the post

damage-initiation softening response can

be either:

• Linear

• Exponential

• Tabular


2 2 2n s t


,f f fn s t

t

max max max,n s t

Linear post-

initiation response

L6.20



• Separation-based damage evolution (cont’d)

• Usage:

*DAMAGE EVOLUTION, TYPE = DISPLACEMENT,

SOFTENING = { LINEAR | EXPONENTIAL | TABULAR },

MIXED MODE BEHAVIOR = TABULAR

L6.21



• Energy-based damage evolution

• As with cohesive elements, the energy-based damage evolution criterion

can be defined as a function of mode mix using either a tabular form or

one of two analytical forms:

Power law Benzeggagh-Kenane (BK)

1I II III

IC IIC IIIC

G G G

G G G

shearIC IIC IC TC

T

shear II III

T I shear

GG G G G

G

G G G

G G G

-

where

L6.22



• Energy-based damage evolution (cont’d)

• Usage:

*DAMAGE EVOLUTION, TYPE = ENERGY,

SOFTENING = { LINEAR | EXPONENTIAL},

MIXED MODE BEHAVIOR = { TABULAR | POWER LAW | BK },

POWER = value

L6.23




• Define damage evolution

• The energy-based damage evolution based on the BK mixed mode

behavior is specified.

4

...


TopSurf, BotSurf




*COHESIVE BEHAVIOR,


5.7e14, 5.7e14, 5.7e14


5.7e7, 5.7e7, 5.7e7

*DAMAGE EVOLUTION, TYPE=ENERGY,

MIXED MODE BEHAVIOR=BK, POWER=2.284

280.0, 280.0, 280.0

GIC GIIC GIIIC

shearIC IIC IC TC

T

GG G G G

G

-

BotSurfTopSurf

bond

L6.24



• Viscous regularization

• Can be specified to facilitate solution convergence in Abaqus/Standard

for surface-based cohesive behavior when stiffness degradation occurs.

• Output:

• Energy associated with viscous regularization: ALLCD


L6.25




• Specify a viscosity coefficient for

the cohesive surface behavior

viscosity coefficient,

...


TopSurf, BotSurf




*COHESIVE BEHAVIOR,


5.7e14, 5.7e14, 5.7e14


5.7e7, 5.7e7, 5.7e7



280.0, 280.0, 280.0


1.e-5

5

BotSurfTopSurf

bond

L6.26




• Summary of the input for the traction-separation response

*COHESIVE SECTION, MATERIAL=cohesive,

RESPONSE=TRACTION SEPARATION,

ELSET=coh_elems, CONTROLS=visco

, 0.02

*MATERIAL, NAME=cohesive

*ELASTIC, TYPE=TRACTION

5.7e14, 5.7e14, 5.7e14


5.7e7, 5.7e7, 5.7e7



280.0, 280.0, 280.0

*SECTION CONTROLS, NAME=visco,

VISCOSITY=1.e-5


*COHESIVE BEHAVIOR,


5.7e14, 5.7e14, 5.7e14


5.7e7, 5.7e7, 5.7e7



280.0, 280.0, 280.0


1.e-5

Cohesive elements Cohesive surfaces

L6.27



• Results

Cohesive elements

Cohesive surfaces

Failed cohesive elements

u2 = 0.006

u2

u2 = 0.006

u2

Element- vs. Surface-based

Cohesive Behavior

L6.29


Element- vs. Surface-based Cohesive Behavior

Preprocessing

• Cohesive elements

• Gives you direct control over the cohesive element mesh density and stiffness properties.

• Constraints are enforced at the element integration points.

• Refining the cohesive elements relative to the connected structures will likely lead to improved constraint satisfaction and more accurate results.

• Cohesive surfaces

• Are easily defined using contact interactions and cohesive interaction properties.

• A pure master-slave in formulation is used.

• Constraints are enforced at the slave nodes.

• Refining the slave surface relative to the master surface will likely lead

to improved constraint satisfaction and more accurate results.

Integration points on an

8-node cohesive element

L6.30



Initial configuration:


• Must be bonded at the start of the analysis.

• Once the interface has failed, the surfaces do not re-bond.


• Can bond anytime contact is established

(i.e., “sticky” contact behavior).

• Cohesive interface need not be bonded at the start of the

analysis.

• You can control whether debonded surfaces will stick or not stick if

contact occurs again.

• By default, they do not stick.

L6.31



Constitutive behavior:


• Allow for several constitutive behavior types:

• Traction-separation constitutive model

• Including multiple failure mechanisms

• Continuum-based constitutive model

• For adhesive layers with finite thickness

• Uses conventional material models

• Uniaxial stress-based constitutive model

• Useful in modeling gaskets and/or single adhesive patches


• Must use the traction-separation interface behavior.

• Intended for bonded interfaces where the interface thickness is negligibly small.

• Only one failure mechanism is allowed.

L6.32



Influence on stable time increment (Abaqus/Explicit only):


• Often require a small stable time increment.

• Cohesive elements are generally thin and sometimes quite stiff.

• Consequently, they often have a stable time increment that is

significantly less than that of the other elements in the model.


• Cohesive surface behavior with the default cohesive stiffness

properties is formulated to minimally affect the stable time increment.

• Abaqus uses default contact penalties to model the cohesive

stiffness behavior in this case.

• You can specify a non-default cohesive stiffness values.

• However, high stiffnesses may reduce the stable time increment.

e

d

Lt

c

L6.33



Mass:


• The element material definitions include mass.


• Do not add mass to the model.

• Indented for thin adhesive interfaces; thus, neglecting adhesive

mass is appropriate for most applications.

• However, nonstructural mass can be added to the contacting

elements if necessary.

L6.34



Summary:


• Are recommended for more detailed adhesive connection modeling.

• Additional preprocessing effort (and often increased computational cost) is compensated for by gaining:

• Direct control over the connection mesh

• Additional constitutive response options

• E.g., model adhesives of finite thickness


• Provides a quick and easy way to model adhesive connections.

• Negligible interface thicknesses only

• Surfaces can bond anytime contact is established (“sticky” contact)

• Model contact adhesives, Velcro, tape, and other bonding agents that can stick after separation.

Workshop 3 (Part 2)

L6.36


Workshop 3 (Part 2)

• Crack growth in a three-point bend specimen using surface-based

cohesive behavior

• Repeat the element-based exercise using surface-based behavior

• Use default traction-separation elastic properties

• Compare with element-based results

Virtual Crack Closure Technique

(VCCT)

Lecture 7

L7.2


Overview

• Introduction

• VCCT Criterion

• Output

• VCCT Plug-in

• Comparison with Cohesive Behavior

• Examples

• Workshop 5

Introduction

L7.4


Introduction

• Motivation is aircraft composite

structural analysis

• To reduce the cost of laminated

composite structures, large

integrated bonded structures are

being considered.

• In primary structures,

bondlines and interfaces

between plies are required to

carry interlaminar loads.

• Damage tolerance

requirements dictate that

bondlines and interfaces carry

required loads with damage.

Modeling debonding along

skin-stringer interface

L7.5


Introduction

• Analysis requirements for composite damage

• Apply Linear Elastic Fracture Mechanics (LEFM) to bondlines and

interfaces

• 2D and 3D delaminations

• Propagation

• Mode separation

• Multiple cracks

• Non-linear behavior (e.g., postbuckling)

• Composite structure

• Practical (CPU time, minimum set of models)

L7.6


Introduction

• VCCT uses LEFM concepts

• Based on computing the

energy release rates for

normal and shear crack-tip

deformation modes.

• Compare energy release

rates to interlaminar fracture

toughness.

• See Rybicki, E. F., and Kanninen,

M. F., "A Finite Element Calculation

of Stress Intensity Factors by a

Modified Crack Closure Integral,"

Engineering Fracture Mechanics,

Vol. 9, pp. 931-938, 1977.

1,6 ,2,5

,2,5

1,6

Nodes 2 and 5 will start to release when:

1

2

where

mode I energy release rate

critical mode I energy release rate

width

vertical force between nodes 2 and 5

vertical

vI IC

I

IC

v

v FG G

bd

G

G

b

F

v

displacement between nodes 1 and 6

Mode II treated

similarly

Node numbers

are shown

Pure Mode IModified VCCT

VCCT Criterion

L7.8


VCCT Criterion

• The debond capability is used to perform the crack propagation analysis

for initially bonded crack surfaces.

• The crack propagation analysis allows for five types of fracture criteria:

• Critical stress criterion

• Crack opening displacement criterion

• Crack length vs. time criterion

• VCCT criterion

• Low-cycle fatigue criterion

• Defining case 4, “VCCT criterion,” is the subject of this lecture.

• The details of cases 1, 2, and 3 are not discussed here. Please

consult the Abaqus Analysis User’s Manual for more details.

• The details of case 5 will be discussed later in Lecture 8 “Low-cycle

Fatigue.”

1

2

3

4

5

L7.9


VCCT Criterion

• When using VCCT to model crack propagation,

• you must:

• define contact pairs for potential crack surfaces;

• define initially bonded crack surfaces;

• activate the crack propagation capability; and

• specify the VCCT criterion.

• you also may:

• define spatially varying critical energy release rates;

• use viscous regularization, contact stabilization, and/or automatic

stabilization to overcome convergence difficulties for unstable

propagating cracks;

• use a linear scaling technique to accelerate convergence for VCCT.

1

2

3

4

L7.10


VCCT Criterion

• Defining the VCCT criterion is not currently supported in Abaqus/CAE.

• However, the VCCT plug-in is available and allows you to interactively

define the debond interface(s).

• The details of the VCCT plug-in will be discussed later in this

lecture.

• Downloaded from “VCCT plug-in utility,” SIMULIA Answer 3235.

L7.11


VCCT Criterion

• Example: Double cantilever beam (DCB)

• Analyze debonding of a DCB model using the VCCT criterion.

• Steps required for setting up the model include:

• Define slave (TopSurf) and master (BotSurf) surfaces along the debond

interface.

• Define a set (bond) containing the initially bonded region (part of TopSurf

in this example).

• The Keywords interface is illustrated in this example.

BotSurf

TopSurf

bond

L7.12


VCCT Criterion

• Define contact pairs for potential crack surfaces

• Potential crack surfaces are modeled as slave and master contact

surfaces.

• Any contact formulation except the finite-sliding, surface-to-surface

formulation can be used.

• Cannot be used with self-contact.

1


1, 121, 1


_TopBeam_S1, S1


_BotBeam_S1, S1

*CONTACT PAIR, INTER=...

TopSurf, BotSurf

slave surface master surface

Note: The frictionless interaction property is assumed.

BotSurfTopSurf

bond

L7.13


VCCT Criterion

• Define initially bonded crack surfaces

• The initially bonded contact pair is identified with the *INITIAL

CONDITIONS, TYPE=CONTACT option.

BotSurfTopSurf

bond


1, 121, 1


_TopBeam_S1, S1


_BotBeam_S1, S1


TopSurf, BotSurf



slave surface master surface a list of slave nodes

that are initially bonded

2

L7.14


VCCT Criterion

• The unbonded portion of the slave surface will behave as a regular

contact surface.

• If the node set that includes the initially bonded slave nodes is not

specified, the initial contact condition will apply to the entire contact pair.

• In this case, no crack tips can be identified, and the bonded

surfaces cannot separate.

• For the VCCT criterion, the initially bonded nodes are bonded in all

directions.

L7.15


VCCT Criterion

• Activate the crack propagation capability

• The DEBOND option is used to activate crack propagation in a given

step.

• The SLAVE and MASTER parameters identify the surfaces to be

debonded.

BotSurfTopSurf

bond


1, 121, 1


_TopBeam_S1, S1


_BotBeam_S1, S1


TopSurf, BotSurf



*STEP, NLGEOM

*STATIC

...

*DEBOND, SLAVE=TopSurf, MASTER=BotSurf

3

L7.16


VCCT Criterion

• Specify the VCCT criterion

• The BK law model is used in this

example. *NSET, NSET=bond, GENERATE

1, 121, 1


_TopBeam_S1, S1


_BotBeam_S1, S1


TopSurf, BotSurf



*STEP, NLGEOM

*STATIC

...


*FRACTURE CRITERION, TYPE=VCCT,

MIXED MODE BEHAVOIR=BK

280.0, 280.0, 0.0, 2.284

GIC GIIC GIIIC

II IIIequivC IC IIC IC

I II III

G GG G G G

G G G

BotSurfTopSurf

bond

BK law:

4

L7.17


VCCT Criterion

• The crack-tip node debonds when the fracture criterion, f,

reaches the value 1.0 within a given tolerance, ftol:

where

Gequiv is the equivalent strain energy release rate, and

GequivC is the critical equivalent strain energy release rate calculated

based on the user-specified mode-mix criterion and the bond

strength of the interface.

• For the VCCT criterion, the default value of ftol is 0.2.

• Use following option to control ftol:

,equiv

equivC

Gf

G

*FRACTURE CRITERION, TYPE=VCCT, TOLERANCE=ftol

1 1 .tolf f

L7.18


VCCT Criterion

• In the DCB model, the tolerance is set to 0.1.


1, 121, 1


_TopBeam_S1, S1


_BotBeam_S1, S1


TopSurf, BotSurf



*STEP, NLGEOM

*STATIC

...



MIXED MODE BEHAVOIR=BK, TOLERANCE=0.1

280.0, 280.0, 0.0, 2.284

BotSurfTopSurf

bond

L7.19


VCCT Criterion

• In addition to the BK law model, Abaqus/Standard also provides two

other commonly used mode-mix criteria for computing GequivC: the Power

law and the Reeder law models.

• An appropriate model is best selected empirically.

• Power law

• Reeder law

• Applies only to three-dimensional problems

am an ao

equiv I II III

equivC IC IIC IIIC

G G G G

G G G G

*FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVIOR=POWER

GIC, GIIC, GIIIC, am, an, ao

*FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVIOR=REEDER

GIC, GIIC, GIIIC,

III II IIIequivC IC IIC IC IIIC IIC

II III i

G G GG G G G G G

G G G

L7.20


VCCT Criterion

• Spatially varying critical energy release rates

• The VCCT criterion can be defined with varying energy release rates by

specifying the critical energy release rates at all nodes on the slave

surface.

• In this case, the critical energy release rates should be interpolated

from the critical energy release rates specified at the nodes with the

*NODAL ENERGY RATE option.

• However, the exponents (e.g., ) are still read from the data lines

under the *FRACTURE CRITERION option.

*NODAL ENERGY RATE

node ID1, GIC, GIIC, GIIIC


...

*STEP

*STATIC

...

*FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVIOR=BK, NODAL ENERGY RATE

GIC, GIIC, GIIIC,

model data

L7.21


VCCT Criterion

• Viscous regularization for VCCT

• Can be used to overcome some

convergence difficulties for

unstable propagating cracks.

• Example: DCB

• Set the value of the viscosity

coefficient to 0.1.


1, 121, 1


_TopBeam_S1, S1


_BotBeam_S1, S1


TopSurf, BotSurf



*STEP, NLGEOM

*STATIC

...

*DEBOND, SLAVE=TopSurf,

MASTER=BotSurf, VISCOSITY=0.1

*FRACTURE CRITERION, TYPE=VCCT, MIXED

MODE BEHAVOIR=BK, TOLERANCE=0.1

280.0, 280.0, 0.0, 2.284

BotSurfTopSurf

bond

L7.22


VCCT Criterion

• In addition, contact and automatic stabilization that are not specific to

VCCT can be also used to aid convergence.

• They are built into Abaqus/Standard and are compatible with VCCT.

• Note that the crack propagation behavior may be modified by the

damping forces.

• Therefore, monitor the damping energy (ALLVD or ALLSD) and

compare it with the total strain energy in the model (ALLSE) to

ensure that the results are reasonable in the presence of damping.

• ALLVD stores the damping energy generated from viscous

regularization.

• ALLSD stores the damping energy generated from contact

stabilization and automatic stabilization.

L7.23


VCCT Criterion

• Linear scaling to accelerate convergence for VCCT

• Abaqus provides a linear scaling technique to quickly converge to the

critical load state. This reduces the solution time required to reach the

onset of crack growth.

• This technique works best for models in which the deformation is

nearly linear before the onset of crack growth.

• Once the first crack-tip node releases, the linear scaling calculations will

no longer be valid and the time increment will be set to the default value.

• Usage:

*CONTROLS, LINEAR SCALING

where is the coefficient of linear scaling.

• For details of linear scaling to accelerate convergence for VCCT, see

“Crack propagation analysis,” Section 11.4.3 of the Abaqus Analysis

User’s Manual.

L7.24


VCCT Criterion

• Tips for using the VCCT criterion

• Crack propagation problems using the VCCT criterion are numerically

challenging.

• To help you create a successful model, several tips for using the VCCT

criterion are provided:

• The master debonding surfaces must be continuous.

• The tie MPCs should NOT be used for the slave debonding surface

to avoid overconstraints.

• A small clearance between the debonding surfaces can be specified

to eliminate unnecessary severe discontinuity iterations during

incrementation as the crack begins to progress.

……

• Note: More tips are provided in “Crack propagation analysis,” Section

11.4.3 of the Abaqus Analysis User’s Manual.

Output

L7.26


Output

• The following output options are

provided to support the VCCT

criterion:

• Abaqus/CAE supports the surface

output requests for VCCT.

*OUTPUT, FIELD, FREQUENCY=freq

*CONTACT OUTPUT, MASTER=master,

SLAVE=slave

*OUTPUT, HISTORY, FREQUENCY=freq

*CONTACT OUTPUT, [(MASTER=master,

SLAVE=slave)|(NSET=nset)]

L7.27


Output

• The following bond failure quantities can be requested as surface output:

DBT The time when bond failure occurred

DBSF Fraction of stress at bond failure that still remains

DBS Stress in the failed bond that remains

OPENBC Relative displacement behind crack.

CRSTS Critical stress at failure.

ENRRT Strain energy release rate.

EFENRRTR Effective energy release rate ratio.

BDSTAT Bond state (=1.0 if bonded, 0.0 if unbonded)

• All of the above variables can be visualized in Abaqus/Viewer.

• The initial contact status of all of the slave nodes is printed in the data (.dat) file.

L7.28


Output

• Example: DCB

• Request surface output:

BotSurfTopSurf

bond

...



*STEP, NLGEOM

*STATIC

...

*DEBOND, SLAVE=TopSurf, MASTER=BotSurf, VISCOSITY=0.1

*FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVOIR=BK, TOLERANCE=0.1

280, 280, 280, 2.284

...

*OUTPUT, FIELD, VAR=PRESELECT

*CONTACT OUTPUT, SLAVE=TopSurf, MASTER=BotSurf

DBT, DBS, OPENBC, CRSTS, ENRRT, BDSTAT

*OUTPUT, HISTORY

*CONTACT OUTPUT, SLAVE=TopSurf, MASTER=BotSurf, NSET=bond

DBT, DBS, OPENBC, CRSTS, ENRRT, BDSTAT

*NODE OUTPUT, NSET=tip

U2, RF2

*END STEP

field output

history output

L7.29


Output

• Results

VCCT

VCCT Plug-in

L7.31


VCCT Plug-in

• VCCT plug-in

• provides an interactive interface to define the debond interface(s).

• supports the following keyword options required for VCCT analysis:

• For details please refer to “VCCT plug-in utility,” SIMULIA Answer 3235.


*DEBOND, SLAVE=slave, MASTER=master, OUTPUT=[fil|dat|both], VISCOSITY=


MIXED MODE BEHAVIOR=[BK|POWER|REEDER], TOLERANCE=ftol,

NODAL ENERGY RATE

*NODAL ENERGY RATE

*CONTROLS, LINEAR SCALING

L7.32


VCCT Plug-in

• Example: Double Cantilever Beam (DCB)

• The VCCT plug-in is discussed in the context of the Keywords interface

presented earlier.

BotSurf

TopSurf

bond

slave surface

master surface

initially bonded region

L7.33


VCCT Plug-in


• Frictionless contact is assumed.

*SURFACE INTERACTION, NAME=IntProp-1

1.

*FRICTION

0.0

*CONTACT PAIR, INTERACTION=IntProp-1

TopSurf, BotSurf

BotSurfTopSurf

bond

1

L7.34


VCCT Plug-in

• Define the VCCT criterion

• Select the fracture criterion, viscosity

coefficient, and cutback tolerance.

...

*STEP, NLGEOM

*STATIC

...

*DEBOND, SLAVE=TopSurf, MASTER=BotSurf,

VICOSITY=0.1

*FRACTURE CRITERION, TYPE=VCCT, TOLERANCE=0.2,


280, 280, 280, 2.284

2

2a

BotSurfTopSurf

bond

L7.35


VCCT Plug-in

• Specify critical strain energy release rates2b

BotSurfTopSurf

bond

...

*STEP, NLGEOM

*STATIC

...

*DEBOND, SLAVE=TopSurf, MASTER=BotSurf,

VICOSITY=0.1

*FRACTURE CRITERION, TYPE=VCCT, TOLERANCE=0.2,


280, 280, 280, 2.284

L7.36


VCCT Plug-in

• The VCCT plug-in also supports defining spatially varying critical energy

release rates.

• Click mouse button 3 to manage the table.

*NODAL ENERGY RATE



...

*STEP

*STATIC

...


MIXED MODE BEHAVIOR=BK, NODAL ENERGY RATE

GIC, GIIC, GIIIC,

L7.37


VCCT Plug-in

• Define the VCCT bonded interface

• Select the initially bonded region,

the crack propagation output file

and frequency, and the debond

initiation step.

• Note: The VCCT plug-in

allows specification of linear

scaling.

3



*STEP, NAME=Step-1

*STATIC, NLGEOM

...

*DEBOND, SLAVE=TopSurf, MASTER=BotSurf, VISCOSITY=0.1

*FRACTURE CRITERION, TYPE=VCCT, MIXED MODE BEHAVOIR=BK

280, 280, 280, 2.284

L7.38


VCCT Plug-in

• The relevant keywords

will be generated when

Abaqus/CAE writes the

input file.

debond

field output

surface interaction

initial contact conditions

fracture criterion

history output

Comparison with Cohesive Behavior

L7.40



• VCCT and cohesive behavior are very similar in their application and

formulation.

• Both theories

• are used to model interfacial shearing and delamination crack

propagation and failure,

• use an elastic damage constitutive theory to model the

material's response once damage has initiated, and

• dissipate the same amount of fracture energy between damage

initiation and complete failure.

L7.41



• The fundamental difference between VCCT and cohesive behavior is in

the way crack propagation is predicted.

• In VCCT an existing flaw is assumed.

• VCCT is appropriate for brittle crack propagation problems.

• However, cohesive behavior can model damage initiation.

• Damage initiation in cohesive behavior is based strictly on the

predefined ultimate (normal and/or shear) stress/strain limit.

• Cohesive behavior can be used for both brittle and ductile crack

propagation problems.

L7.42



• VCCT may be viewed as more fundamentally based on fracture

mechanics.

• The damage initiation and damage evolution are both based on

fracture energy, whereas cohesive behavior use the fracture energy

only during damage evolution.

• Applicability of VCCT is limited to “self-similar” crack propagation

analyses.

• This implies a steady-state running crack.

• Difficult to reproduce in practice.

L7.43



• Summary: Complementary techniques for modeling of debonding

• Both are needed to satisfy general fracture requirements

VCCT Cohesive behavior

Use the debond framework (surface based) Interface elements (element based) or

contact (surface based)

Assumes an existing flaw Can model crack initiation

Brittle fracture using LEFM occurring along a

well defined crack front

Ductile fracture occurring over a smeared

crack front modeled with spanning cohesive

elements or cohesive contact

Requires GI, GII, and GIII Requires E, σmax, GI, GII, and GIII

Crack propagates when strain energy release

rate exceeds fracture toughness

Crack initiates when cohesive traction

exceeds critical value and releases critical

strain energy when fully open

Crack surfaces are rigidly bonded when

uncracked.

Crack surfaces are joined elastically when

uncracked.

Available only in Abaqus/Standard Available in Abaqus/Standard and

Abaqus/Explicit

Examples

L7.45


Examples

• Verification problems

• DCB

• SLB

• ENF

• Alfano-Crisfield

• Alfano, G., and M. A. Crisfield, “Finite Element Interface Models for

the Delamination Analysis of Laminated Composites: Mechanical

and Computational Issues,” International Journal for Numerical

Methods in Engineering, vol. 50, pp. 1701–1736, 2001.

• Also available as Abaqus Benchmark Problem 2.7.1 with cohesive

elements

• NASA Panel

• Reeder, J.R., Song, K., Chunchu, P.B., and Ambur, D.R.,

“Postbuckling and Growth of Delaminations in Composite Plates

Subjected to Axial Compression,” AIAA 2002-1746.

L7.46


Examples

Euler buckling

0

5000

10000

15000

20000

25000

30000

0 0.01 0.02 0.03 0.04 0.05

Displacement (in)

Lo

ad

(lb

)

FEA

closed form

Multiple crack tips

Buckling driven delaminations

• Compression Buckling/Delamination Single Disbond (Unreinforced)

L7.47


Examples

L7.48


Examples

Multiple cracks can also be addressed

• Compression Buckling/Delamination Multiple Disbonds (Unreinforced)

L7.49


Examples

L7.50


Examples

• T-Joint Pull–off Model

L7.51


• Postbuckling Behavior of Skin-Stringer Panels

Examples

• VCCT can be applied to

determine the global

strength and failure mode

for typical aerospace

composite structures like

this skin/stringer panelCourtesy Boeing

L7.52


Examples

Displacement

imposed at corner nodesContact surfaces defined

for region of fracture

L7.53


Examples

Initially bonded nodes

Initially debonded nodes

Crack tip

L7.54


Examples

The Abaqus Tech Brief on skin/stringer bonded joint

analysis can be downloaded from www.simulia.com

http://www.simulia.com/

L7.55


Examples

Workshop 5

L7.57


Workshop 5

• Crack growth in a three-point bend specimen using VCCT

• Repeat the cohesive-based exercises using VCCT and compare results

Low-cycle Fatigue

Lecture 8

L8.2


Overview

• Introduction

• Low-cycle Fatigue in Bulk Materials

• Low-cycle Fatigue at Material Interfaces

Introduction

L8.4


Introduction

• Low-cycle fatigue analysis is a quasi-static analysis of a structure

subjected to sub-critical cyclic loading.

• It can be associated with thermal as well as mechanical loading.

• In Abaqus can simulate low-cycle fatigue in:

• bulk ductile materials

• material interfaces

L8.5


Introduction

• Low-cycle fatigue analysis uses the direct cyclic procedure to directly

obtain the stabilized cyclic response of the structure.

• The direct cyclic procedure combines a Fourier series

approximation with time integration of the nonlinear material

behavior to obtain the stabilized cyclic solution iteratively using a

modified Newton method.

• You can control the number of Fourier terms, the number of

iterations, and the incrementation during the cyclic time period

to improve the accuracy.

• Within each loading cycle, it assumes geometrically linear behavior and

fixed contact conditions.

• Geometric nonlinearity can be included only in any general step

prior to a direct cyclic step

• For more details, please see “Low-cycle fatigue analysis using the direct

cyclic approach,” Section 6.2.7 of the Abaqus Analysis User’s Manual.

L8.6


Introduction

• Defining low-cycle fatigue analysis

where t0: initial time increment

T: time of a single loading cycle

tmin: minimum time increment allowed

tmax: maximum time increment allowed

n0: initial number of terms in the Fourier series

nmax: maximum number of terms in the Fourier series

n: increment in number of terms in the Fourier series

imax: maximum number of iterations allowed in a step

N: total number of cycles allowed in a step

Nmin: minimum increment in N over which the damage is extrapolated forward

Nmax: maximum increment in N over which the damage is extrapolated forward

Dtol: damage extrapolation tolerance

*DIRECT CYCLIC, FATIGUE, [CETOL=tolerance, DELTMX=max]

t0, T, tmin, tmax, n0, nmax, n, imax

Nmin, Nmax, N, Dtol

controls the incrementation

controls the iteration

controls the Fourier

series representations

controls damage

extrapolation in

the bulk material

Low-cycle Fatigue in Bulk Materials

L8.8



• Abaqus/Standard offers a general capability for modeling the

progressive damage and failure of ductile materials due to stress

reversals and the accumulation of inelastic strain energy when the

material is subjected to sub-critical cyclic loadings.

• Damage in low-cycle fatigue is defined within the same general

framework of modeling progressive damage and failure (continuum

damage approach):

• a constitutive behavior of undamaged ductile materials;

• a damage initiation criterion; and

• a damage evolution response.

• The damage initiation and evolution are characterized by the stabilized

accumulated inelastic hysteresis strain energy per stabilized cycle.

• Note: Damage initiation and evolution for low-cycle fatigue analysis is

currently not supported in Abaqus/CAE.

L8.9



• Example: Thermal cycling failure of solder joint

• Solder joint reliability analysis of automotive electronics under cyclic

thermal loading.

The crack propagates forward

L8.10



• Quarter-symmetry model:

• Solder material (63Sn/37Pb)

• Modeled using temperature-

dependent elasticity and

power-law creep.

• Low-cycle fatigue analysis run for

801 cycles.

• Each thermal cycle is 1920

seconds.

• Define the low-cycle fatigue analysis

step

*STEP, INC=800

*DIRECT CYCLIC, FATIGUE

60., 1920.,,, 29, 29,, 100

50, 100, 801, 1.1

Temperature load in once cycle

Quarter-symmetry model

electronic chip

printed

circuit

board

gullwing

leads

solder joints

L8.11



• Damage initiation criterion for ductile damage in low-cycle fatigue

• The onset of damage in low-cycle fatigue is characterized by the accumulated inelastic hysteresis energy per cycle, w, in a material point when the structure response is stabilized in the cycle.

• The cycle number (N0) in which damage is initiated is given by

where c1 and c2 are material constants.

• Note: c1 depends on the system of units in which you are working;

care is required to modify c1 when converting to a different system units.

• The initiation criterion can be used in conjunction with any ductile material.

• Damage initiation criterion output:

CYCLEINI Number of cycles to initialized the damage

20 1

cN c w

L8.12



• Defining damage initiation criterion


*MATERIAL, NAME=SOLDERF

*ELASTIC

31976, 0.4, 273

20976, 0.4, 398

*EXPANSION, ZERO=273

21E-6,

*CREEP,LAW=USER

*DAMAGE INITIATION, CRITERION=HYSTERESIS ENERGY

33.3, -1.52

...

*STEP, INC=800


60., 1920.,,, 29, 29,, 100

50, 100, 801, 1.1

c1 c2


solder joint

bond pad

underneath

solder joint

20 1

cN c w

L8.13



• Damage evolution for ductile damage in low-cycle fatigue

• Once the damage initiation criterion is satisfied at a material point, the damage state is calculated and updated based on the inelastic hysteresis energy for the stabilized cycle.

• The rate of the damage (dD/dN) at a material point per cycle is given by

where c3 and c4 are material constants, L is the characteristic length

associated with the material point, and D is the scalar damage variable.

• The details of choosing characteristic length will be discussed later.

• Note: c3 depends on the system of units in which you are working;

care is required to modify c3 when converting to a different system units.

43

cc wdD

dN L

L8.14



• Defining damage evolution



*MATERIAL, NAME=SOLDERF

*ELASTIC

31976, 0.4, 273

20976, 0.4, 398

*EXPANSION, ZERO=273

21E-6,

*CREEP,LAW=USER

*DAMAGE INITIATION, CRITERION=HYSTERESIS ENERGY

33.3, -1.52

*DAMAGE EVOLUTION, TYPE=HYSTERESIS ENERGY

9.88E-4, 0.98

...

*STEP, INC=800


60., 1920.,,, 29, 29,, 100

50, 100, 801, 1.1

c3 c4

43

cc wdD

dN L

L8.15



• Results

Damage initiation at joint toe

Cycle number 199

Damage evolution

Cycle number 749

Damage evolution

Cycle number 801

L8.16



• Characteristic length associated with an integration point

• The characteristic length implemented in the damage evolution model is

based on the element geometry and formulation:

Element type Characteristic length used in

the damage evolution model

first-order element typical length of a line across the element

second-order element half of the typical length of a line across the element

beam and truss characteristic length along the element axis

membrane and shell characteristic length in the reference surface

axisymmetric element characteristic length in the rz plane only

cohesive element the constitutive thickness

L8.17



• The characteristic length is used because the direction in which fracture

occurs is not known in advance.

• Therefore, elements with large aspect ratios will have rather

different behavior depending on the direction in which the damage

occurs.

• Some mesh sensitivity remains because of this effect, and

elements that are as close to square as possible are

recommended.

• However, since the damage evolution law is energy based,

mesh dependency of the results may be alleviated.

L8.18



• Difficulties associated with element removal and LCF

• When elements are removed from the model, their nodes remain in the

model even if they are not attached to any active elements.

• When the solution progresses, these nodes might undergo non-

physical displacements in Abaqus/Standard.

• For example, applying a point load to a node that is not

attached to an active element will cause convergence

difficulties since there is no stiffness to resist the load.

• It is the user’s responsibility to prevent such situations.

Low-cycle Fatigue at

Material Interfaces

L8.20


Low-cycle Fatigue at Material Interfaces

• Delamination growth in composites due to sub-critical cyclic loadings is a

widespread concern for the aerospace industry.

• The low-cycle fatigue criterion available in Abaqus models progressive

delamination growth at interfaces in laminated composites subjected to

sub-critical cyclic loadings.

• The interface along which the delamination (or crack) propagates

must be indicated in the model.

• The onset and growth of fatigue delamination at the interfaces are

characterized by the relative fracture energy release rate

• The fracture energy release rates at the crack tips in the

interface elements are calculated based on the VCCT

technique.

L8.21



• The onset and fatigue delamination growth

at the interfaces are characterized by using

the Paris Law, which relates crack growth

rates da/dN to the relative fracture energy

release rate G,

G = Gmax – Gmin

where Gmax and Gmin correspond to the

strain energy release rates when the

structure is loaded up to Pmax and Pmin,

respectively.

• The Paris regime is bounded by Gthresh and

Gpl.

• Below Gthresh, there is no fatigue crack

initiation or growth.

• Above Gpl, the fatigue crack will grow

at an accelerated rate.

a: crack length

N: number of cycles

G: strain energy release rate

Gthresh: strain energy release rate threshold

Gpl: strain energy release rate upper limit

GequivC: critical equivalent strain

energy release rate

L8.22



• GequivC is calculated based on the

user-specified mode-mix criterion

and the bond strength of the interface.

• This was discussed in Lecture 7

“VCCT.”

• Onset of fatigue delamination

• The fatigue crack growth initiation

criterion is defined as:

where c1 and c2 are material

constants.

• The interface elements at the

crack tips will not be released

unless the above equation is

satisfied and Gmax Gthresh.

21

1.0,c

Nf

c G

a: crack length

N: number of cycles





energy release rate

L8.23



• Fatigue delamination growth

• Once the delamination growth criterion

is satisfied at the interface, the crack

growth rate da/dN can be calculated

based on G.

• da/dN is given by the Paris Law if

Gthresh< Gmax< Gpl,

where c3 and c4 are material

constants.

43

cdac G

dN

a: crack length

N: number of cycles





energy release rate

43

cdac G

dN

L8.24



• Fatigue crack growth governed by the Paris Law

• Repeat the above process until the maximum number of cycles is reached or until the ultimate load carrying capability is reached.

Calculate the relative fracture

energy release rate, G, when

the structure is loaded between its

maximum and minimum values.

Crack initiation: 21

coN c G

Crack evolution: 43

cdac G

dN

Damage extrapolation: Calculate

the incremental number of cycles,

N, for each crack tip and find

minimum cycles to fail, Nmin

If N + N > No

N + N

43

cN N Na a Nc G

a: crack length

N: number of cycles

N: incremental number of cycles

c1, c2 , c3, c4: material constants

If Gthresh < Gmax < Gpl

Release the most

critical element

G = Gmax(Pmax) – Gmin(Pmin)

1 2

3

L8.25



• The syntax used to define the low-cycle fatigue criterion and the

corresponding output requests is similar to those used for the VCCT

criterion except the following:

• For the low-cycle fatigue criterion, set TYPE=FATIGUE on the

*FRACTURE CRITERION option:

• By default, Gthresh/GequivC = 0.01 and Gpl/GequivC = 0.85.

• Note: Defining the low-cycle criterion is not currently supported in

Abaqus/CAE.

*FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=[BK|REEDER]

c1, c2, c3, c4, Gthresh/GequivC, Gpl/GequivC, GIC, GIIC

GIIIC, , , fv

*FRACTURE CRITERION, TYPE=FATIGUE, MIXED MODE BEHAVIOR=POWER

c1, c2, c3, c4, Gthresh/GequivC, Gpl/GequivC, GIC, GIIC

GIIIC, am, an, ao, , fv

L8.26



• Example: Low-cycle fatigue prediction for the DCB model

• This case consists of the following steps:

• Step 1: VCCT analysis

• This step can be used to check whether the peak loading leads

to static crack propagation.

• Step 2: Low-cycle fatigue analysis

• This step assesses the fatigue life of the DCB model subjected

to sub-critical cyclic loading.

displacement loading in one cycle

0 10.50

=0.001

u2

t

BotSurfTopSurf

bondu2

u2

L8.27



• Partial input:

*STEP, INC=5000


0.25,1,,,25,25,,5

,,1000


MASTER=BotSurf

*FRACTURE CRITERION, TYPE=FATIGUE,

MIXED MODE BEHAVIOR=BK

0.5,-0.1,4.8768E-6,1.15,,,280,280

280,2.284

*OUTPUT, FIELD

*CONTACT OUTPUT

BDSTAT, DBT, DBS, OPENBC, CRSTS,

ENRRT

...

*END STEP

...

*CONTACT PAIR, SMALL SLIDING

TopSurf, BotSurf



*STEP, NLGEOM

*STATIC

...


MASTER=BotSurf



280, 280, 280, 2.284

*OUTPUT, FIELD

*CONTACT OUTPUT, SLAVE=TopSurf,

MASTER=BotSurf


ENRRT

*END STEP

Step 2:

Fatigue

analysis

Step 1:

VCCT

analysis

Model

data

BotSurfTopSurf

bond

L8.28



• The procedure to complete the DCB model through the first step (the VCCT analysis) is exactly the same as that discussed in Lecture 7 “VCCT.”


• Define initially bonded crack surfaces

• Activate the crack propagation capability in the first step

• Specify the VCCT criterion in the first step (a static, general step)

• The details of defining the low-cycle fatigue analysis (the second step) will be discussed next.

...

*CONTACT PAIR, SMALL SLIDING

TopSurf, BotSurf



*STEP, NLGEOM

*STATIC

...


MASTER=BotSurf



280, 280, 280, 2.284

*OUTPUT, FIELD

*CONTACT OUTPUT


ENRRT

...

*END STEP

Step 1:

VCCT

analysis

model

data

BotSurfTopSurf

bond

1

2

3

4

L8.29



• Define the low-cycle fatigue analysis

• The following data are used to define this

low-cycle fatigue analysis:

• Initial time increment: 0.25 sec

• Time of a single loading cycle: 1 sec

• Initial number of terms in the Fourier

series: 25

• Maximum number of terms in the

Fourier series: 25

• Maximum number of iterations

allowed in the step: 5

• Total number of cycles allowed in

the step: 1000

• Default values are used for all other

entries.

...

*STEP, INC=5000

Low-cycle Fatigue Analysis


0.25,1,,,25,25,,5

,,1000

BotSurfTopSurf

bond

5

L8.30



• Activate the crack propagation capability

• Similar to the VCCT analysis, the

*DEBOND option is used to activate the

crack propagation in the low-cycle

fatigue analysis step.

• The SLAVE and MASTER

parameters identify the surfaces to

be debonded.

...

*STEP, INC=5000



0.25,1,,,25,25,,5

,,1000


MASTER=BotSurf

BotSurfTopSurf

bond

6

L8.31



• Specify the low-cycle fatigue criterion

• In this model, the material constants are

assumed to be the following:

• c1 = 0.5,

• c2 = –0.1

• c3 = 4.8768E–6

• c4 = 1.15

• Note: The values of these material

constants should be determined

experimentally.

• The BK model (default) is used.

...

*STEP, INC=5000



0.25,1,,,25,25,,5

,,1000


MASTER=BotSurf



0.5,-0.1,4.8768E-6,1.15,,,280,280

280,2.284

21

1.0c

Nf

c G

43

cdac G

dN

GIC GIIC

GIIIC

BotSurfTopSurf

bond

7

L8.32



• Request output

• The output options for the low-cycle

fatigue criterion are same as those for

the VCCT criterion.

...

*STEP, INC=5000



0.25,1,,,25,25,,5

,,1000


MASTER=BotSurf



0.5,-0.1,4.8768E-6,1.15,,,280,280

280,2.284

*OUTPUT, FIELD

*CONTACT OUTPUT


ENRRT

BotSurfTopSurf

bond

8

L8.33



• Results

N=1 N=11

N=21 N=51

initially bonded nodes delamination

N is the number of cycles

L8.34



• More results

delamination growth after 100

loading cycles

crack length vs. cycle

number

Mesh-independent Fracture Modeling (XFEM)

Lecture 9

L9.2


Overview

• Introduction

• Basic XFEM Concepts

• Damage Modeling

• Creating an XFEM Fracture Model

• Example 1 – Crack Initiation and Propagation

• Example 2 – Propagation of an Existing Crack

• Example 3 – Delamination and Through-thickness Crack Propagation

• Modeling Tips

• Current Limitations

• Workshop 6

• References

Introduction

L9.4


Introduction

• The fracture modeling methods discussed so far only permit crack

propagation along predefined element boundaries

• This lecture presents a technique for modeling

bulk fracture which permits a crack to be located

in the element interior

• The crack location is independent of the mesh

L9.5


Introduction

• This modeling technique…

• Can be used in conjunction with the cohesive zone model or the virtual

crack closure technique

• Delamination can be modeled in conjunction with bulk crack

propagation

• Can determine the load carrying capacity of a cracked structure

• What is the maximum allowable flaw size for safe operation?

• Applications of this technique include the modeling of bulk fracture and

the modeling of failure in composites

• Cracks in pressure vessels or engineering structures

• Delamination and through-thickness crack modeling in composite plies

L9.6


Introduction

• Some advantages of the method:

• Ease of initial crack definition

• Mesh is generated independent of crack

• Partitioning of geometry not needed as when a crack is represented

explicitly

• Nonlinear material and nonlinear geometric analysis

• Arbitrary solution-dependent crack initiation and propagation path

• Crack path does not have to be specified a priori

• Mesh refinement studies are much simpler

• Reduced remeshing effort

• Improved convergence rate for the finite element solution (stationary

crack)

• Due to the use of singular crack tip enrichment

L9.7


Introduction

• Mesh-independent Crack Modeling – Basic Ingredients

1. Need a way to incorporate discontinuous geometry – the crack – and

the discontinuous solution field into the finite element basis functions

• eXtended Finite Element Method (XFEM)

2. Need to quantify the magnitude of the discontinuity – the displacement

jump across the crack faces

• Cohesive zone model (CZM)

3. Need a method to locate the discontinuity

• Level set method (LSM)

4. Crack initiation and propagation criteria

• At what level of stress or strain does the crack initiate?

• What is the direction of propagation?

• These topics will be discussed in this lecture

Basic XFEM Concepts

L9.9


Basic XFEM Concepts

• eXtended Finite Element Method (XFEM) Background

• XFEM extends the piecewise polynomial function space of conventional finite

element methods with extra functions

• The solution space is enriched by the extra “enrichment functions”

• Introduced by Belytschko and Black (1999) based on the partition of unity

method of Babuska and Melenk (1997)

• Can be used where conventional FEM fails or is prohibitively expensive

• Appropriate enrichment functions are chosen for a class of problems

• Inclusion of a priori knowledge of partial differential equation behavior into

finite element space (singularities, discontinuities, ...)

• Applications include modeling fracture, void growth, phase change ...

• Enrichment functions for fracture modeling

• Heaviside function to represent displacement jump across crack face

• Crack tip asymptotic function to model singularity

L9.10


Basic XFEM Concepts

• XFEM Displacement Interpolation

Heaviside enrichment term

H(x) Heaviside distribution

aI Nodal enriched DOF (jump discontinuity)

NG Nodes belonging to elements cut by crack

Crack tip enrichment term

Fa(x) Crack tip asymptotic functions

Nodal DOF (crack tip enrichment)

NG Nodes belonging to elements containing crack tip

b I

a

uI Nodal DOF for conventional shape functions NI

4

1

u (x) (x) u (x (x)b)ah

I I

I

II

I

N

N

I

N

FHN a

aa

G

L9.11


Basic XFEM Concepts

• The crack tip and Heaviside enrichment functions are multiplied by the

conventional shape functions

• Hence enrichment is local around the crack

• Sparsity of the resulting matrix equations is preserved

• The crack is located using the level set method (discussed shortly)

• Heaviside function

• Accounts for displacement jump across crack

*1 if ( ) 0( )

1 otherwiseH x

x x ns

n

x

x*

Here x is an integration point, x* is the closest point to x on the crack face and n is the unit normal at x*

H(x) = 1 below crack

H(x) = 1 above crack

L9.12


Basic XFEM Concepts

• Crack Tip Enrichment Functions (Stationary Crack Only)

• Account for crack tip singularity

• Use displacement field basis functions for sharp crack in an isotropic

linear elastic material

]2

cossinr ,2

sinsinr ,2

cosr ,2

sinr[4]-1 ),([

aa xF

Here (r, ) denote coordinate values from a polar coordinate system located at the crack tip

L9.13


Basic XFEM Concepts

• Phantom Node Approach (Crack Propagation Implementation)

• Implementation of XFEM fitting into the framework of conventional FEM

• Discontinuous element with Heaviside enrichment is treated as a

superposition of two continuous elements with phantom nodes

• Does not include the asymptotic crack tip enrichment functions

• Introduced by Belytschko and coworkers (2006) based on the

superposed element formulation of Hansbo and Hansbo (2004)

L9.14


Basic XFEM Concepts

• Level Set Method for Locating a Crack

• A level set (also called level surface or isosurface) of a real-valued function

is the set of all points at which the function attains a specified value

• Example: the zero-valued level set of f (x, y) : x2 y2 r2 is a circle of

radius r centered at the origin

• Popular technique for representing surfaces in interface tracking problems

• Two functions F and Y are used to completely describe the crack

• The level set F = 0 represents the crack face

• The intersection of level sets Y = 0 and F = 0 denotes the crack

front

• Functions are defined by nodal values whose spatial variation is

determined by the usual finite element shape functions (example

follows)

• Function values need to be specified only at nodes belonging to

elements cut by the crack

L9.15


• Calculating F and Y

• The nodal value of the function F is the signed distance of the node from

the crack face

• Positive value on one side of the crack face, negative on the other

• The nodal value of the function Y is the signed distance of the node from

an almost-orthogonal surface passing through the crack front

• The function Y has zero value on this surface and is negative on the

side towards the crack

Basic XFEM Concepts

Y = 0F = 0

1 2

3 4

0.5

1.5

Node F Y

1 0.25 1.5

2 0.25 1.0

3 0.25 1.5

4 0.25 1.0

Damage Modeling

L9.17


Damage Modeling

• Damage modeling is achieved through the use of a traction-separation

law across the fracture surface

• It follows the general framework introduced in earlier lectures



• Traction-free crack faces at failure

• Damage properties are specified as part of the bulk material definition

Damage initiation

Failure

L9.18


Damage Modeling

• Damage Initiation

• Two criteria available at present

• Maximum principal stress criterion (MAXPS)

• Initiation occurs when the maximum principal stress reaches

critical value

• Maximum principal strain criterion (MAXPE)

• Initiation occurs when the maximum principal strain reaches

critical value

• Crack plane is perpendicular to the direction of the maximum principal

stress (or strain)

• Crack initiation occurs at the center of the element

• However, crack propagation is arbitrary through the mesh

• The damage initiation criterion is satisfied when 1.0 ≤ f ≤ 1.0 + ftol

where f is the selected damage criterion and ftol is a user-specified

tolerance value

max

0

max

f

max

0

max

f

L9.19


Damage Modeling


• Any of the damage evolution models for traction-separation laws

discussed in the earlier lectures can be used

• However, it is not necessary to specify the undamaged traction-

separation response

L9.20


Damage Modeling

• Damage Stabilization

• Fracture makes the structural response nonlinear and non-smooth

• Numerical methods have difficulty converging to a solution

• As discussed in the earlier lectures, using viscous regularization helps

with the convergence of the Newton method

• The stabilization value must be chosen so that the problem definition

does not change

• A small value regularizes the analysis, helping with convergence

while having a minimal effect on the response

• Perform a parametric study to choose appropriate value for a class

of problems

L9.21


Damage Modeling

• Damage stabilization can currently be defined in Abaqus/CAE only

through the keyword editor

Creating an XFEM Fracture Model

L9.23



• Steps

1. Define damage criteria in the material model

2. Define an enrichment region (the associated material model should

include damage)

• Crack type – stationary or propagation

3. Define an initial crack, if present

4. If needed, set analysis controls to aid convergence

• Steps will be illustrated later through examples

• Crack initiation and propagation in a plate with a hole

• Propagation of an existing crack

• Delamination and through-thickness crack propagation in a double

cantilever beam

• The next few slides describe step-dependent enrichment activation

and postprocessing

L9.24



• Step-dependent Enrichment Activation

• Crack growth can be activated or deactivated in analysis steps

*STEP...

*ENRICHMENT, NAME=Crack-1, ACTIVATE=[ON|OFF]

1

2

L9.25



• Output Quantities

• Two output variables are especially useful

• PHILSM

• The signed distance function F used to represent the crack

surface

• Needed for visualizing the crack

• STATUSXFEM

• Indicates the status of the element with a value between 0.0

and 1.0

• A value of 1.0 indicates that the element is completely cracked,

with no traction across the crack faces

• Any other output variable available in the static stress analysis

procedure

L9.26



• Postprocessing

• The crack location is specified by the zero-valued level set of the signed

distance function F

• Abaqus/CAE automatically creates an isosurface view cut named

Crack_PHILSM if an enrichment is used in the analysis

• The crack isosurface is displayed by default

• Contour plots of field quantities should be done with the crack isosurface

displayed

• Ensures that the solution is plotted from the active parts of the

overlaid elements according to the phantom nodes approach

• If the crack isosurface is turned off, only values from the “lower”

element are plotted (corresponding to negative values of F)

• Probing field quantities on an element currently returns values only from

the “lower” element (on the side with negative values of F)

Example 1 – Crack Initiation and

Propagation

L9.28


Example 1 – Crack Initiation and Propagation

• Model crack initiation and propagation in a plate with a hole

• Crack initiates at the location of maximum stress concentration

• Half model is used taking advantage of symmetry

L9.29



Define the damage criteria


1

Damage initiation tolerance (default 0.05)

*MATERIAL...

*DAMAGE INITIATION, CRITERION=MAXPS, TOL=0.05

L9.30



Define the damage criteria (cont’d)


*DAMAGE INITIATION, CRITERION=MAXPS, TOL=0.05

*DAMAGE EVOLUTION, TYPE=ENERGY, MIXED MODE BEHAVIOR=POWER LAW, POWER=1.0

2870.0, 2870.0, 2870.0

1

L9.31



Define the damage criteria (cont’d)

• Damage stabilization

Keyword interface


1.e-5

Coefficient of viscosity m

• Abaqus/CAE interface currently not available

• The keyword editor may be used to add stabilization through

Abaqus/CAE.

1

L9.32



Define the enriched region

Pick enriched region

Specify contact interaction

(frictionless small-sliding contact only)

Propagating crack

2

L9.33



Define the enriched region (cont’d)

Keyword interface

No initial crack definition is needed

• Crack will initiate based on specified damage criteria

3

*ENRICHMENT, TYPE=PROPAGATION CRACK, NAME=CRACK-1,

ELSET=SELECTED_ELEMENTS, INTERACTION=CONTACT-1

Frictionless small-sliding contact interaction

2

L9.34



Set analysis controls to improve convergence behavior

• Set reasonable minimum and maximum increment sizes for step

• Increase the number of increments for step from the default value of

100

*STEP

*STATIC, inc=10000

0.01, 1.0, 1.0e-09, 0.01...

4

L9.35



Set analysis controls to improve convergence behavior (cont’d)

• Use numerical scheme applicable to discontinuous analysis

*STEP

*STATIC, inc=10000

0.01, 1.0, 1.0e-09, 0.01...

*CONTROLS, ANALYSIS=DISCONTINUOUS

4

L9.36



Set analysis controls to improve convergence behavior (cont’d)

• Increase value of maximum number of attempts before abandoning

increment (increased to 20 from the default value of 5)

*STEP

*STATIC, inc=10000

0.01, 1.0, 1.0e-09, 0.01...

*CONTROLS, ANALYSIS=DISCONTINUOUS

*CONTROLS, PARAMETER=TIME INCREMENTATION

, , , , , , , 20

8th field

4

L9.37



• Output Requests

• Request PHILSM and STATUSXFEM in addition to the usual output for

static analysis

L9.38



• Postprocessing

• Crack isosurface (Crack_PHILSM) created and displayed automatically

• Field and history quantities of interest can be plotted and animated as

usual

Example 2 – Propagation of an Existing

Crack

L9.40


Example 2 – Propagation of an Existing Crack

• Model with crack subjected to mixed mode loading

• Initial crack needs to be defined

• Crack propagates at an angle dictated by mode mix ratio at crack tip

L9.41



Define damage criteria in the material model as described in Example 1

Specify the enriched region as in Example 1

Define the initial crack

• Two methods are available to define initial crack in Abaqus/CAE

1. Create a separate part representing the crack surface or line and

assemble it along with the part representing the structure to be

analyzed

2. Create an internal face or edge representing the crack in the part

• Method 1 is preferred as it takes full advantage of the mesh-

independent crack representation possible using XFEM

• Meshing is easier using this method

• Method 2 will create nodes on the internal crack face

• Element faces/edges are forced to align with the crack

1

2

3

L9.42



Define the initial crack (cont’d)3

The crack location can be an edge or a

surface belonging to the same

instance as the enriched region or to a

different instance (preferred)

** Model data

*INITIAL CONDITIONS, TYPE=ENRICHMENT

901, 1, Crack-1, -1.0, -1.5

901, 2, Crack-1, -1.0, -1.4

901, 3, Crack-1, 1.0, -1.4

901, 4, Crack-1, 1.0, -1.5

Element Number

Relative Node Order in Connectivity

Enrichment Name

F Y

L9.43



• The other steps are as described in Example 1 and are in line with

those necessary for the usual static analysis procedure

Example 3 – Delamination and

Through-thickness Crack Propagation

L9.45


Example 3 – Delamination and Through-thickness Crack

• Model through-thickness crack propagation using XFEM and

delamination using surface-based cohesive behavior in a double

cantilever beam specimen

• Interlaminar crack grows initially

• Through-thickness crack forms once interlaminar crack becomes long

enough and the longitudinal stress value builds up due to bending

• The point at which the through-thickness crack forms depends upon the

relative failure stress values of the bulk material and the interface

L9.46


Example 3 – Delamination and Through-thickness Crack

• This model is the same as the double cantilever beam model presented

in the surface-based cohesive behavior lecture except:

• Enrichment has been added to the top and bottom beams to allow

XFEM crack initiation and propagation

Modeling Tips

L9.48


Modeling Tips

• General Information

• Averaged quantities are used in an element for determining crack

initiation and the propagation direction

• The integration point principal stress or strain values are averaged

• A new crack always initiates at the center of the element

• Within an enrichment region, a new crack initiation check is performed

only after all existing cracks have completely separated

• This may result in the abrupt appearance of multiple cracks

• Complete separation is indicated by STATUSXFEM=1

• Cracks cannot initiate in neighboring elements

• Crack propagates completely through an element in one increment

• Only the initial crack tip can lie within an element

L9.49


Modeling Tips

• The enrichment region must not include “hotspots” due to boundary

conditions or other modeling artifacts

• Otherwise, unintended cracks may initiate at such locations

• Damage initiation tolerance

• A larger value may result in multiple cracks initiating in a region

• Small value results in small increment size and convergence difficulty

• Damage stabilization

• As mentioned earlier, judicious use of viscous regularization can aid in

convergence

• Initial crack should bisect elements if possible

• Convergence is more difficult if crack is tangential to element boundaries

• Use displacement control rather than load control

• Crack propagation may be unstable under load control

L9.50


Modeling Tips

• Limit maximum increment size and start with a good guess for initial

increment size

• In general, this is a good approach for any non-smooth nonlinearity

• Analysis controls

• Can help obtain a converged solution and speed up convergence

• Contour plots of field quantities should be done with the crack

isosurface displayed

• Ensures that the solution is plotted from the active parts of the overlaid

elements according to the phantom nodes approach

• If the crack isosurface is turned off, only values from the “lower” element

are plotted (on the side with negative values of F)

L9.51


Modeling Tips

• When defining the crack using Abaqus/CAE, extend the external crack

edges beyond base geometry

• This helps avoid incorrect identification of external edges as internal due

to geometric tolerance issues

Defining a through-thickness crack in a cylindrical vessel

Top View

Current Limitations

L9.53


Current Limitations

• Implemented only for the static stress analysis procedure

• Can use only linear continuum elements

• CPE4, CPS4, C3D4, C3D8 and their reduced integration/incompatible

counterparts

• Element processing is not done in parallel

• On SMP machines, only the solver runs in parallel

• Cannot run in parallel on DMP machines

• Contour integrals for stationary cracks not currently supported

• Cannot model fatigue crack growth

• Intended for single or a few non-interacting cracks in the structure

• Shattering cannot be modeled

• An element cannot be cut by more than one crack

• Crack cannot turn more than 90 degrees in one increment

• Crack cannot branch

L9.54


Current Limitations

• The first signed distance function F must be non-zero

• If the crack lies along an element boundary, a small positive or negative

value should be used

• This slightly offsets the crack from the element boundary

• Only frictionless small-sliding contact is considered

• The small-sliding assumption will result in nonphysical contact behavior

if the relative sliding between the contacting surfaces is indeed large

• Only enriched regions can have a material model with damage

• If only a portion of the model needs to be enriched define an extra

material model with no damage for the regions not enriched

• Probing field quantities on an element currently returns values only

from the “lower” element (corresponding to negative values of F)

Workshop 6

L9.56


Workshop 6

• In this workshop, you will

continue with the analysis of a

cracked beam subjected to pure

bending using XFEM

• This workshop demonstrates:

• The ease of meshing and initial

crack definition compared to the

techniques presented in earlier

lectures

• The use of analysis controls

References

L9.58


References

1. I. Babuska and J. Melenk, Int. J. Numer. Meth. Engng (1997), 40:727-758

2. T. Belytschko and T. Black, Int. J. Numer. Meth. Engng (1999), 45:601-620

3. A. Hansbo and P. Hansbo, Comp. Meth. Appl. Mech. Engng (2004),

193:3523-3540

4. J. H. Song, P. M. A. Areias and T. Belytschko, Int. J. Numer. Meth. Engng

(2006), 67:868-893

Documents

Fracture Lecture of Abaqus