TALAT Lecture 2403: Applied Fracture Mechanics

TALAT Lecture 2403

Applied Fracture Mechanics

49 pages and 40 figures

Advanced Level

prepared by Dimitris Kosteas, Technische Universität München

Objectives: − Teach the principles and concepts of fracture mechanics as well as provide

recommendations for practical applications − Provide necessary information for fatigue life estimations on the basis of fracture

mechanics as a complementary method to the S-N concept Prerequisites: − Background in engineering, materials and fatigue required Date of Issue: 1994 EAA - European Aluminium Association

TALAT 2403 2

2403 Applied Fracture Mechanics Contents

2403.01 Historical Context ..................................................................................... 3 Fracture and Fatigue in Structures ...........................................................................3

2403.02 Notch Toughness and Brittle Fracture ................................................... 5 Notch-Toughness Performance Level as a Function of Temperature and Loading Rates.........................................................................................................................5 Brittle Fracture .........................................................................................................8

2403.03 Principles of Fracture Mechanics............................................................ 9 Basic Parameters ......................................................................................................9

Material Toughness ............................................................................................ 9 Crack Size ........................................................................................................... 9 Stress Level ......................................................................................................... 9

Fracture Criteria .....................................................................................................13 Members with Cracks ............................................................................................14 Stress Intensity Factors ..........................................................................................16 Deformation at the Crack Tip ................................................................................21 Superposition of Stress Intensity Factors...............................................................22

2403.04 Experimental Determination of Limit Values according to Various Recommendations ................................................................................................... 23

Linear-Elastic Fracture Mechanics ........................................................................24 Experimental Determination of KIc - ASTM-E399........................................... 24 Test procedure: ................................................................................................. 25

Elastic-Plastic Fracture Mechanics ........................................................................27 Crack opening displacement (COD) - BS 5762 ................................................ 27 Determination of R-Curves - ASTM-E561........................................................ 28 Determination of JIc - ASTM-E813 .................................................................. 30 Determination of J-R Curves - ASTM-E1152 .................................................. 33 Crack Opening Displacement (COD) Measurements - BS 5762...................... 34

2403.05 Fracture Mechanics Instruments for Structural Detail Evaluation... 36 Free Surface Correction Fs ....................................................................................37 Crack Shape Correction Fe ....................................................................................37 Finite Plate Dimension Correction Fw ..................................................................38 Correction Factors for Stress Gradient Fg .............................................................38 Remarks on Crack Geometry.................................................................................39

2403.06 Calculation of a Practical Example: Evaluation of Cracks Forming at a Welded Coverplate and a Web Stiffener .............................................................. 41

Coverplate ..............................................................................................................42 Web Stiffener .........................................................................................................43

2403.07 Literature/References ............................................................................... 47 2403.08 List of Figures.......................................................................................... 48

TALAT 2403 3

2403.01 Historical Context

• Fracture and fatigue in structures In his first treatise on "Mathematical Theory of Elasticity" Love, 100 years ago, discus-sed several topics of engineering importance for which linear elastic treatment appeared inadequate. One of these was rupture. Nowadays structural materials have been im-proved with a corresponding decrease in the size of safety factors and the principles of modern fracture mechanics have been developed, mainly in the 1946 to 1966 period. Fracture mechanics is the science studying the behaviour of progressive crack extension in structures. This goes along with the recognition that real structures contain disconti-nuities. Fracture mechanics is the primary tool (characteristic material values, test procedures, failure analysis procedures) in controlling brittle fracture and fatigue failures in struc-tures. The desire for increased safety and reliability of structures, after some spectacular failures, has led to the development of various fracture criteria. Fracture criteria and fracture control are a function of engineering contemplation taking into account econo-mical and practical aspects as well. Fracture and Fatigue in Structures Brittle fracture is a type of catastrophic failure that usually occurs without prior plastic deformation and at extremely high speeds. Brittle fractures are not so common as fatigue (the latter characterised by progressive crack development), yielding, or buckling failures, but when they occur they may be more costly in terms of human life and prop-erty damage. Fatigue failures according to statistics is responsible for approx. 7% of failures. Aristotle talked about hooks on molecules, breaking them meant fracture. Da Vinci and Gallileo talked about fracture, too. The big break in fracture mechanics came in 1920 with the Griffith theory, applicable mostly to brittle materials, as well as Orowan and Irwing and Williams in the 1940's. Catastrophic brittle failures were recorded in the 19th and early 20th century. There were several failures in welded Vierendeel-truss bridges in Europe shortly after being put into service before World War II. However, it was not until the large number of World War II ship failures that the problem of brittle fracture was fully appreciated by engineers. 1962 the Kings Bridge in Melbourne failed by brittle fracture at low tempera-tures due to poor details and fabrication resulting in cracks which were nearly through the flange prior to any service loading. Although this failure was studied extensively, bridge-builders did not pay particular attention until the failure of the Point Pleasent Bridge in West Virginia, USA on December 15, 1965. This was the turning point initia-ting the possibilities of fracture mechanics in civil engineering. This failure was unique in several ways, it was investigated extensively and its results were characteristic for the procedures and possibilities of fracture mechanics analysis. Therefore they are mentioned briefly here:

TALAT 2403 4

(a) fracture appeared in the eye of an eyebar caused by the growth of a flaw to a criti-cal size under normal working stresses,

(b) the initial flaw was caused through stress-corrosion cracking from the surface of

the hole, hydrogen sulphide was probably the reagent responsible, (c) the chemical composition and heat treatment of the eyebar produced a steel with

very low fracture toughness at the failure temperature, and (d) fracture resulted from a combination of factors and it would not have occurred in

the absence of anyone of these - the high hardness of the material made it susceptible to stress corrosion

cracking - close spacing of joint components made it impossible to apply paint to high

stressed regions yet provided a crevice where water could collect - the high design load of the eyebar resulted in high local stresses at the inside

of the eye greater than the yield strength of the steel. - the low fracture toughness of the steel led to complete fracture from the

slowly propagating stress corrosion crack when it had reached a depth of only 3.0 mm

It has been shown that an interrelation exists between material, design, fabrication and loading as well as maintenance. Fractures cannot be eliminated in structures by merely using materials with improved notch toughness. The designer still has the fundamental responsibility for the overall safety and reliability of the structure.

TALAT 2403 5

2403.02 Notch Toughness and Brittle Fracture

• Notch-toughness performance level as a function of temperature and loading rates

• Brittle fracture In the following chapters it will be shown how fracture mechanics can be used to de-scribe quantitatively the trade-offs among stress, material toughness, and flaw size so that the designer can determine the relative importance of each of them during design rather than during failure analysis. Notch-Toughness Performance Level as a Function of Temperature and Loading Rates The traditional mechanical property tests measure strength, ductility, modulus of elasti-city etc. There are also tests available to measure some form of notch toughness. Notch toughness is defined as the ability of a material to absorb energy (usually when loaded dynamically) in the presence of a flaw. Toughness is defined as the ability of a smooth, unnotched member to absorb energy (usually when loaded slowly). Notch toughness is measured with a variety of specimens such as the Charpy-V-notch impact specimen, dynamic tear test specimen KIc, pre-cracked Charpy, etc. Toughness is usually characterised by the area under a stress-strain curve in a slow tension test. Notches or other forms of stress raisers make structural materials susceptible to brittle fracture under certain conditions. The ductile or brittle behaviour of some structural materials like steels is well known, depending on several conditions such as temperature, loading rate, and constraint (the latter arising often in welded components among other reasons due to residual stresses and the complexity of welds). Ductile fractures are generally preceded by large amount of plastic deformation occurring usually at 45° to the direction of the applied stress. Brittle or cleavage fractures generally occur with little plastic deformation and are usually normal to the direction of principal stresses. Figure 2403.02.01 shows the var-ious fracture states and the transition from one to another depending on environmental conditions. Plane-strain behaviour refers to fracture under elastic stresses and is essentially brittle. Plastic behaviour refers to ductile failure under general yielding conditions accompanied usually, but not necessarily, with large shear lips. The transition between these two is the elastic-plastic region or the mixed-mode region. Higher loading rates move the characteristic transition curve to higher temperatures. A particular notch toughness value called the nil-ductility transition (NDT) temperature generally defines the upper limits of plane-strain behaviour under conditions of impact loading. In practice the question has to be answered regarding the level of material performance which should be re-quired for satisfactory performance in a particular structure at a specific service tem-perature, see Figure 2403.02.02. In this example and for impact loading the three different steels exhibit either plane-strain behaviour (steel 1) or elastic-plastic behaviour (steel 2), or fully plastic behaviour (steel 3) at the indicated service temperature.

TALAT 2403 6

Although fully plastic behaviour would be a very desirable level of performance, it may not be necessary or even economically feasible for many structures.

alu

Training in Aluminium Application Technologies

Notch-Thoughness Performance Levelsvs. Temperature

2403.02.01

Leve

ls o

f per

form

ance

as

mea

sure

dby

abs

orbe

d en

ergy

in n

otch

ed s

peci

men

s

Plastic

Elastic -Plastic

PlaneStrain

(Macro linear)Elastic

Impact loading

Static loading

Intermediateloading rate

NDT (Nil-Ductility Transition)

TemperatureD. Kosteas, TUM

Notch-Thoughness Performance Levelsvs. Temperature

alu


Relation between Performance and Transition Temperature for three different Steel Qualities

Plastic

Elastic-Plastic

PlaneStrain

Leve

ls o

f Per

form

ance

Service Temperature

Steel 1Steel 2Steel 3

NDT(Steel 3)

NDT(Steel 2)

NDT(Steel 1)

TemperatureD. Kosteas, TUM

2403.02.02

Not all structural materials exhibit a ductile-brittle transition. For example, aluminium as well as very high strength structural steels or titanium do not undergo a ductile-brittle transition. For these materials temperature has a rather small effect on toughness, see Figure 2403.02.03.

TALAT 2403 7

alu


Notch-Toughness vs. Temperature

2403.02.03

+

++

++ +

xx

x xx

x x ISO - ProbelongitudinaltransverseHAZfiller

parent metal

x+

Alloy 5083-H113

Not

ch to

ughn

ess

kpm

/ cm

²

Temperature in °C

6

4

2

0200-100-200

D. Kosteas, TUM

}

Notch-Toughness of Welded Aluminium Alloy 5083vs. Temperature

Notch toughness measurements express the behaviour and respective laboratory test re-sults can be used to predict service performance. Many different tests have been used to measure the notch toughness of structural materials. These include

− Charpy-V notch (CVN) impact, − drop weight NDT − dynamic tear (DT) − wide plate − Battelle drop weight tear test (DWTT) − pre-cracked Charpy, etc.

Notch toughness tests produce fracture under carefully controlled laboratory conditions. Hopefully, test results can be correlated with service performance to establish curves like in Figure 2403.02.01 for various materials and specific applications. However, even if correlations are developed for existing structures, they do not necessarily hold for certain designs or new operating conditions or new materials. Test results are expressed in terms of energy, fracture appearance, or deformation, and cannot always be translated to engineering parameters. A much better way to measure notch toughness is with the principles of fracture mech-anics, a method characterising the fracture behaviour in structural parameters readily recognised and utilised by the engineer, namely stress and flaw size. Fracture mechanics is based on a stress analysis as described in the next chapters and can account for the ef-fect of temperature and loading rate on the behaviour of structural members that have sharp cracks. Large and complex structures always have discontinuities of some kind. Dolan has made the flat statement that "every structure contains small flaws" whose size and distribution are dependent upon the material and its processing. These may range from non-metallic inclusions and microvoids to weld defects, grinding cracks, quench cracks, surface laps, etc. Fisher and Yen have shown that discontinuities exist in practically all structural members, ranging from below 0.02 mm to several cm long. The significant point is that

TALAT 2403 8

discontinuities are present in fabricated structures even though the structure may have been inspected. The problem of establishing acceptable discontinuities, for instance in welded structures, is becoming an economic problem since techniques that minimize the size and distribution of discontinuities are available if the engineer chooses to use them. Whether a given defect is permissible or not depends on the extent to which the defect increases the risk of failure of the structure. It is quite clear that this will vary with the type of structure, its service conditions, and the material from which it is constructed. The results of a fracture-mechanics analysis for a particular application (specimen size, service temperature, and loading rate) will establish the combinations of stress level and flaw size that would be required to cause fracture. The engineer can then quantitatively establish allowable stress levels and inspection requirements so that fracture cannot occur. Fracture mechanics can also be used to analyse the growth of small cracks, as for example by fatigue loading, to critical size. Fracture mechanics have definite advantages compared with traditional notch-toughness tests. The latter are still useful though, as there are many empirical correlations between fracture-mechanics values and existing toughness test results. In many cases, because of the current limitations on test requirements for measuring fracture toughness KIc exist-ing notch-toughness tests must be used to help the designer estimate KIc values. Brittle Fracture The number of catastrophic brittle fractures have been very small generally. Especially for aluminium, exhibiting rather high fracture toughness values over the whole temperature regime, probability of failure is low. Nonetheless, when

− the design becomes complex, − thick welded plates from high strength materials are used − cost minimisation for the structure becomes more significant − the magnitude of loading increases, and − actual factors of safety decrease because of more precise computer designs,

the possibility of brittle fracture in large complex structures must be considered.

TALAT 2403 9

2403.03 Principles of Fracture Mechanics

• Basic parameters • Fracture criteria • Members with cracks • Stress intensity factors • Deformation at the crack tip • Superposition of stress intensity factors

Basic Parameters Numerous factors like service temperature, material toughness, design, welding, residual stresses, fatigue, constraint, etc., can contribute to brittle fractures in large structures. However, there are three primary factors in fracture mechanics that control the susceptibility of a structure to brittle fracture:

1) Material toughness Kc, KIc, KId 2) Crack size a 3) Stress level σ

Material Toughness is the ability to carry load or deform plastically in the presence of a notch and can be described in terms of the critical stress-intensity factor under conditions of plane stress Kc or plane strain KIc for slow loading and linear elastic behaviour or KId under con-ditions of plane strain and impact or dynamic loading, also for linear elastic behaviour. For elastic-plastic behaviour, i.e. materials with higher levels of notch toughness than linear elastic behaviour, the material toughness is measured in terms of parameters such as R-curve resistance, JIc, and COD as described later on.

Crack Size Fracture initiates from discontinuities or flaws. These can vary from extremely small cracks within a weld arc strike to much larger or fatigue cracks, or imperfections of welded structures like porosity, lack of fusion, lack of penetration, toe or root cracks, mismatch, overfill angle, etc. Such discontinuities, though even small initially, can grow by fatigue or stress corrosion to a critical size.

Stress Level Tensile stresses (nominal, residual, or both) are necessary for brittle fractures to occur. They are determined by conventional stress analysis techniques for particular structures. Further factors such as temperature, loading rate, stress concentrations, residual stresses, etc., merely affect the above three primary factors. Engineers have known these facts for many years and controlled the above factors qualitatively through good design (adequate sections, minimum stress concentrations) and fabrication practices (decreased imperfection or discontinuity size through proper welding and inspection), as well as the use of materials with sufficient notch toughness levels.

TALAT 2403 10

A linear elastic fracture mechanics technology is based upon an analytical procedure that relates the stress field magnitude and distribution in the vicinity of a crack tip to the nominal stress applied to the structure, to the size, shape, and orientation of the crack, and to the material properties. In Figure 2403.03.01 are the equations that describe the elastic stress field in the vicinity of a crack tip for tensile stresses normal to the plane of the crack (Mode I deformation).

alu


Elastic Stress Field Distribution near a Crack

2403.03.01

σ

σNominalstress

y

x

Magnitude of stressalong x axis, σy

σy (θ = 0)

σy

σx

θ

Cracktip r

σx

σy

σy = cos (1 + sin sin )KI

(2 π r)1/2

3 θ2

θ2

θ2

σx = cos (1 - sin sin )KI

(2 π r)1/2

3 θ2

θ2

θ2

Elastic Stress Field Distribution near a Crack

Source: Irwin and Williams, 1957

The equations describing the crack tip stress field distribution were formulated by Irwin and Williams (1957) as follows:

σπ

θ θ θx

IKr

= −

2 2

12

32

cos sin sin

σπ

θ θ θy

IKr

= +

2 2

12

32

cos sin sin

τπ

θ θ θxy

IKr

=2 2 2

32

sin cos cos

τ τxz yz= = 0 σ z = 0 Plane Stress (thin sheet) ( )σ µ σ σz x y= ⋅ + Plane Strain (thick sheet) The distribution of the elastic stress field in the vicinity of the crack tip is invariant in all structural components subjected to this type of deformation. The magnitude of the elastic stress field can be described by a single parameter, KI, designated the stress in-tensity factor. The applied stress, the crack shape, size, and orientation, and the struc-tural configuration of structural components subjected to this type of deformation affect the value of the stress intensity factor but do not alter the stress field distribution. This allows to translate laboratory results directly into practical design information. Stress in-tensity values for some typical cases are given in Figure 2403.03.02.

TALAT 2403 11

alu


KI values for various crack geometries

2403.03.02

2 a

σ

σ

Through thickness crack

σ

σ

a

2 c

σ

σ

a

KI Values for Various Crack Geometries

K aI = ⋅ ⋅σ π K aI = ⋅ ⋅ ⋅112 σ πK aQI = ⋅ ⋅ ⋅112 σ π

Surface crack

where Q = f(a/2c, σ)

Edge crack

It is a principle of fracture mechanics that unstable fracture occurs when the stress in-tensity factor at the crack tip reaches a critical value Kc. For mode I deformation and for small crack tip plastic deformation, i.e. plane strain conditions, the critical stress in-tensity factor for fracture for fracture instability is KIc. The value KIc represents the fracture toughness of the material (the resistance to progressive tensile crack extension under plane strain conditions) and has units of MN/m3/2 (or MPa/mm1/2 or ksi√in). 1ksi√in = 34.7597 N/mm3/2 or MPa/mm1/2 {1ksi = 6.89714 N/mm2 or MPa} This material toughness property depends on the particular material, loading rate, and constraint:

Kc critical stress intensity factor for static loading and plane stress conditions of variable constraint. This value depends on specimen thickness and geometry, as well as on crack size.

KIc critical stress intensity factor for static loading and plane strain conditions of maximum constraint. This value is a minimum value for thick plates.

KId critical stress intensity factor for dynamic (impact) loading and plane strain conditions of maximum constraint.

TALAT 2403 12

where K K K C ac Ic Id, , = ⋅ ⋅σ

C = constant, function of specimen and crack geometry σ = nominal stress a = flaw size

Through knowledge of the critical value at failure for a given material of a particular thickness and at a specific temperature and loading rate, tolerable flaw sizes for a given design stress level can be determined. Or design stress levels may be determined that can be safely used for an existing crack that may be present in a structure, see Figure 2403.03.03.

alu

Training in Aluminium Application Technologies2403.03.03

Relation between material toughness, flaw size and stress

σ2a

σ

(Fracture zone)

KC of tougher steelKC

KC = critical value of KI

KI = f(! , a)

! f

! 0

a0 af

Incr

easi

ng s

tress

, σ

Increasing flaw size, 2a

Relation between Material Toughness, Flaw Size and Stress

Increasing materialtoughness

(COD, JIc, R)

In an unflawed structural member, as the load is increased the nominal stress increases until an instability (yielding at σys) occurs. Similarly in a structural member with a flaw as the load is increased ( or as the size of the flaw grows by fatigue or stress corrosion) the stress intensity KI increases until an instability, fracture at KIc, occurs. Another ana-logy that helps to understand the fundamental aspects of fracture mechanics is the com-parison with the Euler column instability, Figure 2403.03.04. To prevent buckling the actual stress and L/r values must be below the Euler curve. To prevent fracture the act-ual stress and flaw size must be below the Kc level.

TALAT 2403 13

COLUMN RESEARCH COUNCILCOLUMN STRENGTH CURVE

EULER CURVE

YIELDING

(a) COLUMN INSTABILITY

P

P

L

L/r

! YS

!

! = ! YS

( )σ π

cE

L r= ⋅2

2/

alu


YIELDING

(b) CRACK INSTABILITY

!

!

a

! YS

!

! = ! YS

σπcCK

C a=

⋅ ⋅2a

Source: Madison and Irwin

Analogy: Column Instability and Crack Instability 2403.03.04

The critical stress intensity factor KIc represents the terminal conditions in the life of a structural component. The total useful life NT of the component is determined by the time necessary to initiate a crack NI and by the time to propagate the crack NP from subcritical dimensions a0 to the critical size ac. Crack initiation and subcritical crack propagation are localised phenomena that depend on the boundary conditions at the crack tip. Subsequently it is logical to expect that the rate of subcritical crack propaga-tion depends on the stress intensity factor KI which serves as a single term parameter re-presentative of the stress conditions in the vicinity of the crack tip. Fracture mechanics theory can be used to analyse the behaviour of a structure throughout its entire life. For materials that are susceptible to crack growth in a particular environment the KIscc-value is used as the failure criterion rather than KIc. This threshold value KIscc is the value below which subcritical crack propagation does not occur under static loads in specific environment. In the relationship between material toughness, design stress and flaw size, Figure 2403.03.03, KIscc replaces Kc as the critical value of KI. Fracture Criteria A careful study of the particular requirements of a structure is the basis in developing a fracture control plan, i.e. the determination of how much toughness is necessary and adequate. Developing a criterion one should consider

− service conditions such as temperature, loading, loading rate, etc. − the level of performance (plane strain, elastic-plastic, plastic) − consequences of failure

TALAT 2403 14

Developing a fracture control plan for a complex structure is very difficult. All factors that may contribute to the fracture of a structural detail or failure of the entire structure have to be identified. The contribution of each factor and the combination effect of dif-ferent factors have to be assessed. Methods minimising the probability of fracture have to be determined. Responsibility has to be assigned for each task that must be underta-ken to ensure the safety and reliability of a structure. A fracture control plan can be defined for a given application and cannot be extended indiscriminately to other applications. Certain general guidelines pertaining to classes of structures (such as bridges, ships, vehicles, pressure vessels, etc.) can be formulated. The fact that crack initiation, crack propagation, and fracture toughness are functions of the stress intensity fluctuation ∆KI and of the critical stress intensity factor KIc (where-by the stress intensity is related to the applied nominal stress or stress fluctuation) de-monstrates that a fracture control plan depends on

− fracture toughness KIc or Kc of the material at the temperature and loading rate of the application. The fracture toughness can be modified by changing the material.

− the applied stress, stress rate, stress concentration, and stress fluctuation. They can be altered by design changes and fabrication.

− the initial size of the discontinuity and the size and shape of the critical crack. These can be controlled by design changes, fabrication, and inspection.

Members with Cracks Failure of structural members is caused by the propagation of cracks, especially in fati-gue. Therefore an understanding of the magnitude and distribution of the stress field in the vicinity of the crack front is essential to determine the safety and reliability of struc-tures. Because fracture mechanics is based on a stress analysis, a quantitative evaluation of the safety and reliability of a structure is possible.

Fracture mechanics can be subdivided into two general categories namely linear-elastic and elastic-plastic. The following relationships and equations for stress intensity factors are based on linear elastic fracture mechanics (LEFM). It is convenient to define three types of relative movements of two crack surfaces. These displacement modes, Figure 2403.03.05, represent the local deformation in an infinitesimal element containing a crack form. Figure 2403.03.06 shows the coordinate system and stress components ahead of a crack tip

TALAT 2403 15

alu


Basic cracking modes

Mode I

Mode II

Mode III

xy

z

xy

z

xy

z

Basic Modes of Cracking

alu


Coordinate System and Stress Components ahead of a Crack Tip 2403.03.06

Leading edge of the crack

r

y

x

z

σ x

σ y

τ yz

σ z

τ xz

τ xy

θ

Mode I is characterized by displacement of the two fracture surfaces perpendicular to each other in opposite directions. Local displacements in the sliding or shear Mode II and Mode III, the tearing mode, are the other basic types corresponding to respective stress fields in the vicinity of the crack tips, Figure 2403.03.07. In any problem the deformations can be treated as one or combination of these local displacement modes. Respectively the stress field at the crack tip can be treated as one or a combination of the three basic types of stress fields. For practical applications Mode I is the most important since according to Jaccard even if a crack starts as a combination of different modes it is soon transformed and continues its propagation to a critical crack under Mode I.

TALAT 2403 16

alu

Training in Aluminium Application Technologies2403.03.07Stress Field Equations in the Vicinity of Crack Tips

σπ

θ θ θx

IKr

=⋅

−

2 2

12

32

cos sin sin

σπ

θ θ θy

IKr

=⋅

+

2 2

12

32

cos sin sin

τπ

θ θ θxy

IKr

=⋅2 2 2

32

sin cos cos

τ τxz yz= = 0

u KG

rI=

− +

2 2

1 22

1 2

2

πθ ν θcos sin

v KG

rI=

− −

2 2

2 22

1 2

2

πθ ν θsin cos

w = 0

σπ

θ θ θx

IIKr

= −⋅

+

2 2

22

32

sin cos cos

τπ

θ θ θxy

IIKr

=⋅

−

2 2

12

32

cos sin sin

σπ

θ θ θy

IIKr

=⋅2 2 2

32

sin cos cos

( )σ ν σ σz x y= +

τ τxz yz= = 0

u KG

rII=

− +

2 2

2 22

1 2

2

πθ ν θsin cos

v KG

rII=

− + −

2 2

1 22

1 2

2

πθ ν θcos sin

w = 0

τπ

θxz

IIIKr

= −⋅2 2

sin

τπ

θyz

IIIKr

=⋅2 2

cos

σ σ σ τx y z xy= = = =0

w KG

rIII=

2 2

1 2

πθsin

u v= = 0

Stress Field Equations in the Vicinity of Crack TipsMode I Mode II Mode III

( )σ ν σ σz x y= +

Dimensional analysis of the equations shows that the stress intensity factor must be linearly related to stress and directly related to the square root of a characteristic length, the crack length in a structural member. In all cases the general form the stress intensity factor is given by

K f g a= ⋅ ⋅( ) σ where f(g) is a parameter that depends on the specimen and crack geometry. Stress Intensity Factors Various relationships between the stress intensity factor and structural component confi-gurations, crack sizes, orientations, and shapes, and loading conditions can be taken out of respective literature [1] C.P. Paris and G.C. Sih, "Stress analysis of cracks" in "Fracture toughness testing

and its applications", ASTM STP No381, ASTM, Philadelphia 1965 [2] H. Tada, P.C. Paris and G.R. Irwing, ed., "Stress analysis of cracks handbook",

Del Research corporation, Hellertown, Pa. 1973 [3] G.C. Sih, "Handbook of stress intensity factors for researchers and engi-neers",

Institute of fracture and solid mechanics, Lehigh University, Bethlehem, Pa. 1973 Stress intensity factor for a through-thickness crack:Figure 2403.03.08

TALAT 2403 17

alu


Through-Thickness Crack

σ

σ

Finite Width Plate

2a

2b

SIF for Through-Thickness Crack 2403.03.08

σ

σ

Infinite Width Plate

Nominal Stress

2a

K a= ⋅σ π( ) ( )( )[ ]K a a bI = ⋅ ⋅ ⋅ ⋅σ π πsec / 2 12

( ) ( )( )[ ]sec /π⋅ ⋅a b2 12

Tangent correction for finite widtha

b0.074 1.000.207 1.030.275 1.050.337 1.080.410 1.120.466 1.160.535 1.220.592 1.29

Stress intensity factor for a double-edge crack: Figure 2403.03.09

alu


Double-Edge Crack

σ

σ

a

2b

SIF for Double-Edge Crack 2403.03.09

a

( ) ( )( )[ ]sec /π⋅ ⋅a b2 12

Tangent correction for finite widtha

b0.074 1.000.207 1.030.275 1.050.337 1.080.410 1.120.466 1.160.535 1.220.592 1.29

( ) ( )( )[ ]K a a bI = ⋅ ⋅ ⋅ ⋅112 2 12. sec /σ π π

TALAT 2403 18

Stress intensity factor for a single-edge crack: Figure 2403.03.10

alu


Single-Edge Crack

σ

σ

2b

SIF for Single-Edge Crack 2403.03.10

KI =1.12 σ(πa)1/2

a

For a semi-finite edge-cracked specimen:

For a finite width edge-cracked specimen:

KI = σ(πa)½ f(a/b)

Correction factor for a single-edge crackin a finite width plate

a/b f(a/b) 0.10 1.15 0.20 1.20 0.30 1.29 0.40 1.37 0.50 1.51 0.60 1.68 0.70 1.89 0.80 2.14 0.90 2.46 1.00 2.86

Stress intensity factor for cracks emanating from circular or elliptical holes: Figure 2403.03.11

alu


!

!

bN

aN

ρ

aFa

( )K FI = ⋅ ⋅λ δ σ π, a

where = aa

and = baN

N

N

λ δ

( )F ,λ δ

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

01.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40

Cracks Emanating from Circular or Elliptical Holes

SIF for Cracks Emanating from Circular or Elliptical Holes

The stress intensity factor in the case of a finite plate is

δ =1.0For a circular hole

2403.03.11

λ = aaN

TALAT 2403 19

Stress intensity factor for a single-edge crack in beam in bending: Figure 2403.03.12

alu


Single-Edge Crack in Beam in Bending

SIF for Single-Edge Crack in Beam in Bending 2403.03.12

a

wM M

( )K M

W af a

WI =−

632

W: Depth of the Beam

Correction factor for notched beams a/W f(a/W) 0.05 0.36 0.10 0.49 0.20 0.60 0.30 0.66 0.40 0.69 0.50 0.72 >0.60 0.73

Stress intensity factor for an elliptical or circular crack in an infinite plate: Figure 2403.03.13

alu


!

!

aa

c

c"

Elliptical or Circular Crack in an Infinite Plate

SIF: Elliptical or Circular Crack in an Infinite Plate 2403.03.13

( )Ka a

bI = +

σ πθ

β β32

0

2

2

2

14

sin cos

θ θ θπ

0

2 2

22

32

0

12

= − −

⋅∫c a

cdsin

The stress intensity factor at any point along the perimeter of elliptical or circular cracks in an infinite body subjected to uniform tensile stress is

The point on the perimeter of the crack is defined by the angle ß and the elliptic integral #o

( )For circular cracks, c = a ⇒ = ⋅K aI 2 1 2σ π

When c and = 2 this case is reducedto a through- thickness crack.

→ ∞ β π ,

TALAT 2403 20

Stress intensity factor for a surface crack: Figure 2403.03.14

alu


Surface Crack"Thumbnail Crack"

SIF: Surface Crack

!

2ca

!

2403.03.14

θ θ θ

π

0

2 2

22

0

2

32

1= − −

⋅∫c a

cdsin

The stress intensity factor can be calculated from the equations of elliptical crack using a free surface correction factor of 1.12 and for the position ß=π/2

KI = ⋅ ⋅

1121

2

. σ π aQ

with and the elliptic integralQ = 0θ 2

Q is regarded as a shape factor because its values depend on a and c. For values of Q see Figure 2402.03.15.

Flaw shape parameter: Figure 2403.03.15

alu


0.5

0.4

0.3

0.2

0.1

00.5 1.0 1.5 2.0 2.5

a/2cRatio

Flaw Shape Parameter, Q

Flaw Shape Parameter, Q 2403.03.15

σ σ ys = 0

σ σ ys = 080.σ σ ys = 0 60.

σ σ ys =10.

2c a

Flaw Shape Parameter, Q

TALAT 2403 21

Deformation at the Crack Tip The stress field equations show that the elastic stress at a distance r from the crack tip (where r<<a) can be very large, see Figure 2403.03.16. In reality the material in this region deforms plastically. A plastic zone surrounds the crack tip.

alu

Training in Aluminium Application TechnologiesDistribution of Stress in the Crack Tip Region 2403.03.16

Crack Tip

Elastic Stress Distribution

Stress Distribution after Local Yielding

PlasticZone

! YS

2rY

x

!

The size of the plastic zone ry can be estimated from the stress field equations for plane stress conditions and setting σy = σys

r Ky

ys

=

12

2

π σ

Following a suggestion by Irwin that the increase in tensile stress for plastic yielding caused by plane strain elastic constraint is of the order of √3, we can estimate the size of the plastic zone under plane strain conditions as

r Ky

ys

=

16

2

π σ

The plastic zone along the crack front in a thick specimen is subjected to plane strain conditions in the center portion of the crack front where w = 0 and to plane strain condi-tion near the surface of the specimen where σz = 0. That means that the plastic zone in the center of a thick specimen is smaller than at the surface of the specimen, Figure 2403.03.17.

TALAT 2403 22

alu


Plane stressmode I

Planestrain

mode I

Cracktip

Midsection

Surface

Crack tip

y

zx

KI²2 $ ! y²

rIp (plane strain)

zrp(Plane stress)

Plane strain region

Machine notch

Fatigue crackPlastic zone

Plastic Zone Dimensions

A - Overall view B - Edge view

Specimen cross-section

Superposition of Stress Intensity Factors Stress components from such loads as uniform tensile loads, concentrated tensile loads, or bending loads, all belonging to Mode I type loads, have the same stress field distribu-tions in the vicinity of the crack tip according to the equations given in Figure 2403.03.07. Consequently the total stress intensity factor can be obtained by algebraically adding the individual stress intensity factors corresponding to each load.

TALAT 2403 23

2403.04 Experimental Determination of Limit Values according to Various Recommendations

• Linear-Elastic Fracture Mechanics

− Experimental determination of KIc - ASTM-E399 • Elastic-Plastic Fracture Mechanics

− Crack opening displacement (COD) - BS 5762 − Determination of R-Curves - ASTM-E561 − Determination of JIc - ASTM-E813 − Determination of J-R Curves - ASTM-E1152 − Crack opening displacement (COD) measurements - BS 5762

Structural materials have certain limiting characteristics, yielding in ductile materials or fracture in brittle materials. The yield strength σys is the limiting value for loading stresses, the critical stress intensity factors KIc, KId or Kc are the limiting values for the stress intensity factor KI. The critical stress intensity factor Kc at which unstable crack growth occurs for conditions of static loading at a particular temperature depends on specimen thickness or constraint, Figure 2403.04.01.

alu


Effect of Thickness on Kc

Plane stress Plane strain240

200

160

120

80

40

05.03.02.01.00.750.500.300.25

Thickness, in.

K c, k

si

in.

K Ic

Thickness at the beginningof plane strain is:

Effect of Thickness on Kc

t = 2.5 KIc

y

2

σ

The limiting value of Kc for plane strain (maximum constraint) conditions is KIc (slow loading rate) or KId (dynamic or impact load). The KIc-value is the minimum value for plane strain conditions. The KIc-value would also be a minimum value for conditions of maximum structural constraint (for example, stiffeners, intersecting plates, etc.) that

TALAT 2403 24

might lead to plain-strain conditions even though the individual structural members might be relatively thin. Design procedures based on the KIc-value provide a "conservative" approach to the fracture problem. The following chapter gives information on experimental require-ments and procedures for the measurement of KIc-values. For thin section problems further procedures based on a Kc or R-curve analysis for elastic-plastic behaviour problems an analysis on the basis of JIc- or COD-procedures will be described in further chapters. Linear-Elastic Fracture Mechanics

Experimental Determination of KIc - ASTM-E399

The accuracy with which KIc describes the fracture behaviour of real materials depends on how well the stress intensity factor represents the conditions of stress and strain in-side the actual fracture process zone. This is the extremely small region just ahead of the tip of a crack where crack extension would originate. In this sense KI is exact only in the case of zero plastic strain as in brittle materials. For most structural materials, a sufficient degree of accuracy may be obtained if the plastic zone ahead of a crack tip is small in comparison with the region around the crack in which the stress intensity factor yields a satisfactory approximation of the exact elastic stress field. The decision of what is sufficient accuracy depends on the particular application. A standardised test method must be reproducible, specimen size requirements are chosen so that there is essentially no question regarding this point. The standardised test method for determining KIc ma-terial values is the ASTM-E399 test method.

alu

Training in Aluminium Application TechnologiesSE Bend Specimen and CT Tension Specimen

2.1W 2.1W B=W/2

W

a

1.25 WW

0.6

W0.

6 W

0.275 W

0.275 W

0.25 W Dia.

a B=W/2

Compact TensionSpecimen

SE(B) Bend Specimen

after ASTM-E399

2403.04.02

TALAT 2403 25

Several different test specimen forms have been proposed in the course of the de-velopment and standardization. Two of the most common for engineering applications, the bend specimen or SE(B) specimen and the compact tension or C(T) specimen are reproduced in Figure 2403.04.02 according to ASTM-E399.

Test procedure:

Determine location and orientation of test specimen in respect to component to be analysed: Specimen orientation L-S and L-T cover the most common crack cases in structural engineering components, such as welded structures, see Figure 2403.04.03. Surface cracks grow initially in the direction of the plate thickness and propagate further on as through-thickness cracks.

alu


Specimen Orientation 2403.04.03

according to ASTM

Determine critical specimen size dimension: To fulfil requirements of maximum constraint and small plastic zone in relation to specimen dimensions the following relations must be observed.

a = crack depth > 2.5 ⋅ (KIc/σys)2

B = specimen thickness > 2.5 ⋅ (KIc/σys)2

W = specimen depth > 5.0 ⋅ (KIc/σys)2 This leads for instance to a specimen thickness of approximately 50 times the radius of the plane strain plastic zone. Even before a KIc test specimen can be machined, the KIc value to be obtained must already be known or at least estimated. Three general rules may be used

TALAT 2403 26

− overestimate the KIc value on the basis of experience with similar materials and judgement based on other types of notch-toughness tests

− use specimens that have as large as thickness as possible, namely a thickness equal to that of the plates to be used in service

− use the following ratio of yield strength to modulus of elasticity to select a specimen size. These estimates are valid for very high strength structural materials, steels having yield strength of at least 1000 MPa and aluminium alloys having yield strength of at least 350 MPa.

σys/E Minimum recommended

thickness and crack length [mm]

0.0050-0.0057 75,0 0.0057-0.0062 63,0 0.0062-0.0065 50,0 0.0065-0.0068 44,0 0.0068-0.0071 38,0 0.0071-0.0075 32,0 0.0075-0.0080 25,0 0.0080-0.0085 20,0 0.0085-0.0100 12,5

0.0100 or greater 6,5

Many low- to medium-strength structural materials in section sizes of interest for most large structures (ships, bridges, pressure vessels) are of insufficient thickness to main-tain plane strain conditions under slow loading and at normal service temperatures. Thus, the linear elastic analysis to calculate KIc values is invalidated by general yielding and the formation of large plastic zones. Alternative methods must be used for fracture analysis as described in further chapters. The following values are given as an example for common aluminium alloys

KI

MPa σ0,2 MPa

KI/σ0,2 2.5(KIc/σ0,2)2

AlMgSi1 50 245 0,0416 104 AlZn4,5Mg1 73 370 0,0389 97

Select and prepare a test specimen: Most probably one of the two standard specimen shapes will be selected, slow-bend test specimen or compact-tension specimen. The initial machined crack length 'a' should be 0.45 ⋅ W so that the crack can be extended by fatigue to approximately 0.5 ⋅ W. Usually the selection of the specimen thickness B is made first. Perform test following requirements of ASTM-E399 procedure: This includes initial fatigue cracking of the test specimen. Measure and plot crack opening displacement ∆v against load P.

TALAT 2403 27

Analyse P-∆v record, calculate conditional KIc (=KQ) values, perform validation check for KIc: If the KQ values meet the above stated requirements, like a or B ≥ 2.5(KQ/σ0.2)

and W ≥ 5.0(KQ/σ0.2)2, the KQ=KIc. If not the test is invalid, the results may be used to estimate the material toughness only. Elastic-Plastic Fracture Mechanics Elastic-plastic fracture mechanics analysis is an extension of the linear elastic analysis. As already mentioned low- to medium-strength structural materials used in the section sizes of interest for large complex structures are of insufficient thickness to maintain plane strain conditions under slow loading conditions at normal service temperatures. Large plastic zones from ahead of the crack tip, the behaviour is elastic-plastic, invali-dating the calculation of KIc values. There are three possible approaches into the elastic-plastic region, through

− crack opening displacement (COD) − R-curve analysis − J-integral

Crack opening displacement (COD) - BS 5762 Proposed by Wells in 1961 the fracture behaviour in the vicinity of a sharp crack could be characterized by the opening of the notch faces, namely the crack opening displace-ment. He also showed that the concept of crack opening displacement was analogous to the concept of critical crack extension force and thus the COD values could be related to the plane strain fracture toughness KIc. COD measurements can be made even when there is considerable plastic flow ahead of a crack. Using a crack tip plasticity model proposed by Dugdale it is possible to relate the COD to the applied stress and crack length. As with the KI analysis the application of the COD approach to engineering structures requires the measurement of a fracture toughness parameter δc, the critical value of the crack tip displacement, which is a material property as a function of temperature, loading rate, specimen thickness, and possibly specimen geometry, i.e. notch acuity, crack length and overall specimen size. Since the δc-test is regarded as an extension of the KIc testing the british standardized test method after BS 5762 is very similar to the ASTM-E399 test method for KIc. Similar specimen preparation, fatigue-cracking procedures, instrumentation, and test procedures are followed. The displacement gage is similar to the one used in KIc testing (Clip-Gage) and a continous load-displacement record is obtained during the test.

On the basis of the British Standard PD 6493 an analysis results is based on the comparison of the critical COD value δc to the actual crack tip opening displacement of the component analyzed and characterized by geometrical dimensions of the component

TALAT 2403 28

and the existing flaw and its location, as well as the material used - for a specific service temperature and loading rate.

Determination of R-Curves - ASTM-E561 KIc is governed by conditions of plane strain (εz=0) with small scale crack tip plasticity. Kc is governed by conditions of plane stress (σz=0) with large scale crack tip plasticity. Kc values are generally 2-10 times larger than KIc. KIc values depend on only two variables, temperature and strain rate. Kc values depend on 4 variables, temperature, loading rate, plate thickness and initial crack length. Plane stress conditions rather than plane strain conditions actually exist in service. Plane stress fracture toughness evaluations using an R-curve or resistance curve analysis as one of several extensions of linear elastic fracture mechanics into elastic-plastic fracture mechanics is envisaged. An R-curve characterizes the resistance to fracture of a material during incremental slow stable crack extension. An R-curve is a plot of crack growth resistance as a function of actual or effective crack extension. KR, also in MPa√m units is the crack growth resistance at a particular instability condition during the R-curve test, i.e. the limit prior to unstable crack growth. In Figure 2403.04.04 the solid lines represent the R-curves for different initial crack lengths. The dashed lines represent the variation in KI with crack length for different constant loads P1<P2<P3. Each line is a function of the crack length, KI = f(P,√a).

alu

Training in Aluminium Application Technologies2403.04.04R-Curves and Critical Fracture Toughness Values KR

for Different Initial Cracks a0

KR

a

P2

P3

AppliedKI Levels

a1 a2

KR = Kc

for a0 = a2

KR = Kcfor a0 = a1

KR < Kc

for a0 = a1

R-Curves

∆aactual ∆aactual

K P,I = f ( )a

P1<P2<P3

The two points of tangency represent points of instability, or the critical plane stress intensity factor Kc = KR, at the particular crack length and, of course, for the given conditions of temperature and loading rate. The KR value is always calculated by using the effective crack length aeff and is plotted against the actual crack extension aact, that takes place physically in the material during the test.

TALAT 2403 29

R-curves can be determined either by load control or displacement control tests. The load control technique can be used to obtain only that portion of the R-curve up to the Kc value where complete unstable fracture occurs. The displacement control technique can be used to obtain the entire R-curve. The evaluation of R-curves for relatively low-strength, high-toughness alloys exhibiting large scale crack tip plasticity σy at fracture, relative to the test specimen in plane dimensions W and a, requires an elastic-plastic approach. Here the crack-opening displacement δ at the physical crack tip is measured and used in calculating the equivalent elastic K value. This elastic-plastic crack model is designated the crack-opening-stretch (COS) method, where δ and COS are equivalent terms. This method can be used with either a load-control or displacement control test. A standardized testing procedure is available after ASTM-E561. Generally the thickness of a test specimen is equal to the plate thickness considered for actual service. The other dimensions are made considerably larger. The advantage of the displacement control technique is partially offset by the necessity for new or unique loading facilities and sophisticated instrumentation, whereas the load-control method in conjunction with relatively simple measuring devices can be used with a conventional tension machine. The limits of application of this technique are for materials with high strength with low toughness, small plastic zone ahead of the crack tip. For materials with high levels of toughness this analysis becomes increasingly less accurate. The R-curve is determined by graphical means. A series of secant lines are constructed on the load displacement record from a test sample. The compliance values δ/P from these second lines are used to determine the associated aeff/W-values that reflect the effective crack length aeff = a0 + ∆a + ry using also an appropriate relationship for the given specimen between crack opening and effective crack length. Each aeff/W-value is then used to determine a respective Keff-value, the latter is plotted against aeff producing the R-curve for the given material. The significance of the critical fracture toughness values obtained from an R-curve ana-lysis is in the calculation of the critical flaw size acr required to cause fracture instability. A normalised plot showing the general relationship of acr to such design parameters, nominal design stress and yield stress, for a large center-cracked tension specimen subjected to uniform tension is given in Figure 2403.04.05. The plot can be used for any material for which valid fracture mechanics results (KIc, KId, Kc, KIscc) are available under a given loading rate, temperature and state of stress.

TALAT 2403 30

alu


Normalized Plot for Critical Flaw Size

2403.04.05

10.08.0

5.0

3.0

2.0

1.00.80

0.50

0.30

0.20

= 0.10( )KC

σYS

100

40

20

10

4

2

1.0

0.4

0.2

0.10

0.04

0.02

0.010

0.004

0.002

0.0011.000.900.800.700.600.500.400.300.200.100

a cr,

criti

cal f

law

siz

e, in

ches

( )σD

σYS

acr

2acr

σD

σD

KI = σD π a

∆acr =1π

2

( )KC

σD

Normalized Plot for Critical Flaw Size

Determination of JIc - ASTM-E813

Another means of directly extending fracture mechanics concepts from the linear-elastic behaviour to the elastic-plastic behaviour is the path independent J-integral proposed by Rice as a method of characterizing the stress strain field at the tip of a crack by an integration path taken sufficiently far from the crack tip to be substituted for a path close to the crack tip region. For linear elastic behaviour the J-interal is identical to the energy release rate G per unit crack extension. Therefore a J-failure criterion for the linear elastic case is identical to the KIc-failure criterion, under linear-elastic plane strain conditions.

( )J GK

EIc IcIc= =

−1 2 2υ

The energy line integral J is defined for either elastic or elastic-plastic behaviour

J W dy T Ux

dxR

= ⋅ − ⋅∫∂∂

where R=any contour around the crack tip. Figure 2403.04.06 shows the crack-tip coordinate system and arbitrary line integral contour. Note the counterclockwise evaluation starting from the lower flat notch surface.

TALAT 2403 31

alu


Note the counterclockwise evaluation starting from the lower flat notch surface.

R

r

y

x

n

θ

J - Integral 2403.04.06

J W dy T Ux

dxR

= ⋅ − ⋅∫∂∂

The energy line integral J is defined for an arbitrary contour R around the crack tip for either elastic or elastic-plastic behaviour.

W = the strain energy densityT = the traction vector according to the outward normal n along R, U = displacement vectors = arc length along R

T ni ij ij= ⋅σ

J - Integral

W = the strain energy density T = the traction vector according to the outward normal n along R, Ti = σij ⋅ nj U = displacement vector s = arc length along R The actual testing procedure is standardized in ASTM-E813 and either a family of load displacement records for different initial crack sizes or a single specimen (i.e. same initial crack size) may be used. Using a compliance method several specimens of varying crack length are used to obtain P vs. ∆ curves. Values of energy per unit thickness (area under the P-∆ curve) are obtained for different initial crack lengths at various values of deflection ∆. The slopes of these curves are the changes in potential energy per unit thickness per unit change in crack length and thus are equal to values of

JB

Ua

= − ⋅1 ∂∂

See Figure 2403.04.07 for an illustrative interpretation of the J integral. Specimen forms and dimensions (bend, bar, C(T) or WOL specimen), testing equipment and procedure are given in ASTM-E813.

TALAT 2403 32

alu

Training in Aluminium Application TechnologiesInterpretation of the J - Integral

PB

∆a da

2403.04.07

P

∆

a

a+da

JBda

J U= − ⋅1B a

∂∂

Using a compliance method several specimens of varying crack length are used to obtain P vs. ∆ curves. Values of energy per unit thickness (area under the P-∆ curve) are obtained for different initial crack lengths at various values of deflection ∆. The slopes of these curves are the changes in potential energy per unit thickness per unit change in crack length and thus are equal to values of J.

Interpretation of the J - Integral

For the data analysis, Figure 2403.04.08, calculate J values from the P vs. ∆ using J=A/(B ⋅ (W-a0)) ⋅ f(a0/W), where f(a0/W) is a correction factor for the given specimen form, for a three-point bending specimen f(a0/W) = 2. Plot J vs. ∆a. Construct the 'blunting' line J=2r' ⋅ ∆a, with σ'=½(σys+σult). Draw the best fit line to the J vs. cack-extension points. Include only the points where actual crack extension has occured, see Figure 2403.04.09. Where crack extension appears only as a stretch zone the point should fall along the blunting line.

alu

Training in Aluminium Application TechnologiesProcedure of J Measurement

Load

Displacement, ∆(Step 1)

PrecrackEnd ∆a

(Step 2)

J

JIcFit of Data Points

(Step 4)(Step 3)

J = ⋅ ⋅2 σ flow a∆

∆a

J

∆a

2403.04.08

TALAT 2403 33

alu

Training in Aluminium Application TechnologiesEvaluation Procedure and Limits of J Measurement

Blunting Line

0.15 mm Offset

J - I

nteg

ral k

Joul

es/m

²

R-Curve Regression Line

1.5mm Offset

( eliminated data)

0 0.5 1.0 1.5 2.0 2.5∆ap [ ]mm

( )∆a minp ( )∆a maxp

2403.04.09

Evaluation Procedure and Limits of J Measurement

Eliminate all data points which lie above Jmax=b0 ⋅ σ'/15. Final verification of valid results is performed by comparing specimen dimensions a, B, b0 = W-a0 as follows

( )α

σ

= ≥a B or b

JQ

,

'

0 25

Determination of J-R Curves - ASTM-E1152 The J-R curve characterizes the resistance of metallic materials to slow stable crack growth after initiation from a preexisting fatigue crack or other sharp flaw. The J-R can be used as an index of material toughness for alloy design, material selection, and qual-ity assurance. The J-R curve from bend type specimens defines the lower bound esti-mates of J-capacity as a function of crack extension, and has been observed to be conservative in comparison with those obtained with tensile loading specimen configurations. The J-R curve can be used to assess the stability of cracks in structural details in the presence of ductile tearing. A testing procedure has been standardized in ASTM-E1152. The resistance curve is measured and the J integral is estimated in a way analogous to ASTM-E813. The maximum J integral for a specimen is given by the smaller of: Jmax = b0 ⋅ σ'/20 or Jmax = B ⋅ σ'/20

TALAT 2403 34

Data points with J values exceeding Jmax should be eliminated. The maximum crack extension capacity is given by ∆amax = 0.1 ⋅ b0

Crack Opening Displacement (COD) Measurements - BS 5762 A standardized testing procedure is given in BS 5762. The crack opening displacement δ is measured on three-point bend specimens, see Figure 2403.04.10, and transformed to a crack tip opening displacement δ. The crack extension values are measured on the specimen fracture surface. A plot of COD δ values vs. these crack extension values ∆a allows the estimation of a critical δ value.

alu


S = 4W4.5W

W

P

a

vp

z

r(W-a) W

a

δ

B = W2

after BS 5762

Hinge Model for a Standard Bending Specimen 2403.04.10

Measurements of COD

Values of δ are estimated by

δc = δu = δm = δi =K

EW a

W a zv

yp

2 212

0 40 4 0 6

( ) , ( ), ,

− + −+ +

υσ

where

K P SB W

f aW

= ⋅⋅

3 2/

TALAT 2403 35

Depending on the form of the respective load-displaced curve 4 different critical values δ are interpreted: δc COD without stable crack extension, instable crack leading to fracture, partial

brittle fracture or pop-in in the P-v curve δu COD with stable crack extension, instable crack leading to fracture fracture,

partial brittle fracture or pop-in in the P-v curve δi COD at initiation of stable crack extension δm COD at maximum load for P-v curves with an extended region of stable crack ex-

tension Use specimen dimensions, especially thickness, similar to dimensions in service.

TALAT 2403 36

2403.05 Fracture Mechanics Instruments for Structural Detail Evaluation

• Free surface correction Fs • Crack shape correction Fe • Finite plate dimension correction Fw • Correction factors for stress gradient Fg • Remarks on crack geometry

In the structural component to be evaluated the existence of a crack-like flaw of given or assumed dimensions is postulated. The stress field at the location of the flaw has to be considered. Information on flaw size and orientation and loading have to be considered through a fracture mechanics parameter, a, K, J, or δ value. For the material and its service conditions, like temperature, environment, static or dynamic loading, the respec-tive characteristic material value has to be estimated. Failures in metal structures are most of the times due to the formation of one or more cracks as a result of the repeated application of loads, i.e. fatigue. Under certain environmental conditions, like corrosion, or due to fabrication conditions a flaw may be present which will act as an initiating crack for a fatigue failure. If a large crack exists in a structure because of fabrication or some other event, the de-sign fatigue resistance curves are no longer applicable. The residual fatigue resistance in these cases must be assessed by fracture mechanics models. Complex details such as those in common use in most structural engineering structures, especially but not only in welded structures, the stress intensity factor for a surface crack of depth 'a', can be conveniently related to the well known expression for a central through crack in an infinite plate by use of correction factors. The correction factors modify σ π⋅ ⋅ a to account for effects of free surface Fs, finite width Fw, non uniform stresses acting on the crack Fg, and the crack shape Fe. The re-sulting stress intensity factor is expressed as

K F F F F ae s w g= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅σ π To evaluate fracture instability, the total sum of stresses due to residual welding or roll-ing stresses, dead load, and live loads must be considered. For cyclic fatigue loading, ∆σ is the live load variation in stress which results in a ∆K stress intensity value range. For the correction factors, solutions both empirical and exact can be found in the literature. For common cases in engineering practice the following expressions are stated.

TALAT 2403 37

Free Surface Correction Fs In a semi-elliptical surface crack in a plate subjected to uniform stress

F a cs = − ⋅1211 0186. . The accuracy is ±1.5% for 0.2 < (a/c) < 0.4. The ratio (a/c)≈0.3 has been observed with the welded aluminium beams of the TUM fatigue program. This value has been reported as a lower limit for steel weldments as well. Crack Shape Correction Fe Integral transformation of a 3-dimensional elliptical crack shape has resulted in the elliptical crack shape correction factor,

Fe = 1/E(k)

for the point of maximum stress intensity on the ellipse, where

( )E k k do

( ) sin= −∫ 1 2 22

θ θ

π

with

k c ac

22 2

2= −

i.e. dependent only upon the ratio of the minor to the major axis semi-diameter ratio a/c. Values of Fe for respective values of k2 can be taken from the curve in Figure 2403.05.01.

alu


1

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0 0.2 0.4 0.6 0.8 1k2

Fe

Crack Shape Correction Factor Fe 2403.05.01

Crack Shape Correction Factor Fe

( )E k k do

( ) sin= − ⋅∫ 1 2 22

θ θ

π

k c ac

22 2

2= −F E ke =1 ( ) where with

TALAT 2403 38

Relationships for a/c have been empirically determined for different structural geome-tries and are given in Figure 2403.05.02. The lower boundary for stiffeners is given with c a= ⋅1403 0 951. . mm. The lower boundary for coverplates is given with c a= ⋅5 451 1 133. . mm. For the above mentioned mean value a/c = 0.3 the respective Fe value is Fe = 0.912.

alu


C = 1.296 a0.946 (mm)

C = 1.403 a0.951 (mm)

C = 3.355 + 1.29a (mm)

C = 1.489 a1.241 (mm)

C = 1.088 a0.946 (in.)

C = 1.197 a0.951 (in.)

C = 0.132 + 1.29a (in.)

C = 3.549 a1.133 (in.)

C = 5.451 a1.133 (mm)

C = 3.247 a1.241 (in.)

StiffenersCoverplates

Crack Depth, a (mm)

a/c

1.0

0.8

0.6

0.4

0.2

0 2 4 6 8

0.1 0.2 0.3 a (in.)

Source: Fisher, steel structures

Crack Shape Measurements 2403.05.02

Finite Plate Dimension Correction Fw For a central crack in a plate of finite width the correction factor is (see also under Lecture 2403.03):

( ) ( )( )[ ]F a bw = ⋅ ⋅sec /π 21 2

or for a double edge crack in a plate of finite width

( ) ( )( ) ( ) ( )( )[ ]F b a a bw = ⋅ ⋅ ⋅ ⋅ ⋅2 21 2

/ tan /π π with an accuracy of 0.3% for (a/b)<0.7. Correction Factors for Stress Gradient Fg Expressions for the stress gradient correction factor Fg can be very complex and often require a procedure involving first determining the stress field with finite elements in the uncracked structure and then removing these stresses from the crack surface by inte-

TALAT 2403 39

gration. An outline for this procedure is given in "Albrecht/Yamada: Rapid Calculation of SIF, Journal Struct. Division ASCE, Vol 103, No ST2, Feb 1977". Approximate equations for the factor Fg have been derived for several details. One approximate method which appears applicable to a number of details such as stiffe-ners, attachments, coverplates and gussets states that

FKGg

tm=+ ⋅1 α β

with α = a/t and Ktm = maximum SIF at weld toe. Characteristic values for G and β and expressions for Ktm can be taken from Figure 2403.05.03.

alu

Training in Aluminium Application TechnologiesRepresentative Values for G, " and Ktm

Source: Albrecht/Yamada

Web Gusset

Type of Detail

Z

tf

tcp

tg

WWg

R > 25 mm

tf L

L

Wf

tf

tw

G "

Gusset Plateswith RadiusTransition

CoverplatedBeam

Stiffener andShort Attachment

2.776 0.2487

6.789 0.4348

0.862 0.60

0.88 0.576

2403.05.03

for web stiffeners:Ktm = 1.621*log(z/tf) + 3.963

for coverplates:Ktm = -3.539*log(z/tf)

+ 1.981*log(tcp/tf) + 5.798

for gussets with R>25mm:Ktm = -1.115*log(R/W)

+ 0.537*log(L/W)+ 0.138*log(Wg/W)+ 0.285*log(tg/tf)+0.68

Ktm

For groove welds (butt welds) with the reinforcement in place (Gurney)

( )F agb= ⋅ −2 5.

where ( ) ( )b h= ⋅ + ⋅1 2 3 1 0 06. log . and a = 2a/t, h is the acute angle between the plate surface and the tangent to the weld profile. Remarks on Crack Geometry Cracks emanating from internal discontinuities in welds quickly assume a circular shape. Even very irregularly shaped discontinuities (pores, inclusions) may be modelled as either an ellipse growing into a circular shaped crack or the initial discontinuity may be considered as a circumscribed circle.

TALAT 2403 40

Cracks forming and growing at the weld toe will be generally modelled as semi-ellipti-cal surface cracks. Multiple cracks will occur along the weld toe of a transverse weld, such as weld toes of coverplates and stiffeners welded to the flange of a girder. These small single cracks are usually located closely to each other and initially tend to grow into a semi-circular shape, but eventually the cracks begin to coalesce. Coalescence of single cracks into a merged crack has been observed to occur at crack depths as small as 1.27 mm. Typical initial discontinuities will be approximately 0.38 mm to 1.52 mm long and up to 0.76 mm deep, see Figure 2403.05.02. The following expression for the crack length 'c' may be used an approximate lower boundary of measured coverplated beams

c a= ⋅5 462 1 133. . where 2c is the width of the coalesced multiple crack. Finally Figure 2403.05.04 shows a plot of the correction factors as a function of crack size for a coverplated beam. It can be seen that small cracks are significantly affected by the stress gradient correction factor Fg. This also contributes to the spread in the crack width and the early coalescence of the individual flaws. Fg tends to decay rapidly and is not a significant factor for larger cracks.

alu

Training in Aluminium Application TechnologiesCorrection Factors vs. Crack Size for Coverplates

Correction Factors vs. Crack Size for Coverplates

Cor

rect

ion

Fact

ors

Fg

Fs

Fw

Fe

7.0

6.0

5.0

4.0

3.0

2.0

1.0

0 0.5 1.0α = a/tf

2403.05.04

TALAT 2403 41

2403.06 Calculation of a Practical Example: Evaluation of Cracks Forming at a Welded Coverplate and a Web Stiffener

• Coverplate • Web stiffener

For an existing crack in a structural component the residual fatigue resistance may be assessed by fracture mechanics. The crack is modelled as above and the stress intensity range estimated as

∆ ∆K F F F F ae s w g e= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅σ π

where ∆σe is the equivalent constant stress range calculated from random or spectrum stress amplitudes through a damage accumulation assumption. The cycles and time required to propagate a crack ai to some larger crack af can be esti-mated as

N N dN daC Kp

N

N

ma

a

i

f

i

f

= = =⋅∫ ∫∆∆

It is also necessary to check the fracture resistance of the large crack. This is given by

K F F F F a Ke s w g cmax max= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ≤σ π

when the crack tip is in a residual tension region, σmax = σy . Two different details of a welded aluminium beam are investigated, a coverplate and a web stiffener, shown in Figure 2403.06.01.

alu


Cover Plate Web Stiffener

Influence of Cover Plates and Web Stiffeners

Source: D. Kosteas, TUM

In both cases the crack extends into the heat affected zone. The crack propagation beha-viour in this zone is assumed to be similar to the upper limit of the typical experimental

TALAT 2403 42

values for base material given in Figure 2404.06.03. Crack propagation da/dN in the region of Kmax = Kc above 10-5 m/cycle is not taken into account, it corresponds to a life portion in the low cycle fatigue region and may be neglected. The limiting value is Kc = KIc = 60 MPa√m in the case of base metal of 10 mm thickness. Actually a reduc-tion of 10% must be taken into account for values in the HAZ in 30 mm plate thickness. Respective values for 15 mm flange thickness as in the above example are not available. The ultimate strength limit for the material is 360 MPa. The size of initial flaws may be taken as 0.1 mm, such as observed oxide inclusions in the fusion zone. Coverplate With the approximation c a= ⋅5 462 1 133. . and a = 0.1 mm at the beginning of crack propagation we get c = 0.402 and a/c = 0.250 and k2 = 0.938 and consequently with Fe = 0.933. With a = 15 mm and c = 117.5 mm at the approximate end of crack propagation a/c = 0.128 and Fe = 0.976. The surface correction F a cs = − ⋅1211 0186. . = 1.118 or 1.144 for a/c = 0.25 and a/c = 0.128. The finite plate width results in the following correction. The flange width is 300 mm and the flange thickness is 15 mm. We assume an initial crack at the coverplate weld toe with a depth a = 0.1 mm into the flange thickness and a width of 2c. In the beginning of crack extension with the average a/c = 0.3, c results in c = 0.33 mm. On the other hand the approximation for coverplates of c a= ⋅5 462 1 133. . gives c = 0.40 mm. Such cracks always develop in the middle of a flange above the web. So we have the configuration of a crack in the middle of a plate of finite width. Since this is not a through crack, the geometric influence is covered already by Fs as above and Fw = 1. At the approximate end of crack propagation we have a through thickness crack of a = 15 mm and a respective c a= ⋅5 462 1 133. . = 117.45 mm. Therefore the correction factor for finite width is given as

Fw = [sec((πc)/(2b))]1/2 = [sec((π ⋅ 117.45)/(300))]1/2 = 1.7295 The stress gradient correction factor Fg depends on the actual crack length a. We assume at the beginning of crack propagation a = 0.1 mm. Hereafter, we have the following values for the constants a = a/tf = 0.1/15 = 0.00667 and G = 6.789 and b = 0.4348. Finally the stress concentration factor

Ktm = ( ) ( )− ⋅ + ⋅ +3539 1981 5798. log . log .Z t t tf cp f and with tcp = 15 mm, tf = 15 mm and z = tcp = 15 mm we have Ktm = 5.798. In the geometrical assumptions for the above constants the width of coverplate and flange was identical. In the case of our example we have bf = 300 mm > 250 mm = bcp as well as a further stress concentration because of the beam shape and the web-to

TALAT 2403 43

flange weldments. Strain gage measurements have indicated a stress concentration of approximately 30 %, so a further global stress concentration factor of 1.3 is assumed. Web Stiffener In a similar way as above the free surface correction factor is calculated. For the beginning of the crack propagation a = 0.1 mm and at the approximate end of crack propagation, with a through thickness crack established a = 15 mm. From the relationship of Figure 2403.05.02 c = 1.403a0.951 for stiffeners we get for the half crack-width c = 0.157 mm or c = 18.43 mm and accordingly a/c = 0.64 or a/c = 0.81; we assume an average value of a/c = 0.8.

F a cs = − ⋅ = − ⋅ =1211 0186 1211 0186 0 8 1045. . . . . .

the accuracy could be lower though, since a/c > 0.4. For the beginning of crack propagation we assume again Fw = 1 since concentration effects are covered by the free surface correction. At the end of crack propagation with a through-flange-thickness crack of a = 15 mm and with a/c = 0.8 as assumed c = 18.75mm. So the correction factor for finite width plate with an edge crack (as the

worst case for the component) is ( ) ( )( ) ( ) ( )( )[ ]F b c c bw = ⋅ ⋅ ⋅ ⋅ ⋅2 21 2

/ tan /π π and with 2b = 300 mm, Fw = 1.013. The shape correction factor Fe is calculated for the beginning of crack propagation with a = 0.1 mm and a/c = 0.64, Fe = 0.767. For the end of the elliptical crack shape, when the crack is through the flange thickness with a = 15 mm and c = 18.75 mm we get Fe = 0.706. The stress gradient correction factor Fg depends on the actual crack length a. We assume at the beginning of crack propagation a = 0.1. Hereafter we have the following values for the constants a = a/tf = 0.1/15 = 0.00667 and G = 2.776 and b = 0.2487.

Finally the stress concentration factor Ktm = ( )1621 3 963. log .⋅ +Z t f with tf = 15 mm and z = tf = 15 mm we have Ktm = 3.963. The stress gradient correction factor for the beginning of crack propagation is calculated as Fg = 1.545 and for the end of crack propagation Fg = 1.050. Detail Crack Size Fs Fg Fw Fe Coverplate Beginning 1,118 3,278 1,000 0,933 End 1,144 0,744 1,730 0,976 Web Stiffener Beginning 1,045 1,545 1,000 0,767 End 1,045 1,050 1,013 0,706 Correction factors can be also estimated through the IIW Recommendation "The Fitness for Purpose of Welded Structures", SST-1141/89. A solution is provided for surface cracks at weld toes in the general form of K M aK= ⋅ ⋅ ⋅σ π .

TALAT 2403 44

alu

Training in Aluminium Application TechnologiesCorrection Factor Development with Progressing Crack

Coverplate FtotWeb Stiffener MK

Ftot, MK

a/B

5

4

3

2

1

00 0,2 0,4 0,6 0,8 1

2403.06.02

Correction Factor Development with Progressing Crack

The full correlation between correction factors Ftot for the coverplate or MK for the web stiffener vs. the crack development expressed with the parameter a/B (where B = 15 mm = flange thickness) is given in the diagram of Figure 2403.06.02. These values will be used together with the appropriate regions of crack propagation diagrams for the respective material zones, Figure 2403.06.03.

alu


Crack Propagation vs. ∆Κ and ∆Κeff


Crack Propagation vs. ∆Κ and ∆Κeff

2403.06.03

For the estimation of number of cycles to failure the upper portion of the da/dN diagram above 10-5 m/cycle is not taken into account. The upper boundary da/dN curve actually used in the calculation is shown in Figure 2403.06.04, where also the limit curve given by IIW recommendation SST-1141-89 is plotted. These lead to the final estimated S-N

TALAT 2403 45

curve for detail D2 = coverplate, Figure 2403.06.05, and detail B2 = web stiffener, Figure 2403.06.06, plotted against respective experimental results.

alu


Calculation Curves for da/dN Lines

2403.06.04Calculation Curves for da/dN Lines

AlZn4.5Mg1 (7020)Base materialR = 0.0, Thickness = 8.0 mm

10-3

10-4

10-5

10-6

10-7

10-8

10-9

10-10

10-11

1 2 4 6 10 20 40 60 100

Boundary curvefor calculation

da/d

N [m

/cyc

le]

∆Keff [MPa m]

KC = 60 MPa m1/2

IIWLimit curve

+ R= 0

x R = -1

o R = +0.5


mI

mII

mIII

mIV

mV

mImIImIIImIVmV

∆Keff da/dN

=15=2.5=0.94=8.0=2.7

0.7812.84.656.1660

1x10-11

4x10-10

5.25x10-9

8.46x10-9

8.08x10-8

3.37.10-5

Boundary curve data for calculation:

alu


S-N Curve for Cover Plate

x+ + +

+

+

+ +

+ +

x x

x x

++

+

x

29.521.319.9

P 50P 50P 50

500

400

300

200

100

8070605040

30

20

10

8765

104 105 106 107 1082 4 6 2 4 6 2 4 6 2 4 6 2 4 6

Stre

ss ra

nge

in M

Pa

Cycles to failure N

S-N Curve for Cover Plate

Detail D2 - TUMAlZn4,5Mg1

R = -1 R = 0,1 R = +0,6

Calculation result

IIW boundary curve

Test results


TALAT 2403 46

alu


S-N Curve for Web Stiffener

54.9

P 50

500

400

300

200

100

8070605040

30

20

10

8765

104 105 106 107 1082 4 6 2 4 6 2 4 6 2 4 6 2 4 6

Stre

ss ra

nge

in M

Pa

Cycles to failure N

S-N Curve for Web Stiffener

Detail D2 - TUMAlZn4,5Mg1

R = -1R = 0,1

Calculation result

IIW boundary curve

Test results


TALAT 2403 47

2403.07 Literature/References Introductory chapters to fracture mechanics have been taken mainly out of the text book by Rolfe/Barsom: "Fracture and Fatigue Control in Structures" further information has been utilised from Hertzberg: "Deformation and fracture mechanics of engineering materials", John Wiley & Sons, New York, 1976 Schwalbe: "Bruchmechanik metallischer Werkstoffe", Hanser Verlag, München, 1980 F. Ostermann (Hrsg.): "Betriebssicherheit von Aluminium Konstruktionen", Seminar Meckenheim, 1990 Graf, U.: "Bruchmechanische Kennwerte und Verfahren für die Berechnung der Ermüdungsfestigkeit geschweißter Aluminium Bauteile", Berichte aus dem konstruktiven Ingenieurbau, 3/92, TU München, 1992 Kosteas: "Vorlesungsmanuskript zur angewanden Bruchmechanik", TU München, 1991 Rossmanith: "Grundlagen der Bruchmechanik", Springer Verlag, Wien/New York, 1982

TALAT 2403 48

2403.08 List of Figures Figure Nr. Figure Title (Overhead) 2403.02.01 Notch-Toughness Performance Levels vs. Temperature 2403.02.02 Relation between Performance and Transition Temperature for Three

Different Steel Qualities 2403.02.03 Notch-Toughness of Welded Aluminium Alloy 5083 vs. Temperature 2403.03.01 Elastic Stress Field Distribution near a Crack 2403.03.02 KI Values for Various Crack Geometries 2403.03.03 Relation between Material Toughness, Flaw Size and Stress 2403.03.04 Analogy: Column Instability and Crack Instability 2403.03.05 Basic Modes of Cracking 2403.03.06 Distribution of Stress in the Crack Tip Region 2403.03.07 Stress Field Equations in the Vicinity of Crack Tips 2403.03.08 SIF for Through-Thickness Crack 2403.03.09 SIF for Double-Edge Crack 2403.03.10 SIF for Single-Edge Crack 2403.03.11 SIF for Cracks Emanating from Circular or Elliptical Holes 2403.03.12 SIF for Single-Edge Crack in Beam in Bending 2403.03.13 SIF: Elliptical or Circular Crack in an Infinite Plate 2403.03.14 SIF: Surface Crack 2403.03.15 Flaw Shape Parameter, Q 2403.03.16 Distribution of Stress in the Crack Tip Region 2403.03.17 Plastic Zone Dimensions 2403.04.01 Effect of Thickness on Kc 2403.04.02 SE Bend Specimen and CT Tension Specimen 2403.04.03 Specimen Orientation 2403.04.04 R-Curves and Critical Fracture Toughness Values KR for Different Initial

Cracks a0 2403.04.05 Normalized Plot for Critical Flaw Size 2403.04.06 J-Integral 2403.04.07 Interpretation of the J-Integral 2403.04.08 Procedure of J Measurement 2403.04.09 Evaluation Procedure and Limits of J Measurement 2403.04.10 Hinge Model for a Standard Bending Specimen

TALAT 2403 49

Figure Nr. Figure Title (Overhead) 2403.05.01 Crack Shape Correction Factor Fe 2403.05.02 Crack Shape Measurements 2403.05.03 Representative Values for G, β and Ktm 2403.05.04 Correction Factors vs. Crack Size for Coverplates 2403.06.01 Influence of Cover Plates and Web Stiffeners 2403.06.02 Correction Factor Development with Progressing Crack 2403.06.03 Crack Propagation vs. ∆K and ∆Keff 2403.06.04 Calculation Curves for da/dN Lines 2403.06.05 S-N Curve for Cover Plate 2403.06.06 S-N Curve for Web Stiffener

Education

TALAT Lecture 2403: Applied Fracture Mechanics