Lecture #6: 3D Rate-independent Plasticity (cont ... · Department of Mechanical and Process Engineering, Chair of Computational Modeling of Materials in Manufacturing Lecture #6:

2/15/2016 1 1Lecture #6 – Fall 2015 1D. Mohr

151-0735: Dynamic behavior of materials and structures

by Borja Erice and Dirk Mohr

ETH Zurich, Department of Mechanical and Process Engineering,

Chair of Computational Modeling of Materials in Manufacturing

Lecture #6: • 3D Rate-independent Plasticity (cont.)

• Pressure-dependent plasticity

© 2015



Three-dimensional Rate-independent Plasticity



Additive strain rate decomposition

The strain rate is decomposed into an elastic and a plastic part,

pe εεε

The corresponding algorithmic decomposition of the strain increment associated wit finite time increments Dt reads

pe εεΔVε DDD )ln(

The above decomposition is an approximation of the well-established multiplicative decomposition of the total deformation gradient,

peFFF

The approximation (*) of (**) yields reasonable results in finite strain problems when the elastic strains are small compared to unity.

(*)

(**)



Elastic constitutive equation

The linear elastic isotropic constitutive equation reads

eεCσ :

with C denoting the fourth-order elastic stiffness tensor. For notational convenience, the above stress-strain relationship is rewritten in vector notation

e

e

e

e

e

e

E

23

13

12

33

22

11

23

13

12

33

22

11

21

021

0021

0001

0001

0001

)21)(1(

with the Young’s modulus E and the Poisson’s ratio n.

Sym.



Equivalent stress definition

The yield function is often expressed in terms of an equivalentstress, i.e. a scalar measure of the magnitude of the Cauchy stresstensor. The most widely used scalar measure in engineering practiceis the von Mises equivalent stress:

SSσ :2

3][

with the deviatoric stress tensor

1σ

σσS3

][][

trdev

Note that the von Mises equivalent stress is a function of the deviatoricpart of the stress tensor only. It is thus pressure-independent, i.e. it is insensitive to changes of the trace of .



Equivalent stress definition

The von Mises equivalent stress is an isotropic function, i.e. it isinvariant to rotations of the Cauchy stress tensor:

] [][ TRσRσ for any rotation

})()(){( 222

21

IIIIIIIIIIII

R

As an alternative it may also be expressed as a function of the stresstensor invariants or the principal stresses, e.g.

23J SS :21

2 Jwith

Von Mises plasticity models are therefore also often called J2-plasticity models.



Yield function and surface

With the von Mises equivalent stress definition at hand, the yieldfunction is written as:

][][],[ pp kf σσ

I

II

III

The yield surface is

0],[ pf σ



Flow rule

In 3D, it has been demonstrated that the direction of plastic flow is aligned with the outward normal to the yield surface,

σε

fp with

S

σσ 2

3

f

σ

f

0f

In other words, the ratios of the components of the plastic strain rate tensor are the same as the deviatoric stress ratios

p

kl

p

ij

p

kl

p

ij

S

S



Flow rule

The proposed associated flow rule also implies that the plastic flow is incompressible (no volume change),

0][

2

3][

Sε

trtr p

σ

f

0f

The magnitude of the plastic strain rate tensor is controlled by the non-negative plastic multiplier . It is also called equivalent plastic strain rate.

0



Isotropic strain hardening

The flow stress is expressed as a function of the equivalent plastic strain,

][ pkk ][

32

pk with

dttp ][

It controls the size of the elastic domain (diameter of the von Mises cylinder in stress space).



Isotropic hardening

The same parametric forms for are used in 3D as in 1D.

n

pS Ak )( 0

][ pkk

]exp[10 pV Qkk

0.00E+00

5.00E+01

1.00E+02

1.50E+02

2.00E+02

2.50E+02

3.00E+02

3.50E+02

4.00E+02

0.00E+00

5.00E+01

1.00E+02

1.50E+02

2.00E+02

2.50E+02

3.00E+02

3.50E+02

4.00E+02

0.00E+00

5.00E+01

1.00E+02

1.50E+02

2.00E+02

2.50E+02

3.00E+02

3.50E+02

4.00E+02

SV kkk )1(

Swift Voce

Qkkd

dk

p

0 ,0

Hardening saturation

p p p

k k k



Loading/unloading conditions

0f0 if

0f0 if

0f0 if

0fand

0fand

The same loading and unloading conditions are used in 3D as in 1D:



Isotropic hardening plasticity (3D) - Summary

i. Constitutive equation for stress

)(: pεεCσ

ii. Yield function][][],[ pp kf σσ

iii. Flow rule

iv. Loading/unloading conditions

0f0 if

0f0 if

0f0 if

0fand

0fand

v. Isotropic hardening law

][ pkk with dtp

σε

fp



Abaqus/explicit user material (VUMAT) interface

nV

1nF

nR

nF

1nR

1nV

FD

RDINITIAL

DEFORMED @ tn

DEFORMED @ tn+1

STRETCHED @ tn STRETCHED @ tn+1

(stretched & rotated)




Kinematics: application

][cos][sin

][sin][cos][

tt

ttt

R

Rotation

][5.010

0][1][

tu

tutU

RUF

Stretching

1

1

1.2

0.9

1.4

0.8

1

1

20⁰ 40⁰

u[t] increases linearly from 0 to 0.4 [t] increases linearly from 0 to 40°



Kinematics: application (cont.)

2211 8.04.18.00

04.1eeeeU

223.00

0336.0]8.0ln[]4.1ln[ln 2211 eeeeU

766.0643.0

643.0766.0

40cos40sin

40sin40cosR

Spectral decomposition:




VRRUF TRURRURV 1

766.0643.0

643.0766.0

8.00

04.1

766.0643.0

643.0766.0V

048.1295.0

295.015.1

766.0643.0

643.0766.0

613.0900.0

512.0073.1

2211

2211

1

8.04.1

8.04.1

ReReReRe

ReeeeRRURRURV

TT

Spectral decomposition:

413.0492.0

492.0587.011 ReRe

587.0492.0

492.0412.022 ReRe




048.1295.0

295.0152.1

587.0492.0

492.0412.08.0

413.0492.0

492.0587.04.1

8.04.1 2211 ReReReReV

Spectral decomposition (cont.):

413.0492.0

492.0587.011 ReRe

587.0492.0

492.0412.022 ReRe

008.0276.0

276.0105.0

587.0492.0

492.0412.0]8.0ln[

413.0492.0

492.0587.0]4.1ln[

8.04.1ln 2211 ReReReReV




613.0900.0

514.0072.1

766.0643.0

643.0766.0

048.1295.0

295.0152.1VRF

Check:

613.0900.0

514.0073.1

8.00

04.1

766.0643.0

643.0766.0RUF



][cos][sin

][sin][cos][

tt

ttt

R

Rotation

][5.010

0][1][

tu

tutU

RUF

Stretching

1

1

1.2

0.9

1.4

0.8

1

1

20⁰ 40⁰

u[t] increases linearly from 0 to 0.4 [t] increases linearly from 0 to 40°

Kinematics in Abaqus/explicit



Kinematics in Abaqus/explicit

We apply the displacement boundary condition to a singleelement such that the average deformation gradient seen at theintegration point of a C3D8R element is

][][][ ttt URF

We then monitored the integral of the variable D (strainInc) inthe user subroutine (about 30’000 time steps performed);

0

223.0

337.0

]4[

]2[

]1[~

strainInc

strainInc

strainInc

Δεε

223.00

0336.0ln U

which is almost the same as




However, the Abaqus CAE output variable LE is different. For thesame simulation, we find

551.0

008.0

105.0

12

22

11

LE

LE

LE

which is almost the same as

008.0276.0

276.0105.0ln V

Note that the shear component of LE is twice that of lnV whichis due to the difference between the mathematical andengineering definition of shear strains.




For the same simulation, the Abaqus CAE output variable Sreads

2684.5

4879.1

4900.1

12

22

11

e

e

e

S

S

S

While the stress variable supplied by the subroutine is

101.3

4832.1

4948.1

]4[

]2[

]1[

e

e

stressNew

stressNew

stressNew




If we rotate the stress tensor supplied by the user subroutine, i.e.

4879.1057.0

057.04900.1

766.0643.0

643.0766.0

4832.1101.3

101.34948.1

766.0643.0

643.0766.0

R4832.1101.3

101.34948.1R T

e

e

e

e

e

e

057.0

4879.1

4900.1

12

22

11

e

e

S

S

S

we obtain good agreement with the Abaqus CAE variable S:




1nF

1nU

1nR

INITIAL

DEFORMED @ tn+1

STRETCHED @ tn+1


Stress tensor to be provided by VUMAT subroutine

1~

nσ

n~

nσt ~~~1 n

nRn ~1 n

nσt 1 n

Cauchy stress tensor (variable S in CAE)

t

nnnn 1111~

RσRσ

Strain tensor in user subroutine

D1

1

1 ]ln[~n

i

in Uε

Strain tensor (Variable LE in CAE)T

nnnn 1111~

RεRε

*The above relationships are still schematic approximations of the “real” kinematics computations performed by Abaqus/explicit



Exercise: total strain integration

The total strain obtained after summing the strain increments,

D1

1

1 ]ln[~n

i

in Uε

may be different from that obtained from the total strain definition

]ln[~11 in Uε

If the principal axes of the stretch tensor remain unchanged duringthe entire loading path, both definitions are identical, i.e.

]ln[]ln[ 1

1

1

D i

n

i

i UU



Now, consider for example a rotation-free two-step loadingcomprised of

D8.00

011U

D12.0

2.012Ufollowed by

The total stretch applied is

DD8.02.0

16.0112 UUUtot

1

1

1

0.8

1UD 2UD




The spectral decompositions of the stretch tensors are

D1

0

1

08.0

0

1

0

10.1

8.00

011U

D707.0

707.0

707.0

707.08.0

707.0

707.0

707.0

707.02.1

12.0

2.012U

886.0

465.0

886.0

465.0695.0

548.0

836.0

548.0

836.0105.1

8.02.0

16.01totU




and the corresponding logarithmic strain tensors are

D

223.00

00]ln[ 1U

D

020.0203.0

203.0020.0]ln[ 2U

]ln[]ln[255.0195.0

195.0009.0]ln[ 21 UUU DD

tot

DD

243.0203.0

203.0020.0]ln[]ln[ 21 UU

which provides an illustration of the small error associated with theincremental strain definition .




This small difference between the incremental and total straindefinition is worth noting in the context of elasticity. Many materialmodel implementations make use of a so-called hypoelastic lawwhich provides the incremental elastic relationship:

εCσ ~:~ DD

According to our previous considerations, we can have two loadingscenarios that lead to the same total stretch , i.e.

• Loading path #1: 12 UUU DDtot1UD

totU

followed by 2UD

• Loading path #2: direct application of totU

Load path independence of elasticity



Even though both loading scenarios lead to the same stretch, theapplication of the incremental hypoelastic law would result indifferent stresses:

]ln[]ln[:]ln[:]ln[:~21211 UUCUCUCσ DDDD

This inequality violates the basic principle of loading pathindependence of elasticity. Note that this error is associated withthe integration of the total strain. However, for the sake ofcomputational efficiency, this small error is widely accepted inindustrial practice (and even in academia).

• Loading path #1:

• Loading path #2:

12~]ln[:~ σUCσ tot




To avoid any artificial path dependency of the elastic response in FEcomputations, it is recommended to compute the total stretchtensor through polar decomposition of the current deformationgradient.

RUF

And then compute the associated stress tensor


3

1

22 )()(i

iii

TuuUFF

3

1

)](ln[ln~

i

iii uuUε

)~~(:~2 pεεCσ

Note that the plastic strain is just included for completeness in the above elasticconstitutive equations. It has been zero in the examples considered above.



Pressure-dependent plasticity



In solid mechanics, the pressure is defined as

)(3

1

33

][ 1IIIIII

Itrp

σ

Stress tensor decomposition

It characterizes the hydrostaticpart of the Cauchy stress tensor:

= +

HYDROSTATIC PART(average stress)

DEVIATORIC PART(differences among stresses)

𝛔𝐈

𝛔𝐈𝐈

𝛔𝐈𝐈𝐈

−𝒑

(𝛔𝐈𝐈𝐈+𝒑)

(𝛔𝐈 + 𝐩)

(𝛔𝐈𝐈+𝒑)

−𝒑

−𝒑

1Sσ p



Isotropic yield functions can be conveniently expressed in terms ofthe pressure (or first invariant of the stress tensor) and theinvariants J2 and J3. of the deviatoric stress tensor:

Von Mises yield function

],,[ 321 JJIff

Due to the proportionality of p and I1, the pressure dependence of ayield function is characterized through its dependence on the firstinvariant I1. The von Mises function is a typical example of apressure-independent yield function.

22 3][ JJff vMvM



The Drucker-Prager yield function is a first extension of the vonMises yield function assuming a linear pressure dependence:

Drucker-Prager yield function

It is applicable to mildly porous metals (cast iron), concrete or steelat very high pressures.

1221 3],[ aIJJIff DPDP



Illustration

Initial

ep

Hydrostatic pressure Hydrostatic pressure plus axial loading

ep

ep

ep

• Compression experiments under hydrostatic pressure



Effect of hydrostatic pressure on the stress-strain response of 4330steel:

Illustration: Results from Richmond and Spitzig (1980)



• Effect of hydrostatic pressure on the stress-strain response of4330 steel:

Illustration: Results from Richmond and Spitzig (1980)

kaIJJIfDP 1221 3],[

)( 1 MPaI

)(

2M

Pa

J



Reading Materials for Lecture #6

• M.E. Gurtin, E. Fried, L. Anand, “The Mechanics and Thermodynamics of Continua”, Cambridge University Press, 2010.

• Abaqus Theory Manual abaqus.ethz.ch:2080/v6.11/pdf_books/THEORY.pdf

• O. Richmond and W.A. Spitzig,,“Pressure dependence and dilatancy of plastic flow”, IUTAM Conference, ASME, 1980, p. 377

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Lecture #6: 3D Rate-independent Plasticity (cont ... · Department of Mechanical and Process Engineering, Chair of Computational Modeling of Materials in Manufacturing Lecture #6: