Lecture #6: 3D Rate-independent Plasticity (cont.) Pressure-dependent plasticity

Transcription

1 Lecture #6: 3D Rate-independent Plasticity (cont.) Pressure-dependent plasticity by Borja Erice and Dirk Mohr ETH Zurich, Department of Mechanical and Process Engineering, Chair of Computational Modeling of Materials in Manufacturing 05

2 Three-dimensional Rate-independent Plasticity

3 Additive strain rate decomposition The strain rate is decomposed into an elastic and a plastic part, ε ε e ε p The corresponding algorithmic decomposition of the strain increment associated wit finite time increments Dt reads D ε ln( ΔV) Dε e Dε p (*) The above decomposition is an approximation of the well-established multiplicative decomposition of the total deformation gradient, F F e F p (**) The approximation (*) of (**) yields reasonable results in finite strain problems when the elastic strains are small compared to unity

4 : Dynamic behavior of materials and structures Elastic constitutive equation The linear elastic isotropic constitutive equation reads ε e C σ : 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 ) )( ( with the Young s modulus E and the Poisson s ratio n. Sym.

5 Equivalent stress definition The yield function is often expressed in terms of an equivalent stress, i.e. a scalar measure of the magnitude of the Cauchy stress tensor. The most widely used scalar measure in engineering practice is the von Mises equivalent stress: 3 [ σ ] S : S with the deviatoric stress tensor S dev[ σ] σ tr[ σ] 3 Note that the von Mises equivalent stress is a function of the deviatoric part of the stress tensor only. It is thus pressure-independent, i.e. it is insensitive to changes of the trace of

6 Equivalent stress definition The von Mises equivalent stress is an isotropic function, i.e. it is invariant to rotations of the Cauchy stress tensor: T [ σ] [ R σ R ] for any rotation As an alternative it may also be expressed as a function of the stress tensor invariants or the principal stresses, e.g. R 3J with J S : S {( I II ) ( I III ) ( II III ) } Von Mises plasticity models are therefore also often called J- plasticity models

7 Yield function and surface With the von Mises equivalent stress definition at hand, the yield function is written as: III f [ p p σ, ] [ σ] k[ ] The yield surface is f [ σ, p ] 0 II I 7 7 7

8 Flow rule In 3D, it has been demonstrated that the direction of plastic flow is aligned with the outward normal to the yield surface, ε p f σ with f 3 S σ σ In other words, the ratios of the components of the plastic strain rate tensor are the same as the deviatoric stress ratios f σ f 0 p ij p kl S S p ij p kl 8 8 8

9 Flow rule The proposed associated flow rule also implies that the plastic flow is incompressible (no volume change), 3 tr[ S] tr[ ε ] p The magnitude of the plastic strain rate tensor is controlled by the non-negative plastic multiplier 0. It is also called equivalent plastic strain rate. 0 f σ f

10 Isotropic strain hardening The flow stress is expressed as a function of the equivalent plastic strain, with k [ t] p k[ p ] dt It controls the size of the elastic domain (diameter of the von Mises cylinder in stress space). 3 k[ p ] 0 0 0

11 Isotropic hardening The same parametric forms for k k[ p ] are used in 3D as in D E E E+0 Hardening saturation 4.00E+0 k k dk k 0, k k0 Q d p 3.50E E E+0.50E E+0.00E E+0.50E k A ) S Swift ( p 0 n.00e E+0 p 0.00E+00 k V k Voce Q exp[ ] 0 p.00e E+0 p 0.00E+00 k ( ) k V k S p

12 Loading/unloading conditions The same loading and unloading conditions are used in 3D as in D: 0 if f 0 0 if f 0 and f 0 0 if f 0 and f 0

13 Isotropic hardening plasticity (3D) - Summary i. Constitutive equation for stress ii. Yield function iii. Flow rule f σ C : ( ε ε p) [ p p iv. Loading/unloading conditions σ, ] [ σ] k[ ] f ε p σ 0 if f 0 0 if f 0 and f 0 0 if f 0 and f 0 v. Isotropic hardening law k k[ p ] with p dt 3 3 3

14 Abaqus/explicit user material (VUMAT) interface F n DF t n+ (stretched & rotated) t n (stretched & rotated) V n F n V n INITIAL R n DR t n t n+ R n 4 4 4

15 : Dynamic behavior of materials and structures Kinematics: application 0.8 U[ t] u[ t] 0 Stretching 0 0.5u[ t] F RU Rotation cos[ t] R[ t] sin[ t] sin[ t] cos[ t]..4 u[t] increases linearly from 0 to 0.4 [t] increases linearly from 0 to 40 0⁰ 40⁰ 5 5 5

16 Kinematics: application (cont.) Spectral decomposition: U e e 0. 8 e e ln U ln[.4] e e ln[0.8] e e R cos40 sin 40 sin 40 cos

17 Kinematics: application (cont.) F RU VR V RUR T RUR V V Spectral decomposition: RUR Re Re RUR T R.4e.4Re e Re 0.8e e 0.8Re Re Re R T Re

18 Kinematics: application (cont.) Spectral decomposition (cont.): Re Re Re Re V.4Re Re 0.8Re Re ln V.4Re Re 0.8Re Re ln[.4] ln[0.8]

19 Kinematics: application (cont.) Check: F VR F RU

20 : Dynamic behavior of materials and structures Kinematics in Abaqus/explicit F RU Stretching Rotation U[ t] u[ t] u[ t] R[ t] cos[ t] sin[ t] sin[ t] cos[ t]..4 u[t] increases linearly from 0 to 0.4 [t] increases linearly from 0 to 40 0⁰ 40⁰ 0 0 0

21 Kinematics in Abaqus/explicit We apply the displacement boundary condition to a single element such that the average deformation gradient seen at the integration point of a C3D8R element is F[ t] R[ t] U[ t] We then monitored the integral of the variable D (straininc) in the user subroutine (about time steps performed); straininc[] ~ ε Δε straininc[] straininc[4] which is almost the same as ln U

22 Abaqus/explicit user material (VUMAT) interface However, the Abaqus CAE output variable LE is different. For the same simulation, we find which is almost the same as LE 0.05 LE LE ln V Note that the shear component of LE is twice that of lnv which is due to the difference between the mathematical and engineering definition of shear strains.

23 Abaqus/explicit user material (VUMAT) interface For the same simulation, the Abaqus CAE output variable S reads S.900e 4 S.879e 4 S 5.684e While the stress variable supplied by the subroutine is stressnew[] stressnew[] stressnew[4] e 4 e

24 Abaqus/explicit user material (VUMAT) interface If we rotate the stress tensor supplied by the user subroutine, i.e..948e 4 R e R.83e e e 4 T e we obtain good agreement with the Abaqus CAE variable S: S.900e 4 S.879e 4 S

25 Abaqus/explicit user material (VUMAT) interface F n t n+ (stretched & rotated) Cauchy stress tensor (variable S in CAE) σ R σ ~ R t n n n n n R n~ n σ n t n Strain tensor (Variable LE in CAE) ε R ~ ε R T n n n n R n n ~ ~ t σ ~ n~ n INITIAL U n t n+ Stress tensor to be provided by VUMAT subroutine ~ σ n Strain tensor in user subroutine n ln[ D ] i ~ ε n U i *The above relationships are still schematic approximations of the real kinematics computations performed by Abaqus/explicit 5 5 5

26 Exercise: total strain integration The total strain obtained after summing the strain increments, n ln[ D ] i ~ ε n U i may be different from that obtained from the total strain definition ~ ε n ln[ U ] i If the principal axes of the stretch tensor remain unchanged during the entire loading path, both definitions are identical, i.e. n i ln[ DU i ] ln[ Ui ] 6 6 6

27 : Dynamic behavior of materials and structures Exercise: total strain integration Now, consider for example a rotation-free two-step loading comprised of D U followed by D U The total stretch applied is U tot DU DU DU DU 7 7 7

28 Exercise: total strain integration The spectral decompositions of the stretch tensors are D U D U U tot

29 and the corresponding logarithmic strain tensors are : Dynamic behavior of materials and structures Exercise: total strain integration 0 ln[ DU ] ln[ DU ] ln[ DU ] ln[ DU] ln[ Utot ] ln[ DU ] ln[ DU] which provides an illustration of the small error associated with the incremental strain definition

30 Load path independence of elasticity This small difference between the incremental and total strain definition is worth noting in the context of elasticity. Many material model implementations make use of a so-called hypoelastic law which provides the incremental elastic relationship: Dσ ~ C : D~ ε According to our previous considerations, we can have two loading scenarios that lead to the same total stretch, i.e. U tot Loading path #: DU followed by DU Utot DU DU Loading path #: direct application of Utot

31 Load path independence of elasticity Even though both loading scenarios lead to the same stretch, the application of the incremental hypoelastic law would result in different stresses: Loading path #: σ~ C : ln[ DU] C : ln[ DU] C : ln[ DU] ln[ DU Loading path #: σ ~ C : ln[ U ] σ ~ tot This inequality violates the basic principle of loading path independence of elasticity. Note that this error is associated with the integration of the total strain. However, for the sake of computational efficiency, this small error is widely accepted in industrial practice (and even in academia) ]

32 Load path independence of elasticity To avoid any artificial path dependency of the elastic response in FE computations, it is recommended to compute the total stretch tensor through polar decomposition of the current deformation gradient. F RU F T ~ ε i 3 u And then compute the associated stress tensor F σ~ U ln U C : ( ~ ε 3 i ( ) i ( u ln[ ]( u ~ ε p ) i i i i u Note that the plastic strain is just included for completeness in the above elastic constitutive equations. It has been zero in the examples considered above. ) i ) 3 3 3

33 Pressure-dependent plasticity

34 Stress tensor decomposition In solid mechanics, the pressure is defined as tr[ σ] I p ( I It characterizes the hydrostatic part of the Cauchy stress tensor: : Dynamic behavior of materials and structures II σ S III ) p σ III HYDROSTATIC PART (average stress) p DEVIATORIC PART (differences among stresses) (σ III +p) σ I σ II p = + p (σ I + p) (σ II +p)

35 Von Mises yield function : Dynamic behavior of materials and structures Isotropic yield functions can be conveniently expressed in terms of the pressure (or first invariant of the stress tensor) and the invariants J and J 3. of the deviatoric stress tensor: f f [ I, J, J 3] Due to the proportionality of p and I, the pressure dependence of a yield function is characterized through its dependence on the first invariant I. The von Mises function is a typical example of a pressure-independent yield function. f vm f vm [ J J ]

36 Drucker-Prager yield function The Drucker-Prager yield function is a first extension of the von Mises yield function assuming a linear pressure dependence: f DP f DP [ I J ai, J ] 3 It is applicable to mildly porous metals (cast iron), concrete or steel at very high pressures

37 Illustration : Dynamic behavior of materials and structures Compression experiments under hydrostatic pressure p e p e p e p e Initial Hydrostatic pressure Hydrostatic pressure plus axial loading

38 Illustration: Results from Richmond and Spitzig (980) Effect of hydrostatic pressure on the stress-strain response of 4330 steel:

39 Illustration: Results from Richmond and Spitzig (980) Effect of hydrostatic pressure on the stress-strain response of 4330 steel: [ I, J ] J ai k f DP 3 J ( MPa ) I ( MPa)

40 Reading Materials for Lecture #6 M.E. Gurtin, E. Fried, L. Anand, The Mechanics and Thermodynamics of Continua, Cambridge University Press, 00. Abaqus Theory Manual abaqus.ethz.ch:080/v6./pdf_books/theory.pdf O. Richmond and W.A. Spitzig,, Pressure dependence and dilatancy of plastic flow, IUTAM Conference, ASME, 980, p