A semi-implicit integration scheme for rate-dependent and rate-independent plasticity

Computers d S~rurrures Vol. 63, No. 3, pp. 579-600, 1997 0 1997 Ekvier Science Ltd

PII: SOO45-7949(%)00352-5 Printed in Great Britain. All rights reserved

0045-7949/97 Sl7.00 + 0.00

A SEMI-IMPLICIT INTEGRATION SCHEME FOR RATE-DEPENDENT AND RATE-INDEPENDENT

PLASTICITY

E. B. Marint and D. L. McDowell George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta,

GA 30332-0405. U.S.A.

(Received 18 July 1995)

Abstract-A semi-implicit constitutive integration procedure for rate-independent and rate-dependent inelastic flow of metals is presented. This integration scheme, originally proposed by Moran et al. [Formulation of implicit finite element methods for multiplicative finite deformation plasticity. Inr. J. Numer. Melh. Engng 29, 483-514 (1990)], has the feature of being explicit in the plastic flow direction and hardening moduli but implicit in the incremental plastic strain. Two approaches to this scheme are devised for rate-dependent constitutive relations, denoted herein as the kinetic equation and the dynamic yield condition approaches. Details of the integration scheme are developed and applied to both incompressible and compressible inelasticity, including pure isotropic and combined kinematic-isotropic hardening theories. Both rate- and temperature-dependent and rate- and temperature-independent constitutive laws are considered. In addition, the explicit version of this scheme is obtained, resulting in the rate tangent modulus method. 0 1997 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

The numerical solution of nonlinear initial-boundary value problems involving the inelastic behavior of metals requires the time integration of the constitutive equations governing the material response. Current constitutive models of inelasticity (plasticity and viscoplasticity) typically describe the material behavior by a sel. of first-order coupled ordinary differential evaluation equations for the internal state variables. In principle, conventional integration techniques (one step methods, linear multistep methods [2]) which account for particular features of these constitutive equations such as their lack of smoothness (plasticity) or their stiff behavior (viscoplasticity) can be applied. However, a method based on the functional values at several time steps may have serious limitations since the character of the constitutive equations quite routinely changes abruptly over a single timestep. Hence, single step methods have been primarily used in inelasticity.

For prescribed displacement increments (e.g. displacement-based finite element method), the goal of a constitutive integration scheme is to obtain the stress and other internal state variables from a known strain (or strain rate) history. For this purpose, many (single step) explicit [3-S], semi-implicit [l, g-121 and implicit [ 13-201 ti:me integration procedures for rate-independent [S-5,7-10, 13, 14, 19,211 and rate-

* Presently with Beam Technologies, Ithaca, New York.

dependent [ 1,&8, 11, 12, 1 S-201 plasticity have been proposed in the literature. Among these are the forward gradient schemes [ 11, 121, asymptotic integration schemes [20], the plastic predictor-elastic corrector method [8] and the family of generalized trapezoidal and generalized midpoint operators (which includes the forward Euler and backward Euler schemes as particular cases) [9, 10,211. Most of these schemes have been used with deviatoric (incompressible) inelasticity. A few have been extended to integrate porous (compressible) inelastic models [ll, 141.

In this paper, we present a semi-implicit integration scheme, recently proposed by Moran et al. [l], as applied to integrate incompressible and compressible inelastic constitutive models for rate-independent and rate-dependent inelasticity of initially isotropic, ductile metals. This integration procedure, which can be interpreted in the context of the generalized mid-point integrators, is explicit in the plastic flow direction and the hardening functions but implicit in the incremental plastic strain. This special feature reduces the equation-solving effort during the update of the state variables to the solution of a single nonlinear scalar algebraic equation for the incremental effective plastic strain which can be solved by an iterative procedure such as Newton method. Two approaches to this constitutive integrator are developed and the corresponding expressions for the constitutive Jacobian matrix [ 13, 181, resulting in quadratic convergence of the (implicit) finite element

579

580 E. B. Marin and D. L. McDowell

equilibrium iterations, are derived. These approaches are denoted herein as the kinetic equation and dynamic yield condition approaches.

The details of the integration scheme are developed using a finite strain generalization of the classical J2 (deviatoric) rate-independent and rate-dependent associative plasticity models. For comparison purposes, the purely explicit version of this integration procedure is also considered here, resulting in the so-called tangent modulus method (forward gradient scheme) [I 1, 121. The semi-implicit scheme, together with the constitutive model, has been implemented in the implicit displacement-based finite element code ABAQUS [22] through the user subroutine UMAT. To illustrate the performance of this integration scheme, this ABAQUS-UMAT code is used to solve the case of a thick cylinder subjected to a prescribed velocity at the inner surface.

The time integration procedure is then extended to treat porous (compressible) inelastic constitutive equations. Two typical structures of a porous constitutive mode1 are used in the development: a power law model based on isotropic hardening and an internal state variable model with combined kinematic-isotropic hardening with recovery effects. These constitutive equations are also implemented in ABAQUS using the semi-implicit scheme. This code is then used to solve some application problems based on particular cases of these porous constitutive models.

2. THE SEMI-IMPLICIT INTEGRATION SCHEME

The details of the semi-implicit integration procedure are presented using rate-independent and rate-dependent J2 (deviatoric) inelasticity models for finite deformation problems. In this paper, we will confine our consideration to strain rate and temperature effects in the context of isothermal behavior.

2.1. A J2 (deviatoric) rate-dependent associative inelasticity model

This section will consider a very simple form of combined nonlinear kinematic-isotropic hardening for a rate-dependent, power law-hardening material. This model is a generalization of the classical rate-dependent J2 flow theory for small strains obtained by replacing the ordinary material time derivative by the Jaumann stress rate with the implicit assumption of small elastic stretch.

The inelastic flow rule prescribes the evolution of the inelastic rate of deformation DP

DP = &‘%I, (1)

where Sp is the effective inelastic strain rate, i.e.

;P = ,/; ~[DP[] = & (2)

with l/DPll = (DP:Dp)“Z = rj and n, is a deviatoric (for inelastic incompressibility) unit vector in stress space which defines the direction of inelastic flow. For an associative flow rule, n, is normal to the (rate-dependent) flow potential Fdr

aF,jas s-a “=izEim=~=i (3)

where s and a are the deviatoric Cauchy stress and the deviatoric backstress tensors, respectively. Here, s = u - l/3 tr(a)I with tr the Cauchy stress tensor and I the second-order identity tensor; the symbol tr(.) denotes the trace of a tensor.

The evolution equation for Zr is given by the kinetic equation

ip = g(c?, c”) (4)

where 6 is the effective von Mises overstress, i.e.

d = &(a, a) = 411.9 - all. (5)

The deviatoric back stress tensor, a, evolves according to a Prager’s type hardening rule

Z=oi-W.a+ a.W=6DP (6)

where W is the continuum spin and the scalar function b is assumed to be of the form b = 6(p, Zr). This genera1 form for b has been mainly used in the context of power law inelasticity [l] and admits nonlinear kinematic hardening of a specific, simple form. Plastic spin effects have been neglected here in co-rotational derivatives. The evolution of the Cauchy stress tensor, u, is given by the hypoelastic relationship

v u = ti - W. u + u W = (;: (D - DP) (7)

where the additive decomposition of the rate of deformation tensor D into elastic and plastic components is implied, i.e. D = De + DP. In eqn (7), C is the fourth-order constant elastic stiffness tensor for isotropic elastic response, C = 2fi* + I(I@I), where p and I are the Lame constants, 9 is the fourth-order identity tensor, and the symbol @ denotes the tensor product.

It is important to note that by inverting the kinetic equation, eqn (4), we obtain

Fd = &(a, P, P) = d - h(P, i”) = 0 (8)

during plastic flow, where the function h(.), the inverse of the function g(.), represents the dynamic yield strength of the material [23]. Equation (8) can be interpreted as a rate-dependent (dynamic [23]) yield condition (Mises plastic potential surface). Functional forms for g (and hence for h) that have been suggested in the literature are of exponential,

Rate-dependent and rate-independent plasticity 581

power law and hype&olic form [24]. A typical power law form which will be used in the present work is given by [l]

where & is given by

rq = a&*) := tpJ0

( > 1 + z ” + (1 - rl)ao. (10)

Here, co and &, are a reference strain and strain rate, respectively; n is the strain hardening exponent, m is the strain rate sensitivity exponent, and a0 is the initial static yield strength. The parameter q is a constant in the range [O, 11. The limiting cases q = 1 and q = 0 correspond to pure isotropic and pure kinematic hardening, respectively. This model pro- vides for nonlinear kinematic hardening of Prager- type. Rate-dependence is partitioned between the dynamic yield strength, h, and the rate of kinematic hardening. For this power law inelastic model, the coefficient b in the Prager-type kinematic hardening rule, eqn (6), is given by [l]

b=&~,z”)=;(l-?/) 0 ’

z “--L$ (11)

Note that cY in eqn (10) represents the static yield strength [23]. Furthermore, in the limit as rn tends to 0 in eqn (9), the rate-independent limit is reached wherein: l Equation (8) will represent the (static) von Mises yield condition Fd’(d, ?, Z*)-+F(b, p) with h = /;(?‘), i.e.

F(d, p”) = d - h(P) = 0. (12)

l The evolution eqn (4) (kinetic equation) will be replaced by

$(4:D) ‘* = ?x 2~ + b + ;(dh/dP) (13)

which is obtained by using the consistency condition that the effective stress d is constrained to remain equal to the static yield strength eY (= h(p)) during plastic deformation. MacCauley brackets () in eqn (13) define the loading/unloading condition.

For the state update algorithm, it is convenient to write this rate-dependent constitutive model for inelastic incompressible solids with an associative flow rule and comibined isotropic-kinematic hardening as follows:

Dynamic yield condition: Fd = R(u - a, P, Z’) = 0

Kinetic equation: 2’ = g(a - a, p’) = &i

Evolution of u, a: z = C:(D - pq)

z = b$n,. (14)

In this formulation, a and 8? represent kinematic and isotropic hardening, respectively. Note here that rigid body rotation effects are embedded in eqns (14) through the continuum spin in the expression of the Jaumann rate of u and a.

2.2. Remarks about the integration scheme

Remark 1. For problems involving large stretch and rotation, the constitutive time integration scheme must ensure numerical objectivity of the tensorial variables over finite time steps [19,25-281. A particular procedure to enforce objectivity typically starts with the decoupling of deformation and rotation effects [19,25,26,28] in eqn (14) using the time-dependent incremental rotation tensor Q(t) (= AR(t)) defined by the initial value problem

Q(t) = W(t) . Q(t), t. < t < t, + I

Q(t.) = I. (15)

This rotation tensor specifies the orientation of a reference frame spinning at a rate W(t). In this reference frame, the Jaumann rate of u and a can be expressed as a material time derivative of the transformed tensorial variables d and a’ according to

8(t) = Q’(t) . z(t) . Q(t), d(t) = QT(t) . z(t). Q(t)

(16)

where ti=Q’.u.Q,S=Q’.a.Q. Any superim- posed rigid body motion can then be decoupled from the numerical solution by transforming eqn (14) to this reference frame,

Dynamic yield condition:

Fd = A(6 - d, P, Z”) = 0

Kinetic equation: ;p = g(d - d, q = J-&i

Evolution of d, a’: $ = C:@ -pi&)

d = bj%, (17)

where fi = QT 9 D. Q, ti = QT. n,. Q. Here, we have used the isotropic form for C and the isotropic properties of the functions (Fd, g, b). Note that the driving quantities of the initial value problem (15) and the constitutive eqn (17) are W and 8, respectively. Hence, a proper numerical


approximation (kinematic discretization [ 19,281) of these quantities in At ensures objectivity of the incremental approximation for (a, a).

Remark 2. In general, the updating procedure for the Cauchy stress and back stress tensors using the constitutive eqn (17) requires first solving the initial value problem (15) to obtain the rotation tensor Q(t. + ,). Then after integration of eqn (17) this tensor is used to rotate the transformed quantities (B(t, + I), a(t. + I)) to update the variables (a(r, + I), cr(r, + I)), i.e.

a(&+,) =Q(t.+~).d(f,+~).Q~(t,+,),

a(t~+~)=Q(t.+l).07(t.+l)~QT(t.+l). (18)

An alternative approach, which yields identical results [28], is to un-transform eqn (17) using Q(t.+ ,) before the integration procedure, i.e.

Dynamic yield condition: Fd = gd(u, a, P, P) = 0

Kinetic equation: ip = g(a, a, P) = Jip

Evolution of u, a: 6 = C:(D - @n,)

ci = b& (19)

where

Q = a(t) = Q(L+ I) . t?(t) . QT(t.+ I),

a = a(t) = Q(L+ I) . 6(t) * QT(tn+ ,), (20)

and similarly for D and n+ Then, after the time integration of eqn (19), the variables (a(t, + ,), a(t. + ,)) are directly recovered. This can be clearly seen if we write the following general expression for the integrated form of eqn (19),,

= Q(t.+ I) . a(&) . QYt.+ I)

I

In+ I + c: (D - pn,) dt

1.

= Q(t.+J. oi(t,). QYfn+l)

s

t.+ I + obn, dt (21)

1”

where C(L) = QT(t.) U(L) . Q(L) = U(G) (eqn (15)) and similarly for G?(L). The latter procedure will be assumed in the present work.

Remark 3. To develop the integration scheme, we will consider the configuration of the body at times t. and t,+ ,, with t. + , = t, + At. Accordingly, subscripted variables “n” and ‘?r + 1” indicate that they are evaluated at t, and t. + , , respectively. Also, we will assume that:

(9

(ii)

(iii)

Numerical approximations to the spin tensor W and the stretching tensor b have been obtained [ 19,281 using the incremental displacement field computed from the global Newton’s method. The tensors W and fi have been used to obtain (a) the rotation tensor, Q.+ ,, by solving the initial value problem (15); (b) the increment of strain AC, by integrating fi in At, i.e.

s I.+ I

AE = Ddt=Q.+,. r.

. Qf;+ 1. (22)

The initial values (a,,, o?., c) are known from the previous time step, where the tensors (&, oi.) need to be rotated to the (current) configuration at t,+l using Q.+, to account for rigid body rotation during the increment At, as indicated by eqn (21).

Hence, the input to the integration scheme will consist of A6 and (Q.+,.t%.Q:+,,Q.+, . 6, . Q7;+ ,, t;“,) (with d and oi already rotated to the current configuration) and the goal is to determine the updated values

(UW+ I,, a(,+ I), %+ I) ). In what follows, we simplify the notation by denoting the rotated tensorial variables

(Qn+l .a,.Q~+,,Q.+,.a?..Q~+,) as (u(.),a&

2.3. Semi-implicit time integration procedure

As mentioned in Section 2.2, the constitutive eqns (19) are used to discuss the semi-implicit integration scheme. This scheme is a member of the generalized mid-point algorithms. Its main feature is that it is explicit in the flow direction n, and the hardening function b (i.e. n, = nNn) and b = b,., during the time step Ar) and implicit in the incremental effective plastic strain, Ap. This feature reduces the computational effort during the state update to the solution of a single nonlinear scalar equation for

AP PI. Two approaches using this integration scheme will

be considered. They are based on either the kinetic equation or the dynamic yield condition to compute Ap. The state update procedure in these two approaches is essentially the same, with the main difference between them being the nonlinear algebraic equation needed to solve for Ap and the expression for the stress Jacobian.

In general, the optimal approach for the solution of specific rate-dependent problems depends on the


structure of the co:nstitutive model. In particular, for where this 52 rate-dependent model, the approach will be selected based on the power law form that is used to ag, agu 3 agO characterize the material response. Typically, ab=-aol= 2ad J

--II, (28)

the function g is specified in creep problems whereas the function h (or Fd) is given in high strain rate applications (dynalmic plasticity). On the other hand, with nS = nNo,, and the derivatives da/dAp, da/ in rate-independent models, the function dAp and dP/dAp are evaluated from eqns (24) Fd(d, c”, P)+F(b, i?) is specified and, therefore, the and (25). The resulting expression for Newton (static) yield condition approach must be used. method is

1

(29)

2.3.1. Solution for Ap based on the kinetic equation. Integrating the kinetic equation, (eqn ( 19)2, from t. to t., I we obtain

s ‘n+ I

A.?= g(cr - a, P) dt = At go(a - a, P) (23) 1.

where ACT = CC (n + I1 - $, and the notation go means that the arguments of the function g are evaluated at time to = t, + 0 At (generalized mid-point scheme), i.e.

u = U(0) = (1 - e)l?,,, + BU(. + 1) = u +@Au (“)

with A(.) = (.)(.+ ,) - (.)(.,, AP = (2/3)“* Ap, and 0 < 0 < 1. The expressions for Au and Aa in terms of Ap are obtained by integrating their evolution equations, eqn (19)-, assuming b = b(,, and n, = ng.,, i.e.

~~ := (;:A6 - AppC: nrCn)

Aa =: Apb(,,n+,

where Ap = fiCBj At = p+ 1 fi dt; k (or P = g) is taken constant in At. Based on relations (24) and (25), we can write eqn (23) as

G(Ap) = & Ap - Atgo = 0 (26)

which is a nonlinear algebraic equation in Ap. This equation is solve,d using Newton method with G’ = dG/dAp evaluated as

G’(Ap)=&At$=,/j

(27)

where the superscript (i) and (i + 1) indicates values obtained at the ith and (i + I)th iterations, respectively. Once Ap is found, the updated values of the vanables (a(. + I), a(” + ,), ~7” + ,, ) are obtained from eqns (24) and (25) by setting e= 1.

2.3.2. Solution for Ap based on the rate-dependent (dynamic) yield condition. The dynamic yield condition, eqn (19)(, can be expressed as a nonlinear equation in Ap by evaluating the arguments of the function Fd at time to, and using ip = OS, = (2/3)‘/*IjCo, = (2/3)“* Ap/At where, again, fi (or C’) is assumed constant in At,

Fd(Ap) = Fd,(u - a, P, P)

= &(u - a) - h@, ip) = 0. (30)

This nonlinear equation can also be solved by Newton method. In this case, Fi = dFd/dAp is evaluated by the chain rule as

with

ado abo 3 x=-z= Tn,,

d

dZp 2 1 &= 3At J

-- (32)


and the derivatives da/dAp, da/dAp and dP/dAp also 2.41. Explicit approach based on the kinetic evaluated from eqns (24) and (25). The resulting equation. The kinetic equation can be expressed as expression for Newton method is: (generalized mid-point scheme)

(33)

With Ap found from eqn (33), the updated values of the variables (Q(,+ ,), a,,+ ,), t;“.+ ,)) are obtained from eqns (24) and (25) with 6 = 1.

2.3.3. Constitutioe (stress) Jacobian matrix. The stress Jacobian .J is required in the global equilibrium

iP = i$, = g(a(., + 6’ Au - a,,, - 0 Aa, %, + 0 AP).

(37)

iterations in an implicit displacement-based finite element scheme [13, 181. For this purpose, we must The first-order Taylor series expansion of g about

obtain ,! for the constitutive model, eqn (19), and (etn), a(.), 6%) yields

consistent with the semi-implicit integration algorithm described above. For this specific case, the stress Jacobian is computed from t),=g.+R(~l”:A~+~b:Aa+~l”AC>

an,+, Iz = a<,+, - = f& = C - (C:n,,.J@~ (34)

(38)

where eqn (25), has been used. The expression for dAp/aAc is derived from either eqn (26) (kinetic equation approach) or eqn (30) (dynamic yield where g, = g(a,,,, - a(,), q,‘.,) = g(&.,, C&J. With ex- condition approach), along with eqns (24) and (25). pressions (25), (28) and i$, = (2/3)“* Ap/At, Ap The resulting expressions for J are: follows as

(i) Kinetic equation approach:

J=C-- (35)

(ii) Dynamic yield condition approach:

All quantities with subscript 0 in eqns (35) and (36) are evaluated at the last Newton iteration. Note that

J ig + i 6 2 (ns:QAc)

both stress Jacobians are unsymmetric. Ap = At

2.4. Explicit time integration scheme 1 + 0 At + $ (n&n, + b&:4) - $$

Explicit versions of the previous schemes are 1 obtained by using Taylor series expansion of the (39) functions g, in eqn (23), and Fd, in eqn (30), about the values of the corresponding arguments evaluated at

where the subscript ‘n’ has been dropped with the

t = tn. Note that this expansion a priori limits the understanding that all variables and derivatives are

accuracy of the results to a small time increment. It evaluated at t,. It is important to note that this

is of interest to ultimately compare such schemes with relation could have also been obtained using a

the semi-imulicit scheme nresented in the nrevious generalized trapezoidal scheme, i.e.

section. . .

zb, = (1 - e)2;, + ezp. + ,, = (1 - ek. + egn + , (40)


where g. + , is expanded using first-order Taylor series about (%, %)r G;“n,) (forward gradient method [I 1, 121).

2.4.2. Explicit approach based on the rate-dependent (dynamic) yield condition. The dynamic yield condition can be expressed as

+ 0 BP, Z&,) = 0. (41)

First-order Taylor series expansion of F about

(ec(.,, qn), $,, %) yields

where FdCnj = Fd(a(,,, - a(.), c?& Z&,) = 0 (dynamic yield condition satisfied in the previous time step). If we use Fd = a(u - a) - h(P, P), relations (25), (32), and i& = (2/3)“* Ap/At in eqn (42), we can obtain

J 61 ah 2 e sip ip + i (n,:QAc)

Ap = 3 2 (n& n, + bn,:n,) + &, + & $

(43)

where all the varhbles and derivatives are evaluated at time t,. This expression agrees with the one developed by Yoshimura et al. [12] using the forward gradient method.

2.4.3. Rate tlzngent modulus. The explicit expressions for A,o can be used to derive tangent modulus stress-strain relations in incremental form. We express the flow rule and the evolution of Cauchy stress in incremental form, i.e.

A6p = Apn,

Au = Q(Ac - A@‘) (44)

where A@ = Dip At, and use n,:n, = 1, n&:n, = 2/1, n,:lc:AE = 2p(&:A~). Then, substitut- ing the expressions for Ap from each of eqns (39) and (43) into eqn (44), we obtain:

(i) kinetic equation approach

I= 1+0At ;(2jz+b)$-g 1 , (45)

(ii) dynamic yield condition approach

(46)

In the previous expressions, C” and ct are the elasto-plastic tangent stiffness for rate-dependent materials based on the kinetic and dynamic yield condition approaches, respectively. Note that both Cp and <;I depend on the magnitude of At. Also, they can be used as an approximation to the stress Jacobian ,! for the implementation of this J2 plasticity model and the corresponding explicit integration scheme within implicit finite element codes which require J. 2.5. A rate-independent model and time integration procedures

Previous sections have considered a rate-dependent flow rule (kinetic equation). In this section, we discuss treatment of the limiting case of a rate-independent theory.

2.5.1. J2 associative plasticity model. The rate-independent version of the previous model (Fd(b, .C’, ip)+F(b, 67)) is written as

F=f(u-a,P)=O

C = c:D -@Cm,

ai = bpn, (47)

where h = IlDPII. Both the plastic multiplier A(=?) and the plastic modulus H are determined from consistency condition, i.e. F = P = 0. The material time derivatives of tensorial quantities in eqn (47) are subject to the same remarks with regard to objectivity as eqn (19).

2.52. Semi-implicit time integration scheme and stress Jacobian. For rate-independent models, it is common to specify the yield function, F. Therefore, a natural approach to integrate these constitutive models using the semi-implicit algorithm will be


based on this function. For this case, the equation for Ap necessary to apply Newton method is

1

- (48)

which is a rate-independent counterpart of eqn (33). The updated values of the state variables are computed from eqns (24) and (25) with 0 = 1. With this scheme, the yield criterion will be satisfied exactly at time t = ts. The stress Jacobian for this rate-independent model consistent with the semi- implicit algorithm is

which is the rate-independent counterpart of eqn (36).

2.5.3. Explicit time integration scheme and tengent modulus. The explicit expression for Ap is given by

Ap = n,:c:AE

2 ah (50) n&n, + bns:ns + - -

3 acQ

(a) (b)

2

where all the quantities are evaluated at time t = t.. When using this scheme, the stress point will not be exactly on the yield surface at the end of the time step. Therefore, a consistent end-of-step correction [29] will be needed in order to satisfy the consistency condition. Also, when taking relatively large time steps, a sub-incrementation procedure for the excess stress [3,4] is recommended for an accurate prediction of the updated state.

The (rate-independent) tangent modulus stress- strain relation can be written as

Au = cdp: AE

2.6. Applications of rate-dependent theory

The constitutive models and semi-implicit time integration scheme described in earlier sections have been implemented in the implicit finite element code ABAQUS 1221 through the user material subroutine, UMAT. This subroutine is called once for each integration point in the model, at each global iteration of every increment. The input provided to UMAT consists of the rotation tensor Qn + , = AR. + , , the strain increment AC = A4+ ,, and the variables (Q.+,.k'.QQ:+,, &, ~7")) at the beginning of the time step, where oi need be rotated to the current

b/a = 2

A b

. . ,.

,. . .

Fig. 1. Schematic showing: (a) thick-wall cylinder subjected to a prescribed velocity at inner surface; and (b) finite element mesh used for the analysis (IO axisymmetric eight-node quadrilateral elements, type

ABAQUS-CAXIR [22]).


I.2 r

1.0

0.8

s p. 0.6

0.4

at . :

Thick wall cylinder

. I, rate dependent plasticity Explicit method, ti I a& = 1.0 i

l

0.2 . .

A K.E., At = 0. I25 set -I .

0 . K.E., At = I .OO set l D.Y.C.. At =0.125 MC l D.Y.C.. At = 0.100 ICC

0.0 1 I 1 I I I I I 0 2 4 6 8 10 12 14

u /aeo

Fig. 2. Normalized pressure as a function of deformation for axisymmetric deformation of a thick-wall cylinder: solution convergence of the explicit (forward gradient) method using both kinetic equation (K.E.) and dynamic yield condition (D.Y.C.) approaches for At = 0.125 and

1 .oo.

configuration. The output from UMAT consists of the variables (uy,. ,), a(,+ ,), &+ ,)) at the end of the increment plus the contribution J (stress Jacobian) to the global Jacobian matrix used in the overall Newton iteration .scheme. For purposes of comparison, the explicit (forward gradient) scheme has also been implemented in ABAQUS-UMAT.

As an application of this implementation, we analyze the quasistatic axisymmetric viscoplastic small deformation of a thick-wall cylinder under prescribed velocity conditions at its inner surface [ 121. This example is used to compare the performance of both the semi-implicit and the explicit (forward gradient) time integration schemes for the two approaches (kinetic equation and dynamic yield condition). In general, the properties of the integration scheme (stability, accuracy and convergence) will depen,d on the time step At (which is determined by the .material rate sensitivity [6]) and the parameter 8 in the midpoint representation for g and Fd. Here, we have selected 0 = 1 to study the effect of At on the performance of the numerical schemes.

A schematic of the problem is shown in Fig. 1. The finite element mesh, which consists of ten axisymmet-

eight-noded quadrilateral GAQUS-CAXSR [22]), .

elements (type is also shown in Fig. 1. The

material response of the cylinder is represented by the 5r rate-dependent plasticity model, eqn (24), with power-law stress-plastic strain and stress-plastic strain rate relationships, eqns (9)-(10). Also, we will assume r~ = 1 (pure isotropic hardening model) and the following material constants [ 121

Elasticity: E = 20 x 10’ MPa, v = 0.3

(p = 7.7 x 10’ MPa, 1 = 11.5 x lo3 MPa)

1.2

1.0

0.8

s Q. 0.6

0.4

0.2 . . 0.0 -

: :

Thick wall cylinder

. 12 rate dependent plasticity

. Semi-implicit method, i I a% = 1.0

. * K.E.. At = 0.125 set . 0 K.E.. At = 1.00 see . l D.Y.C.. At = 0.125 MC

l D.Y.C.. At = 0.100 aec

0 2 4 6 8 10 12 14

u taco

Fig. 3. Normalized pressure as a function of deformation for axisymmetric deformation of a thick-wall cylinder: solution convergence of the semi-implicit method using both kinetic equation (K.E.) and dynamic yield condition

(D.Y.C.) approaches for At = 0.125 and 1.00.

Viscoplasticity: m = 0.01, n = 0.1, i0 = 0.002 s-r,

c0 = 0.002, a0 = 40 MPa

where E and v are the Young’s modulus and the Poisson ratio, respectively.

We compare the numerical solutions obtained using both the kinetic and dynamic yield condition approaches in Fig. 2 (forward gradient scheme) and Fig. 3 (semi-implicit scheme), where plots of the pressure ratio (p/aO) vs the nondimensional displacement (u/ate) are presented for a material point located at the inner surface of the cylinder. In each

1.8 I I

I

s 0.9 lb

0.6 -

0.3 -

0.0 L 0.00

Thick wall cylinder J2 rate dependent plasticity KiMtiCqn,Q/a~=1.0

* Explicit, At = 0. I25 ccc c Explicit, At = 0.50 set * Explicit, At = I .OO xec l Semi-implicit, AI E 1 .oO xec

I I 0.01 0.02 0.03

EP

Fig. 4. Normalized effective stress as a function of deformation for axisymmetric deformation of a thick-wall cylinder: solution convergence of the explicit and semi-implicit method using the kinetic equation (K.E.) approach,

for different time steps.


case, time steps of Ar = 0.125 and 1 .OO have been used, with a prescribed velocity at the inner surface of zi/u& = 1. For this range of time steps, we observe that no oscillatory behavior is exhibit by the explicit (forward gradient) scheme. These numerical results show that both the kinetic equation and the dynamic yield condition approaches give similar results. Moreover, we see that the accuracy and convergence of the semi-implicit scheme are superior to the explicit one. We note here that for the semi-implicit scheme, the number of iterations for computing Ap using the (local) Newton method varied in both approaches. For the dynamic yield condition, the Newton method converged in two iterations for the whole range of time steps used. On the other hand, for the kinetic equation approach this number increased from two (for At = 0.125) to seven (for At = 1.00) iterations.

The performance of the semi-implicit and explicit integration schemes is compared in Figs 4 and 5, where plots of effective stress ratio (~?/a~) vs effective plastic strain (P) for a material point located at the inner surface of the cylinder are presented. Figure 4 is for the kinetic equation approach and Fig. 5 for the dynamic yield condition approach. A velocity of ti/& = 1 is prescribed at the inner surface of the cylinder. These figures clearly show the accuracy of the semi-implicit scheme (using either approach) for relatively large time steps. On the other hand, the solution accuracy of the forward gradient method depends on the size of the time step. In this case, the predicted stress response converges to the solution of the semi-implicit scheme as the time step is reduced.

Figure 6 presents plots of (p/a,, - u/ao,) and (a/o,, - P) to show the performance of the semi-implicit scheme using different velocity conditions (strain rates) at the inner surface of the cylinder.

lx4 I I

1.5

1.2

1

J? 0.9 - lb

0.6 -

0.3 -

Thick wall cylinder J2 rate dependent plasticity

Dynamic yield condition, ti I a% = 1.0

a Explicit. At = 0.125 see 0 Explicit, At = 0.50 set 0 Explicit, At = 1 .OO set l Semi-implicit_ At = I .O see

0.0 & I 1 I 0.00 0.01 0.02 0.03

EP

Fig. 5. Normalized effective stress as a function of deformation for axisymmetric deformation of a thick-wall cylinder: solution convergence of the explicit and semi-implicit method using the dynamic yield condition (D.Y.C.)

approach, for different time steps.

(a) 1.8 I

b”

:*:;i

. Thick wall cylinder

;;, 0.9 - J2 rate dependent plasticity

Semi-implicit method, D.Y.C.

0.6

0.00 0.01 0.02 0.03

P

(b) 1.2 I 1 I I ’ II

t

i- $06 ; Thick wall cylinder

a . :

12 rate dependent plasticity Semi-implicit method, D.Y.C.

u I act

Fig. 6. Normalized: (a) effective stress; and (b) pressure as a function of deformation for axisymmetric deformation of a thick-wall cylinder: solution convergence of the semi-implicit method using the dynamic yield condition (D.Y.C.) approach for different strain rates and different time steps

(number of time increments).

These results were obtained using the dynamic yield condition approach. This figure shows that the accuracy of the semi-implicit scheme is maintained with relatively small (96 increments) and relatively large (12 increments) time steps, for a wide range of strain rates. Although similar numerical results were also found using the kinetic equation approach for zi/a&, = 1 x lO-2, 1 x 100, 1 x lOI, the computations (Newton method) did not converge for the highest velocity (li/& = 1 x lo+).

These numerical results demonstrate that the semi-implicit scheme has better convergence and accuracy properties than the forward gradient method for a range of time steps. In general, the time step used with the semi-implicit scheme can be an


order of magnitude greater than that of the forward gradient method while preserving accuracy in the numerical predictions. However, limitations on the time step size are still in order and are reflected on the iterative procedure to compute Ap using Newton method. The number of iterations increases as the time step size is increased, and for sufficiently large time steps, the Newton method will not converge. The restriction in the time step size of the semi-implicit scheme is due to its explicit character on the plastic flow direction and the hardening functions. In addition, the numerical results have also shown that the kinetic equation and the dynamic yield condition approaches give similar results for both the semi-implicit and explicit integration schemes. However, in the semi-implicit procedure, the Newton method converges faster for the dynamic yield condition approach.

3. POROUS ASSOCIATIVE INELASTICITY: SEMI-IMPLICIT SCHEME

This section describes the time integration procedure of constitutive equations for compressible rate-dependent associative inelasticity using the semi-implicit integration algorithm presented in the previous section. These constitutive equations are integrated using both the kinetic equation and the dynamic (rate-dependent) yield condition approaches.

In contrast to the constitutive model considered in Section 2, which is based on the incompressibility of inelastic deformation (tr(DP) = 0), the structure of the constitutive equations for a porous ductile material reflects a non-zero volumetric strain and a marked influence of hydrostatic stress. The basis for formulating these constitutive equations is an expression for the void growth rate which, in the context of associative flow rules, is implied by specifying a pressure- and porosity-dependent flow potential. Two tylpical functional forms for this low potential have bee:n principally used in the literature. These are denol:ed here as elliptic [30-351 and Gurson [36-371 forms. In this work, a strain-controlled void nucleation law is also considered in the formulation [38].

Two typical structures of the constitutive model with isotropic damage are considered. These models differ primarily in the description of the hardening behavior of the matrix material. The first model considers hardening rules prescribed by power law functions, while the second one describes the hardening process using saturation-type hardening laws (hardening minus recovery format). In this work, we will denote these models as “power law” and “internal state variable” inelastic models, respectively.

The differential evolution equations of the constitutive models which involve the Jaumann stress rate will be written here in terms of the material time

derivative. This assumes that a numerically objective transformation of the constitutive equations has been carried out, as was explained in the previous section. Also, in the development of the constitutive integrator, we will assume that AE and AR plus the variables at t, are known quantities to determine the variables at t,, ,, with t. + I = t, + At.

3.1. Power law porous inelasticity

3.1.1. Power law porous inelastic constitutive

model. To simplify the exposition, the constitutive equations of this model will be formulated in the context of isotropic hardening. Then, the set of variables which characterize the mechanical response of a material point will be given by (a, 9, Rd), or equivalently (a, 9, p), with 9 representing the void volume fraction (scalar measure of damage). In this model, Rd represents the dynamic yield strength of the matrix material.

The structure of this model considers a dynamic yield condition of the form

Fd = &r, Rd, 9) = &I,, Jz, &, 9) = 0 (52)

where F*, the viscoplastic potential function for a porous material, is given by either of the following forms.

Elliptic model: Fd=d%%?&h,&

Gurson model:

Fd = J

where Rd = i?,(p, ip) represents the dynamic yield strength of the matrix material, and

Z, = tr(a), J2 = f tr(s2). (54)

In general, hi = &(9, m) for i = 1, 2,3. Particular expressions of the coefficients hi for the elliptic model are listed in Table 1. For the Gurson model, h, = 1, h2 = 2q,9, h> = (1 + (q19)2)“2, with a typical value of q, = 1.5 [37]. Note that by virtue of eqn (53), Z& can also be functionally represented as & = Z&(u, 9). Note also that Fd = 0 during viscoplastic deformation.

The associative viscoplastic flow rule is expressed as

DP = /I aa = IIDPlln,

where /i or /lDPll are formally specified by the “dynamic” consistency condition Fd = #d = 0, and the unit director vector n, is given by

dFd dF* -= ,r+$s. aa , 2

(56)


Cocks [30]

Table 1, Coefficients of the elliptic model A. Rate-dependent porous plasticity

11 9 -- h,=1+$3, h2=Zl+ml+9’ h, = (1 - @l/l.i+1,

Michel and Suquet [31] and Duva and Crow [32]

Sofronis and McMeeking [33]

h1=(1+9)2/(m+l), hr=;(+@T+‘i, h,=(l-@WV+”

B. Rate-independent porous plasticity

Doraivelu et al. [34]

h, = 1 h2 _ ~(2 + 9) 3[2(1 - 9)*- 11 “2

2 + (1 Q)Z’ h, =

- [ 2 + (1 - 9)2 1 Oyane et al. [35]

h,=l, hz=;+$, hl=l-9

Cocks [30]

1 9 h,=l, h2=Zm+ h,=l-9

The evolution equations for the Cauchy stress u and the damage variable (void volume fraction) 9 are [37, 381

b = C($):(D - D”)

4 = ,,$c,ip + (1 - S)tr(DP) (57)

where c(9) = 2~(9)p+ 1(9)1 I; ~(9) and d(S) are the damaged Lame constants which are expressed in terms of those of the matrix (superscript m) as ~(9) = (1 - 9)~“, I(9) = (1 - S)nm. The coefficient c3 in eqn (57h defines a strain-controlled nucleation process; the second term of the equation reflects void growth. The equivalent plastic strain rate of the matrix material, ip, is defined by [37, 381

(58)

The stress rate in eqn (57) is subject to the same considerations as discussed for eqn (19) with respect to incremental objectivity. The isotropic hardening behavior of the matrix material is described by either of the functions

Zp = g(Rd, P) or Rd = h(P, P) (59)

where g and h are taken to have power law forms, e.g. eqn (9) (with (7 replaced by &) and (10) (with t] = 1).

Note that eqn (59), is the kinetic equation in this model. Define

Xs = &WS + (1 - S)tr(n,)

(60)

Then, the equations of this power law porous inelasticity model can be succinctly written as follows.

Dynamic yield cond.: Fd(a, 9, h(P, ip)) = 0 Kinetic equation: ip = g(a, 9, P)

Evolution of (a, 9, P): d = C(S):D -pc(9):&

ip = X<@ (61)

where fi = ]]DPI]. The semi-implicit integration scheme described in the previous section assumes that the X-factors (~0, xc). the damaged elastic stiffness tensor (2(g), and the unit director vector n, in eqn (61), are constant throughout Ar, with their values defined by those at t = t. kw,, m, WC,) = Cwr kd

Newton method:

Rate-dependent and rate-independent plasticity

Box I Rate-dependent power law porous plasticity

Kinetic equation approach

591

where:

Stress Jacobian:

where HS is given above (without the superscript (i)) and all quantities with subscript 0 are evaluated at the last Newton iteration.

3.1.2. Semi-implicit time integration scheme. As ~(I# + I, = a(., + f&r: Ac - Ap &,:nw before, the integrakion scheme will be based on either eqn (61)2 (kinetic equation approach) or eqn (61), 9 <n+ I) = 4”) -t xen, Ap (dynamic yield condition approach). Since the main steps of this integration procedure have already been rp,+ I) = c7j + x‘(n) AP. outlined in Section 2, we present here only the equations needed to iterate for Ap using Newton method and the corresponding stress Jacobians. They 3.2. Znternaf state variable porous inelasticity

(62)

are listed in Boxes 1 and 2. Once Ap is found using We next consider extensions of a specific internal either of the equations for Newton’s method, the state variable model due to Bammann [39] which is updated state is obtained using representative of the class of unified creep-plasticity

Box 2

Newton method:

Rate-dependent power law porous plasticity Dynamic yield condition approach

where:

Stress Jacobian:

where HO is given above (without the superscript (i)) and all quantities with subscript B are evaluated at the last Newton iteration.


constitutive equations. This particular model has The unit director vectors n, and e are defined by been developed in the context of deviatoric (incompressible) inelasticity and for high temperature applications but has been applied along with explicit ns = ,,; 7 ;;;z,,* e=-I-

void growth relations [40] to admit compressibility J5’ (67)

effects (partially coupled model) [41]. In this work, we It is important to note that DP can be expressed as

will extend this model to deal with porous (compressible) inelasticity in an associative frame- work, restricting the analysis to the isothermal case. DP= Ddp + D: = mg

In this model, the internal state variables are given by (( >> +& n, (68)

the set (A, R, 9), where A (non-deviatoric backstress) and R are the kinematic and isotropic hardening where [42] variables, respectively.

3.2.1. Extension of Bammann rate-dependent state variable porous plasticity model [39]. The inelastic flow rule is given by

aqaa 1 -J- n, = IlaFjaa = m lrS + JW e. (69)

The evolution equations for the Cauchy stress u, the internal variables A and R, and the damage variable

Dz = g((&))n,, D: = Bp((&))e (63) 9, are given by [4L421

d = C(@:(D - DP) where D$ and DC are the deviatoric and volumetric components of the plastic rate of deformation tensor DP, with DP = Dz + De. In this model, the constitutive function g is sinh(.) form and, for our purposes, the inelastic potential function, F = $((Q - A, R; 9) = $(Z,, 3, R, 9), is assumed to

(1 -S)bnA-i& IIAIIA zp 1 be given by an elliptic form, i.e. - J ; & IAllA

3h,$’ + hzfl* - h,(R + g = f,,$ sinh(”

Y) h3 V >

(64)

where hi = &(9, m) (see Table 1) with m = V/Y; f, Y and V are temperature-dependent matrix material parameters (see Table 2), and fl and Jj are defined as

fl = tr(a - ;A), 3 = ftr(s - ia)’ (65)

9 = (70) J

i csip + (1 - S)tr(DP)

where a = A - l/3 tr(A)I. Note that in this model F > 0 for viscoplastic deformation and R + Y can be

where b, kd, k,, B, Kd, KS are temperature-dependent

interpreted as a “quasi-static” yield strength of the matrix material parameters (see Table 2). Note the

matrix. In eqn (63), j? is called the void growth factor presence of both dynamic (kd, &) and static thermal

which for an elliptic functional form of F is given (kS, K,) recovery terms in eqn (70)2_,. As in eqn (19), material time derivatives are taken with resnect to

by 1421

Table 2. Internal state

rotated tensorial quantities to approximate the Jaumann derivative in eqn (7O)1_2. The directional

(66) index nA gives the direction of the kinematic

variable inelastic model temperature-dependent material functions [39,41]

V = CI exp( - G/T),

k, = C7 exp( - G/T),

Ih = CIJ exp( - C14/i?,

where T is absolute temnerature

Y = C3 exp( - G/T),

b = CS exp( - G/T),

B = G~exp(-G6/T),

f = Gexp(--G/T)

k, = CII exp(-C12/T)

K, = Gexp(-C18/T)


translation of the yield surface. Two definitions have been proposed [42]. Here we use

nA = -

’ (71)

F 350 2 + 3pz_&_lim!?! 2 ’ 3h, 8-1 h2 1 “*

This definition assumes that lim(hl/h2) exists as 9+ 1. Note here that nA is a unit vector and as 9-0, hl/h,-+O (see Table 1) resulting in nA = R. The equivalent plastic strain rate of the matrix material, ip, for an elliptic viscoplastic potential is [38]

ip = (tr(Dp))2

(72) The rate-dependent (dynamic) yield function, Fd,

can be obtained by inverting the flow rule, eqn (63),, i.e.

Fd = @a - A, R, 9, ]lD$II)

= F(sa - A, R, 9) - h(9, ]]D:1/) (73)

where the function h is given by

h = h,Vg-I(.) = hsVsinh-’ 2 lIDSIt (J >

-- 3 f . (74)

Note here that using the elliptic form for F, eqn (64), the dynamic yield strength of the matrix can be expressed as Rd = Y + R + Vg-It). Denote

XA = (1 -- 9)bnA - f&j IL-M xc, 1 xi = f&g IIAIIA

J

x8 =

XI= &&JI +&(1 -@tro(tr())‘.

(75)

Then, the state variable porous inelasticity model can be succinctly expressed as follows.

Dyn. yield cond.:

F&r - A, R, Q,@d) = F(u - A, R, 9) - h(9, k) = 0 Kinetic equation:

jh=g & =g(a-A,R,9) ( >

Evol. eqn of (a, A, R, 9):

t = C(S):D - J1+B2d&(S):n,

9: = ~x8dd (76)

where dd = ]]Ddp I]. Again, the semi-implicit integration scheme is based on the assumption that the X-factors (xA, xi, XR, XX, xs, x6), the damaged elastic stiffness tensor, c(8), the void growth factor ji and the unit director n, in eqn (76h are constant during At, defined by their values at t,

tXA(n)? X&n,, XRWr X&n,, x4+ Xr(n)r c(%,) = &r &n,, %,).

3.2.2. Semi-implicit time integration scheme. For this internal state variable model for porous inelasticity, the equations to iteratively compute Ap using Newton method as well as the stress Jacobians for both approaches (kinetic equation and dynamic yield condition) are given in Boxes 3 and 4. Once Appd is found, the updated state is obtained using

R( n+ I) = Rw + JmxRW APT - x&n, At

%+I, = 9,~ + J’i?&wAppd. (77)

3.3. Applications

The previous porous inelastic models and the corresponding equations for the semi-implicit integration scheme were also implemented in the ABAQUS user-material subroutine, UMAT. In this section, we will use this code to solve three examples considering particular cases of the porous inelasticity models presented. Specifically, the first two problems are single element tests which are solved using the rate-independent version of the power law porous plasticity model (see Box 5). Here, we use the static yield condition approach. The numerical results will be compared with an accurate reference solution in order to check the performance of the semi-implicit


Newton method:

Box 3

Rate-dependent state variable porous plasticity Kinetic equation approach

where go(,) = dg,/d(.), and

Stress Jacobian

where He is given above (without the superscript (i)) and all quantities with subscript 0 are evaluated at the last Newton iteration.

scheme. The third problem is the necking develop- steps of the order of the yield strain. In all these ment in a plane strain specimen under tension and it examples, we again use 0 = 1. is treated using the modified Bammann rate-dependent internal state variable model. The dynamic yield 3.3.1. Rate-independent power law porous condition approach is used in this case. This example plasticity (Box 5). The examples presented here are will show that the semi-implicit scheme can be readily taken from Ref. [14] and are solved using the applicable to solve complex problems with strain (static) yield condition approach (see Box 5). The

Box 4

Rate-dependent state variable porous plasticity Dynamic yield condition aneroach

Newton method:

where

1 J@=-._-- 2 aql [II II a@ aa’ 3 ae

II II

- - ab”:nac., - dffXR(“j -

-G Stress Jacobian:

where HO is given above (without the superscript (i)) and all quantities with subscript 0 are evaluated at the last Newton iteration.


Box 5

Rate-independent power law porous plasticity

&(a, Rdr g)--rF(u, R, 9)

Constitutive equations:

F = &. 9, h(P)) = 0

DP= &+:a,) IL,=& ( >

where

Q = (;@):D - p(3(9):n,

4 = x3@

P = ,y‘b

CJ = h(P) = R = uo n,

Semi implicit integration scheme (static yield condition approach):

with:

Stress Jacobian: same as the one given in Box 2, but with HO given above (without the superscript (i)).

first example is the homogeneous uniaxial tensile where test of an ax.isymmetric single element type ABAQUSCAX4 [22] (see Fig. 7a). The matrix H* = ;(2$ + d)H

material properties used are given by E = 200 x 10’ MIPa, v = 0.3, a0 = 310 MPa, n = 0.256, 6. = 0.01. The test is displacement

= “?-‘9”) [

w(q + o) g - &,yR 1 controlled with etqual increments of 0.0025 lo (strain increments of the order of the yield strain). Only - 3(1 - S)tjyR voidgrowth is considered with an initial void volume fraction is 1.5%. This problem is solved for Gurson yield function. Using the equations in Box 5, one w=:,

finds that the equations describing the problem (with cl # 0) are:

___=___E_ da dc (t] + a,)&

y = qI cash - 4,s

H*

C=(~+cu)coE+%+l' (79)

3$I(l - 9) + This set of nonlinear equations is integrated

dp r + (1) -- z = (1 - 8)C

(with c3 = 0) using an explicit Euler forward

(78) integration procedure with small, equal strain increments of 2.5 x lo-’ (l/62 of the yield strain).


(a) 2

b

(b) z I

-- Y

/ ‘X

Fig. 7. Homogeneous deformation of a single element under: (a) uniaxial tension (element type ABAQUS- CAX4 (221); and (b) hydrostatic tension (element type

ABAQUS-C3D8 [22]).

This accurate reference solution with small strain steps is compared with the finite element computations. Figure 8 presents plots of the stress ratio r~/cr~ vs uniaxial strain c and void volume fraction 9 vs uniaxial strain L. It is seen that the results of the finite element analysis agree well with the “exact” solution.

The second example is the hydrostatic tension test on an &node brick element, type ABAQUS- C3D8 [22]. The element and the boundary conditions are shown in Fig. 7(b). Equal displacement increments of 0.002510 are imposed on the three faces of the element. The material properties and the initial void volume fraction are the same as those used in the previous example. In this case the problem is solved using the elliptic yield function. The

(4

b 1.5

1.0

0.5

Uaiaxial tenrion Power law Porour plasticity Gurson-Tvergaard mode1

c, = 0.0

- Exact o F.E.M.

0.0 1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

(b) f

1 I I I 1 1

Uniaxial tension Power law Porous plasticity

Guraon-Tvergaard model es = 0.0

- Exacct Q F.E.M.

0% 0.0 0.2 0.4 0.6 0.8 1.0 1.2

l

I.1 t

Fig. 8. Rate-independent power law porous plasticity: (a) true stress vs true strain response; and (b) porosity evolution using Gurson-Tvergaard void growth model [36,37] for homogeneous uniaxial tension: comparison of “exact” integration using an explicit forward Euler scheme with very small strain increments with the single element solution using the semi-implicit algorithm (static yield condition

approach).

corresponding equations describing the problem (with c3 # 0) are

do uH* dCkk= H* + 9h2K

dp w 3hzK dCkP=- 1 - t? H+ + 9h$


where

- 3(1 - S)JI;;$

3a aF Ro dhz Rdhs w=R’ K2JI;Id9 Cl9 (81)

The expressions for h2 and hS will depend on the particular model chosen. In particular, Cocks’ model (see Table 1B) was chosen for this example. These equations are also integrated “exactly” (with C) = 0) using the forward Euler scheme with\ equal volumetric strain increments of 2.5 x 10e5. These results are compared with the finite element solution in Fig. 9, where plots of stress ratio a/u0 vs volumetric strain ekk and void volume fraction 9 vs volumetric strain tkk are presented. Again, it is seen that both solutions agree well.

3.3.2. Rate-dependent state variable porous plasticity. The modified Bammann viscoplastic model is used to study numerically the initial stages of the necking process in a plane strain specimen under tension. The equations of this model are integrated with the semi-implicit scheme using the dynamic yield condition approach. The value of the material constants CI-C,s (see Table 2) used in these calculations cor:respond to those of 6061-T6 aluminum [41], as listed in Table 3. The elastic constants for this material are E = 69 x IO3 MPa, v = 0.33. A constant temperature of 21°C is assumed for the computations. Here, we select Cocks model for the coefficients h,, h2, hj in eqn (64) (see Table l.A). Only void growth is considered (c, = 0) with a uniform initial void volume fraction of & = 1 - p. = 10m4 (typical value for ductile metals). Necking is initiatled using an initial perturbed value of goi = 1.5 x 10M4 at a specific location in the specimen.

The finite element mesh and the element where the initial imperfection soi is introduced are shown in Fig. 10(a). The symmetry of the problem permitted consideration of only l/4 of the tensile specimen. The mesh consists of 330 four-node quadrilateral elements type ABAQUS- CPE4 [22]. The original length and width of the specimen are 210 (lo = 101.6 x lo-*m) and 2a, respectively, with an aspect ratio lo/a = 4. A unit thickness is assumed (th = 1.0). We applied displacement-rate boundary conditions on S, such that constant displacement increments of 0.0002510 are imposed at constant time steps of 0.01 s. This corresponds to an applied velocity of 2.54 x lo-* m/s (average strain rate in the order of 2 x lo-* s-l). The deformed mesh at an average

extensional strain of c = ln(1 + u/lo) = 0.180 is presented in Fig. 10(b).

The computed load-average extensional strain curve of the specimen is presented in Fig. 11(a). The maximum load is reached at a strain of approx. 0.1066. Note that at the onset of necking, the material response starts to deviate from that under a homogeneous deformation (H.D.) path. The porosity evolution response obtained at a material point at the center of the specimen (point “d” in Fig. 10) is presented in Fig. 11(b), where the void volume fraction has been plotted against the cross-sectional area (logarithmic)

4

3

S b

2

Unixxial tension Power law porous plasticity

Cocks model es = 0.0

- Exact 0 F.E.M.

“0.0 0.1 0.2 0.3 0.4 0.5 0.6

%k

@I

40

30

Hydrostxtic tcnxion Power Ixw porous pluticity

Cocks model

20

10

0 0.0 0.1 0.2 0.3 0.4 0.5 t 5

%k

Fig. 9. Rate-independent power law porous plasticity: (a) pressure vs volumetric strain response: and (b) porosity evolution using Cocks’ void growth model [30] for homogeneous hydrostatic tension: comparison of “exact” integration using an explicit forward Euler scheme with very small strain increments with the single element solution using the semi-implicit algorithm (static yield condition

approach).


Table 3. Constants for Bammann rate-dependent state variable porous plasticity 6061-T6 AI

C, = 6.90 IO-* MPa, C, = 0.00 IO0 K, CJ = 1.60 10’ MPa

Cd = 1.62 IO2 K, CS = 1.00 loo s-1, G = 0.00 loo K

CT = 1.91 10” MPa-‘, Cs = 6.94 IO* K, G = 1.03 10’ MPa

C,o = 0.00 lo0 K, CI, = 0.00 IO0 MPa-’ s-l, C,, = 0.00 IOr’ K

C,, = 4.42 lo-? MPa-‘, C,4 = 8.56 10’ K, C,S = 8.34 10 MPa

C,6 = 0.00 lo0 K, Cl7 = 0.00 lOa MPa-’ s-l, C,s = 0.00 100 K

strain. The predicted porosity at the onset of necking is of the order of 0.015%. After necking initiates, the formation of the neck produces a tri-axial stress state at the center of the neck that accelerates void growth in that region. This can be clearly seen in Fig. 10(b), where the rate of void growth for the necked specimen is greater than that corresponding to the specimen subjected to a homogeneous deformation path.

(a)

P

(b)

f

L- 3 1 Element with initial

imperfection e6,, I 3

1

Fig. 10. Plane strain specimen: (a) finite element mesh and element with imperfection 90i used for example of necking process. Mesh consists of 330 four-node quadrilateral elements, type ABAQUS-CPW [22]. Element aspect ratio along symmetry plane at the center of specimen is approx. 3: I. Displacement-rate boundary conditions are applied on S.; (b) deformed mesh at an average extensional strain of L = In(1 + u/

(a) 1.6

1.2

S

b” 0.8 2

0.4

0.0 -

I I , ,

o Muimumload

c

Plane strain specimen Bammatm viroplartic model

p,, = 0.9999. p,,, = 0.99985

I I I I 0.00 0.05 0.10 0.15 0.20 0.25

In(1 +u/l&

(b) 0.4 I I I I I

Plane atrain specimen Banunano viscoplastic model

3 p,, = 0.99990. p,,t = 0.99985

0.3 - 0 Onset of 8 neckiog

3 P h #) 0.2 - E 3 P $ p 0.1 -

0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6

Fig. 11. Bammann rate-dependent state variable porous plasticity model [41,42]. Effect of necking on: (a) load-average extensional strain curve; and (b) porosity evolution at material point at the center of plane strain specimen. The integration of the constitutive equations was carried out using the dynamic yield condition approach of the . . . .

semt-tmplictt algontnm. la) = 0.180.


4. CONCLUSIONS

A semi-implicit constitutive integration procedure was introduced in this paper for rate-dependent and rate-independent plasticity. This procedure was proposed in principle by Moran er al. [l] and can be interpreted in the context of generalized mid-point integration schemes. This semi-implicit scheme has the feature of being implicit in the incremental effective plastic strain AP (or Ap) but explicit in the plastic flow direction and hardening functions. This feature simplifies the solution procedure for the state update but imposes restrictions on the time step.

Two approaches to this integration scheme and the corresponding consistent tangent operators (stress Jacobians) were -presented. These approaches were denoted as the (dynamic yield condition and the kinetic equation approaches. An incompressible (deviatoric) finite deformation rate-dependent model and its rate-independent version were used to present the details of the algorithm. The explicit version of this integration scheme was developed and resulted in the so-called rate-tangent modulus formulation [l l-121.

Computations using this J2 rate-dependent model revealed that the semi-implicit scheme has better convergence and accuracy properties than the explicit scheme and that both the dynamic yield condition and the kinetic equation approaches give similar results. In general, the time step used with the semi-implicit scheme can be an order of magnitude greater than that used by the explicit method. The explicit character of the scheme, however, limits the time step size. In the example solved in this paper, time steps corresponding to strain increments of the order of the yield strain were taken. In addition, the numerical results showed that the dynamic yield condition approach converges faster than the kinetic equation approach.

The semi-impl:icit scheme was then readily extended to integrate the constitutive equations of porous inelasticity. The pertinent equations for the two approaches (dynamic yield condition and kinetic equation) were also presented. Two typical structures of a porous constitutive model were used in the development: the power law and the internal state variable inelasticity models. The first one assumed only isotropic hardening. The second one was the extension of the combined isotropic-kinematic hardening developed by Bammann [39,41] to deal with associative inelasticity of the damage-coupled rate- and temperature-dependent response. Some problems solved with these models illustrated the good performance of the semi-implicit integration scheme.

This paper has addressed the theoretical and implementation details of the semi-implicit integration algorithm. A relatively wide range of inelastic constitutive equations have been considered, including rate- and temperature-independent plasticity, rate- and temperature-dependent viscoplasticity,

incompressible and compressible inelastic flow, finite stretch and rotation, simple power law hardening with strain rate dependence, and combined kinematic-isotropic hardening using internal state variable concepts representative of unified creep-plasticity approaches [38,43-49]. Hence, most constitutive relations relevant to the response of initially isotropic, ductile metallic polycrystals have been considered explicitly. Several potentially important phenomena have been neglected in this particular work, including the plastic spin at the macroscale, anisotropic shape change in the static or dynamic yield/viscosity functions, and nonisothermal history effects.

Acknowledgements-The authors are grateful for the support of the U.S. Army Research Office for this research, as well as support of the U.S. National Science Foundation. The authors are also grateful to Dr D. J. Bammann and others in the Continuum Mechanics groups at Sandia National Laboratories in Livermore, CA for stimulating and helpful discussions. The experimental contributions of C. Bertoncelli from IRSID are also gratefully acknowledged.

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A semi-implicit integration scheme for rate-dependent and rate-independent plasticity

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