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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

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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 eserved
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, srcinally 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 constitu- tive 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 inte- gration 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 incremen- tal 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 pur- poses, 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-depen- dent) 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: 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) The evolution eqn (4) (kinetic equation) will be replaced by $(4:D) ‘* = ?x 2~ +
b + ;(dh/dP)
(13) which is obtained by using the consistency condition that the effective stress d is constrained to remain equal to the static yield strength eY = h(p)) during plastic deformation. MacCauley brackets () in eqn (13) define the loading/unloading condition. For the state update algorithm, it is convenient to write this rate-dependent constitutive model for inelastic incompressible solids with an associative flow rule and comibined isotropic-kinematic harden- ing as follows: Dynamic yield condition: Fd = R(u - a, P, Z’) = 0 Kinetic equation: 2’ = g(a - a, p’) = &i Evolution of u, a: z = C:(D - pq) z = b$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 particu- lar procedure to enforce objectivity typically starts with the decoupling of deformation and rotation effects [19,25,26,28] in eqn (14) using the time-de- pendent incremental rotation tensor
Q(t) (=
AR(t)) defined by the initial value problem
Q(t) = W(t) . Q(t), t. < t < t, + 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
582
E. B. Marin and D. L. McDowell 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+ + c: (D - pn,) dt 1. = Q(t.+J. oi(t,). QYfn+l) s t.+ +
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 ob- tained
[
19,281 using the incremental displace- ment 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
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+ ,, 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)
nd
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 alge- braic 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
Rate-dependent and rate-independent plasticity 583
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 t = p+ 1 i 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 iter- ations, 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 or Ap based on the rate-dependent (dynamic) yield condition.
The dynamic yield con- dition, 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)

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