J. Mech. Phys. Solids, Vol. 44, No. 4, pp. 5255558, 1996 Copyright 0 1996 Elsevia Science Ltd Printed in Great Britain. All rights reserved
Pergamon
SOO22-5096(96)000014
A COMPUTATIONAL INDEPENDENT
0022-5096/96 $15.00+0.00
PROCEDURE FOR RATECRYSTAL PLASTICITY
L. ANAND and M. KOTHARI Department
of Mechanical
Engineering, Massachusetts Institute MA 02139, U.S.A.
(Received 20 July 1995 ; in revisedform
of Technology,
Cambridge,
25 November 1995)
ABSTRACT In the rate-independent theory of crystal elasto-plasticity there have been three long-standing problems. The first is to determine which slip systems are active, and the second is to determine the increments of shear on the active slip systems. Third, because of the typical multiplicity of slip systems in ductile crystals, the selection of slip systems required to produce an arbitrary deformation increment is not necessarily unique. The purpose of this paper is to present a robust calculation scheme which determines a unique set of active slip systems and the corresponding shear increments in a rate-independent theory. We show by comparing the predictions from our computational procedure for the rate-independent theory against corresponding predictions from a procedure for a similar but rate-dependent theory (with a low value of the rate sensitivity parameter) that the results from the two procedures are essentially indistinguishable.
1.
INTRODUCTION
The quantitative description of plastic flow by crystallographic slip may be traced back to the early work of Taylor and Elam (1923,1925), and Taylor (1938a, 1938b). Constitutive equations for elasto-plastic behavior of ductile single crystals from the standpoint of modern continuum mechanics were first formulated by Mandel (1965) and Hill (1966), and extended to finite deformations by Rice (1971), Hill and Rice (1972), Asaro and Rice (1977), and Asaro (1983a, b). For a recent review see Havner (1992). In these models, intended for low homologous temperatures, the plastic deformation of single crystals by crystallographic slip is idealized to be rate independent. There have been three long-standing problems in the rate-independent theory of crystal plasticity. The first is to determine which slip systems are active, and the second is to determine the increments of shear on the active slip systems. Third, because of the typical multiplicity of slip systems in ductile crystals, the selection of slip systems required to produce an arbitrary deformation increment is not necessarily unique. These features of the rate-independent theory of plasticity pose a problem when the constitutive theory is applied to the numerical solution of boundary value problems. The first numerical calculations for a two-dimensional boundary value problem for a non-homogeneously deforming rate-independent elastic-plastic single crystal, whose geometry was idealized in terms of a planar double slip model, are those of Peirce et 525
526
L. ANAND
and M. KOTHARI
al. (1982). Because of a lack of a robust solution strategy to determine the active slip
systems and the amount of slip on these systems, the element stiffness matrices in the finite element scheme of Pierce et al. became singular for a particular choice of a slip system hardening rule, and their numerical scheme broke down. To overcome the limitations of crystal plasticity models based on a rate-insensitive idealization of crystalline slip, Asaro and Needleman (1985) developed a simple ratedependent crystal plasticity model which uniquely predicts constitutive response for arbitrary deformation histories. Specifically, Asaro and Needleman (1985) proposed that the shearing rate Jo”on a slip system CIis uniquely specified by 3”
=
y.
” s’
II
sign(Y).
(1)
In the equation above, r’ is the resolved shear stress on the slip system, and sa( > 0) is the slip system deformation resistance. The parameter Jo,is a reference rate of shearing, and the parameter wzcharacterizes the material rate sensitivity. The rate-independent limit is m + 0. The slip system shear rate is uniquely specified by this equation, and is non-vanishing as long as the resolved shear stress ril on that system is not identically zero. In crystal plasticity theories the slip system resistance parameters soLare taken to evolve according to
where i)” is the shearing rate on slip system /?, and the matrix [A@]describes the rate of increase of the deformation resistance on slip system CIdue to shearing on slip system j3; it describes both self-hardening and latent-hardening of the slip systems. The use of the absolute value of ps in the hardening equation reflects the assumption that the hardening behavior is not significantly affected by the direction of shearing on a slip system. Each element h”” depends on the deformation history. It is fair to state that at present these instantaneous hardening moduli /r@ are the least wellcharacterized part of the constitutive equations for crystal elasto-plasticity.7 In the rate-dependent formulation of Asaro and Needleman (1985) there is no division of slip systems into “active” or “inactive” sets, instead all slip systems always slip at a rate which depends on the current stress and slip system deformation resistance. Thus once the stress state is known (which it is in their formulation), then the slipping rates on all possible slip systems are uniquely determined ; and in this, no restrictions on the form of the hardening matrix [/@I are required.1 Although there has been considerable recent progress that has been made in the extension and application of the rate-dependent crystal plasticity model of Asaro and t In
an f.c.c. crystal there are 144 elements that need to be specified! f There has been a long history of considerations of possible mathematical restrictions on the hardening matrix [h”B] in order to formulate a complete theory of rate-independent crystal plasticity. Early in the development of the theory, Hill (1966) noted that with finite strain effects neglected (see Section 2 for a more complete discussion), positive definiteness of the hardening matrix was a sufficient condition to ensure unique slipping rates. Experiments on the other hand suggest that latent-hardening rates are generally somewhat larger than the self-hardening rates which means that the matrix [h’q may be indefinite. As noted earlier, at present the instantaneous hardening moduli [/@I are not well characterized.
Rate-independent
crystal
plasticity
527
Needleman (1985) to the solution of important boundary value problems (see, for example Bronkhorst et al., 1992; Kalidindi et al., 1992; Anand and Kalidindi, 1994), the rate-dependent constitutive equations for low values of the rate-sensitivity parameter m are very stiff, and the calculations are very time consuming. Accordingly, it is still of substantial technical interest to develop a robust calculation scheme to determine the active slip systems and the corresponding shear increments for the rateindependent theory of crystal plasticity, for physically realistic, but essentially arbitrary hardening matrices [@‘I. The purpose of this paper is to present such a scheme. In Section 2 we formulate a rate-independent constitutive model of crystal plasticity. In Section 3 we develop a similar incremental formulation for the model which is suitable for direct numerical implementation in displacement-based finite element programs, and briefly describe the details of our implementation of the constitutive model in a finite element program. In Section 4 we show results of the calculation of stress-strain curves, slip system activity, and evolution of crystal lattice orientation for tension tests on crystals initially oriented for single-slip and multi-slip for both non-hardening, h@ = 0, and a simple hardening rule. In order to obtain a tractable description of crystal hardening, several simple forms for the hardening matrix have been proposed in the past, and these have been reviewed by Peirce et al. (1982), and more recently by Havner (1992), and Bassani (1993). In their numerical calculations Peirce et al. (1982) used the following simple form for the hardening moduli hmP= [q+ (1 -q)P]hB
(no sum on fi),
with h” denoting the self-hardening rate and the parameter q, with values in the range 1 < q < 1.4, representing a latent-hardening parameter. The latent-hardening parameter q is not necessarily a constant, and may of course be history dependent just as the self-hardening parameter h is. This simple form for h@ yields an acceptable description of the physical phenomena of latent-hardening.? We present our calculations for q = 1.4, which is in the range of values of q for which the calculation procedures of Peirce et al. (1982) broke down. In Section 4 we also show the results of the calculation of stress-strain curves and crystallographic texture evolution during simple compression of a strain-hardening polycrystalline aggregate. In 1982 Peirce et al. conjectured that, “It may well be that no unique description of texture development, and therefore of polycrystalline behavior at finite strains, is possible within a rate-independent idealization, at least within one that attempts to describe latent-hardening in a realistic way. ” We shall show that our new formulation and calculation procedure for rate-independent plasticity overcomes this long-standing conjectured limitation of the rate-independent theory. Indeed, we shall show by comparing the predictions from our computational procedure for the rate-independent theory against corresponding predictions from a procedure for the ratedependent theory (Kaldindi et al., 1992) for a low value of the rate sensitivity parameter m, that the results from the two procedures are essentially indistinguishable. t However, see the papers by Wu er al. (1991) and Bassani latent-hardening in single crystals.
and Wu (1991) for some recent work on
528
L. ANAND and M. KOTHARI
2.
CONSTITUTIVE
MODEL
We shall use notation which is now common in modern continuum mechanics (e.g. Gurtin, 1981). In particular, the inner product of two vectors u and v will be denoted by u * v. The tensor product of two vectors u and v will be denoted by u 0 v ; it is the tensor which assigns to each vector w the vector (v. w) u. The symmetric and skew parts of a second rank tensor S are denoted by symS = (1/2)(S+ ST) and skwS = (l/2) (S - ST), respectively. The inner product of two tensors S and T is defined by S * T = trace (STT). At each time, the variables governing the response of our isothermal constitutive model are as follows. (i) The velocity gradient, L. (ii) The Cauchy stress, T. (iii) Crystal slip systems, labeled by integers ~1.Each slip system is specified by a unit normal na to the slip plane, and a unit vector ma denoting the slip direction. The slip systems (ma, na) are assumed to be known. (iv) The slip system deformation resistance scL> 0, with dimensions of stress. The quantities sa represent averaged slip system resistances to crystallographic glide offered by the underlying strengthening mechanism such as intrinsic lattice resistance, solid solution strengthening, forest dislocation densities, subgrains, etc. We assume that the velocity gradient L is specified, and the constitutive model that we shall consider consists of a coupled set of evolution equations for the state variables (T,(m”, 0, s”}. The incremental deformation of a crystal is taken as the sum of contributions from two independent atomic mechanisms : (i) an overall “elastic” distortion of the lattice, and (ii) “plastic” simple shears that do not disturb the lattice geometry. Accordingly, the macroscopic velocity gradient L is taken to be additively decomposable into an elastic and a plastic part L = Le+LP,
(2)
and that Lp is given as the sum of the shearing rates on all the slip systems Lp = 19’ sign(z”)S, CI
with 3” Z 0,
(3)
where SCr=mrr@na
(4)
rz = ma-T& = T-S
(5)
is the Schmid tensor, and
denotes the resolved shear stress, or the Schmid stress on the slip system ~1.With this, the elastic and plastic stretching and spin tensors are denoted by D” = symL’,
(6)
W” = skw L",
(7)
DP = ~~” sign(r”
with P” = symS”,
(8)
Rate-independent
crystal plasticity
with Q” = skw s”,
Wp = 19” sign(r’)v,
respectively. The lattice corotational
529
(9)
rate of the Cauchy stress is denoted by TVe = T-w=T+Twe,
(10)
and assuming that the elastic stretches are small, the evolution equation for T is taken as TV’ = %?[D”]= %[D] --c 9” sign(r”)%?[P”], with y” 3 0,
(11)
where %?is a fourth order elasticity tensor. The conditions under which a slip system is inactive, p” = 0, or active, i)” > 0, are based on the following yield and unloadingloading criteria. The actual magnitudes of i)”for the active systems are determined by the consistency condition to be considered shortly. The yield conditions of the rate-independent crystal plasticity problem are ]r’l < s’. The planes ]ra] = s” denote the facets of the current pyramidal yield surface in stress space, and {sign(r”)P”} denotes the outward unit normal to each facet. If ]z’] < s’, or if jr’1 = scLand a trial stress rate %[D] points to the interior side of the yield facet, that is, if {sign(r”)P”} * %[D] < 0, then the slip system is inactive and 9” = 0. A slip system is potentially active, that is i)” > 0, if ]?I = soLand {sign(r” *%Y[D]> 0. To summarize d” =
0
if )T’I < s’,
or if ]ral = S’ and if Jr’I = s’
i 20
and
{sign(z”)P”} * %?[D]< 0, {sign(z”)P”} *%?[D]> 0. 1
(12)
We denote by LY’d= {ala=
l,...,nJ
the n potentially active slip systems. Of the n potentially active systems the m < n systems for which the shear rates are actually non-zero are the active systems, and we denote the set of active systems by &=
{ala=
l,...,m
The orientation of the slip systems is taken to evolve as m’ = Wema,
and
n” = We&,
(13)
and the evolution equations for the slip resistances are taken as (14) where ha” are the hardening moduli which describe the rate of increase of the deformation resistance on slip system tl due to shearing on slip system /I. The matrix [h”“] describes both self-hardening and latent-hardening of the slip systems. Finally, during plastic flow the active systems must satisfy the consistency condition
530
L. ANAND
and M. KOTHARI
and for inactive systems the condition A
(16)
lZUl < S”
must hold. Using (5), (lo), and (13) we obtain fi
= Ima .‘Tn’] = {sign(r”)P”} * TVe.
(17)
Substituting (17) and the evolution equations for the stress (11) and the slip resistances (14) in the consistency condition (15), gives the following set of equations for the evolution equations for the state variables (T,(mC(,n’), sa} to be “consistent” c AaBxB = b”, BE.d
and,
(18)
with A@ = VB+ sign(?) sign(rb)P” - %?[Pfi],
(19)
b” = {sign(t)P”} * W[D] > 0,
(20)
xp = i)” > 0.
(21)
Equation (18) is a system of linear equations for xs z pa > 0. However, we do not know the elements of the set d. These are determined in the following iterative fashion. Since we do know the potentially active systems, we assume that all the potentially active systems are active, extend the equations in (18) to
(22) and solve this linear system of equations Ax = b, with A an n x n matrix, for the vector of shear rates x, looking for elements xfi = 3” > 0. If for any system the solution y$ = 38 Q 0 2 then this system is inactive, we remove it from our list of active systems, redefine our reduced system Ax = b, and solve the linear system of equations until all xfi = 3a > 0. This iterative solution procedure determines the active slip systems and the corresponding slip increments. At each stage of our iterative procedure the linear equations Ax = b are solved as follows. Recall that the range, null space, and rank of a matrix AE 9Txn are, respectively, defined by : R(A) = {y E w”Jy = Ax for some x E B’}, N(A) = {x E %!“IAx = O},
Rate-independent
crystal plasticity
531
r = rank(A) = dim[R(A)]. If A has rank r = n, then it is non-singular. If A has rank Y< n, then it is singular, and in this case N(A) # {0}, and dim[N(A)] = n-r. The problem concerning us is how to find x E @ such that Ax = b, where A E L%?“I and b E W” are given. As is well known (e.g. Golub and Van Loan, 1983), this problem has a solution if and only if b E R(A), and it is unique if and only if N(A) = (0). For the rate-independent crystal plasticity problem formulated above, we assume that b E R(A), that is a solution exists. However, because of the typical multiplicity of slip systems in ductile crystals we expect that for certain crystal orientations and loading conditions the matrix A may be singular, that is N(A) # (0). In this case the singular set of equations Ax = b does have a solution x, but it is non-unique. We examine this case next with the aid of the powerful singular value decomposition theorem. A singular value decomposition (SVD) of an n x n real matrix A is a factorization of the form (e.g. Golub and Van Loan, 1983 ; Press et al., 1986 ; Strang, 1988)
A = Q,%Z:,
(23)
where Q:Q, = QTQz = I,,
and
C = diag(a,, . . . , a,)
with cr, > c2 > .. 3 cr,,3 0. (24)
The columns of Q, are orthonormalized eigenvectors of AAT, and the columns of Q2 are orthonormalized eigenvectors of ATA. The diagonal elements of C are the nonnegative square roots of the eigenvalues of both AAT and ATA, and are called the singular values of A. The singular value decomposition is unique. In terms of the singular value decomposition, the rank r of a matrix A is the number of non-zero singular values. Thus if rank (A) = r, then or+, = cr,+? = . . = gn = 0. Let A = Q,XQ: be the singular value decomposition of A, then the pseudoinverse of A is defined by A+ = QJ+Q:,
with C+ = diag(oT),
(25)
where l/a,
for 0, > 0,
0
for ci = 0.
(26)
If A is square and non-singular, then all its singular values are non-zero. In this case A is invertible with inverse A-’ = A+, and Ax = b has one and only one solution x = A+b = A-lb. That is, the pseudoinverse is the same as the inverse when A is invertible. Let b E R(A) be a given vector, and suppose we wish to determine a vector x such that Ax = b when A is singular. In this case there is no unique solution. However, the solution given by x+ = A+b,
(27)
532
L. ANAND
and M. KOTHARI
is unique, and it possesses the important satisfy Ax = b it has the smallest length
characteristic
that amongst all x which
/ix+ /I = (XTX)“2 = min. All other solutions x have some component from the null space N(A). The length squared of any such x will be equal to the length squared of x+ added to length squared of the nullspace component-which can only increase in length over that of x+. We adopt x+ as our solution to the linear problem (22) at each stage of our iterative procedure to determine the active slip systems and the corresponding slip increments. Remark 1 : If instead of (13), which corresponds to the requirement that (ma, n’) be orthogonal unit vectors which rotate rigidly with the lattice spin, one adopts the evolution equations s
= Lema = WemE+ pm”,
which correspond lattice slip plane as the “reciprocal the linear system
h” = _ ~e’rna = Wena _ Dena,
(28)
to an alternative assumption that the vectors m” convect with the so that their length does not remain constant, and the choice of n’ base vector” normal to the deformed slip plane, then one obtains of equations Ax = b with
A@ = hXB+ sign(Y) sign(@)P” *%JPB]+ sign(r”) sign(@)B” * P”,
(29)
b” = {sign(z”)P”} *V[D]+ {sign(z”)B”} *D > 0,
(30)
xfl = p” > 0,
(31)
B” = Q*T-TQ”.
(32)
where
Also, in this case the loading-unloading criteria (12) needs to be modified to use [ { sign(z”)P”) - %?[D]+ { sign(z”)B”} * D] instead of [ { sign(z”)P”) * W[D]]. Note that since terms involving B” are directly attributable to the small lattice stretching of the slip vectors (ma, n’), their effect on the matrix A and the vector b (which enters the loadingunloading conditions) is also expected to be small. Hill and Rice (1972) have shown that if the matrix A with elements (29) defined over the active systems is positive definite, then one always obtains a unique set of slip rates. There have been numerous attempts at formulating equations for the hardening moduli /z@to ensure that A is positive definite to guarantee a unique set of slip rates (e.g. Bassani and Wu, 1991 ; Havner, 1992). However, since the slip system hardening moduli /z@ are typically three orders of magnitude smaller than the incremental elastic moduli %’and the effect of the terms involving B” is small, the second term in (29) dominates, Thus, the uniqueness (or lack thereof) of the p”, is dominated by the multiplicity of the slip systems and the corresponding inter-dependence of the tensors P” and PB. Any robust solution scheme for rate-independent crystal plasticity should not depend heavily on a precise knowledge of the hardening moduli VB which, as mentioned previously, are the least well-characterized part of the constitutive model. We shall show that our computational scheme for calculating
Rate-independent
crystal plasticity
533
the shear rates i)”works for essentially arbitrary, h”“, including the technically important non-hardening case haB= 0. Remark 2: The smallest length solution x + is reminiscent of Taylor’s early but incomplete method of determining the amount of slip rates p” on the active slip systems. In the rigid-plastic, non-hardening theory, Taylor made the assumption that each crystal will slip on that combination of systems which makes the incremental work the least. Then from his additional assumption of equal positive slip resistances for every slip system he deduced his principle of minimum incremental shears. His most concise statement of this for f.c.c. crystals is in Taylor (1938b) “Of all the possible combinations of the 12 shears which would produce an assigned strain, only that combination is operative for which the sum of the absolute values of the shears is the least.” Now, the p-norms of any vector XEP 1983)
are defined by (e.g. Golub and Van Loan,
llxllp =(1x, Ip+ “’ +Ix,Jp)“p,
p 3 1,
of which II-d, =O,l+
‘~.+l47l)v
and IIxl12= (Ix, I*+ . . . + lx,12)‘i2 = (x=x)1’2 are two important special cases. Also, these two norms satisfy the inequality II412 6 llxll1.
Thus the l/x/l2 = (xTx)‘12norm of our solution x+ is also smaller than or equal to the I/x(/, = (Ix,1 + . . + /x,l) norm of Taylor’s. In Taylor’s original formulation it was not clear that the yield criterion was satisfied on all active slip systems, and not violated on the inactive systems. Later, Chin and Mammel (1969) rephrased Taylor’s method for determining the active slip systems and the amount of shear rates Jo”> 0 on these systems in terms of an optimization problem of linear programming. In particular, with scL> 0 denoting the deformation resistance of a slip system, and y” 2 0 the corresponding shearing rate, they showed that Taylor’s method of determining the active slip systems consists of minimizing the “internal rate of work”
subject to the constraints C(y) = D--c?” 1
sign(z”)P” = 0,
where D is the imposed stretching tensor. The minimum rate of work is obtained as the saddle value of the Lagrangian function
m
A) = W(3) + A * C(P),
534
L. ANAND and M. KOTHARI
where A is a Lagrange multiplier. To obtain the saddle point one finds (j,,, A,) such that &FiO = D-13; I ?F] ay
0 = sc(- sign(r
sign(z”)P’ = 0
- p” =
>O
(33)
always ‘.
i =0
if?*,>0 1
(34)
The inequalities (34) are equivalent to Sa-JrCrg]=
20 i =0
always if?“, >O I ’
(35)
with the Lagrange multiplier A,, identified as the stress T,. Hence in order to obtain a set of shear rates i)”that minimizes the internal rate of work W(y), there exists a stress T which obeys the yield conditions, and which attains the critical value sa on the systems where slip occurs (y; > 0), and does not exceed it on the inactive systems (-3: = 0). Although it was unrecognized by Taylor, this restriction is part of the minimum rate of work analysis. Note that such a solution is still non-unique, whereas the minimum length solution x+ obtained by using the SVD method to solve our previously formulated linear system Ax = b is unique.
3.
AN INCREMENTAL
FORMULATION MODEL
OF THE CONSTITUTIVE
Here, we present an incremental version of the rate-independent constitutive model for crystal elasto-plasticity. The major differences between the model considered here and that presented in the previous section are that : (i) instead of the additive decomposition of the velocity gradient into an elastic and plastic part, L = Le+LP, we use a multiplicative decomposition of the deformation gradient into an elastic and plastic part, F = FeFP, and (ii) instead of the rate constitutive equation TV”= %?[D’] for the stress, we shall use a total hyperelastic constitutive equation for the stress. For infinitesimal elastic stretches, the two formulations may be shown to be equivalent. However, the incremental formulation developed here is suitable for direct numerical implementation in displacement-based finite element programs, and it has the additional advantage that one does not have to develop an “incrementally objective time integration procedure” for it (see, for example Weber et al., 1990), the time integration scheme arising naturally from the incremental model automatically satisfies this important requirement. The governing variables of the model are as follows. (i) The Cauchy stress, T. (ii) The deformation gradient, F. (iii) Crystal slip systems, labeled by integers CI.Each slip system is specified by a unit normal nol,to the slip plane, and a unit vector rn; denoting the slip direction. The slip systems (m& ng) are assumed to be known in the reference configuration. (iv) The plastic deformation gradient, Fp, with detFP = 1.
Rate-independent
crystal plasticity
535
The local configuration defined by F* is relaxed, that is T = 0, and is chosen such that the crystal lattice in this configuration has the same orientation with respect to a fixed orthonormal basis in space as it had in the reference configuration. Following Mandel (1974), such a local configuration is called an isoclinic relaxed configuration. (v) The slip system deformation resistance ,sa> 0, with units of stress. In terms of F and F* the quantity F” E
FFP-’
(36)
defines an elastic deformation gradient. Let t denote the current time, At an infinitesimal time increment, and r = t + At. We take as given (F(t), F(r)), and (m;, II:), {T(t), F*(t), s”(t)}. Then the incremental problem is to calculate {T(r), F*(r), sa(r)}, and the orientation of the slip systems in the deformed configuration at time r from m; = F”(r)m”,, n,l = F”(r))=n”,, and march forward in time. For metallic materials the elastic stretch is usually infinitesimal, and accordingly the constitutive equation for stress may be taken as a linear relation T*(r) = %[E”(r)],
(37)
where V is a fourth order elasticity tensor. With C(r) = (F”(r))=F”(r), defining an elastic right Cauchy-Green
(38)
tensor
E’(r) =(1/2){c”(r)-1)
(39)
is an elastic strain measure, and T*(r) = F”(r))’ {(det F”(r))T(r)}F”(r))T
(40)
is a stress measure conjugate to the strain measure (39). The tensor product SCr,= m: @ n;
(41)
is called the Schmid tensor for the slip system CI,and the scalar r’(z) = {Ce(z)T*(r)} .S;
(42)
is the resolved shear stress, or the Schmid stress, on the slip system CIat time r. For infinitesimal elastic stretches the resolved shear stress r’(r) may be approximated by r”(r) 5 T*(r) *S;
(43)
Next, we define a trial elastic strain and stress at time r as follows. We fix the value of F* at time t, and define a trial elastic deformation gradient by
L. ANAND
536
F”(z)”
and M. KOTHARI =
F(7)FP(t)-‘,
(44)
in terms of which we define a trial elastic right Cauchy-Green (Ye’
tensor by
= (F’(7)“)TF”(7)“,
(45)
a trial elastic strain by Ee(7)tr =(1/2){C’(r)“-l},
(46)
a trial stress by T*(z)” and a trial resolved
= %?[E”(z)“],
(47)
shear stress by ~~(7)~’ = T*(z)“*S*,.
(48)
We make the important physical assumption that sign(z”(7)) = sign(z”(7)“).
(49)
With these definitions, the incremental flow rule is taken as 1 +c Ay” sign(7”(7)“)!3”, FP(t),
FP(7) =
a
I
0
if 17’(7)trI < s”(t),
i 20
if 17Z(7)trI > s”(t).
i
(50)
with Ay’ =
(51)
Systems for which 17x(7)trI 6 s”(t),
are inactive, and those for which 17a(7)fTI> s”(t), are potentially
active. As before, we denote by
9?d=
{or(cr= l,...,n}
the n potentially active slip systems. Of the n potentially active systems, the m < n systems for which the shear increments are actually non-zero are the active systems, and we denote the set of active systems by d=
{ala=
l,...,m
Next, the evolution equation for the slip resistances is taken as s’(7) = s”(t)+ where h@ are the hardening
moduli,
1 PP(t)AyB, pt.4
tl = 1,. . . ,N,
and N is the total number
(52) of slip systems.
Rate-independent
crystal
537
plasticity
Finally, during plastic flow the active systems must satisfy the consistency condition 17x(7)l
=
f(7),
(53)
<
f(7),
(54)
and for inactive systems the condition I7”(7)l
must hold. Retaining terms of first order in AyB,it is straightforward
to show that
[z”(7)] = ]7a(7)tr] - &{sign(7”(7)“)
sign(7a(7)‘r)
SE -%?[sym(C”(~)“S{)]}Ay~‘,
(55)
use of which in the consistency condition (53) gives
(56) with AxB= h”“(t) +sign(z”(z)“) sign(zB(7)“)S; ~V[sym(C’(z)“S[)], 6” = ]70L(7)“I -s”(t)
> 0,
xB 3 Ay” > 0.
(57) (58) (59)
Equation (56) is a system of linear equations for xB = Ay” > 0. Apart from the presence of a small elastic stretch related term Ce(7)tr, the system of linear equations (56) is similar to the system (18), and the iterative solution procedure to determine the active slip systems and the corresponding slip increments developed in the previous section applies here as well. The procedure is summarized below. Let t denote the current time, At an infinitesimal time increment, and 7 = t+At. Then, Given : (1) F(t), F(7) ; (2) (mz, ni)-time-independent quantities; and (3) {FP(t), s”(t), T(0). Calculate: (a) {FP(7),.P(7),T(7)} ; and (b) the orientation of the slip system in the deformed configuration at time 7 from rn; = F'(T)m",,
and march forward in time. The steps used in the calculation procedure are :
538
Step 1.
L. ANAND
and M. KOTHARI
Calculate the trail elastic strain Earn
:
Fe(7)tr = F(7)FP(t), Cafe
= (F”(~)“)=F”(z)~~,
Ee(7)tr = (1/2){C”(7)“-1). Step 2.
(60)
Calculate the trial stress T*(z)“: T*(z)” = %?[Ee(7)t’].
Step 3.
Calculate the trial resolved shear stress ~‘(7)‘~on each slip system S; =mol,@n$, ~~(7)~~ = T*(z)” -S;.
Step 4.
Determine the set YZXZ’ of the n potentially active slip systems which satisfy 17a(7)frj
-f(t)
>
0.
Step 5. Calculate the shear increments from the consistency condition using the SVD method as X+ =
A+b,
where A+ is the pseudoinverse of the matrix of the n x n matrix A, defined over all the potentially active slip systems, with elements AEB= h@(t) + sign(z”(7)“) sign(78(7)“)S; *%?[sym(C”(z)“S{)] b” =
IZa(7yI
xD E
AyB > 0.
-s”(t),
If for any system the solution xB = Ays < 0, then this system is inactive; remove it from the set of potentially active slip systems by removing the /?th row and column of A and the Pth row of b, redefine the reduced system Ax = b, and solve the linear system of equations until all xB = AyB> 0. The size of A at the end of this step is m x m, where m is the number of active slip systems. Step 6.
Update the plastic deformation
gradient FP(7)
FP(7) = l+ 2 sign(z’(z)“)Ay”S*, FP(t). a=l 1 { Step 7.
Check if detFP(7) = 1. If not, normalize FP(7) as FP(7) = [detFP(7)]-“3FP(7).
Step 8.
Compute the elastic deformation F”(7) = F(7)FP+(7)
gradient F’(7) and the stress T*(7)
Rate-independent
crystal
539
plasticity
T*(z) = Te(7)tr- f$ {Ay’ sign(r”(z)“)}%?[[sym(C”(z)“S”,)]. a=1 Step 9.
Update the variables {T(7), f(7)) T(7) = f(7)
Fe(z){[detFe(r)]-‘T*(r)}F”‘(z)
= f(t)+
f
B=I
PBAy”,
a=1 ,“‘> N.
Here, N denotes the total number of slip systems. For example, N = 12, for an f.c.c. crystal. Step 10. Check if consistency is satisfied for all the slip systems, i.e. l7Y7)l = f(7),
if the slip system is active, or I7”(7)l <
f(7)
(61)
if it is inactive. If a particular system, which has previously been deemed to be inactive, does not satisfy this check,? then include it in the set of potentially active slip systems, go back to step 5 and recompute the shear increments on the active slip systems. Step 1I. Calculate the “texture” (m:, n,“) rn; = F”(z)m;, n; = FeeT(7)n;.
The constitutive equations and the time-integration procedures for the rate-independent crystal plasticity model described in this section have been implemented in the finite element programs ABAQUS/Standard (1994) and ABAQUS/Explicit (1994) by writing “user material” subroutines. ABAQUS/Standard is a static implicit finite element program which requires a “Jacobian matrix” for the Newton-type iterative method it uses for calculating the equilibrated configuration at the end of a time step. In our implementation we used a quasi-analytical1 Jacobian employing the ratedependent constitutive equation (1) for the slip increments (Kalidindi, 1992). ABAQUS/Explicit is a dynamic explicit finite element program which does not require a “Jacobian matrix”. The single crystal calculations shown in the following section were carried out using our implementation in ABAQUS/Standard, whereas the polycrystal calculations reported in what follows were carried out using our implementation in ABAQUS/Explicit. t This occasionally occurs at the knee of a load-displacement curve. 1 Note that an approximate Jacobian has no effect on the accuracy of the solution computation time by affecting the rate of convergence of the solution.
but only increases
the
540
L. ANAND
4. EVALUATION
and M. KOTHARI
OF THE RATE-INDEPENDENT
MODEL
In this section we evaluate the rate-independent model by comparing the predictions of the stress-strain response and the evolution of the crystal orientations against the predictions of the rate-dependent model of Kalidindi et al. (1992) for tension tests on f.c.c. single crystals and a compression test on a f.c.c. polycrystal. For f.c.c. crystals the anisotropic elasticity tensor %?may be specified in terms of three stiffness parameters, C, ,, C,, and C, defined as follows C, , = (e; 0 e;) *W[e; 0 e;],
(62) (63)
Cd4 = (e; 0 e;) - %?[2sym(e; 0 e;)],
(64)
where {ep]i = 1,2,3} denotes an orthonormal basis associated with the crystal lattice. In what follows we will consider tests on copper. The values of the elastic parameters for copper are taken as (Simmons and Wang, 1971) C, , = 170 GPa,
C, z = 124 GPa,
C,, = 75 GPa.
For f.c.c. crystals crystallographic slip is assumed to occur on the 12 (11 l}( 110) slip systems. The components of the slip plane normals and slip directions with respect to an orthonormal basis associated with the crystal lattice for these slip systems are presented in Table 1. The Schmid and Boas (1935) convention of labeling the slip systems is also shown in this table. 4.1. Single crystals Here we evaluate the rate-independent model by comparing the predictions of the stress-strain response, the slip system activity and the evolution of the crystal lattice orientation for tension tests on f.c.c. crystals against corresponding predictions of the rate-dependent model of Kalidindi et al. (1992) with a low value of the rate sensitivity parameter m. We consider two types of crystal hardening behavior : (1) Non-hardening
: ha0 = ()
(2) A simple hardening model with moduli : h@ = [q+ (1 - q)P8]hS
(no sum on B),
(65)
where hS is a single slip hardening rate, and q is the latent-hardening ratio. The parameter q is taken to be equal to 1.O for coplanar slip systems, and 1.4 for noncoplanar slip systems (Peirce et al., 1982 ; Asaro and Needleman, 1985). For the single slip hardening rate, the following specific form is adopted? t The hardening behavior
equations (65) and (66) do not model the classical Stage I, Stage II, etc. hardening of single crystal single slip, but are intended to represent polyslip behavior.
Rate-independent
crystal plasticity
541
Table 1. Components of m; and n; with respect to an orthonormal basis associated with the crystal lattice for f.c.c. crystals. The slip systems are also labeled according to the Schmid and Boas (1935) convention
Label
a
I
A2
2
A3
3
A6
4
D4
5
Dl
6
D6
8
B2
B4
B5
(66) where h,, a and S, are slip system hardening parameters which are taken to be identical for all slip systems. The values for the slip hardening parameters for single crystal copper are taken as h, =
180 MPa,
S, = 148 MPa,
a = 2.25
for f.c.c. copper single crystals (Bronkhorst et al., 1992).
542
L. ANAND and M. KOTHARI
ii 1
001
011
Fig. 1. Primary orientation triangle in a [OOl] stereographic projection. The symbol * denotes the initial orientations of the axes of the three tension tests considered.
The initial value of the slip system deformation hardening and hardening cases is taken as
resistance for both the non-
S, = 16 MPa. In order to compare the predictions of the rate-independent model against the ratedependent model, we used the rate-dependent constitutive equation (1) for the slip rates, which we repeat below for convenience m l/m i)”
=
3,
sign(Y).
L I
sa I
(67)
Here p, and m are material parameters, respectively, representing a reference shearing rate and the rate sensitivity of slip. The reference shearing rate jamwas taken to be equal to 0.001 s-‘, and the value of the strain rate sensitivity parameter, m, is set equal to a low value of m = 0.012. Recall that the rate independent limit is m -+ 0. The single crystal orientations considered are the two corner multi-slip orientations [OOl] and [ill], and a single-slip orientation [236] inside the primary stereographic triangle [OOl]-[Ol l]-[il l] as shown in Fig. 1. These initial orientations are specified by three Euler angles (e, 4 and o} shown in Table 2 (Kalidindi et al.,1992). A single eight-noded brick ABAQUS C3D8 element was used to perform the finite Table 2. Single crystal orientations specified by three Euler angles
Orientation $Y; [2361
9 54.7356 0.0 -31.0
4 135.0 0.0 33.7
CfJ 0.0 0.0
Rate-independent
-rd -rd .._.... ri
543
crystal plasticity
0.05
ri
-
b B4, B5, C5, C3,
-
D4, D6, A3, A6 0.1
0.2
0.3
E
(b)
(a>
001
011
Fig. 2. Comparison of the results from a rate-independent (ri) crystal plasticity model against those of a rate-dependent (rd) model for simple tension of a non-hardening crystal initially oriented in the [OOI] direction, (a) Stress-strain curve ; (b) slip shears ; (c) inverse pole figure of the change in the orientation of the tensile axis (* denotes initial orientation and + denotes the final orientation).
element simulations using ABAQUS/Standard. The tensile direction is aligned with one of the element axes. The two faces of the cube initially perpendicular to the tensile axis are constrained to remain parallel to each other and perpendicular to the tensile axis throughout the deformation. These boundary conditions simulate a stiff loading machine. The simulations were carried out for an axial strain rate of 0.001 s-’ to a final true strain of 0.25. 4.1.1. Non-hardening case. Figures 24 show the stress-strain curves, the slip system activities, and the evolution of the crystal orientations obtained from the rateindependent model and the rate-dependent model during tension tests on single
L. ANAND
and M. KOTHARI
60
2 z
40
b 20
E
(a>
Fig. 3. Comparison of the rate-dependent (rd) model direction. (a) Stress-strain the tensile axis
results from a rate-independent (ri) crystal plasticity model against those of a for simple tension of a non-hardening crystal initially oriented in the [I 111 curve ; (b) slip shears ; (c) inverse pole figure of the change in the orientation of (* denotes initial orientation and + denotes the final orientation).
crystals with initial orientations [OOl], [ill], and [236], respectively. The results from the rate-independent model are almost indistinguishable from the rate-dependent model. There are eight slip systems activated in the [OOl] orientation. The slip activity on all the systems is equal, and the orientation is stable, Fig. 2. There are six slip systems activated in the [il l] orientation. The slip activity on all the systems in equal, and the orientation is also stable, Fig. 3. At first there is only slip system A3 = (111) [TOl] which is activated in the specimen initially oriented in the [236] direction. During this single slip period the orientation of the tensile axis rotates on a great circle towards the slip direction [TO11until it
Rate-independent 50
crystal
plasticity
I
40
i--, b
20
10
\ /’ -rd
ri
\
Fig. 4. Comparison of the rate-dependent (rd) model direction. (a) Stressstrain the tensile axis
results from a rate-independent (ri) crystal plasticity model against those of a for simple tension of a non-hardening crystal initially oriented in the [236] curve ; (b) slip shears ; (c) inverse pole figure of the change in the orientation of (* denotes initial orientation and + denotes the final orientation).
reaches the [OOl]-[il I] boundary of the stereographic triangle around a strain of -0.07, Fig. 4(c). At this point, the conjugate slip system B5 = (lli)[Oll] becomes active.? If the crystal were to slip only on the slip system B5, the tensile axis would now t There is a small difference in the results from the rate-independent and rate-dependent model with regard to the activation of the conjugate slip system B5. In the rate-independent model the activation of the conjugate slip system B5 is abrupt as compared to the smooth activation in the rate-dependent model, Fig. 4(c). In the rate-dependent model all the slip systems with non-zero resolved shear stress are active. On the other hand, in the rate-independent model a slip system is potentially active only if the trial resolved shear stress exceeds the slip deformation resistance on the slip system. This difference in the activation criteria of a slip system is reflected in the lag in the activation of the conjugate slip system B5 in the rateindependent model when compared with the rate-dependent model. The slip system BS is activated at a strain level of 0.06 in the rate-dependent model when compared to a strain level of 0.07 in the rateindependent model, Fig. 4(c). The predicted slip shear increments on the conjugate slip system are identical thereafter in both the models.
546
L. ANAND
and M. KOTHARI
rotate towards the conjugate slip direction [Ol 11. However, since there is simultaneous equal slip on the two slip systems A3 and B5, the net result is a slow rotation of the tensile axis along the [OOl]-[i 1l] boundary of the stereographic triangle towards the limit orientation [i12] (which is the vector sum of [iOl] and [Oil]), Fig. 4(c). It is to be noted that for a non-hardening material model the rate-independent and ratedependent models predict no overshoot of the tensile axis beyond the [OOl]-Ii1 11 boundary. As we shall see in the next section, this is consistent with observations that it is only strong latent-hardening that gives rise to overshoot and continued single slip on the original system beyond the [OOl]-[il l] boundary (Havner, 1992). Note also, that even though the slip systems are non hardening, Fig. 4(a) shows that there is hardening of the macroscopic stress-strain response of the crystal. This macroscopic hardening is entirely due to the change in the orientation of the crystal. This analysis shows that the predictions of the rate-independent crystal-plasticity model match those from the rate-dependent model accurately for the orientations inside the primary triangle, as well as the corner multi-slip orientations where the matrix [ALla]is singular. In what follows next, the case of a strain-hardening crystal with strong latent-hardening is examined. 4.1.2. Strain-hardening case. Here we consider the strain hardening model contained in (65) and (66), with the listed material parameters. Recall that we are considering the case of a rather strong latent-hardening with q = 1.4. Figures 5-7 show the stress-strain curves, the slip system activities, and the evolution of the crystal orientations obtained from the rate-independent model and the rate-dependent model during tension tests on single crystals with initial orientations [OOl], [ill], and [236], respectively. Again, the results from the rate-independent model are almost indistinguishable from the rate-dependent model. As before, there are eight slip systems activated in the [OOl] orientation. The slip activity on all the systems is equal, and the orientation is stable, Fig. 5. For the specimen in the [f 111 orientation there are six activated slip systems, the slip activity on all the systems is equal, and the orientation is also stable, Fig. 6. For the specimen initially oriented in the [z36] direction, as in the non-hardening case, at first there is only slip system A3 = (11 I)[iOl] which is activated, and during this single slip period the orientation of the tensile axis rotates on a great circle towards the slip direction [TOI]. However, the conjugate slip system B5 = (1 li)[Ol I] does not get activated when the tensile axis reaches the [OOl]-[il l] boundary of the stereographic triangle, and the single slip regime overshoots it. The slip system B5 does not become active until an axial strain of - 0.12, which compares with an axial strain of - 0.07 for the non-hardening case. The physical reason for the overshoot is that since the latent-hardening ratio q has been set to greater than unity, slip on primary slip system A3 has hardened the latent slip system B5 much more than itself. Once the conjugate slip system does becomes active, there is an abrupt change in the direction of rotation of the tensile axis as seen in Fig. 7(c). Since the orientation [?36] is not stable relative to the underlying lattice, there is macroscopic hardening due to a change in the orientation of the crystal lattice, and due to the hardening of the slip systems Fig. 7(a). The predictions of the rate-independent model are in excellent agreement with those
Rate-independent
20°
crystal
O.1°
I
547
plasticity
7
- -rd .._.... ri - s
0.05 -
B4, B5, C5, C3, D4, D6, A3, A6
0.0
0.1
0.2
0.0
0.3
0.2
0.1 E
f
W
(a> -
Fig. 5. Comparison of the rate-dependent (rd) model direction. (a) Stress-strain the tensile axis
results from a rate-independent (ri) crystal plasticity model against those of a for simple tension of a strain-hardening crystal initially oriented in the [OOl] curve ; (b) slip shears ; (c) inverse pole figure of the change in the orientation of (* denotes initial orientation and + denotes the final orientation).
of the rate-dependent model, Also, for the hardening case the predictions are in qualitative agreement with existing experimental observations of tension tests on f.c.c. materials which have been recently reviewed by Havner (1992). His summary findings are paraphrased? below : (1) “In virgin crystals carefully oriented in high-symmetry (six- or eight-fold) multiple-slip positions : (a) the loading axis is stable relative to the underlying t Havner here.
also comments
on the evolution
of dislocation
structures,
but we do not go into that matter
L. ANAND and M. KOTHARI
548
0.15 - -
c
0 0.0
0.2
0.1
0.10
0.3
rd ri
-
0.0
E
0.1
0.2 E
(4
@I
ill
A -rd
001
ri
011
w Fig. 6. Comparison of the results from a rate-independent (ri) crystal plasticity model against those of a rate-dependent (rd) model for simple tension of a strain-hardening crystal initially oriented in the [l 1 l] direction. (a) Stress-strain curve ; (b) slip shears ; (c) inverse pole figure of the change in the orientation of the tensile axis (* denotes initial orientation and + denotes the final orientation).
(2)
lattice ; (b) the deformation is ordinarily axisymmetric ; and (c) the hardening of all systems (both active and latent) is very nearly equal. In virgin crystals oriented for single slip either in tension or in compression : (a) the intersecting latent systems in general harden more than the active slip system ; (b) significant overshooting is the norm in tests carried to that stage of axis rotation; and (c) typical deformation beyond the symmetry line is consistent with dominant single slip on the original system.”
4.2. Polycrystals Here we evaluate the rate-independent constitutive model for polycrystalline materials by comparing the predictions for stress-strain curves and the evolution of
Rate-independent
0.1
crystal
549
plasticity
0.2 E
(b)
(a>
iii
w Fig. 7. Comparison of the rate-dependent (rd) model direction. (a) Stress-strain the tensile axis
results from a rate-independent (ri) crystal plasticity model against those of a for simple tension of a strain-hardening crystal initially oriented in the [?36] curve ; (b) slip shears ; (c) inverse pole figure of the change in the orientation of (* denotes initial orientation and + denotes the final orientation).
crystallographic texture in simple compression against corresponding calculations using a rate-dependent model of Bronkhorst et al. (1992). The calculations were carried out using our implementation of our constitutive models in ABAQUS/Explicit. First we model an initially isotropic polycrystalline material as an aggregate of 343 single crystals, with each crystal being represented by a single finite element. In this numerical simulation a set of initially “random” crystal orientations was assigned to the elements. In this model of a polycrystalline material both compatibility and equilibrium are satisfied in the weak (finite element) sense, throughout the aggregate. The initial mesh of 343 ABAQUS-C3D8R elements used in the calculations is shown in Fig. 9(b). As sketched in Fig.9(a), this mesh represents one-eighth of a
550
L. ANAND
and M. KOTHARI
X*
Ill11
Fig. 8. Initial random
texture
of polycrystalline
OFHC
copper.
Rate-independent
Fig. 9. Initial mesh for simple compression
simulation.
crystal
plasticity
(a) Octant
of a cube
551
; (b) initial finite element mesh.
rectangular parallelepiped specimen. Each element represents a single crystal, and a set of 343 initially “random” grain orientations taken from the set of orientations shown in Fig. 8 was assigned to these elements. The values for the slip hardening parameters for single crystals of copper comprising the polycrystal are taken as h, = 180 MPa,
S, = 148 MPa,
a = 2.25,
and the initial value of the slip system deformation resistance is taken as
552
Fig. 10. Deformed
L. ANAND
and M. KOTHARI
finite element mesh for simple compression simulation 100%. (a) Rate-dependent model ; (b) rate-independent
after a compressive model.
strain
of
s, = 16 MPa. The macroscopic (1,2), (2,3) and (3,l) planes of the meshed octant which are embedded in the full model are constrained to remain plane. The top (1,2) plane was constrained to remain plane, prescribed to be free from shear traction, and subjected to displacement boundary conditions which resulted in an axial true strain rate of -0.001 s-‘. Deformed finite element meshes at late stages of the numerical experiments are shown in Fig. 10(a) for the rate-dependent model and in Fig. 10(b) for the rate-independent model. The calculations had to be terminated when the grains became too distorted. The macroscopic stress-strain response and crystallographic texture are computed as volume averages over the entire aggregate. The crystallographic texture is arrived at by direct equal-area projection of the orientations of all the grains. Figures I1 and 12, respectively, show a comparison of the stress-strain curves and the evolved { 111)) { 1lo}, { lOO}equal-area pole figures after an axial compressive strain of - 1.O, obtained from the numerical simulations using the rate-independent model and the rate-dependent model. The results from the two models are close. The small deviations between the results are attributed to the differences in the computational procedures. Figures 13 and 14 show the corresponding results when a polycrystalline material element is modeled using the classical Taylor hypothesis of uniform deformation gradient within each grain (e.g. Bronkhorst et al., 1992). In order to compare the results between the calculations using the Taylor model with those obtained using the
Rate-independent
-
0
553
crystal plasticity
rd ri
’
I
I
0.0
0.5
1.0
Strain Fig. 11.Comparison of the stress-strain response from a finite element polycrystalline rate-independent (ri) crystal plasticity model against those of a finite element rate-dependent (rd) polycrystalline model for simple compression of OFHC copper.
finite element model, the Taylor simulations were carried out with the same set of Euler angles as were used in the finite element model. A single C3D8 element was used to perform these simulations using ABAQUS/Standard. Again, the results from the rate-independent and the rate-dependent models are very close. The results from the finite element model are in good agreement with the experimental measurements of Bronkhorst et al. (1992), and those from the simpler Taylor model are also quite acceptable.
5. CONCLUDING
REMARKS
Although plastic deformation of ductile crystalline solids by dislocation glide is inherently a rate-dependent phenomena, it is of substantial technical interest to have a general, robust calculation scheme to be able to compute using the rate-independent idealization of the response of these materials. The critical issues in rate-independent crystal plasticity have been l l l
determination of active slip systems ; determination of the amount of shear on active slip systems ; non-uniqueness due to multiplicity of slip systems.
L. ANAND
0a
and M. KOTHARI
@I
Fig. 12. Comparison of the crystallographic texture at a strain of - 1 for simple compression of polycrystalline OFHC copper. (a) finite element rate-independent polycrystalline model ; (b) finite element ratedependent polycrystalline model.
Rate-independent
0.0
crystal plasticity
555
0.5
Strain Fig. 13. Comparison of the stress-strain response from a Taylor-type rate-independent (ri) polycrystalline model against those of a Taylor-type rate-dependent (rd) polycrystalline model for simple compression of OFHC copper.
We have developed and evaluated a new calculation procedure which determines a unique set of active slip systems and corresponding shear increments. Our computational procedure overcomes these long-standing limitations to the application of the rate-independent theory of crystal elasto-plasticity. Also, implementations of our computational procedure for rate-independent polycrystal elasto-plasticity in dynamic explicit finite element programs should be of utility in speeding up simulations of essentially quasi-static deformation processes by “timescaling”, that is by artificially increasing the speeds of the deformation processing operations in the simulations.
ACKNOWLEDGEMENTS Financial support for this work was provided by the U.S. Army DAAH04-94-G-0060, and the U.S. National Science Foundation The ABAQUS finite element code was made available under from HISS Inc., Pawtucket, RI. Discussions with G. Strang and acknowledged.
Research Office under Grant under Grant DMI-9215246. an academic license to MIT A. Staroselsky are gratefully
556
L. ANAND
0a
and M. KOTHARI
(b)
Fig. 14. Comparison of the crystallographic texture at a strain of - 1 for simple compression of polycrystalline OFHC copper. (a) Taylor-type rate-independent polycrystalline model ; (b) Taylor-type ratedependent polycrystalline model.
Rate-independent
crystal plasticity
551
REFERENCES ABAQUS (1994) Reference Manuals. Hibbit, Karlsson & Sorensen Inc., Pawtucket, RI. Anand, L. and Kalidindi, S. R. (1994) The process of shear band formation in plane strain compression of fee metals : effects of crystallographic texture. Mech. Muter. 17, 223243. Asaro, R. J. (1983a) Micromechanics of crystals and polycrystals. A&. Appl. Mech. 23, 1-1 IS. Asaro, R. J. (1983b) Crystal plasticity. ASME J. Appl. Mech. 50, 921-934. Asaro, R. J. and Needleman, A. (1985) Texture development and strain hardening in rate dependent polycrystals. Acta Metall. 33, 923-953. Asaro, R. J. and Rice, J. R. (1977) Strain localization in ductile single crystals. J. Mech. Phys. Solids 25, 309-338. Bassani, J. L. (1993) Plastic flow of crystals. Adu. Appl. Mech. 30, 191-258. Bassani, J. L. and Wu, T.-Y. (1991) Latent hardening in single crystals II. Analytical characterisation and predictions. Proc. Royal Sot. London A 435, 2141. Bronkhorst, C. A., Kalidindi, S. R. and Anand, L. (1992) Polycrystal plasticity and the evolution of crystallographic texture in face-centered cubic metals. Phil. Trans. Royal Sot. London A 341,443477. Chin, G. Y. and Mammel (1969) Generalization and equivalence of the minimum work (Taylor) and maximum work (BishopHill) principles for crystal plasticity. Trans. Metall. Sot. AZME 245, 1211-1214. Golub, G. H. and Van Loan, C. F. (1983) Matrix Computations. The Johns Hopkins University Press, Balitmore, MA. Gurtin, M. E. (1981) An Introduction to Continuum Mechanics. Academic Press, New York. Havner, K. S. (1992) Finite Plastic Deformation qf Crystalline Solids. Cambridge University Press. Hill, R. (1966) Generalized constitutive relations for incremental deformation of metal crystals by mutislip. J. Mech. Phys. Solids 14, 95-102. Hill, R. and Rice, J. R. (1972) Constitutive analysis of elastic-plastic crystals at arbitrary strain. J. Mech. Phys. Solids 20,401413. Kalidindi, S. R. (1992) Polycrystal plasticity : constitutive modeling and deformation processing. PhD Thesis. MIT, Cambridge, MA. Kalidindi, S. R., Bronkhorst, C. A. and Anand, L. (1992) Crystallographic texture evolution during bulk deformation processing of fee metals. J. Mech. Phys. Solids 40, 537-569. Mandel, J. (1965) Generalisation de la theorie de la plasticite de W. T. Koiter. ht. J. Solids Struct. 1,273-295. Mandel, J. (1974) Thermodynamics and plasticity. Proc. Int. Symp. Foundations ofContinuum Thermodynamics (ed. Delgado Domingos, J. J., Nina, M. N. R. and Whitelaw, J. H.), pp. 283-3 Il. McMillan, London. Peirce, D., Asaro, R. J. and Needleman, A. (1982) An analysis of nonuniform and localized deformation in ductile single crystals. Acta Metall. 30, 1087-I 119. Press, W. H., Flannery, B. P., Teukolsky and Vetterling, W. T. (1986) Numerical Recipes. The Art of ScientiJi’c Computing. Cambridge University Press, Cambridge. Rice, J. R. (1971) Inelastic constitutive relations for solids : an internal variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19,433455. Schmidt, E. and Boas, W. (1935) Plasticity of Crystals. Chapman and Hall, London. Simmons, G. and Wang, H. (1971) Single Crystal Elastic Constants and Calculated Aggregate Properties. The MIT Press, Cambridge, MA. Strang, G. (1988) Linear Algebra and Its Applications. Harcourt Brace Jovanovich College Publishers, Forth Worth. Taylor, G. I. (1938a) Plastic strain in metals. J. Institute of Metals 62, 307-324. Taylor, G. I. (1938b) Analysis of plastic strain in a cubic crystal. Stephen Timoshenko 60th Anniversary Volume, pp. 218-224. McMillan Co., New York. Taylor, G. I. and Elam, C. F. (1923) The distortion of an aluminum crystal during a tensile test. Proc. Ro_val Sot. London A 102, 643-667.
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Taylor, G. I. and Elam, C. F. (1925) The plastic extension and fracture of aluminum single crystals. Proc. Royal Sot. London A 108,28-51. Weber, G., Lush, A. M., Zavaliangos, A. and Anand, L. (1990) An objective time-integration procedure for isotropic rate-independent and rate-dependent elastic-plastic constitutive equations. Znt. J. Plast. 6, 701-744. Wu, T.-Y., Bassani, J. L. and Laird, C. (1991) Latent hardening in single crystals I. Theory and experiments. Proc. Royal Sot. London A 435, l-19.