Computational plasticity for composite structures based on mathematical homogenization: Theory and practice

ELSEVIER Comput. Methods Appl. Mech. Engrg. 148 (1997) 53-73

Computer methods in applied

mechanics and engineering

Computational plasticity for composite structures based on mathematical homogenization: Theory and practice

Jacob Fish*, Kamlun Shek, Muralidharan Pandheeradi, Mark S. Shephard Departments of Civil and Mechanical Engineering and Scientific Computational Research Center,

Rensselaer Polytechnic Institute, Troy, NY 12180, USA

Received 29 April 1996; revised 10 January 1997; accepted 20 February 1997

Abstract

This paper generalizes the classical mathematical homogenization theory for heterogeneous medium to account for eigenstrains. Starting from the double scale asymptotic expansion for the displacement and eigenstrain fields we derive a close form expression relating arbitrary eigenstrains to the mechanical fields in the phases. The overall structural response is computed using an averaging scheme by which phase concentration factors are computed in the average sense for each micro-constituent, and history data is updated at two points (reinforcement and matrix) in the microstructure, one for each phase. Macroscopic history data is stored in the database and then subjected in the post-processing stage onto the unit cell in the critical locations.

For numerical examples considered, the CPU time obtained by means of the two-point averaging scheme with variational micro-history recovery with 30 seconds on SPARC 10151 as opposed to 7 hours using classical mathematical homogenization theory. At the same time the maximum error in the microstress field in the critical unit cell was only 3.5% in comparison with the classical mathematical homogenization theory.

1. Introduction

Metal matrix composites reinforced by aligned continuous fibers may exhibit a significant amount of elastic-plastic defolrmation caused by plastic flow of the matrix. Although the fibers strengthen the matrix substantially, the local stress concentration caused by the presence of fibers may trigger plastic flow of the matrix at a lower overall stress level than the yield stress of the unreinforced matrix. The total strains seen in fibrous system are up to 0.01-0.02, seldom exceeding the failure strain of the fiber. Consequently, the utilization of high strength of the metal matrix composites requires a consideration of the elasto-plastic deformation primarily in the small strain regime [13].

In modeling heterogeneous media it is tempting to adopt a macroscopic point of view, which considers the composite as a homogeneous medium with anisotropic properties that have to be determined. Several authors [28] applied anisotropic yield criterion developed by Hill [21] for elasto- plastic analysis of fibrous composites. On the other hand Dvorak and Rao [13] and Lin et al. [25] have shown that Hill’s ;anisotropic yield criterion (originally designed for metals) assuming that hydrostatic stress does not influence yielding and that there is no Bauschinger effect is not valid for fibrous composites.

Under these circumstances micromechanical analysis of composite materials, which provides the overall composite behavior from the properties of individual constituents (reinforcement and matrix) and their interaction, seems to be the only viable alternative. Most of the current micromechanical

* Corresponding author.

0045-7825/97/$17.00 c3 1997 Elsevier Science S.A. All rights reserved PII SOO45-7825(97)00030-3

54 J. Fish et al. I Comput. Methods Appl. Mech. Engrg. 14X (1997) K-73

methods approximate the local fields within each phase either as uniform [l ,11,12], or as piecewise constant [2,3,33]. Micromechanical approaches can be also classified into the following two categories: (i) those that require evaluation of instantaneous concentration function by solving the rate form of an inclusion or unit cell problem [1,2,3,11,12,33], and (ii) those that require evaluation of elastic concentration function only. The latter is based on the transformation field analysis recently developed by Dvorak [9,10]. An excellent survey of these micromechanical approaches for plasticity of fibrous composites is given in [9,11] and references therein provide good insight into this approach.

Methods based on a multiple scale asymptotic expansion, which are parallel to micromechanical approaches for linear systems appeared in the mid 70s by the name of mathematical homogenization. The mathematical homogenization method is advantageous in the sense that it provides convergence characteristics of certain norms of interest in addition to only bounds of equivalent material properties. The fundamentals of mathematical homogenization theory can be found, among others, in [5,20,29]. Various higher order mathematical homogenization theories and their finite element application were described in [16,18]. A variant of the classical mathematical homogenization that abandons the classical hypothesis of uniformity of the macroscopic fields within the unit cell domain has been developed in [14,15]. By this approach the solutions obtained from the mathematical homogenization theory are only used to simulate the global response of the discrete heterogeneous medium, whereas the classical relaxation techniques are employed to capture the oscillatory response.

For small deformation plasticity a mathematical homogenization theory has been established by Suquet [31,32]. Finite element application issues were addressed in [19]. In these attempts the uncoupling of scales was carried out for a linearized system in a similar fashion to that of the linear problem. Refs. [2,3,33] represent the micromechanical approaches parallel to the mathematical homogenization for plasticity [19,31,32] in terms of the formulation and computational complexity involved.

One of the objectives of the present work is to develop a mathematical homogenization theory with eigenstrains-a counterpart to the micromechanical approach based on the transformation field analysis recently developed by Dvorak [9,10]. In Section 2.1 we derive a closed form expression relating arbitrary transformation fields to mechanical fields in the phases, which for the case of piecewise constant transformation fields reduce to the form derived by Dvorak [S] on the basis of the uniform fields concept. This is followed by the derivation of return mapping stress update procedures in Section 2.3 and consistent tangent operators for heterogeneous systems in Section 2.4.

From the practical stand point, solution of large scale nonlinear structural systems with accurate resolution of microstructural fields is not yet feasible. For linear problems a unit cell or a representative volume problem has to be solved only once, whereas for nonlinear history dependent systems it has to be solved at every increment and for each macroscopic (Gauss) point. Furthermore, history data has to be updated at a number of integration points equal to the number of Gauss points in the macro problem multiplied by the number of Gauss points in the unit cell. To illustrate the computational complexity involved we consider elasto-plastic analysis of the composite (titanium matrix) flap problem shown in Fig. l(a). The macrostructure is discretized with 788 tetrahedral elements (993 degrees of freedom), whereas microstructure, as shown in Fig. l(b), is discretized with 98 elements in the reinforcement domain and 253 in the matrix domain, totaling 330 degrees of freedom. The CPU time on SPARC lo/51 for this problem was 7 hours, as opposed to 10 seconds if metal plasticity was used instead,, which means that 99.96% of CPU time is spent on stress updates. Under these circumstances it seems that the statement made by Hill [22] that ‘. . . for non-linear systems, the computations needed to establish any constitutive law are formidable indeed . . .’ is still valid almost 30 years later.

The second objective of the present work is to devise an efficient computational scheme that will enable one to solve large scale structural systems in heterogeneous media at a cost comparable to problems in homogeneous media without significantly compromising on solution accuracy. The proposed computational procedure consists of the following three steps: l Compute the overall structural response using two-point mathematical homogenization scheme

with eigenstrains (Section 2) by which the concentration factors are computed in the average sense for each phase, i.e. history data is updated only at two points (reinforcement and matrix) in the microstructure, one for each phase.

1. Fish et al. I Comput. Methods Appl. Mech. Engrg. 148 (1997) 53-73 55

(4 Fig. 1. FE meshes for (a) the nozzle flap problem and (b) the unit cell model.

l Store the history of macroscopic fields in the database. l Post-process the microscopic solution (Section 3) by subjecting the unit cell to the macroscopic

solution history in the critical locations. Numerical experiments conducted in Section 6 indicate that when the proposed computational

scheme is combinecl with the state of the art solution technique for large scale nonlinear system [17] one can solve large scale nonlinear structural systems in the range of 60 000-90 000 unknowns in less than 3 hours of CPU time on SUN SPARC station 10/51.

Throughout this paper, we will use boldface letters to represent vectors and matrices and lightface letters to scalar quantities or tensor components. Besides, Einstein’s summation convention for repeated indexes is assumed unless indicated explicitly. In the contracted notation the indexes confirm with the definition 11~ 1, 22 = 2, 33 = 3, 12 = 4, 13 = 5 and 23 = 6. Strain and eigenstrain vectors are with engineering components; stress and back stress vectors are with tensor components. AT and A-’ denote the transpose and inverse of matrix A, respectively.

2. Mathematical homogenization with eigenstrain

In this section, we generalize the classical mathematical homogenization theory [4,5] for heterogeneous medium to account for eigenstrains. We will regard all inelastic strains as eigenstrains in an otherwise elastic body. Attention will be limited to small deformations.

The microstructure of a composite is assumed to be periodic (Y-periodic) so that the homogenization process can be performed in a unit cell domain, denoted by 0. This implies that all the response functions, such as displacement and stress, are also periodic with periods proportional to the ratio of the characteristic lengths, F, between the micro representative volume element and the macroscopic structure. Let x be a macroscopic co-ordinate vector and y =X/E be a microscopic position vector. For any periodic response function f, we have f(x, y) =f(x, y + kY) in which vector Y is the basic period of the microstructure and k is a 3 x 3 diagonal matrix with components of any finite integers. Adopting classical nomenclature any Y-periodic function f can be represented as

56 J. Fish et al. I Comput. Methods Appl. Mech. Engrg. 148 (1997) 53-73

where superscript E denotes that the corresponding function f is Y-periodic and is a function of macroscopic spatial variables. The indirect macroscopic spatial derivatives off ” can be calculated by the chain rule as

where subscripts followed by a comma denote partial derivatives with respect to the subscript variables (i.e. fX, = af/axi).

In Section 2.1 we will derive closed form expressions relating arbitrary eigenstrains to mechanical fields as well as to the overall eigenstrain in a multi-phase composite medium.

2.1. Formulation for multi-phase composite medium

In modeling a heterogeneous medium, micro-constituents are assumed to possess homogeneous properties and satisfy equilibrium, constitutive, kinematics and compatibility equations as well as jump conditions on the interface boundary between the micro-phases. The corresponding boundary value problem is governed by the following equations:

mz,xj + bi = 0 in 0 (equilibrium equations) (2)

az = Lijk/(EJI - Pi/) in 0 (constitutive equations) (3)

E; = ZA~,~~) in 0 (kinematic equations) (4)

ZL; = zii on r, (displacement boundary conditions) (5)

w;nj = t; on 4 (traction boundary conditions) (6)

in which a>, E$ are stress and strain tensors; Lijkr and & are elastic stiffness and eigenstrain tensors, respectively; bi is a body force; U: denotes a displacement vector; the subscript pair with parentheses denotes the symmetric gradient defined as u;,,,,) = (uFX, + uyTXj) /2; 0 is the macroscopic domain of

interest with boundary r; &” and < are prescribed displacement and traction non-intersected boundary portions such that r, U c = r; zii and 6 are prescribed displacements on r, and traction on 4; ni denotes the normal vector on r. Here we assume that the interface between the phases is perfectly bonded, i.e. [m;Gj] = 0 and [Iu”] = 0 on the interface boundary c”, where ri, is the normal vector on 1_i”, and [*I is a jump operator.

The displacement uE(.x) and eigenstrain &J(X) are then replaced by U&C, y) and E_L~~(x, y) and approximated by the corresponding double scale asymptotic expansions:

L$(x, y) = up@, y) + EU:(X, y) + . . . (7)

PijCx> Y) z P:Cx? Y) + EPl:(X7 Y) + ’ ’ . (8)

Expansion for strain tensor can be obtained by substituting (7) into (4) with consideration of the indirect differentiation rule (1)

‘ij(‘, Y) -~a,‘(x.y)+E~(X,y)+EE:I(X,y)+...

in which the strain tensor for various orders of E is given as

(9)

l ,;’ = l ,,(u”)

4~.=Eij~(Us)+Eiiy(Usf1), s=O,l,...

where

'ijx('") = u;i,xj) Y Eijy(""> = ‘Ti,yj)

The stress and strain tensors for different O(&) are related by the constitutive rules (3)

(10)

J. Fish et al. I Comput. Methods Appl. Mech. Engrg. 148 (1997) 53-73

CT,;’ = Lijk,E ;,’

urj = Lijk,(& - pi,) ) s = 0, 1, . . .

57

(11)

and thus the stress tensor can be written as

uijtx? Y) ~~u,i~(x,y)+up,(x,y)+Eu~(x,y)+... (12)

Inserting the stress tensor asymptotic expansion (12) into equilibrium equations (2) and making the use of (1) yields the following equilibrium equations for various orders

0(E-2) : a,;; = 0 ’ I (13)

O(E-‘):c$ +& =o (14) ’ I

O(EO) : uf.j i- a;j,,j + b; = 0 (15)

O(&s):cr;j,,,tcT;;;,=O, s=1,2 )... (16)

Consider the O(E-‘) equilibrium equations (13) first. Pre-multiplying it by up and integrating over a unit cell domain @ yields

and subsequent integration by parts results in

in which r, denotes the boundary of 0. The boundary integral term in the above equation vanishes due to the periodicity of boundary conditions on r,. Since the elastic stiffness tensor Lijkl is positive definite, we have

Next, we proceed to the O(E-*) equilibrium equations (14). From Eqs. (11,10) follows

(Lijkl(Eklx(Uo) + Ekly(“‘) - I*~,)),y, = O On @ (17)

To solve for Eq. (17) up to a constant we introduce the following separation of variables

uT(x, Y) = &JY)(%,X(UO) + d:,(x))

in which Hi,,,,, is a !r’-periodic function with symmetry on indexes m and IZ, dz,, is a macroscopic portion of the solution resulting from eigenstrains, i.e. if &x, y) = 0 then d:,(x) = 0. With this form of U: we rewrite 0(&-l) equilibrium equation (17) as

(Lijkl~(Gkmsln + %klrnn)%~nx(~~) + %rklmnd~n - PYcI)).y, = O

where

GAY) = H~ri,y,jmn(~)

(18)

and 6,, is the Kronecker delta. Since Eq. (18) is satisfied for arbitrary macroscopic strain field E,,,(u’), and eigenstrain field p!, one may first consider pi, = 0, l ,,,(u”) # 0 and then E,,,(u’) = 0, pi, Z 0 which yields the followmg two governing equations in a unit cell domain


Eq. (19) is the standard linear unit cell equation [5] subjected to periodic boundary conditions that can be solved in 0. Finite element method can be used for solving this problem [18,20]. In absence of eigenstrains, ekrX(uo) is related to the total strain by

‘&I =A klmnEmnx(uo) + Ots)

where

A klmn = ; @kmsh + skns,rn) + *khn

A klmn is often referred to as elastic strain concentration function. It should be noted that Aklmn and V k,mn possess minor symmetry such that Aklmn = Alkmn = Aklnm and FkCklm,, =z!kmn = q&lam, but not the major symmetry in general. The elastic homogenized stiffness tensor Ljjk, follows from 0(&O) equilibrium equation [ 161

Astk, d@ (21)

in which 101 is the volume of a unit cell. In the following derivations, we will adopt matrix notation. The matrix representation of lYklmn is

given as

W=

- Fill %I22 %I33 ?I12 ~1113 ?I23 F 2211 %222 %233 %I2 %I3 %23 F 3311 1y3322 l-4333 $312 ?3313 %323

2%21, 2?*22 2Fl233 292,* 2F,2*3 21k;**3

2%3,, 2q1322 29,333 2q3;312 2~1313 2?323

.2%3~, 2%322 2%333 2%3~2 2%3~3 2%323

The stiffness matrix L is arranged similarly although it does not have the multiplier two in the last three rows. The strain concentration function can be written in matrix notation as follows

A=Z+!P

where Z is a 6 x 6 identity matrix. After solving Eq. (19) for T, we now proceed to Eq. (20) for finding d’ subjected to periodic

boundary conditions. Pre-multiplying (20) by H,,, and then integrating the resulting equation by parts with consideration to the periodic boundary conditions yields

QTL(Wd” - p”) d@ = 0

Rewriting this equation in terms of strain concentration function A and manipulating it with Eq. (21) yields

&‘ =-+-,)-‘I, tyTLpodO I@1

where

and thus the O(E’) approximation to asymptotic strain field (9) reduces to

E = AC+‘) + ‘Pd’” + O(E) (23)

Let J, = M,(Y))? b e a set of C-’ continuous functions, then the separation of variables for eigenstrains is assumed to have the following decomposition:

.I. Fish et al. i Cornput. Methods Appl. Mech. Engrg. 148 (1997) 53-73 59

The resulting asymptotic expansion of the strain field (9) can be expressed as follows:

E(K Y) = A(Y)+~) + $ DJY)PO,(~ + O(E) (24)

in which D,(y) are the eigenstrain influence functions given in terms of strain concentration function F(y) as follows:

D,(~) = -!- !P(i -L)-’ I, QTL&, d@ I@1

In particular, if + is a set of piecewise constant functions such that

cG;(Y,) = { 1 ify,EO,

0 otherwise

(25)

(26)

and 0, is the domain of element 77 within the unit cell, c, the element volume fraction given by c,, = \@?I/ 101 and satisfying C, cV = 1, then Eq. (25) reduces to the expression given in [8]

Dp,, = c,?@ --~)-‘W;Lq (27)

in which

Finally, we consider O(E’) equilibrium equation (15) and integrate it over 0. The Is v:,,,; d@ term vanishes due to periodicity and we obtain

Substituting the constitutive relation (11) and the asymptotic expansion of strain tensor (23) into the above equation yields

Lijkt(" ~,rnn~&~) + %tmnCn - &) d@ + b, = 0 ,x I

If we define the macroscopic stress tensor 6, and corresponding macroscopic strain tensor Ei, as

Eli = E,,(u”)

then the equilibrium equation (28) can be simplified as follows:

a,,,; + bi = 0 or L,,tGkt -&t>,xi +

where p is the overall eigenstrain tensor

p= -+,? @L(Wdp -p’)dO I

bi = 0

given by

(29)

Replacing ?P by A - 1 and manipulating the above equation with Eqs. (21) and (22), the expression for overall eigenstrain field can be simplified as

p =+, o BT,podO, I

B(y) = L(y)A(y)? (30)


Eq. (30) represents the well-known Levin’s formula [24] relating the local and overall eigenstrains in which B(y) is often referred to as elastic stress concentration function.

2.2. Two-point averaging scheme for two-phase composite medium

Consider a composite medium consisting of two phases, matrix and reinforcement, with respective volume fractions c, and cf such that c, + cf = 1, where subscripts m and f represent matrix and reinforcement phases, respectively. In this section, we develop a fast computational scheme by which only the overall structural response is sought, whereas local fields are computed by post-processing which will be presented in Section 4. For this purpose, we will define average phase concentration factors as

(31)

where 0, and 0, denote the matrix and reinforcement domains such that 0, U 0, = 0. Since the elastic stiffness of each phase is constant, the overall elastic material properties are given by

i = c,L,A, + cf LfAf

Furthermore, in a two-phase composite with uniform phase eigenstrains, i.e. I/J = ]$m, I,$] T and I/J,,,, & defined as in (26), under assumption of E+O the average phase strains can be written, in analogue to

(24) as

E,=A,E+D,,,,P,,,+D,~+, r=m,f

The resulting eigenstrain influence factors reduce to

(32)

D,, = (Z - A,)(L, - LJ’L,

Drf=(Z-A,)(L,-L,)-‘Lf , r=m, f

where the following relations have been used

(33)

hi (c,A, + cfAr) =Z , Lf-=LT, E=ET

These results are identical to those obtained by Dvorak in [S] based on a uniform fields concept. Finally, for the two-phase composite, Eq. (29a) reduces to

a=c,u,+cfuf (34)

2.3. Zmplicit integration of the constitutive equations

Since the plastic strains (or, equivalently the relaxation stresses) are nonlinear function of stresses (or strains), it is necessary to integrate the constitutive equations along the prescribed loading path in order to obtain the current stress state. It is possible to advance the solution through the plastic deformation process by adopting a simple explicit integration scheme that uses instantaneous compliance/stiffness of the phases and instantaneous stress concentration factors. This approach has been used in [9] for certain assumed yield function, flow rule and hardening laws for multi-phase systems.

Explicit schemes usually require small time or load steps, In this section we present simple, computationally efficient implicit procedures for the elasto-plastic stress updates for two-phase composite media. We assume general anisotropic reinforcement material which remains elastic throughout the loading history, and elasto-plastic matrix phase with isotropic elastic properties and following von Mises yield function with a linear combination of isotropic and kinematic hardening.

We assume an elastic reinforcement, i.e. ccf = 0, and a single source of eigenstrain due to matrix plasticity, denoted by CL,. Combining (3) and (32) one obtains the following relations for matrix and reinforcement stresses:

J. Fish et al. I Comput. Methods Appl. Mech. Engrg. 148 (1997) 53-73 61

u,=R,E-Q,,P,,, , r=m,f (35)

where

R,=LP,, Q,, = L,(%J -DA , r = my f (36)

Consider the yield function of the following form:

(37)

where &m is the yield stress of the matrix material in uniaxial test, which evolves according to the hardening laws assumed. cym corresponds to the center of the yield surface in the deviatoric stress space, also called “back stlress”. Evolution of q,, is assumed to follow the kinematic hardening rule. For von Mises plasticity P is defined as follows:

2 -1 -1 0 0 0 2 -10 0 0

P=iTTI;,

in which T is a 6 X 6 diagonal matrix with 1 in the first three locations and 2 in the remaining three locations; 6 is a projection operator satisfying k = $F and P = Pp = kP, which transforms a vector from non-deviatoric space to deviatoric space.

For simplicity we assume that matrix plastic strain rate follows the associative flow rule:

We adopt a modified version of the hardening evolution law [23] in the context of isotropic, homogeneous, elasto-plastic matrix phase. A scalar, material dependent parameter /3 (0 s p G 1) is used as a measure of the proportion of isotropic and kinematic hardening and A,,, is a plastic parameter determined by the consistency condition. Accordingly, the evolution of the yield stress &m and the back stresses q,, can be (expressed in the following rate forms:

While ,f3 = 0 refers to pure isotropic hardening, p = 1 is merely the widely used Ziegler-Prager kinematic hardening rule [34] for metals without isotropic hardening. H is a hardening parameter defined as the ratio between effective stress rate to effective plastic strain rate. Other laws for kinematic hardening have also been proposed in the literature, for instance [6,27].

Integration of (38), (39) is carried out by the backward Euler scheme:

I Pnl = j-l/~,,, + ‘Ah,P(‘a, - ‘qJ (40)

(41)

(42)

where the left superscript refers to the load step count, i being the current step. The proposed implicit procedure for the evaluation of ia, and iaf is described in what follows. Here we omit the left superscript for the current step i, such that all variables without left superscripts refer to the current load step.

Using the backward Euler scheme for the rate form of a, in (35), and (40) one obtains the following relation for matrix stresses:

u?n za;- 4,Q,,2-‘(~, - cy,) (43)


where 0: is a trial stress defined as

0;s i-1a, + R, AC

Subtracting (42) from (43) we arrive at the following result:

~,-a,=(l+A*~(Qm,,P+5(1-P)~~)}-1((r::-i-101,) (44)

The value of Ah, must be obtained by satisfying the consistency condition which assures that the stress state lies on the yield surface at the end of the current load step. To this end, (44) and (41) are substituted into the yield condition (37) so that ~@~(a,, ar,, c?~) = 0, which produces a nonlinear equation for Ah,. A standard Newton’s method is applied to solve for AA,:

(45)

It can be easily shown that the derivative a@,/a(AA,) required in (45) has the following form:

The converged value of AA,,, is then used in combination with (44), (42) and (43) to compute the back stresses and matrix stresses, whereas (40) and (35) provide the stresses in the reinforcement. The relation (34) is used to obtain the overall stresses. In addition, the overall plastic strain 2 can be calculated from the Levin’s formula (30) as

1= c,$fc~, , B, = L,Aj-’ (46)

2.3.1. Initial estimate of AA,,, The iterative scheme (45) for the solution of AA,,, might be slow or even diverge if the initial value

chosen is far from the actual value. For this reason it is necessary to come up with a better starting estimate than the obvious choice of zero for the evaluation of AA,,,. We propose an initial guess computed, based on an approximation that makes the integration procedure equivalent to that used for the isotropic homogeneous case, where AA, may be obtained in closed form for linear hardening. The procedure is explained below:

Pre-multiplication of Eq. (44) by 4 yields

@(a: - j-l **~={PiAA,(~~,,p+5(1_P)~~~)}~~,-ff~) (47)

Next, we introduce the approximation that @Q,,P =kL,P = 2G,@ (where G, is the shear modulus of the matrix material), which is analogous to the exact relation in isotropic homogeneous case where Q,, is simply the linear elastic stiffness tensor (see (35)). With this approximation, and the identity @F = i, (47) becomes

{1+AA,(2G,,+&3)H)}&~~-a,)=~(a~-‘-’a~) (48)

Let [lullA = (uTAu)“* be the A-norm of an arbitrary vector u in R”, then taking the T-norm on both sides of (48) yields

where (37) has been used. This expression, together with the relation (41) gives the initial approximation for AA,,, as


(49)

2.4. Consistent tangent operator

While an accurate integration of the constitutive equations is necessary to update the stresses given the kinematics and the initial state, and thereby to compute the residual or gradient vector on the finite element mesh, the formation of a tangent stiffness matrix that is consistent with the integration procedures followed is essential to maintain quadratic rate of convergence if one is to adopt a standard Newton’s method for the solution of the global nonlinear system [30].

We start by expressing the incremental form of the relations (35) in terms of total quantities:

0, - i-1a, = R,(E - ‘-‘Z) - Q,,P(a, - cu,)(A, - i-1h,) , r = m, f (50)

where (40) has been used. Taking the material time derivative of (50) for the matrix phase and keeping in mind that all quantities are known at the previous load step i - 1 one obtains

c;, = R,f-Q,,,,(h'&, +P(c+,-&,J AA,) (51)

In order to compute i, we take material time derivative of the yield surface (37), i.e. @,,, = 0, and make use of the relations in (39), which yields

Consequently, Eq. (39b) can be expressed as follows:

(52)

(53)

Once again substituting (52) and (53) into (51) we get the following expression relating ; and a,:

0, = a,; (54)

where

9 - 6(1- P)H Ah,

>

-1 a,,, = Z -t AA,Q,,P +

4Hc;; QmrnJ\r,~; 4, (55)

A similar result can be obtained which relates the rate of the reinforcement stress and the overall strain rate, starting from (50) and proceeding in an analogous manner. The final expression can be shown to be

Of = a,; (56)

where

(57)

Finally, the overall consistent tangent stiffness matrix 3 is obtained from the rate form of (34), (54) and (56):

&=s;, !D = crar + c,(D, (58)

64 J. Fish et al. I Comput. Methods Appl. Mech. Engrg. 148 (1997) S-?-73

3. Incremental mathematical homogenization

This section deals with the incremental form of the mathematical homogenization theory. The solution found by this approach will serve as a reference solution to the two-point averaging approach discussed in the previous section. The derivation is similar to that in Section 2, but the phase constitutive relation is now in rate form and the elastic stiffness as well as the elastic strain concentration function become instantaneous quantities represented by script letters corresponding to the Roman letters of their elastic counterparts. Moreover, the incremental approach will provide a systematic way to post-process the solution obtained from two-point averaging scheme to a general unit cell domain and consequently the critical stress level in the unit cell can be assessed. An algorithm combining the two-point averaging approach presented in the previous section and the micro-history recovery procedure developed in this section is given in Section 4.

3.1. Formulation for multi-phase medium

The rate form of governing equations is identical to that given in Section 2.1 except that the constitutive relation relates instantaneous quantities as

a” = ei.7 , s= -l,O,. . .

where % denotes the instantaneous stiffness matrix. Following the procedure outlined in Section 2.1 and considering the rate form of the O(E-‘)

equilibrium equation (13), pre-multiplying it by ti 0 and integrating over 0 and then taking integration by parts one obtains

(59)

where the first term vanishes due to periodic boundary conditions. Since the phase material may exhibit softening, the instantaneous stiffness tensor Qijkl may not be positive definite and thus

’ Fi.y,) =o * tip=tip(x)

may not be the only possible solution, but nevertheless will be considered here. Introducing separation of variables

Qx, Y) = 6,,,(Y)LZ,(~o)

to the rate form of the O(E-‘) equilibrium equation (14), we obtain an equation analogous to (18)

(2ijkl(skms~n + ~~mn)),y, = 0 on @ (60)

where

Eq. (60) resembles its linear elastic counterpart (19), with the only exception that tiijk[ is history dependent. A weak form of (60) can be obtained by pre-multiplying (60) by &Qis, and integrating it over a unit cell domain, and subsequently integrating it by parts which yields

(61)

for arbitrary S!P& E C-‘. The corresponding discrete system of equations is given by

K/&,,nB = fmnA (62) where

J. Fish et al. I Comput. Methods Appl. Mech. Engrg. 148 (1997) S-73 65

KAB = N~.j)aQij~,Nf3k,,~B d@ 2 I fmna = - 8 I 0 Ng,j)ac;jrnn ‘@

6 kmn = %‘,A,,n~ 9 Finkl = N;,,,Azn~

and N@ is the matrix containing appropriate shape functions for the unit cell problem. The tistantaneous strain concentration function U(y) and the local strain rate are given by

U(y) = z + !P(y) ) +, Y) = U(Y)&) (63)

3.2. Implicit stress update procedure for incremental homogenization

The implicit stress update procedure derived in this section can be used within a macroscopic iterative scheme to obtain a reference solution for comparison to the two-point averaging procedure. Alter- natively, if the strain increment is given for certain Gaussian points in the macroscopic domain, the current stress update scheme can be used as a stand-alone post-processing module to recover local fields in a unit cell domain.

We start from the constitutive relation for a typical Gaussian point y, E 0,

Substituting for the: local strain rate (63b) and local plastic strain rate (38) yields

a, = L,(l$j - i,P(a, - cu,))

Applying the backward Euler integration scheme to the above equation gives

up = ‘-‘up + Z$lP AZ - AhpLpP(ap - Q;)

and together with (42) yields

(64)

Note that the instantaneous strain concentration factors UP are obtained from the solution of Eq. (62) which depends on the instantaneous stiffness at each microscopic Gaussian point. Therefore, U,, is a function of AA which is a vector containing all the plastic parameters in O,,,, AA = LAA,, AA,, . . . , AhEy] *. Substituting (65) into yield function (37) for each Gaussian point in 0, yields a set of ny nonlinear equations @ = [@, , CD*, . . . , Q,,,,] * with IZ~ unknown plastic parameters which can be solved by using Newton’s method

A typical term in the Jacobian matrix is given as

(66)

(67)

In the above expression, the evaluation of alt,/a(AA,) is not following approximation

trivial, and hence we introduce the

66 _I. Fish et al. I Comput. Methods Appl. Mech. Engrg. 148 (1997) 53-73

such that the above Jacobian matrix (67) is approximated by the following diagonal matrix

(68)

At each modified Newton iteration of (66), we need to evaluate the residue vector @ which in turn requires a solution of (62). The iterative process continues until the residual norm II@ 11 2 vanish up to a certain tolerance. The updated stress at each Gaussian point in 0, can then be calculated with (64). For the Gaussian point in the reinforcement domain O,, the updated stress can also be obtained using (64) with AA, = 0. The macroscopic stress then follows from Eq. (29).

4. Two-point averaging scheme with variational micro-history recovery

As discussed in the previous sections, the computation of nonlinear composite systems using multi-phase incremental homogenization scheme is computationally expensive since the number of nonlinear equations to be solved at each macroscopic Gaussian point is equal to the number of plastic Gaussian points in a unit cell. Furthermore, for every microscopic iteration (66) a solution of the linear unit cell problem (62) is required. Nevertheless, a multiphase incremental homogenization scheme can be used as a stand-alone post-processing tool to recover the micro-history for a selected number of points. In Section 6 we will show that the computational complexity of the two-point averaging approach is comparable to that of homogeneous case, but the resulting average phase strains are sufficiently accurate to predict critical microscopic fields of interest.

In this section, we merge the two approaches discussed in Sections 2 and 3. In the combined scheme the overall structural response is computed using the two-point averaging scheme and the micro-history in the unit cell domain corresponding to the critical regions in the macroscopic domain is recovered by means of the incremental homogenization method. The critical regions may be defined as those where the macroscopic effective stress (maximum stress criterion) or the effective overall plastic strains (maximum plastic strain criterion) exceed certain criteria such that

in which aC, and p,, are prescribed critical threshold values, and Z and E are computed from (34) and the Levin’s formula (46), respectively.

The overall analysis procedure is divided into two stages. In the first, denoted as macroscopic finite element analysis, a nonlinear composite structural problem is solved using a finite element method based on the two-point averaging approach developed in Section 2. The macroscopic analysis of the composite structure is carried out and the macroscopic strain histories are stored in a history database at Gaussian points in the critical regions. In the second stage, referred to as micro-history recovery, the microstress distribution in the unit cell is sought. The strain history at macroscopic Gaussian points for critical regions are extracted from the database. Then, the macroscopic strain history is applied to the unit cell through the incremental homogenization procedure discussed in Section 3.2. Since the micro-history recovery is performed only at a select number of Gaussian points of interest without affecting macroscopic analysis, the computational cost is low.

The finite element formulation of a composite problem that incorporates the nonlinear constitutive model presented in Section 4 yields a discrete system of equilibrium equations schematically denoted as

r =f,,, -A,, = 0 (70)

where r is a macroscopic residual vector, f,,, is the external force vector, and X,,(Z) is the vector of internal forces. A macroscopic iterative solution procedure is employed for this set of nonlinear equations.


Note that for small to medium size problems, it is possible to apply a classical Newton’s method for the solution of (70). The consistent tangent operator developed in Section 2.4 ensures that the resulting iterative scheme retains the quadratic rate of convergence provided that the standard requirements for the Jacobian and the initial approximation are maintained [7]. However, Newton’s method becomes expensive because it requires the computation and factorization of the tangent stiffness matrix at every iteration for large-scale problems. An efficient solution algorithm for large scale problems based on multigrid technology is briefly discussed in Section 5. The overall analysis procedure is summarized as below:

5. Macroscopic finite element analysis

1. Solve for the elastic strain concentration factors in a specified unit cell domain using (62) with elastic stiffnesses replacing the instantaneous quantities.

2. Calculate the 2-point average elastic strain concentration factors A, from (31). 3. Construct eigenstrain influence factors D, from (33) and then R, and Q,, as shown in (36). 4. Initialize macroscopic displacements; microscopic stresses and state variables. 5. Define the macroscopic load increment

if end-of-load-step then stop

else continue

endif

6. Evaluate consistent tangent operators Fb from (58) if Newton’s method is adopted. 7. Solve for macroscopic displacement increments and then compute macroscopic strain increments. 8. Carry out implicit stress update at each macroscopic Gaussian point.

l Compute initial estimate of AA, by Eq. (49). l Solve AA,,, by Newton’s iteration with (45). l Update matrix stress and state variables using (44) and (40)-(43). l Calculate macroscopic stress with (34).

9. Update displ,acements, stresses and state variables. 10. Calculate the L, norm of macroscopic residue vector jjr//, in (70) and check convergence

if Ilrjj2 <Tolerance then upda.te equilibrium history write macroscopic strain history to database return to step 5

else return to step 6

endif

6. Micro-history recovery

1. Identify critical1 regions using maximum stress criterion or maximum plastic strain criterion (69). 2. Extract macroscopic strain increment for Gaussian points in critical regions from history database. 3. Carry out implicit macroscopic stress update for critical unit cells.

l Compute modified tangent operator (68). l Calculate instantaneous stiffness tensor for each phase. l Solve unit cell problem for U(y)


0 Evaluate L, norm of residue vector ]I@ ]I2 in the unit cell. l Perform iteration for AA until I]@ ]I2 <Tolerance.

4. Recover microstress distribution in 0 using the converged plastic parameter AA (64). 5. return to step 2.

7. Solution procedures for large scale systems

As discussed in the previous section, Newton’s method is not an optimal choice for large-scale problems. The popular alternatives to the Newton’s method include the BFGS quasi-Newton method and multigrid methods. Even though the BFGS method is known to be robust and stable for general nonlinear systems, its convergence is greatly affected by the choice of the initial approximation to the tangent stiffness matrix. Usually it is necessary to start the procedure from the exact linear stiffness matrix, which again becomes expensive for large-scale problems. On the other hand, the Newton- multigrid method, where a standard linear multigrid method is used to solve the linearized system within the Newton’s method, suffers from its sensitivity to the conditioning of the linearized system. This limitation becomes pronounced for problems with a significant amount of plastic flow and small hardening slope.

A novel nonlinear hybrid solver was introduced in [17], which builds on the advantages of multigrid- and BFGS-like methods and at the same time eliminates the undesirable characteristics of each to the maximum extent possible. It combines the full approximation storage (FAS) version of the multigrid method and the BFGS quasi-Newton method to form a fast, efficient solution method for nonlinear systems. Here we only summarize the fundamental features of the new FAS-BFGS solution procedure, and refer to [17] for a complete description of the method as well as the detailed algorithms. The basic characteristics of the new solution approach include: l Full approximation storage. Solve the nonlinear system directly, instead of solving a sequence of

linearized problems within the Newton method, using the FAS philosophy. l Continuous inter- and intra-cycle BFGS updates. The conventional nonlinear relaxation procedure

on the finest and all the intermediate grids is replaced by a continuous BFGS procedure which exploits the information from previous cycles. This is schematically illustrated in Fig. 2. The continuous BFGS smoothing procedures on the finest and all the intermediate grids are started from the diagonal of the Jacobian matrix for the particular grid. A restart of the BFGS procedures is done periodically after a fixed number of cycles. A restart of the BFGS procedures is done periodically after a fixed number of cycles. The handling of the solution in the coarsest grid is also different in the sense that continuous BFGS updates are performed on the coarsest grid as well. This eliminates the need for the computation and factorization of the coarsest grid Jacobian matrix at every multigrid cycle.

l Consistency of history data. Update the history variables on each grid level directly using their values from the previous load (time) step and the displacements from the current step by performing the integration of the appropriate constitutive relations. As noted in [17], such independent history updates on all the grids are essential for the successful application of FAS method for history-dependent problems such as plasticity.

0, ;o, ;o, ;o Level0

jo/ \o/ o( Level 1

Fig. 2. Schematic illustration of continuous BFGS updates.


8. Numerical examlples and discussion

In the following, we present numerical examples to test the accuracy and efficiency of th,: proposed numerical procedures. Although the numerical procedures are general enough to handle complicated microstructures, such as weave fabric or particulate composite, the microstructure of the composite chosen for the following model problems, is the most widely used continuous fibrous system. The phase properties are given as follows:

Sic Fiber: Young’s modulus = 379.2 GPa, Poisson’s ratio = 0.21, fiber volume fraction C~ = 0.27. Titanium matrix: Young’s modulus = 68.9 GPa, Poisson’s ratio = 0.33, yield stress &m = 24 MPa,

isotropic hardening modulus H = 14 GPa, p = 1.

8.1. The nozzle jklp problem

The nozzle flap problem is solved using the two-point averaging scheme with micro-history recovery and multi-point incremental homogenization procedures for the reference solution. The objective of this numerical study is to investigate the two-point averaging scheme with variational micro-history recovery in terms of accuracy (in both macro- and micro-scales), computational efficiency and memory consumption. The finite element mesh for one-half of the flap (due to symmetry) is shown in Fig. l(a). The flap is subjected to an aerodynamic force (simulated by a uniform pressure) on the back of the flap and a symmetric boundary condition is applied on the symmetric face. Furthermore, we assume that the pin-eyes are rigid and a rotation is not allowed so that all the degrees of freedom on the pin-eye surfaces are fixed. There is about 15 percent of plastic zone in the model due to the loading.

For this problem, we adopt the classical Newton’s method for both procedures. The CPU time is 30 s for the macroscopic analysis using two-point averaging scheme, plus approximately 160 s for each Gaussian point where a micro-history recovery procedure is applied. On the other hand, the CPU time

0.001

0.0008

5 B 0.0006 .k

i

s 0.0004 .c E

E 8 $ 0.0002 - .P c3

point 2

0 ,/’ _

I

point 3

r 1 1 I

Multi-Point incremental Homogenization + Two-Point Averaging Approach +

-0.ooo2 I- I I I I 0 0.2 0.4 0.6

Loading Parameter 0.6 1

Fig. 3. Displacement comparison.


for the incremental homogenization approach takes about 7 h. Moreover, memory requirement ratio for these two approaches is roughly 1: 250.

Solutions obtained by both approaches are compared using the relative error measure defined as

Relative error of U = U”P _ U’P

VP

where U mp is a solution obtained using the multi-point incremental homogenization procedure and U tp denotes the corresponding solution computed using the two-point averaging scheme. The vertical displacement for points 1, 2 and 3 shown in Fig. l(a), as obtained by both approaches are depicted in Fig. 3. The maximum displacement appears at point 1 where relative error is less than 1 percent. The maximum effective stress appears at the pin-eye of the middle flap (Gaussian point A). Micro-histories

0.02

0.01

0.00

-0.01

-0.02

(a) Gaussian P&t A

hlax. Effective Stress = 551.6 MPa

Min. Effective Stress = 38.7 hlPa

Max. R.elative Error = 2.8 cil

(b) Gaussia,n Point B

Max. Eff&ive Stress = ,227.T hlPa

Min. Effective Stress = 33.0 hIPa

Max. Relative Error = 3 5 %, t .

2.5e-02

2 .Oe-Q2

1.5e-02

l.Oe-02

5.Oe-03

O.Oe#C)

-5.os-03

(c) Gaussiau Point C

Max. Effective Stress = 2S6.4 hIPa

Min. Effective Stress = 30.1 MPa

h/lax. Relative Error = 2.8 %a

Fig. 4. Unit cell relative error for effective stress.

J. Fish et al. I Comput. Methods Appl. Mech. Engrg. 148 (1997) 53-73

Fig. 5. FE mesh for the two-cylinder problem.

Fig. 6. FE mesh for the machine part problem.

are recovered for the unit cells corresponding to the macroscopic Gaussian points A, B and C. The maximum relative error at Gaussian points A, B and C are 2.8, 3.5 and 2.8 percent, respectively, as shown in Fig. 4.

8.2. Large scale problems

Two 3-dimensional example problems are chosen for numerical experimentation on large scale systems, both modeled using 4-node tetrahedral elements. Two auxiliary grids are used for both the problems for solution through FAS-BFGS and Newton-multigrid methods.

The first example is a two-cylinder problem [17] as shown in Fig. 5. All the degrees of freedom on one end are fixed a:nd uniform loadings are applied along all the three directions on the remaining three place faces. The finest grid for this problem carried a total of 63 918 degrees of freedom. The loading takes the solution well beyond elastic limit.

The second model problem is a machine part with 94 953 degrees of freedom at the finest mesh shown in Fig. 6. We apply uniform loading along all three directions at the end-face and fix the cylindrical hole at the other end. Again, a large portion of the part is in the plastic region.

Tables 1 and 2 summarize the performance of the various solution schemes on the two composite model problems. A comparison of the CPU times presented reveals that while the FAS-BFGS method appears to be clearly superior to the other methods. The efficiency of the FAS-BFGS algorithm even in

Table 1 Comparison of solution schemes for the two-cylinder problem

Method No. of iterations cpu hours Remarks

Newton 8 32 BFGS 23 4.5 Newton-multigrid 8 2.67 FAS-BFGS 23 1.37

Projected time Projected time

Table 2 Comparison of solution schemes for the machine part problem

Method

Newton BFGS Newton-multigrid FAS-BFGS

No. of iterations

8 28

8 27

cpu hours

46 9.3 6.70 2.75

Remarks

Projected time Projected time


the case cf composite problems with computationally complex stress update procedures should be encouraging considering the fact that this method requires many more gradient vector evaluations than say, the classical Newton’s method and by extension the Newton-multigrid method. In general, this behavior of FAS-BFGS can be expected for large-scale problems modelled in such a way that the computational effort is dominated by the formation and factorization of the tangent stiffness matrix. For unit cell models with hundreds of elements in the cell, e.g. the major part of the computation will be to update the stresses and evaluate the gradient or residual force vector, in which case the FAS-BFGS procedure may not be the optimal choice.

9. Conclusions and future research directions

An alternative to the classical mathematical homogenization theory for nonlinear problems, which provides a comparable accuracy to the classical theory but at a fraction of computational cost, has been developed. For the numerical example considered, the speedup factor was over three orders of magnitude as compared to the classical theory, whereas the maximum error in stresses was less than 3%.

The present work by no means represents a complete account of all theoretical and numerical issues related to inelastic analysis of heterogeneous media. First, the present theory is no more accurate than the classical mathematical homogenization theory, but provides a comparable accuracy at a greater speed. It is important to note that assumptions of periodicity and uniformity of macroscopic fields within a unit cell domain, which are embedded within the two theories, may yield inaccurate solutions in the vicinity of boundary layers or areas of high stress/strain concentration such as cracks or shear bands. The remedies to this phenomenon, ranging from changing the size of the unit cell to carrying out an iterative global-local analysis, have been recently reported in [26] for linear elastic problems. Secondly, various failure modes, other than matrix plasticity, such as delamination, debonding, or matrix cracking have not been accounted for in the present manuscript. These aspects, together with issues of stability and uniqueness, multiple time scales in rate sensitive constitutive equations and large deformation effects at the micromechanical and macromechanical levels, are among the topics of our future investigation.

Acknowledgment

The authors express their sincere appreciation to Prof. George J. Dvorak for his constructive suggestions. The support by the ARPA/ONR, under grant NOO014-92-J-1779, Mechanism-Based Design of Composite Structures program in Rensselaer Polytechnic Institute is gratefully acknowl- edged.

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Computational plasticity for composite structures based on mathematical homogenization: Theory and practice