Elastic-plastic analysis of arbitrary heterogeneous materials with the Voronoi Cell finite element method

ELSEVIER

Computer methods in applied

mechanics and englneerlng

Comput. Methods Appl. Mech. Engrg. 121 (1995) 373-409

Elastic-plastic analysis of arbitrary heterogeneous materials with the Voronoi Cell finite element method

Somnath Ghosh*, Suresh Moorthy Department of Engineering Mechanics, The Ohio State University, Columbus, OH 43210, USA

Received 10 February 1994; revised manuscript received 9 May 1994

Abstract

In this paper, a Voronoi Cell finite element method is developed to solve small deformation elastic-plasticity problems for arbitrary heterogenous materials. Dirichlet Tessellation of microstructural representative materials results in a network of arbitrary-sided polygons called Voronoi cells. Each Voronoi cell encompasses one second phase heterogeneity at most. These are natural elements for the microstructure, representing the basic structural elements of the material. In this paper, formulations are developed for directly considering Voronoi cells as elements in a finite element model without any further dissection. Furthermore, a composite Voronoi Cell finite element method is developed to account for the presence of the second phase within each polygonal element. Various numerical elastic-plastic examples are executed for validating the effectiveness of this formulation. Finally, studies are conducted to understand the effect of size, shape and distribution of second phase on the averaged and true local responses of representative material elements.

1. Introduction

The last two decades have seen a surge in the utilization of functionally superior heterogeneous materials, like composites and alloy systems, for engineering applications. This has necessitated the development of robust analytical / numerical models for predicting effective material properties and evolving state variables in the microstructure. Of particular interest are the effects of microstructural morphology such as shapes, sizes, orientations and location of second phase inclusions on deformation characteristics, like plastic strains and stresses, and developments of material properties like strain hardening. Recent experimental studies have indicated that variations in inclusion geometry can produce different plastic deformation characteristics for different second phase distributions, even though the elastic deformation may be similar.

A large number of analytical micromechanical models have been proposed for predicting effective constitutive response at the macroscopic level. For small deformation elasto-plasticity and viscoplasticity, a majority of these models are based on (a) variational estimates leading to Hashin-Strikman type bounds [l-3], (b) self-consistent schemes [4] and (c) Mori-Tanaka methods [5-71. Variational estimates are obtained with rigid inclusions (e.g. [3]) by assuming a non-linear elastic behavior for the plastic constituents to produce upper and lower bounds. These estimates produce only a lower bound if the second phase is a void but are ineffective when both phases are deformable. Self-consistent and Mori Tanaka methods are based on the concept of the equivalent inclusion method introduced by Eshelby [8], where the actual microscopic non-linear response is derived from an average macroscopic counterpart. These methods provide reasonably good estimates for the overall macroscopic behavior

* Corresponding author.

0045-7825/95/$09.50 0 1995 Elsevier Science S.A. All rights reserved SSDI 0045-7825(94)00687-3

374 S. Ghosh, S. Moorthy / Comput. Methods Appl. Mech. Engrg. 121 (1995) 373-409

when the heterogeneities occupy low volume fraction. However, the self-consistent and Mori-Tanaka methods produce widely discrepant results [9] at higher volume fractions, even in the elastic range. These models are also deficient in treating arbitrary distributions of shapes, sizes and location of the second phase, that are frequently encountered in real heterogeneous materials. Furthermore, true depiction of microscopic stresses and strains in the constituent phases cannot be obtained from these models.

For a better representation of state variables in the microstructure (and consequently the overall response) various numerical schemes have been developed. Broadly classified as Unit Cell models, they have used semi-numerical bounding methods [ll], local field approaches [12,13] and finite element methods [14-171 to solve the micromechanics problem. These methods assume that the material is constituted of periodic repetition of unit cells that are identified as representative volume elements (RVE) of the microstructure. Semi-numerical techniques have used displacement based Galerkin and assumed stress hybrid finite element methods, to generate estimates on the actual overall response of unit cells. In local field approaches, the unit cells are decomposed into subcells containing exactly one constituent phase, and non-linear response is predicted based on averaged state variables within subcells. Complete finite element methods resort to a detailed discretization of the unit cell or RVE to generate near exact representation of the microscopic morphology and thereby, compute local variables and the corresponding macroscopic response. Most of these models also assume local periodicity conditions implying uniform dispersion of simple shaped heterogeneities in the microstructure. This assumption essentially reduces the representative volume element to its basic structural element (BSE), which is often a single inclusion with its neighboring matrix. Consequently, the analysis is performed on very simplified unit cells. Despite overcoming most of the limitations of analytical methods, the local periodicity assumptions in these methods grossly idealize actual microstructures for many engineering materials. Micrographs of real heterogeneous materials often exhibit arbitrariness in shape, size and location of the second phase and prediction of macroscopic response based on analysis of a single BSE is inappropriate. To circumvent these deficiencies, unit cells have been constructed to encompass sufficiently large portions of the microstructure [18]. However, tremendous efforts are expended in representing microstructural details, and prohibitively large and inefficient computational models result.

In the pursuit of developing an effective computational model for real heterogeneous materials with arbitrary microstructural distribution, Ghosh and coworkers [19-221 have developed a material based Voronoi Cell Finite Element Method (VCFEM). In this method, the finite element mesh evolves naturally by Dirichlet Tessellation of a representative microstructure. This is a process of subdivision of space, determined by a set of points, such that each point has associated with it a region that is closest to it than to any other. A mesh generator accounting for arbitrariness in shape, size and spatial distribution of inclusions has been developed by Ghosh and Mukhopadhyay [23]. Tessellation of a microstructural representative material element (RME) discretizes the domain into a network of multi-sided convex ‘Voronoi’ polygons or cells. Each Voronoi cell contains one second phase inclusion at most, as shown in Fig. 1. Formulations have been developed for directly treating multiple phase Voronoi polygons as elements in a finite element model by Ghosh and Mukhopadhyay [19] for linear elasticity and by Ghosh and Liu [20] for micropolar thermo-elasticity. Tessellation methods have been used by Richmond and coworkers [25,26] for quantitative characterization of micrographs obtained from automatic image analysis systems. The Voronoi Cell finite element method can therefore be used effectively to provide a direct link between quantitative metallography and deformation analysis. This method has also been coupled with asymptotic homogenization theory for simultaneous prediction of global and local responses of elastic structures in a recent paper by Ghosh et al. [24].

The objective of the present paper is to develop a Voronoi Cell finite element model for small deformation elasto-plastic analysis of heterogeneous microstructures in two dimensions. The paper consists of a systematic development of the VCFEM for homogeneous media followed by the extension to composite VCFEM, to account for the presence of material heterogeneity within each Voronoi cell. Several numerical experiments are conducted and compared with conventional and analytical solutions to validate the model. Finally, the effect of shape, size, orientation and distribution of second phase inclusions on the macroscopic response and microscopic evolution is studied.

S. Ghosh, S. Moorthy I Comput. Methods Appl. Mech. Engrg. 121 (1995) 373-409 315

(4 (b)

Fig. 1. (a) Example of a heterogeneous Representative Material Element discretized by Dirichlet tessellation and (b) a Voronoi Cell element.

2. Elastic-plastic VCFEM for homogeneous materials

As discussed in [19,20], arbitrary-sided Voronoi cells make rather unconventional elements due to the fact, that different elements can have different number of sides. It is very difficult to ensure interelement displacement compatibility for these element with the conventional displacement-based finite element methods. Additionally, rank deficiencies in the stiffness matrix may result in these formulations [20]. To avoid these difficulties and represent Voronoi cells as conforming elements, the assumed stress hybrid method introduced by Pian [27] is invoked. In this method, element stiffness matrices are derived by assuming an equilibriated stress field in the interior of each element and a compatible displacement field on the element boundary. Since interpolation of the displacement field is needed only along the element boundary, it is appropriate for n-sided polygonal elements with variations in the number of nodes. The element formulation is based on the principle of minimum complementary energy which can be derived from the generalized Hu-Washizu principle (see [28]). Within the framework of small deformation elasto-plasticity, an assumed stress hybrid formulation for rate independent J2 flow theory is developed in this section. This follows the derivation of a hybrid functional presented by Atluri [28,29].

Consider that body is discretized into N Voronoi cell elements, each encompassing a region Q with the boundary 6Q having outward normal n (see Fig. l(b)). In general, the element boundary ,X$ is assumed to be the comprised of the interelement boundary r., the prescribed displacement boundary r,,,, and the prescribed traction boundary c,,,, i.e. cK?~ = r, U rum U cm. It is further assumed that these are mutually disjoint, i.e. r, II r,,,, = 0, c,,, fl r,,,, = 0, and r, n c, = 0. An incremental formulation is invoked to account for the evolutionary constitutive equations in rate independent plasticity. At the beginning of an increment or step (say increment p), let q be an equilibriated stress field with E(U, loading path) as the corresponding strain field in Q and u be a compatible displacement field on the element boundary. The element boundary conditions can be delineated as

u=tc on r,,,, (Displacement boundary)

u-n+ = -a.n- on r, (Interelement traction reciprocity) (I) o.?l=t on q,,, (Traction boundary)

376 S. Ghosh, S. Moorthy I Conzput. Methods Appl. Mech. Engrg. 121 (1995) 373-409

where superscripts + and - denote values at opposite sides of the interelement boundary r, . In the pth step, Au corresponds to an equilibriated stress increment in Q, i.e. V* ha = 0 in the absence of body forces, where V denotes a gradient operator. Also, Au corresponds to a compatible displacement increment on the cell boundary a& such that Au+ = Au- on r’ and Au = AU on &“,. A complimentary energy density increment AB(a, Au) may be derived from the strain energy density increment AA(a, AC) for the pth step, by using the Legendre contact transformation as

AB(a, Au) = -AA(a, AE) + Au : Ae

which leads to the condition

aAB -=AE aha (2)

At the end of the increment, for which the boundary traction is incremented from T to t + Ai, an energy functional can be constructed for a Voronoi element as (Atluri [28,29]):

Q(u, Au, Au) = - I

AB(u, Au) do - I

e:Aud0 0, %

I I

(3)

+ (u+Au)*n*(u+Au)aO- (t+At)(u+Au)dr Jfb r fin

Assuming that variables are specified at the beginning of an increment, stationarity of equation (3) with respect to stress increment Au, when combined with Eq. (2), yields the strain-displacement relation

E + he = + {V’(u + Au) + V(u + Au)} in Q (4)

The total energy functional for the entire domain is obtained by adding the element functionals as

II(u, Au, Au) = i 17, (5) e=l

Stationarity of 17 with respect to displacement increment Au, results in the interelement traction

reciprocity conditions and the traction boundary conditions

(u+Au).n+=-(u+Au)*n- on r, (6)

(u+Au).n=t+At on r;, (7)

Euler equations (4), (6) and (7), together with assumed equilibriated stress field and stress-strain relations in Q, and compatible displacement fields in a&, completely specify the initial value problem for the pth increment under the application of traction increments Ai and displacement increments AU.

In applying the assumed stress finite element method, the equilibriated stress field u in the interior of an element is expressed as a polynomial at the beginning of the increment as

{u) = VW% in f4 (8)

where, for plane problems, {a} is a (3 X 1) vector of stress components, {p}, is a (m X 1) column of m undetermined polynomial coefficients pl, &, . . . , p,,, and [P] is a (3 x m) matrix containing functions of coordinates x and y. Elements of the [P] matrix can be readily obtained by assuming variable degree Airy’s stress functions. For example, stresses resulting form a third-order Airy stress function with no body forces can be delineated as

%

{IF

1yooox 0 pi UY = 001x0 0 y : 7

XY 0 0 0 0 1 -y --x II I p7

Equilibriating stress increments during the step are interpolated, using the same [P] as

{Aa> = F’l{AP)e in J4

(9) (10)


Independent assumptions on compatible displacements {u} on the boundary at the beginning of the increment are required in this formulation. This is accomplished by interpolating {u} from the generalized nodal displacements {q} e as

(~1 = WHqL in aQ (11)

In this work, a linear interpolation function is assumed for the boundary displacements. Specifically, the displacement field on a boundary segment between the ith and i + lth node is expressed as

q2i-1

{I={ } 1; =[Ll{qIe= () [ l-all, 0 all, 0 qzi

U l-all, 0 alli I{ I hi+1

q2i+2

(12)

where li is the length of segment and a is the distance from the ith node. The corresponding boundary displacement increments are written as

{Au} = [L]{Aq}, in afle (13)

Substituting Eqs. (S), (lo), (11) and (13) in Eq. (3) and setting its first variation with respect to {A/?}, to zero leads to the weak form for the strain-displacement relation (4) as

{Saj?)‘([G],{q + Aq}, - I0 [P]‘{E + Ae} do) =0 in Q (14) P

where [G], is a (m x 2n) matrix given as

PI, = j-, PYTbl[~l cm P (15)

and [n] is a (3 x 2) matrix defined in terms of direction cosines of the boundary normal %(4 0 [n(x)] = 0 n&) L 1 44 %(4

Similarly, substituting Eqs. (S), (lo), (11) and (13) in the total energy functional (5) and setting its first variation with respect to {Aq}, to zero, yields

egl @W:([Gl:{P + AP), - We) = 0 (16)

where the force vector {f}, is given by

{f}, = I, [LIT{t+ At) dT fin

For arbitrary variations in the stress coefficients {SAP},, solution of Eq. (14) gives the values of {Aj3}, for a specified {Aq},, to satisfy the element kinematic relations. Solution of Eq. (16) on the other hand, for arbitrary variations in nodal displacements {6Aq},, yields nodal displacements to satisfy traction continuity conditions. For an elastic-plastic material, the strain increments AE in Eq. (14) are non-linear functions of the current state of stress u as well as of stress increments Au. Details of an implicit method to solve for strain increments are presented in the next subsection.

2.1. Numerical integration of constitutive relations

Numerical integration schemes for evaluating plastic strain increments corresponding to specified values of stress increments are presented. Though only J2 flow theory with isotropic hardening is considered in this paper, the same method is applicable to more complex models. In this paper, a stress-space based formulation is implemented for evaluating strains. A treatment of such a method can

378 S. Ghosh, S. Moorthy I Comput. Methods Appl. Mech. Engrg. 121 (1995) 373-409

also be found in Attaway [30]. For rate-independent elasto-plastic materials, the strain increments AE are additively decomposed into elastic AE e and plastic Aep’ components as

be = AC” + Aep’ (17)

For isotropic elastic response with Young’s modulus E and Poisson’s ratio V, the elastic part of the strain increment can be written as

Ace = 1+u

~Aaa-+Arrs (18)

where Au is the hydrostatic component of the stress increment Aa and 6 is the Kronecker delta. The yield surface in stress-space at the beginning of the pth increment is expressed as

where u’ is the stress deviator and Y(E, Ae) is the radius of the flow surface. The plastic strain increment Aep’ is obtained by integrating the flow rule according to

I

P+l

AE PI = CT’ dh (20)

P

where dh is a non-negative differential flow parameter. A numerically stable backward Euler method suggested by Ortiz and Popov [31] is used to integrate Eq. (20). Modifications in this method for implicit stress integrations with specified strain have been suggested by Nikishkov and Atluri [32]. The algorithm uses a generalized mid-point rule with stress deviators at the end of the increment as

ArP’ = AA(u + Au)’ (21)

Since AE is in general a function of AA, u and Au, the hardening variable can be expressed in the form

Y ‘+’ = YP+‘(Ah, u, Au)

Consequently, the flow parameter AA has the values

AA=0 if o$:’ < Yp (elastic unloading) (22)

(T p+l=yP+l eff if v:,:’ 2 Yp (neutral and plastic loading) (23)

Both linear isotropic hardening and power law hardening are considered in this paper. For linear hardening law

Y P+‘-YP=HA;P’=H -Ae i=

where H is a constant, the flow parameter AA for plastic loading takes the form

(24)

3 aE,:l- YP AA=%

(+p+l eff

(25)

For power law hardening

YP+’ = Y” + kc;” = Y” + k (26)

where Y” is the initial yield stress, E is the effective total strain and k is a proportionality constant. Substitution of Eqs. (17), (21) and (26) in Eq. (23) results in a quadratic equation

AA*@:,:’ * ) + 3AA(e + A&)’ : (a + Au)’ + ; ((E + A&) : (E + bee)’ - ; (( 6:;’ - Y”)/k)zim) = 0

(27)

which should be solved for AA.


Numerical implementation requires computation of tangent operators through linearized forms of the constitutive relations. Mathematically, if dr is the first-order correction to the current strain increment AE, and da is the corresponding correction to the stress increment Au, the fourth-order elastic-plastic compliance tensor (or tangent operator) S is given by the relation

de=S:da

The elastic part of this equation is easily obtained from Eq. (18) as

dEe= 1+v -du-+d&=

1+v ~Z+CG :da=S, :da (28)

where Z is the fourth-order identity tensor. The plastic part of the strain correction dEp’, however, requires a first-order approximation dh to the current flow parameter Ah. In the calculation of tangent operators for the power law model (26), an approximated linear hardening law (24) is used as suggested by Tvergaard [15]. The linearized hardening modulus H is then expressed as

H= a?1 1 -_- aY E

from which the linearized plastic response takes the form (see [33])

d6P, _ (a + Au>’ @ (a + Au)’ 9 4H p+l 2

(geff ) :do=S,,:du

The elasto-plastic tangent operator S is obtained by adding Eqs. (28) and (29) as

s = s, + s,, (30)

A linearized form of the incremental complementary energy functional A/3 in Eq. (2) can now be expressed in terms of the tangent operator as

dB(rr,du)=+du:S:du (31)

Note that the elastic-plastic tangent operator S in Eq. (30) is positive definite since its components are individually positive definite. Perfect plasticity models can be generated as a limiting condition of this formulation as H+ 0.

2.1.1. Plane problems Both plane stress and plane strain conditions have been investigated in this paper. Modifications in

the integration schemes are needed for evaluating plastic strains corresponding to specified stress increments for these cases.

Plane stress. In this case, the out-of-plane stress increments Aa3i for i = 1, . . . ,3 are identically equal to zero. Consequently, all non-trivial components of a”+’ is completely known and hence the adaptation of the integration algorithm is straight forward. The out-of-plane strain increments are given

by

he;, = -~(AQ + A~,,)

The elastic-plastic tangent operator in Eq. (30) can be calculated under the assumptions du3,i = 0 for i=l > . . * 7 3.

Plane strain. Under these conditions, the out-of-plane strain increment Aesi for i = 1, . . . ,3 are equal to zero and consequently AUK,, is a non-zero out-of-plane, stress increment. Using the constraint condition

380 S. Ghosh, S. Moorthy / Comput. Methods Appl. Mech. Engrg. 121 (1995) 373-409

A& + AE$ = 0

and substituting in Eqs. (18) and (21), the out-of-plane stress increment may be obtained as

(32)

The resulting a:;;’ is a non-linear function of the flow parameter AA and hence Eqs. (25) and (27) are non-linear functions of AA. AA is calculated using an iterative scheme summarized below l Set i = 1 and as an initial guess set AA’ = 0. l Calculate the out-of-plane normal stress cj33 from Eq. (32) and hence the effective stress (a:;‘)‘. l Calculate the corresponding flow parameter AA’+’ from either of Eqs. (25) or (27) depending on

the hardening law. l Set i=i+l and if (AA’+’ - AA’] > prescribed tolerance TOL return to Step 2. l Set AA = AA’+’ and terminate iteration.

2.2. Numerical implementation of the assumed stress hybrid method

Numerical implementation for the weak form of the assumed stress energy functional involves solution of two sets of non-linear equations. They are:

(i) Element level solution of the kinematic relations (ii) Global level solution of interelement and boundary traction continuity conditions.

2.2.1. Kinematic relations in each element Non-linear dependence of strain increments AE on stress coefficients {A/3}, in element e, make the

strain-displacement relations of Eq. (14) non-linear. This problem therefore involves determining {A@}, for given nodal displacements {Aq}, such that,

[Gl,{q + &I, - i^, [PIT@ + AE) da = (0) (33) e

is satisfied. In an iterative solution process, the actual strain increments AE are linearized about an initial value of the stress increments Au’ as

AE = A+, Aa’) + S : da’

where superscript i corresponds to the iteration number. Calculations of be’ and S have been discussed in the previous subsection. Using Eq. (10) the correction to the stress increment da’ is written as

da’= [P]{dP};

Substitution of these in Eq. (33) results in

FWP): = [G,l{q + &I, - I n P’l’{~ + Ad dfi e

which is a set of linear algebraic equations of the form Ku' =f. The m X m [II] matrix in Eq. (34) is expressed as

WI = j-- PITPI[~l cm (35) ‘2

A quasi-Newton iterative solver with a BFGS update method [34] is used to solve this system.

REMARK 1. The elastoplastic tangent operator S is positive definite and consequently [H] is also positive definite and invertible.


REMARK 2. A requirement for the stress interpolation matrix [P] is that the resulting stiffness matrix be objective with respect to the choice of coordinate axes. This implies that the stress interpolation polynomials should have the same form under any coordinate transformation X, Y-+X,, y,. This ensures the uniqueness of the finite element solution in all coordinate systems. Cook [35] has suggested the use of a special coordinate system embedded in quadrilateral elements. Atluri and coworkers [36,37] have suggested the use of group theory to obtain irreducible representations of the stress functions. This method works well for rectangular elements, but elements are seldom rectangular in a Voronoi Cell mesh. Ghosh and coworkers [19] have used complete forms of Airy stress functions to result in an invariant representation of the stiffness matrix. In the present paper, Airy stress functions of up to order 6 are used.

REMARK 3. As the order of Airy stress function increases, the diagonal terms in [H] become higher-order functions of the global coordinates x and y. This results in an ill-conditioned [H] matrix and its inversion becomes numerically unstable. To avoid this problem, Atluri and coworkers [36,37] have suggested scaling of stress functions in a locally embedded coordinate system 5,~. A linear scaling is used in this paper to produce equilibriated stresses both in the (X - y) and (5 -71) systems. Ideally, the best conditioning for [H] result when 5 E [-1, l] and n E [ - 1, 11, corresponding to the master coordinates in a quadrilateral element. However, in the case of polygonal Voronoi cell elements, this can only be enforced in an approximate sense. The scaling parameter is based on a characteristic length measure of each Voronoi cell. If the centroidal coordinates for a Voronoi element in the global coordinate system are (x,, y,), the characteristic length L is chosen as

L = maxI&, Y,) - 6, y)I W, Y> E ffQ

and thereby 5,~ are represented by the transformation

5= x -x,

L

+-yc L

The 5 - n system is aligned with the x - y system and hence a complete Airy stress function in the 5 - n system ensures an invariant representation of stresses in the x - y system. A candidate second-order stress interpolation

[

I 77

PI21 = 0 0 0 0

matrix may be written as

0 0 0 5 0 v2 0 5’ &I

1 5 0 0 77 0 t2 n2 0 577 0 0 1 -7 -5 0 0 -2&I -n2/2 0 1 -g2/2

and the corresponding fourth-order stress interpolation matrix is constructed as

[

5” n’ 312q .$q’ 0 0 0 0 5” t3q fX2q2 45q3 q4

PI, 35712 0 n3 0 5’7 5’ 5” 4t3n 65%’ 5n3 q4 0 0

-3g-2q 0 -3[q2 -q3/3 -t3/3 0 0 -5” -4t3q -3.$=q2/2 -4.$q3 -q4 0 1 REMARK 4. Though the focus of this paper is only on small deformation plasticity, a generalized mid-point rule is selected for temporal integration of the [H] and [G,] matrices, as suggested in Hughes and Wingett [38]. In this algorithm, time integration is performed in the midstep configuration fl:+i’*. Other methods for objective temporal integration can be found in [39-411.

REMARK 5. Spatial integration of [H] is carried out numerically by subdividing each Voronoi cell into quadrilaterals, for identifying Gauss quadrature points only. Analytical integration of [G,] is carried out using the MAPLE symbolic manipulation package.

2.2.2. Global traction continuity conditions For arbitrary {6Aq},, the global traction continuity conditions in Eq. (16) reduces to the form

382 S. Ghosh, S. Moorthy I Comput. Methods Appl. Mech. Engrg. 121 (199.5) 373-409

jI Wl:{P + APIe - WeI = 0 (36)

An iterative solver is implemented to solve this non-linear equation for { Aq} e. Non-linearity arises from the relation between the stress coefficients {Ap}, and the displacement increments {Aq},. From the knowledge of variables in the ith iteration, variables in the i + lth iteration are calculated as

{Aq};+l = {Aq]6 + {dq)6

where {dq}L’s are calculated by substituting a linearized form equation (34), viz.

{Ap}j+’ = {A& + [W’KWd6 into Eq. (36). This yields the linearized traction continuity

: [Gl:W1[WW~ = !I (We - [Gl:W + Ak%J e=l

The element stiffness matrix [K], is therefore given by

WI, = [Gl:W’[Gl, and the element residual load vector {f}: takes the form

{f):, = W, - [Gl:{P +@L

condition

(37)

(38)

(39)

A quasi-Newton iteration scheme with a BFGS update is used to solve Eq. (37) for nodal displacements and stress coefficients.

REMARK 6. The internal energy (IE) of each Voronoi element, {dq}z[K],{dq},, can be simplified, using linearized kinematic relations, to the form

IE = CW~~[WWL (40)

Since matrix [H] is positive definite, zero internal energy implies that the stress coefficients are identically equal to zero (i.e. {dp}, = (0)). Furthermore, since [H] is invertible, zero stress coefficients implies that the displacement increments {dq}, are in the null space of the matrix [Cl,, i.e.

[Gl,{dqI, = (01 In a n noded Voronoi cell for which (xi, yi) represents coordinates of the ith node, rigid body modes of deformation can be expressed as

WqRL =

‘1 0 -y,-

0 1 X1

1 0 -Y2 a1

0 1 x2 iI ff2 = [4Jl{a) (41) . . . . . .

(y3

; 0 -;,

.o 1 x, -

The product [G],{dqR}, is identically equal to zero for all values of {(Y}. Spurious zero-energy (corresponding to a singular stiffness matrix) can emanate if kinematic modes, other than the rigid body modes, are not properly constrained. In this formulation, the matrix [G] has a dimension ~fz X 2n. Since there are 3 rigid body modes in two-dimensional problems, its rank Y s m - 3. Thus, the condition on rank sufficiency of [K] is that the rank of [G] be at least r = 2n - 3. This governs the necessary choice of the number of {p} parameters according to

m>2n-3 (42)

Sufficiency of the above condition is ensured by checking the invertability global stiffness up to a specified numerical tolerance.


3. Numerical validation of VCFE formulation for a homogeneous solid

Two examples are considered to test the single phase Voronoi Cell finite element formulation for elasto-plasticity. Results are compared with solutions generated by a displacement-based FEM code (DFEM) and published data. DFEM uses isoparametric QUAD4 elements. In this program, plastic strains are integrated using an implicit cutting plane algorithm [42,45].

EXAMPLE 1. Plane strain analysis of a thick homogeneous cylindrical pressure vessel subjected to internal pressure, is conducted. The inner radius of the cylinder is Ri = 100 mm and its outer radius is R, = 200 mm. The cylinder has a Young’s Modulus E = 2.1 X lo4 dN/mm*, Poisson’s ratio v = 0.3, Yield stress Y” = 24 dN/mm*. Perfect plasticity is assumed for post yield behavior. The internal pressure is varied from zero to pi = 18 dN/mm2 and the incremental problem is solved in 36 equal pressure increments. Fig. 2 shows the finite element meshes used by the VCFEM and DFEM programs, respectively. The VCFEM mesh is generated by Dirichlet tessellation of the cylindrical domain using a set of randomly scattered generator points. Perfect plasticity is enforced in VCFEM in a limiting sense with E/H-+ 1000. Fig. 3(a) shows a comparison of evolution of inner radius as predicted by VCFEM and DFEM. Both results are compared with the numerical results of Owen and Hinton [43]. Excellent agreement is obtained between these results. Fig. 3(b) depicts the distribution of hoop stress along the radial section, at the end of loading. These results also match very well considering the fact that there is a slight difference in the representation of perfect plasticity (H = 0 for DFEM). Change in slope of the graph indicates a transition from plasticity (inside) to elasticity (outside).

EXAMPLE 2. A thin rectangular plate with a hole is analyzed for plane stress. The plate is subjected to an end load that varies from zero to 12.15 kgf/ mm*. Dimensions of the plate are 10 mm X 18 mm and the hole has a radius of R = 5 mm. Young’s modulus of the plate is E = 7000 kgf/mm*, Poisson’s ratio is v = 0.2 and the initial yield strength is Y” = 24.3 kgf/ mm*. Post yield behavior of the material is characterized by a linear hardening law with a hardening modulus H = 225 kgf/mm*. The corresponding meshes for DFEM and VCFEM are depicted in Fig. 4. Both methods solve the problem using 10 equal loading steps. Results are compared with experimental data presented in [44]. First yielding takes place at the point A shown in Fig. 4, at which the total tensile strain is denoted as l y”y. Average

Fig. 2. Elastoplasticity problem for pressure vessel with prescribed internal pressure. (a) DFEM mesh, (b) VCFEM mesh.


1

18.00 -

17.00 -

16.00 -

15.00 -

14.00 -

B 13.00 -

$12.00 -

gj 11.00 --

2.00 -

0.00 -/

1.00 -

1 I I 100.00 100.10 loo.20

Radius of inner face (-)

(4

25.00

24.00

23.00 :

22.cKl -

21.00 -

20.00 -

19.00 -

18.00 -

17.00 -

16.00 -

15.00 -

100.00 120.00 140.00 160.00 180.00 200.00 Radial distame (mm)

(b)

Fig. 3. (a) Evolution of inner radius and (b) hoop stress distribution at the end of loading for the pressure vessel problem.

p=12.1 kgf/mm2

Fig. 4. Elastoplasticity problem for plate with a circular hole. (a) DFEM mesh, (b) VCFEM mesh.


1.10

1.05

1.00

0.95

0.90

0.85

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0.30

0.25

0.20

0.15

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0.05

0.00

-0.05 0.00 1.00 2.00 3.00 4.00

E&h

Fig. 5. Evolution of maximum tensile strain with loading.

tensile stress (u,,,,) variation along a section AB is plotted as a function of the tensile strain eGY in Fig. 5. The results of VCFEM and DFEM show excellent agreement, though the experimental results deviate a little in the plastic region. Good comparison of stress and strain distributions along AB are obtained by these methods and are shown in Figs. 6(a) and (b), respectively.

These examples conclusively prove the effectiveness of VCFEM for homogeneous problems.

4. Elastic-plastic VCFEM for heterogeneous media

In this section the VCFEM is further enhanced for heterogeneous media, in which microscopic heterogeneities are arbitrarily dispersed in the matrix. Dirichlet tessellation of such a representative material element generates a network of Voronoi cells. A composite finite element formulation is developed for these Voronoi cells with embedded second phase heterogeneities, as shown in Fig. l(c). Within each Voronoi cell &, a heterogeneity occupies a region @ C Oe with a matrix-inclusion interface denoted by a@. Consequently, the formulations presented in Section 3 must be modified to incorporate the effect of the second phase. Two alternative approaches are considered for this purpose. The first approach introduces a transformation strain in regions of material discontinuity and is based on the original work by Eshelby [S]. Ghosh and coworkers [19] have used a variation of this formulation. The second approach directly introduces traction continuity at the matrix-inclusion interface by the


I / I , I

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6.00 7.00 8.00 9.00 10.00 Radial distace (mm)

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28.00

26.00

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22.00

20.00

18.00

3 I 16.00

zc g 14.00

8.00

t

(4 (b) Fig. 6. (a) Tensile strain and (b) tensile stress distribution along AB at the end of loading, for the plate with a hole.

application of Lagrange multipliers. A comparison between the two methods is made within the framework of linear elasticity problems.

4.1. Transformation strain method

The main idea behind the transformation strain method [S] is that, subject to a prescribed stress field u in a heterogeneous domain, the inclusion occupying the domain flc C @, exhibits an additional non-stress causing eigenstrain E*, given as

S(x):{a}=S”:a+~* inQ (43)

where S(x) is the location dependent elastic compliance of the inhomogeneous medium (matrix + inclusion) and So is the elastic compliance of the homogeneous matrix. E* is non-zero only in the inclusion Q. This equation may be rewritten as

{a}=[S(x]-S”]-‘:~*=AS-‘:~* in& (44)

With the transformation strain method, the assumed stress energy functional for the Voronoi cell finite element in Eq. (3) is modified as

@‘(a, U, E*) = &(a, U) - I 1 a:E*dO+ -e* : AS-‘(x) : E*

f% 2 (45)

Stationarity of nF1 with respect to the first variation of u results in a modified kinematic relationship as

So : u + E* = i (VTu + VU) in Q (Modified strain-displacement relation) (46)

Eq. (46) is a correct representation of the strain field since the transformation strain E* is assumed to exist in the heterogeneity alone. Stationarity of II:’ with respect to the transformation strain E” produces the stress-strain relationships within the heterogeneity (44). Furthermore, stationarity of the global energy functional defined as

(47)


with respect to boundary displacements U, results in the Euler equations (6) and (7) which correspond to interelement and boundary traction continuity conditions, respectively. A priori assumptions on equilibriated stress fields, compatible displacement fields and stress-strain relationships in the matrix together with posteriori Euler equations (46), (44) and (7) completely define the deformation problem in the heterogeneous domain flc for applied tractions t on Cm and prescribed displacements U on r,,.

In addition to approximating the stresses and displacements, transformation strains in each Voronoi element are interpolated as

{e*>, = Pl{~I, (48)

where {A} is a column of s unknown coefficients A,, A,, . . . , A, and [R] is a 3 x s interpolation matrix. For constant transformation strains as considered in this paper, [R] is a 3 x 3 identity matrix. Substituting Eqs. (S), (11) and (48) in L!:’ in Eq. (45) and setting its first variation with respect to {p}, equal to zero, results in the weak form of modified stress-strain relations

[HI@], = [Gle{qI, - [Ql{A>, (49)

where [Q] is a m x s matrix given by

[Ql = i, Pl=[4 dfl c Similarly, stationarity of n:* with respect to {A}, gives the weak form of the constitutive equations in the second phase 0c

[QI’OV, = [W% (50) where [T] is a s x s matrix

PI = I,, [~l=PW’[4 dfl L Matrices [HI, [Cl, are defined in Eqs. (35) and (15), respectively. Solving Eq. (49) at the element level yields the stress coefficients as

{P], = [~l-‘([Gl,{qL - [Ql{A>,) (51)

Substituting Eq. (51) in Eq. (50) reduces the constitutive relations in the heterogeneity to the form

-[QIT[fT’[Gle~d, + WI + [QITIJT’[Ql)W, = (01 (52)

or

Putting this in the global energy functional (47) and evaluating stationarity condition with respect to the nodal displacements {q} leads to

[‘AfW-l~GleW - [‘4:[~l-‘[QlTW = if>, (53)

or

[K],(q) - WI.,(A) = W,

Combining Eqs. (52) and (53) leads to a coupled system of matrix equations

(54)

which must then be solved for displacements and transformation strains. The transformation strain method has only been formulated for linear elasticity in this paper.


4.2. Method for direct implementation of interface traction continuity

Along a bonded matrix-inclusion interface X&, the stress and strain fields may be discontinuous, while the displacement and traction fields are continuous. In the transformation strain method, strain fields are discontinuous while stress fields are continuous across the interface. To enhance the accuracy of solution, a discontinuous stress field is now introduced into the modified complimentary energy by way of permitting jumps in the coefficients of stress polynomials. In this formulation, traction continuity constraint at the interface is directly introduced into the complimentary energy functional 17, through the use of Lagrange multipliers. This method will henceforth be referred to as the direct constraint method.

Consider again the initial value problem defined at the pth step in Section 2. For the heterogeneous Voronoi cell, the stress field u at the beginning of the pth step is assumed to be discontinuous across the interface a& but continuous and equilibriated in both the heterogeneity Q and matrix flm = fie - Q. Analogously, the strain field E(U, A@) at the beginning of the pth step is also discontinuous at the interface X& but continuous in Q. and Q,,. The displacement field on aOc is represented by 1~‘. Element boundary conditions (1) are appended with an additional interface traction continuity condition

uC.n=o”.n on aJJ

where n is the unit outward normal on &f& and the superscripts c and m correspond to inclusion and matrix phases, respectively. During the pth step, the stress increment Au is also discontinuous across the interface boundary but equilibriated in Q and the complimentary energy density AB(x, u, Aa) is an explicit function of the position X. Displacement increments Au and Au’ are also compatible across the element and interface boundaries. Consequently, n, in Eq. (3) is modified to

n,“’ = fl, - (a” + Au” - uc - Au’). n. (u’ + AU’) a0 (55)

Stationarity with respect stress increments in fit and flm produces the kinematic relations (4), while that with respect to the Au’ produces the interface traction constraint

(a’ + Au’) * n = (a” + Au”). n on IX$ (56)

Stationarity of the total energy functional UC2 = (Cr=, n,““) with respect to u yields Eqs. (6) and (7). As before, the stress field u and its increment Au are expressed as

{a> = [WP + ww’), on Q (57)

{Au} = [P]{Ap + L(X) Ap’}, on Q, (58)

where L(X) is defined as a step function representing the discontinuity, i.e.

L(x) = 0 for x C Q - Q

=l forxCQ

Similarly, the interface displacement and its increment are interpolated as

{u’> = Wl{dI, in I@

{Au’} = [L]{Aq’}, in a@

(59)

(60)

where {q’}, and {Aq’}, are, respectively, the generalized nodal displacements and its increments on the interface aflcn,. Element boundary displacements and its increment on aQ are interpolated as before. Eulers equations with respect to variations in stress coefficients {Ap}, lead to element level kinematic relations (33), while those corresponding to variations in {Ap ‘}, results in the inclusion kinematic relations

[G],.{q’ + Aq’}, -IO [P]‘{E + Ae} d0 = (0) in Q (61)

S. Ghosh, S. Moorthy I Comput. Methods Appl. Mech.

[G], is a m x 2n’ matrix, where n’ is the number of nodes on

[Glc = i,,, [~lTbwl &a c

Engrg. 121 (1995) 373-409

the interface, and is given as

389

(62)

Similarly, stationarity of IF* with respect to {Aq}, results in interelement traction reciprocity conditions (36), while that with respect to {Aq’}, leads to the interface traction constraint condition

C.I Fl:{P + W’I, = (01 (63)

As with the homogeneous problem, non-linear equations evolve at element and global levels because of element level kinematic relations and global level traction continuity conditions. The solution procedures for these two sets of equations are presented next, following the developments in Subsection 2.2.

4.2.1. Element level solution scheme with the direct constraint method Eqs. (33) and (61) should be solved for the stress coefficients {Ap}, and {A/?‘}, from given values of

nodal displacements {Aq}, and {Aq’} C. These two equations are coupled due to the fact that strain increments Ah4 in OC depend on both stress coefficients {A/3}, and {Ap ‘}, . In an iterative solution process, the actual strain increments AE are linearized about the stress increment Au as before, to give

AEi+l = AE’(u, Aa) + S : da’

The correction da’ to the current stress increment is written from Eq. (58), as

{da}’ = [P]{dP + L(x) dj?‘}: in Q.

Substituting this in the linearized forms of (33) and (61) yields a coupled equation of the form

G:(q + 4, - I PT(e + A&) dJ2 0,

G&L + A@) - I % PT(e + A&) dR

(64)

where [HI, is a m x m, matrix given as

mn = i, PITNIPl dfl m and [HI, is a m x m matrix given as

WI, = j--, [~lTM~l dfl

A quasi-Newton iterative solver is used to solve Eq. (64).

4.2.2. Displacement increments from global traction continuity Displacement increments are obtained by solving the global traction continuity conditions (36) and

(63), rewritten together as

,ii (PI:@ + AP), - We) = 0 (65)

5 [G]:{P’ + AP’}, = 0 e=l (66)

In an iterative solution process, the displacement increments are updated in the same way as discussed for the homogeneous problem as

390 S. Ghosh, S. Moorthy I Comput. Methods Appl. Mech. Engrg. 121 (1995) 373409

{Aqt}i+l = {Aq’}’ + {dq’}’

These displacement correctors are obtained by solving a linearized form of equations (65) and (66) using Eq. (64)

G:H,‘G, -G;H,‘G,

-G:H,‘G, G;(H,’ + H,l)G, (67)

REMARK 7. The internal energy (IE) of the linearized stiffness matrix in Eq. (67) for a Voronoi element can be simplified using Eq. (64) to the form

IE = $ WP)TIHl,W) + {W + dP’lTIHl,hW + dB’1)

with

{dfi] = [Hl,‘MMz~ - ~Hl,‘[GlcWz’~ and

Cd/? + da’) = [Hl,‘PA,W’~ For positive definite [HI, and [HI,, the internal energy is zero only if each of the stress coefficients are individually zero. Setting {dj? + dp’} to zero and using arguments in Section 2.4, it can be shown that

ma2n’-3

is a necessary condition for + {dp + dfi’}[H],{d/? + dp’} # 0 if {dq’} is a kinematic displacement mode. Further, if {dq’} is a rigid body displacement, then m 3 2n - 3 is the necessary condition for the resulting internal energy IE, = i{dfl}TIH],{dfl} of a heterogeneous element to be non-zero if {dq} is also a kinematic displacement mode. Thus, the necessary conditions for non-spurious zero energy modes is written as

m > max(2n - 3,2n’ - 3) (68)

As in the case of the homogeneous problem, sufficiency of the above condition is ensured by ascertaining the invertibility of the global stiffness matrix in the solution phase of the program.

REMARK 8. In general, the interface displacements {dq’} are not connected to the element boundary displacements {dq}. This implies that for [K], to be invertible, 3 additional constraints should be imposed. These constraints are derived by equating rigid body modes on the interface to the rigid body modes on the element boundary. In order to calculate the interface rigid body modes, consider the interface displacement field {dq’} to be the sum of a rigid body mode and a purely deformation mode. As discussed in Section 2.4, any rigid body mode can be expressed in the form [~]{cu}, and hence

{dq’] = [41’{~) + Wz’Idef (69)

where {d’l’ > def is the pure deformation part of {dq’}. Since the pure deformation mode is orthogonal to the space spanned by rigid body modes, {dqr}def lies in the null space of [+I’, i.e.

[4]‘T{dq’}d”f = (0)

Here [4]’ corresponds to the [4] matrix in Subsection 2.2.2 with components in terms of interface coordinates (xi, y:). Written explicitly, this corresponds to the following set of algebraic equations

u;def + &f + . . . + U;pef = 0

u;def + @f + . . . + v;$f = ()

(-U;defy; + u ;““‘x,) + (-U;de’y; + @fX2) + * * * + (-u;““‘y;. + v~?f_x,,) = 0

The first two constraints imply that there is at least one point on a& for which the x- and y-components

S. Ghosh, S. Moorthy / Comput. Methods Appl. Mech. Engrg. I21 (1995) 373-409 391

of pure deformation are zero. The third constraint postulates that in a pure deformation mode, at least one point on &$, has a zero rotation with respect to the origin. The equation for expressing the equivalence of rigid body modes on the interface a@ and element boundary a@, is then obtained from Eq. (69) as

(70)

The constraint needs to be imposed for making the element stiffness [K], non-singular. This is achieved by adding the constraint (70) by means of a Lagrange multiplier to the modified global energy 17” as

Addition of the constraint modifies the original stiffness matrix to the form

[:T :I{:}={;:} (71)

which is solved using a penalty formation as used in [46].

4.3. Comparison of transformation-strain and direct-constraint methods

To compare the accuracies of the composite Voronoi Cell finite formulation using the transformation strain method and the direct-constraint method, an elastic problem of a single circular inclusion in a square composite domain is considered. The inclusion has a radius R = 0.05 m and the dimensions of the domain are 1 m x 1 m. The matrix material has a Young’s Modulus E = 69 GPa, Poisson’s ratio Y = 0.33 and the inclusion has a Young’s Modulus E = 137 GPa, Poisson’s ratio v = 0.285. The square domain is subjected to a uniform stretch of 0.009 m under the assumptions of plane stress. The same problem is also solved by a general purpose finite element code ANSYS, where a fine mesh with 224 elements is used to discretize the composite domain. Fig. 7 shows the meshes used by the VCFEM and ANSYS programs. The VCFEM mesh is generated by scattering a set of random point seeds in the

(a) (b) Fig. 7. A single inclusion in a square domain is discretized into (a) QUAD4 mesh, (b) VCFEM mesh


domain. A complete 5th order Airy stress function corresponding to 1Sp terms in the stress polynomial is used for VCFEM analysis. In the transformation strain method, E* within the inclusion is assumed to be a constant, which is consistent with Eshelby’s [8] results for a small ellipsoidal inclusion. Fig. 8 shows the tensile stress distribution along a section through the middle of the inclusion, perpendicular to the loading direction. The transformation strain based VCFEM clearly underestimates stresses in the inclusion. This is attributed to the fact that a continuous stress field is used to represent a discontinuous stress distribution within the heterogeneous element. When compared with ANSYS predictions, the direct-constraint method produces a much better representation of the stress distribution (2.5% difference in the inclusion and 7% at the interface). The superiority of the direct-constraint based VCFEM has led to its use for all subsequent computations.

4.4. Analysis of different representative material elements with composite VCFEM

In this section the direct-constraint based composite VCFEM is validated for elasto-plastic deformations in different representative material elements. Representative material elements (RME) character- ize a microstructure at a given point in a structure. Macroscopic response of these RMEs corresponds to the volume average of the microscopic response. In particular, macroscopic stress components Gij are obtained from the true microscopic stresses gij following the relation

1 “11 = V,,, nRME qj dV i

where V,,, is the volume of the representative material element. Correspondingly, the macroscopic strains are given as

Three separate RMEs are analyzed and results are compared with results from a displacement-based FEM package ANSYS and with analytical results published in literature. Unless explicitly stated, in all

860.00 -’ I I_

840.00 -m- _____ _________--.-------_-_ _

820.00 a rect Constran-lt .---___-_.

- ___- 800.00 3 -&-&;; -sL_Hin

780.00 -

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q 720.00 - 3 700.00 ‘. -

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a 640.00 - :

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560.00 -

540.00 -

520.00 -

500.00 -

480.00 -

460.00 -

440.00 -,

0.00 0.50 1.00 Distance through the inclusion <mm)

Fig. 8. Distribution of tensile stress along a section through the inclusion.

S. Ghosh, S. Moorthy I Comput. Methods Appl. Mech. Engrg. 121 (19Y5) 373-409 393

examples considered henceforth, the matrix material is 6061-O Al with Young’s modulus E = 69 GPa, Poisson’s ratio v = 0.33, Yield stress Y” = 43 MPa and the post yield response given as oeeff = Y” + 0.14 x l ;;;‘” GPa and the inclusion is assumed to be brittle Boron fibers with Young’s modulus E = 410 GPa, Poisson’s ratio v = 0.2. The mechanical response under applied loading on the macrosurface

=RME of the RME is analyzed. Appropriate displacement constraints are imposed on the boundaries of representative material

element X&, to remove rigid body modes. Consequently, in the case of uniaxial tension (Examples 1 and 2 in Section 4.4 and examples in Section 4.5.1 and 4.5.2), two adjacent sides are roller supported and a uniform stretch is applied on the third side. For biaxial tension (Example 3(b) in Section 4.4 and all examples in Section 4.5.3), uniform stretches are applied on the third and fourth sides. Though periodicity boundary conditions have not been considered in this paper, its implementation in VCFEM is straightforward, as has been demonstrated by the authors in [24].

EXAMPLE 1. Representative material elements consisting of square edge and square diagonal packing arrangements, loaded in simple tension are considered in this example. Fi g. 9(a) and (b) shows

Fig. 9. ANSYS and VCFEM meshes for (a) square edge packing, (b) square diagonal packing, V, = 46%


the ANSYS and VCFEM meshes for these arrangements. The VCFEM mesh consists of 9 elements for the square edge packing and of 13 elements for the square diagonal packing. The ANSYS mesh consists of 289 QUAD4 elements for square edge packing and 512 elements for the square diagonal packing. even though the same RME could be used for both VCFEM and ANSYS, a larger portion of the microstructure was chosen for the former to avoid edge effects. For ANSYS however, this would lead to an exorbitantly large mesh and has therefore been abandoned. The RME has a dimension of 1 m X 1 m for ANSYS mesh, 6 m X 6 m for the square edge VCFEM mesh and 4 m X 4 m for the square diagonal VCFEM mesh so as to maintain the same dimensions of basic structural elements. The inclusions are circular with a volume fraction of V, = 46%. The RME is loaded incrementally to a maximum macroscopic tensile strain of E,, = 0.5%. The VCFEM in this problem, uses a 6th order Airy stress function giving 25 p parameters in the stress interpolation. Fig. 10(a) and (b), respectively, show the macroscopic stress-strain response for plane stress and plane strain conditions. These plots show an excellent agreement between the VCFEM and ANSYS results. Fig. 11 shows a comparison of the microscopic stress distribution along the mid-section through the inclusion for three different macroscopic strains. This chosen section corresponds to x/L = 0 for the ANSYS mesh, where x and L are shown in Fig. 1. Though these distributions follow a close pattern, a slight discrepancy is noticed at the matrix-inclusion interface. This may be attributed to the rather coarse mesh used in VCFEM calculations. For a different volume fraction of square diagonal packing (V, = 20%), Fig. 12 shows the true microstructural stress distribution in the direction of applied strain. The results are plotted for three different sections corresponding to x/L = 0.2, x/L = 0.3, x/L = 0.5 in the ANSYS mesh, at the end of loading. These results show generally good agreement in pattern even with the coarse VCFEM mesh. The peak values of stresses are considerably different for x/L = 0.2.

EXAMPLE 2. In this example, a microstructural RME with 29 randomly distributed circular inclusions is considered. Fig. 13 shows the discretization for ANSYS and VCFEM models. The ANSYS mesh consists of 8000 QUAD4 elements while the VCFEM mesh has only 29 elements corresponding to the number of inclusions. The dimensions of the RME are 1 m x 1 m and the radius of inclusions is 0.0468 m, corresponding to a volume fraction V, = 20%. The locations of these inclusions were generated by a random number generator suggested in [23]. Histograms in Figs. 14(a) and (b) indicate features of

I I I I I

100.00 - 95.00 -

90.00 -

85.00

80.00 I’ /;

Fig. 10. Comparison of macroscopic stress strain response in (a) plane stress, (b) plane strain for square edge and diagonal packings (V, = 46%).


80.00

70.00 3

si

B

65.00

B 60.00

f .Y

B

55.00

*s$ 50.00

45.00

40.00

35.00

r

t L

L

0.00 0.30 0.60 0.90 YK

Fig. 11. Comparison of microscopic tensile stress distribution for square edge packing (V, = 46%, plane strain) along a section through the inclusion.

quantitative characterization for the microstructure. The RME is loaded incrementally under plane strain conditions to a maximum macroscopic tensile strain of cX, = 0.5%. Fig. 15 shows the overall macroscopic tensile stress-strain response by the two methods. A near perfect match is obtained for all stages of the loading curve. Fig. 16 shows the true microscopic stress distribution at two different sections (x/L = 0.4 and x/L = 0.7). While the stress distributions by the two methods are in general agreement, discrepancies in peak stresses inside the inclusions (larger for VCFEM) and at the Voronoi cell boundary (smaller for VCFEM). One reason for this may be the fact that ANSYS results are obtained by smoothing integration point stresses to the nodes. Another reason may be that the biquadratic stress interpolation function used in VCFEM are not sufficient in representing large oscillations in the actual stress field. The latter effect is seen in the interelement discontinuities at y/L = 0.397 in Fig. 16(a) and at y/L =0.191 in Fig. 16(b). Fig. 17 shows the distribution of the microscopic effective stresses for square edge, square diagonal and random packing RMEs, each having an inclusion volume fraction of V, = 20%. The contour plots are at a macroscopic strain of E;, = 0.5%. These figures show that a major portion of the distortional energy is stored in the inclusion. The effective stresses are considerably smaller in the randomly packed RME. This is probably due to the fact that a larger number of inclusions were used for random packing.

EXAMPLE 3. The final set of examples compare the VCFEM results with results from a semi- analytical method [lo] and a Mori-Tanaka estimate [7]. Instantaneous bounds on stress-strain responses have been obtained by Accorsi and Nemat-Nasser [lo] for square inclusions loaded in simple


2 82.00 -

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fj :z-

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73.00 -

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ANSYS ’ VCFEM

/ 0.80

I i

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79.00 -

$ 78.50 -

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93.00 VCFEM

-

92.00 -

91.00 -

90.00 -

77.00 -

76.00 -

75.00 - 1 / I / , I?

0.00 0.20 0.40 0.60 0.80 1.00

-

77.00 - ‘;. \ /

‘..._, / :_..c

76.50 - I I I I I

0.00 0.20 0.40 0.60 0.80 1.00

Yn.

(4 Fig. 12. Comparison of microscopic tensile stress for square diagonal packing (V, = 20%, plane stress), at (a) x/L = 0.2, (b) x/L = 0.3, (c) x/L = 0.5.

shear. The RME, considered for VCFE analysis, has a square edge packing arrangement with inclusion volume fraction of V, = 25%. The matrix is ductile, with Young’s modulus E = 2.5 GPa, Poisson’s ratio v = 0.25, initial yield stress Y” = 10 MPa, and its post yield behavior is characterized by a,,, = Y” + 0.2 X l err GPa. The inclusion is elastic, with Young’s modulus E = 50 GPa and Poisson’s ratio v = 0.25. Fig. 18(a) shows a comparison of the macroscopic response produced by VCFE analysis with those computed from [lo]. Even though the bounds are for a three-dimensional model, the plot shows that plane strain VCFE analysis results are within the bounds for both elastic and elasto-plastic regions of the response curve.

S. Ghosh, S. Moorthy I Comput. Methods Appl. Mech. Engrg. 121 (1995) 373-409

Fig. 13. (a) ANSYS and (b) VCFEM meshes for random packing composite (V, = 20%).

A second comparison is made with the Mori-Tanaka estimates by Zhao and Weng. [7]. Circular inclusions are distributed in a square edge packing arrangement with an inclusion volume fraction V, = 30%. The RME is loaded in biaxial tension under plane strain conditions. The aluminum matrix has Young’s modulus E = 55.8 GPa, Poisson’s ratio v = 0.32, initial yield stress Y” = 85.8 MPa, and a post yield behavior uerr = Y” + 0.972 (e$,)“.“” GPa. The inclusions are elastic Boron fibers with Young’s modulus E = 379 GPa and Poisson’s ratio v = 0.2. The comparison of the macroscopic response is shown in Fig. 18(b). The loading conditions considered, produce a large amount of triaxiality in the microstructure. As a result, elastic response is the dominant deformation mechanism. The comparison shows an accurate representation with a maximum error of 2.5%.

I I I 6

7

6

F-5 e s 84 L

3

2

1

1 8.13 0.152 0.’

5-

4-

6 = 3-

-II-l l

2 I

1

0.196 0.218 0.240 0.262 0.284 6116 II

0.17

Particulate to Polygon Area Ratio

(a)

Mean Near Neighbour Distance

(b)

Fig. 14. Histograms for quantitative characterization of the random packing composite (V, = 20%). Frequency of occurence of (a) area ratio, (b) mean neighbor distance.

I- ll


135.00 ’ -

130.00 -

125.00 -

120.00 -

115.00 -

110.00 -

105.00 -

100.00 -

95.00 -

90.00 85.00

-

B -

w 80.00 -

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’

If 65.00 70.00 - -

60.00 -

55.00 -

50.00 - ;j!

45.00 -

40.00’ ’ 0.00

I / I I 1 I 0.20 0.40 0.60 0.80 1.00

Fig. 16 (Contd.)

Min.-r L 7.509E.02

Fig. 17. Contour plot of microscopic effective stress (in GPA) at E,, = 0.5% for (a) square edge, (b) square diagonal and (c) random packing (V, = 20%) plane strain).


9.00

8.50

8.00

7.50

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4.50

I I / I_ I 1 I I 0.00 0.200 0.400

.,“f% 0.800 lXXJ0

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700.00 -

6so.w -

6W.w -

550.w -

sW.cKI -

b =45o.w -

400.00 -

lpp35o.w -

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200.00 -

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lw.w -

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0.0 0.1 0.2 0.3 0.4 0.5

e,= E,(W)

(b)

Fig. 18. Macroscopic response comparison. (a) square inclusion in simple shear (V, = 25%), (b) circular inclusion in biaxial tension (V, = 30%).

5. Effect of size, shape and distribution on response

In this section the effect of size, shape and distribution of heterogeneities, on the macroscopic as well as true microstructural response of RMEs is studied. The Voronoi Cell finite element model uses 25 p parameters in the stress interpolation for all cases.

5. I. Effect of size

Various volume fractions of circular inclusions are considered for (a) square diagonal packing (b) square edge packing and (c) random packing distributions. The VCFEM meshes for the three configurations are similar to those shows respectively in Fig. 9(a, b) and Fig. 13. Averaged macroscopic stress-strain curves for an applied macroscopic strain are shown in Fig. 19(a-c). It is seen that an increase in volume fraction of inclusions cause a rise in the material hardening. This is expected since a larger volume of the brittle inclusions will cause the overall response to move in the direction of its individual response.

5.2. Effect of distribution

The square diagonal, square edge packing and random packing RMEs are analyzed for understanding the effect of distributions on the response. Resulting uniaxial tensile curves for three different volume fractions are shown in Fig. 20. While the macroscopic response of the random packing lies between the square edge and square diagonal packing response at higher volume fractions (V, = 0.4 and V, = 0.6), the random packing RME exhibits a higher strength for lower fractions (V, = 0.2). The former result has been observed at a V, = 46% volume fraction for three-dimensional analysis of continuous cylindrical fibers by Brockenbrough and coworkers [18]. The higher strength exhibited by the randomly distributed RME at lower volume fractions (V, = 20%) seems to be a characteristic of the plane problem considered.

-

402 S. Ghosh, S. Moorthy I Comput. Methods Appl. Mech. Engrg. 121 (1995) 373409

140.00

130.00

120_00

110.00

100.00

90.00

2

8

80.00

B 70.00

.a 60.00

P -& 50.00

8 4 40-*

30.00

20.00

10.00

0.00

F

I

0.00

I I I I I

0.10 0.20 0.30 0.40 Macroscopic tensile strain(%)

0.50

Fig. 20. Effect of distribution of circular inclusions on the macroscopic tensile stress-strain response.

Fig. 21. VCFEM meshes for randomly packed elliptical RMEs (V, = 20%). (a) major axis in the horizontal direction, (b) major axis along the vertical direction, (c) major axis arbitrarily oriented, (d) size, shape and orientation arbitrary.


85.00

80.00

75.00

70.00

65.00

60.00

55.00

50.00

45.00

40.00

35.00

30.00

25.00

20.00

15.00

10.00

5.00

0.00 c

Fig. 21 (Contd.)

I I I I I I

Case (a) . . . . . . . . . . . . . . . . . . . . . . Case (b) __________- Case (c) _ _ _ _ _ _ _ _ Case (d)

L 0.00 0.10 0.20 0.30 0.40 0.50

Macroscopic strain (9%)

Fig. 22. Effect of orientation and shape on the macroscopic tensile stress-strain response of elliptical RMEs.


5.3. Effect of shape and orientation

In this study, the effect of shape and orientation of inclusions on the behavior of microstructure is investigated. Thirty nine elliptical inclusions of various shapes, sizes and orientation but with a constant volume fraction of V, = 20%, are distributed randomly in the RME. Fig. 21 shows the VCFEM meshes for 4 different distributions. They are

(a) major axis in the horizontal direction (b) major axis in the vertical direction (c) major axis arbitrarily oriented and (d) arbitrary shapes, sizes and orientation

The ratio of minor to the major axis, for ellipses in the first three cases is 0.6. Fig. 22 shows the effective response curves under plane strain conditions. True microscopic stresses at the mid-section (x/L = 0.5) are depicted in Fig. 23. Fig. 24 shows contour plots of the microscopic plastic strains when all four RMEs are loaded to a macroscopic far-field strain of E,, = 0.5% under plane strain conditions.

These figures show that overall stresses, and also peak microscopic stresses and microscopic plastic strains are higher when the shapes, sizes and orientations are completely random, i.e. case (d). Very slight differences are produced by changing the inclusion orientation alone. It can therefore be concluded that plastic response is highly sensitive to the distribution of shapes and sizes of heterogeneities, even though, the elastic response is relatively unaffected. This fact has also been noted by Brockenbrough et al. [X3]. Evolution of the yield surface in the in-plane stress space (cXX X fl,) is

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4

70.00

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Case (a)

c

L

L I I I I I

0.00 0.25 0.50 0.75 1.00 Y/L

Fig. 23. Effect of orientation and shape on the microscopic tensile stress distribution of elliptical RMEs.


Min.

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(cl (4 Fig. 24. Contour plots of microscopic effective plastic strain at E,, = 0.5% for the elliptical RMEs. (a) case (a), (b) case (b), (c)

case (c) and (d) case (d).

also examined. The RME is assumed to harden according to the power law, consistent with its ductile constituent. Yield surfaces are generated by joining yield points (a,, the effective macroscopic strain E,rr

a,) in the stress space, for which is the same. Fig. 25 shows the evolution of the yield surfaces for

each of the four cases. A comparison of yield surfaces corresponding to an effective macroscopic

E,ff = 0.5% is given in Fig. 26. These two figures conclusively demonstrate that the random microstructure (case (d)) exhibits a significant difference in plastic flow behavior, in comparison with the others.

6. Conclusions

In this paper, a new Voronoi cell finite element method is developed for small deformation 2-D elastic-plastic analysis of arbitrary heterogeneous materials. Arbitrary sided Voronoi cells result from Dirichlet tessellation of a heterogeneous domain, based on the location, shape and size of second phase. Each Voronoi polygon, containing one heterogeneity at most, is treated as an element in the finite element formulation. Arbitrariness in the number of sides in each element is accounted for by an


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@=(GPa)

w

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0.00 0.20 0.40 0.60 0.80 1.00 1.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20 C=tGPa) CmK-m

(4 (4 Fig. 25. .Evolution of yield surface for ellipsoidal RMEs. (a) case (a), (b) case (b), (c) case (c) and (d) case (d).

assumed stress formulation, that makes independent assumptions on the stress and boundary displacement fields in each element. Furthermore, a composite Voronoi Cell finite element method is developed to incorporate the effect of heterogeneities in each Voronoi cell. Two alternative methods, namely the transformation strain method and the direct constraint method are considered. Comparisons with a displacement-based finite element for elastic problems show that the direct constraint method gives superior results when compared with the transformation strain method. Several numerical examples


0.00 0.50 6 xx (GPa)

1.00

Fig. 26. Comparison of yield surfaces at a macroscopic effective strain Qf = 0.5% for various orientations and shapes of ellipsoidal RMEs.

conducted, conclude that the accuracy of VCFEM predictions is extremely competitive with that of the displacement-based finite element models like ANSYS, which require a high degree of refinement leading to prohibitively large numerical models.

Effect of size, shape and distribution of heterogeneities on the global and local responses of sample representative material elements are studied by VCFEM. Both regular and random packing microstructures are considered for this purpose. It is observed that with the increase in inclusion volume fraction, the overall response moves towards the inclusion behavior. A study conducted with various orientations of identically shaped elliptical inclusions show that orientation has negligible effect on the overall response. It is also noticed that the response relation between different distributions of plane RMEs change considerably with different second phase volume fractions. An important observation made is that, though elastic response is relatively unaffected by the distribution of second phase shape and size, the plastic response is significantly altered by this. The effect on plastic response is moderate on the effective stress-strain response but is considerable for true stress evolution in the microstructure.

In conclusion, it can be said that the Voronoi Cell finite element model has a tremendous potential for analyzing arbitrary microstructures, which frequently occur in real materials. For these materials, conventional Galerkin finite element models require tremendous efforts in modeling and also require large and highly refined meshes. Though an exact CPU comparison has not been made due to the different levels of softwares being used, it can be generally said that even a research code was 15 times more efficient than the highly optimized commercial code ANSYS. Multiple scale coupling between structural and microstructural scales can be done easily with VCFEM for simulating material


microstructural evolution in actual structures. Such an analysis has been done by the authors for elastic problems with success [24].

Acknowledgment

The authors would like to thank Dr. 0. Richmond of ALCOA Technical Center for his invaluable suggestions and discussions on this work. Support of this work by the United States Army Research Office through grant No, DAAL03-91-G-0168 to the Ohio State University (Program Director: Dr. K.R. Iyer), is gratefully acknowledged. Computer support of CRAY-YMP usage by the Ohio Supercomputer Center is also gratefully acknowledged.

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Documents

Elastic-plastic analysis of arbitrary heterogeneous materials with the Voronoi Cell finite element method