A material based finite element analysis of heterogeneous media involving Dirichlet tessellations

Computer Methods in Applied Mechanics and Engineering 104 (1993) 211-247 North-Holland

CMA 336

A material based finite element analysis of heterogeneous media involving Dirichlet

tessellations Somnath Ghosh

Department of Engineering Mechanics, The Ohio State University, Columbus. OH 43210, USA

Sankar N. Mukhopadhyay Department of Engineering Mechanics, The University of Alabama, Tuscaloosa, AL 35487, USA

Received 17 January 1992 Revised manuscript received 23 April 1992

In this paper, a new finite element formulation has been developed for analysis of heterogeneous media, in which the second phase is randomly dispersed within the matrix. It introduces a tessellation based mesh generation technique to account for the arbitrariness in location, shape and size of the second phase. An assumed stress hybrid formulation has been implemented to utilize the resulting Voronoi polygons as elements in a finite element model. The effect of the second phase has been accounted for by introducing a transformation strain based constitutive formulation. Numerical examples have been conducted to validate the model.

1. Introduction

In recent years, rapid developments have taken place in processing technology and industrial utilization of new materials such as polymer, ceramic or metal-matrix based composites. Strong and stiff reinforcements in the form of particulates or fibers are dispersed in a matrix to improve the physical and mechanical properties such as strength, stiffness, etc, of these composite materials. Typical examples of these composites include glass, graphite, boron or silicon fibers or particulates in steel or aluminum matrices. However, when composite structures subjected to high loads undergo large localized deformations, reinforcements can adversely affect their failure properties, e.g., ductility or fracture toughness. Voids may nucleate at or near the particle-matrix interface by particle rupture, matrix fissuration or debonding, grow and finally coalesce to cause failure in the structure. Such detrimental behavior has spurred critical examination of the mechanical properties and deformation characteristics of these composites, resulting in a number of deformation models emphasizing failure mechanisms.

A number of computational models use the homogenization theory for reflecting the

Correspondence to: Dr. Somnath Ghosh, Department of Engineering Mechanics, The Ohio State University, Columbus, OH 43210, USA.

0045-7825/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved

212 S. Ghosh, S.N. Mukhopadhyay. A material based finite element analysis

influence of material microstructure on macroscopic behavior [1, 2]. These methods are predominantly based on asymptotic analyses with assumptions on periodic repetition of microstructures. However, in practice, there exists a wide variety of composite samples with a stochastic dispersion of heterogeneities. These consist of particles or fibers of various shapes and sizes that are randomly scattered or clustered in a matrix. The distribution of shapes, sizes and the spatial coordinates of the second phase has a profound influence on the mechanical behavior of the overall structure and should be considered in their analysis.

It is therefore necessary to accurately characterize a heterogeneous medium with respect to distribution of shapes, sizes, locations and material properties of the constituent phases. Material characterization of composites has been achieved in several ways that include substitution of the actual complex structures by simplified geometrical models. Among these, use of spherical inclusions, correlation functions of material properties and random set representations are noteworthy. Rigorous quantitative characterizations of second phase dispersions have been performed by Wray et al. [3] and by Spitzig et al. [4]. Using a scanning electron microscope, they have conducted automatic image analysis to record the size, morphology and centroidal coordinates of the second phase in material samples. Distributions of near and nearest neighboring particles and local volume fractions have been achieved through Dirichlet tesselations of plane sections. These articles indicate that computer generated models can project material behavior with sufficient accuracy once the particle density, mean size and standard deviations of real materials have been established.

Motivation for the present work is derived from techniques used in [3, 4]. A new finite element method that uses a microstructure based computational mesh, generated by Dirichlet tessellation of the domain, is developed in this paper. Using this method, information obtained from SEM generated automatic image analysis of a simple material section can be directly processed for computer modeling. Once a sample composite microstructure has been digitized, an automatic mesh generator can be implemented to discretize the domain based on its morphology, The special finite element technique can then be directly implemented to perform stress and deformation analysis. It should be emphasized that the present modeling does not account for macro-micro scale interactions but considers one scale problems only. This implies that either microstructural samples or macrostructural components with low volume fraction of reinforcements can be analyzed by this method.

This paper det:~h the development of a two-dimensional finite element method in three separate sections.

(a) A tessellation based mesh generator is developed in Section 2 to adequately account for the presence of the randomly dispersed second phase. The resulting mesh consists of a network of Voronoi polygons with each polygon containing only one second phase inclusion at most. The number of sides in these polygons can vary with six being the most probable.

(b) In Section 3, the network of Voronoi polygons is treated as a finite element mesh with each polygon serving as an element in the analysis. Element formulation is conducted based on the assumed stress hybrid finite element method introduced by Pian [6], to incorporate elements with a variable number of nodes. Numerical tests are performed to establish the validity of this formulation.

(c) Lastly, a transformation strain method is introduced in Section 4 to evaluate effective material properties in elements containing second phase inclusions. The performance of the proposed model is checked by comparison with the results of a displacement based model.

S. Ghosh, S.N. Mukhopadhyay, A material based finite element analysis 213

2. Mesh generation

Generation of a robust mesh that will adequately account for the morphology of a heterogeneous domain is a fundamental requirement in the development of finite element models for their analysis. Most of the conventional mesh generators that use coordinate transformation methods, discrete transfinite mapping methods, drag mesh methods or quad- tree/octree approaches among others, may not provide adequate means for automatic discretization of such domains. The limitations of these conventional generators arise out of morphological incompatibility of the generated mesh with the physical domain. For example, a high probability exists that element boundaries of the generated mesh will interfere with the inclusions and intersect them into two or more segments. This may lead to unsurmountable computational and bookkeeping difficulties in the analysis phase due to the fact that the volume fraction and shape of each inclusion in the sharing elements have to be recorded. Additionally, the number of inclusions contained within each element is likely to vary significantly, depending upon the local distribution density. Computational algorithms have to account for this, prior to the evaluation of effective element properties.

Delaunay triangulation methods, based on triangulating a set of points in n-space to produce n-dimensional simplexes, have been recently introduced in automatic grid generation. This method requires locating an initial set of points in space and thereby generating a triangulation that satisfies Delaunay properties, The method has been thoroughly explored for arbitrary two- and three-dimensional continuous domains by Schroeder and Shepherd [7, 8] and for non-convex planar domains by Lo [9]. Although the Delaunay triangulation may not be directly advantageous in modeling composite domains, its geometric dual, the Dirichlet tesselation seems to provide an excellent foundation for natural evolution of the discretization process from the microstructure. In this paper, a two-dimensional mesh generator is developed on the basis of Dirichlet tessellation of a domain to yield a network of Dirichlet cells or 'Voronoi polygons'. This is a method of subdividing an Euclidean space into n.dimensional bounded convex polytopes. It may be perceived as the production of a network of interfaces formed by the impingement of expanding hyperspheres about random nuclei that are growing at a uniform rate from zero. If second phase inclusions are realized to be points in space, the convex polytopes (polygons in two dimensions) resulting from this discretization would encompass one inclusion each, at most.

2.1. Dirichlet tesselations in a plane

Dirichlet tessellation is defined as the subdivision of a plane, determined by a set of points such that each point has associated with it a region of the plane that is closest to it than to any other. Mathematically speaking, let Pl(xl), P2(x2), . . . , P,,(x,,) be a set of n distinct random points in plane. Then the interior of the Voronoi polygon associated with the ith labeled point P, is the region D~ defined as

D , - < xjl, i) . (i)

The aggregate of all such regions D i constitute the Dirichlet tessellation in the plane. Each region may be perceived as the intersection of open half planes bounded by the perpendicular

214 S. Ghosh, S.N. Mukhopadhyay, A material based finite element analysis

bisectors of lines joining the point Pi with each of its neighbors Pj. It is easy to see that this property renders the Voronoi polygons convex. The boundary segment B o is common to the polygons for P~ and Pj and nearer to them than to any other labeled points in the plane, i.e.

Bq = {x: Ix - xil = Ix - xjl < Ix - xtl, V l ~ i, j } . (2)

Also the vertex Vqk of the polygons is equidistant from three generating points Pi, P~, Pk as

"- {X: IX-- X,I -- Ix - - xjI -- Ix- - XkI < Ix- - x,I, VI i, j , k } . (3)

The Voronoi polygons that share boundary segments are called contiguous elements or polygons and the corresponding generating points are called contiguous points.

The tessellation properties above are for an unbounded region. However, for most practical simulations, a finite domain is of interest and the domain boundaries should be taken into consideration in the discretization process. Generation of tessellation within a restricted 'window' has been discussed in [10]. In this paper they define tessellation for a finite number of points P t , / ' 2 , . . . , P,, inside a window W to be made up of regions or tiles given as

D,= (x W: Ix- x,I < Ix- xjl, i, Pj W}. (4)

The labeled points considered in their algorithm are all assumed to lie within the window. The edges of the window may be represented by effective linear inequality constraints, which are used in unambiguous tile creation near the domain periphery.

Several alternate algorithms have been suggested in the literature to delineate Voronoi polygons in a plane. The algorithm of Rhynsburger [11] proceeds by tracing a pathway in the same rotational direction for all polygons. Green and Sibson [10] have presented a more efficient algorithm in which the Voronoi diagram of n points can be computed in O(n log n) time by tracing boundary adjustments as a new polygon is fitted into a previously generated set. Other methods have been presented by Bower [12], Watson [13] and Boots and Murdoch [14]. The present mesh generator has been structured on the algorithm proposed by Green and Sibson [10].

2.2. Mesh generator algorithm

A two-dimensional mesh generator is devised for plane sections of mdltiphase materials which are assumed to consist of unidirectional fibers or particulates. Based on information regarding the boundary of the domain, locations, shapes and sizes of the inclusions, discretization takes place automatically. Following the representation of the domain boundary, the tessellation algorithms create convex elements within the entire region. The effects of nonconvex regions, multiply connected domains, and shapes and sizes of inclusions are taken into account and these have been detailed in [5],

The coordinates of a set of points are first generated by a random number generator in a convex domain with a point density equal to that observed in a given material. These forming points are called objects. The domain boundary generated by the processor is represented as a union of simple closed loops of straight line segments. These segments intersect at nodal


points that are designated as the corners of the domain. The tessellation algorithm begins with a recursive method for establishing a contiguity list for each of the objects as outlined below.

(1) Starting with a corner point, a search is conducted to record the coordinates of the object that is closest to it. In Fig. 1, E is such an object, adjacent to the comer V~.

(2) The domain boundary at the comer is indicated with dotted fines in Fig. 1 as V~S l and V~S 2. To construct an effective representation of the boundary constraints at the comer, two virtual or image points H and A are introduced. These points are the reflections of the object E about the boundary segments V~Si and V~S 2, respectively, and they project the effect of the window edges in the tessellation procedure. Representative virtual points like H and A start the contiguity list for the comer object E. This procedure of replacing boundary constraints by virtual points is continued for all corner objects.

(3) The contiguity list for each corner object of the entire domain is then completed. By virtue of reflection properties, the boundary corner V~ becomes the first vortex of the tile belonging to object E, i.e., I,'1 is the center of a circle passing through E and the first two points on its contiguity list H and A. The search for the next contiguous object continues in the closed half plane on one side of the infinitely extended line joining the object E and the last point entered in its contiguity list. In this operation, the center V 2 of the circle with points E, A and each candidate contiguous point is derived. It is evident that this search process could lead to expensive computations. Hence, a sorting routine for locating N closest candidate points in the search process may be employed to reduce the computational effort. The object B, for which the distance between the vertex V~ and the corresponding center V 2 is smallest, becomes the next point in the contiguity list for E. Simultaneously, E and A begin the contiguity list for B if, thus far, it is empty. Addition of objects on the contiguity list for E proceeds counterclockwise in this manner until the first and last points on the contiguity list become

H (Image pc • of S )

Do,-,,iq Boundary

Yl " " " "Sl ,G

- - - ..,m~"~-"~'plsme • F of search

D (Image pc

orB)

's 2 Fig. 1. Preparing the contiguity list.

C•t?•uhy List for point E

2, A 3. B (complete) 4.1= $.G

C~I ?~guitY List for poim B

2. A (parda]) 3. 4.


identical. A list of coordinates of the corresponding centers of the circles that determine the contiguous points should be recorded simultaneously as the contiguity list is updated.

(4) Once the contiguity list for the corner objects have been completed, other objects near the boundary (e.g., B) are considered, starting from those that are adjacent to the corner objects. It is expected that the tiles of these boundary objects will have edges coincident with segments of the domain boundary. At this stage, the first two points in the contiguity list for these objects have already been entered from the prior search procedure. Addition to the contiguity list begins by effective replacement of the boundary constraint. The reflected image D is therefore designated as the third contiguous point of B. A natural degeneracy could evolve in this case due to the fact that more than three points (B, E, A and D) lie on the circumference of a circle. However, this does not affect the discretization process as the virtual points lie outside the domain under consideration. The contiguity list is then completed for all boundary points in a way identical to the corner objects. Also the coordinate table for the centers of the corresponding circles is maintained.

(5) The same procedure is now implemented for establishing contiguity lists for all interior objects until the entire domain is exhausted.

(6) For each object, Voronoi polygons or Dirichlet cells are created by joining the centers of the circles (vertices) that generate the contiguity list, in the same order as they are enlisted.

The generation of a well behaved mesh for heterogeneous media should give consideration to a number of additional aspects. For example, special techniques should be incorporated for non-convex boundaries and multiply connected domains. Also, the second phase inclusions can seldom be mmiifested by points and therefore the mesh generator should account for their shapes and sizes. A systematic approach with these considerations has been discussed in [5].

2.3. A numerical example

An example of discretizing a gear toot section made of random particulate/fiber composites is presented. In reality, the sizes of the particulates are microscopic and their number per unit volume is very large. However, for the purpose of demonstrating the ability of the mesh generator, the volume fraction of inclusions in the matrix assumed to be very low. The dimensions of the discretized section are

Tip diameter: 0.476 m,

Root diameter: 0.350 m,

Circular pitch: 0.088 m,

Tip width: 0.030 m.

The section contains 239 randomly distributed particulates with randomly varying radii between 0.1 mm and 1.2 mm. The locations and radii of the circular I~articulates have been generated by a random number generator using a uniform distribution function. Figure 2 shows the discretized domain for the gear tooth section. Histograms for the frequency distribution of the local area fraction corresponding to each element and the mean near neighbor distance are presented in Figs. 3(a) and 3(b) characterizing the material constitution.

S. Ghosh, S.N, Mukhopadhyay, A material based finite element analysis 217

=.x

Fig. 2. Discretization of a gear tooth.

3. Finite element formulation with Voronoi polygons

Conventional two-dimensional finite element analyses commonly use elements that are shaped as triangles or quadrilaterals. Typical element formulation procedures include adop- tion of three or four term polynomial functions of position coordinates for interpolation of the dependent variable, e.g., displacement or velocity. The requirement that the nodal values of the dependent variable uniquely determine the coefficients of its approximating polynomial is then imposed. With any assumed choice of terms in the approximating polynomial, this condition may not be satisfied for Voronoi cell elements of arbitrary shapes, having n corner nodes (n > 4). This is because, although n linear equations are obtained for the n polynomial coefficients, the equations may not be independent. This would occur whenever an element's nodes lie on any curve of the form p(x, y) = 0, where p(x, y) is the type of approximation polynomial. In addition, the polynomials should be chosen to insure rotational invariance of the element and satisfaction of interelement displacement compatibility. If the first condition is satisfied by including symmetric terms in the polynomial, so that no preferred coordinate direction occurs, the number of coefficients may not match the number of nodes in an element. Also, interelement compatibility cannot be satisfied unless the values of p(x, y) along each side of an element depend only on its values at the nodes on that side. A conventional


0.18

0.16

0.14

0.12 i io., "; 0.08

0.06 ! m 0.04

0.02

!

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Pmiculam to PolylPm Anm

0.14

0.1:

0.]

0,041-

0,02

0.5 ! 1.5 2 2.3

Me~n Near Nei0zl)mr l:dmnce

Fig. 3, (a) Area fraction distribution; (b) mean near neighbor distance distribution for a gear tooth.

formulation can also lead to rank deficiencies in the stiffness matrices. These difficulties have triggered the search for alternative methods to establish element formulation. Pian [6] introduced an assumed stress hybrid finite element method for four node quadrilateral elements. This formulation has been extended in the present paper for polygonal elements.


3.1. Assumed stress hybrid method for Voronoi polygons

In Pian's paper [6] with four node quadrilateral elements, element stiffness matrices are derived by assuming compatible displacement fields along interelement boundaries and a stress distribution in the interior of each element. In this method, assumptions on displacement fields are made on the element boundary (as opposed to the entire element) to satisfy compatibility with neighboring elements, while stress polynomials are chosen to satisfy equilibrium within the element. The element formulation is based on the principle of minimum complementary energy. As this method necessitates the interpolation of the displacement field only along the element boundary, it becomes easy to handle n-sided polygonal elements having a varying number of nodes. Tong and Pian [15] have proved the convergence of this method for four-node quadrilateral elements.

3. I. 1. Element formulation In the principle of minimum complementary energy, the functional to be varied is given as

fD L Tiu ids (5)

where S~kt are components of the elastic compliance tensor, ~2~ are the prescribed displacements along the boundary 0n and T~ are the boundary tractions. Stress components ¢r~s are assumed to satisfy the equilibrium condition

0% 0x--7. + fi =0 (6)

and be compatible with prescribed boundary tractions. In applying the finite element method, the assumed stress field need not be continuous across the interelement boundaries, but equilibrium must be maintained with surface tractions T~ defined by

T i = oronj, (7)

where n s are components of the unit vector normal to the boundary. In the application of this variational principle, the equilibrating stress field is expressed as a

polynomial in the interior of an element as

(8)

where o" is a column vector of stress components ({ ¢r x, (ry, ~'xy } in two-dimensional analysis), jO is a column of m undetermined stress coefficients/31,/32,...,/3m and P is an m x m matrix containing functions of coordinates x, y corresponding to the chosen polynomial. From (7) and (8), the surface tractions T~ can be expressed in terms of the undetermined stress coefficients/3 as

T = R ~ . (9)

220 S. Ghosh, S, N. Mukhopadhyay. A material based finite element analysis

The prescribed boundary displacements u~ can be interpolated from the generalized displacement d at the nodes, in the form

(lO)

where the elements of the matrix L are functions of the boundary coordinates. Rewriting (5) in matrix form and substituting the expressions for o', T and u, the functional becomes

~ = ½ 1 3 t i l l 3 - f l 'Gd , (11)

where H = J'n P'SP dv and G = Y,~n RtL ds. Stationarity of ~ with respect to stress coefficients, i.e.,

gives

Ofl~ =0 f o r i = l , 2 , . . . , m , (12)

HI3 = Gd or 13 = H-tGd. (13)

Substitution of/~ in the expression for the complementary strain energy for an element with volume £J¢ yields

where U= fn ~.o"$o'dv- ½fl'HO = ½d'GtH-tGd = ~d'Kd,

¢

K ~- G'H~IG

(14)

(15)

is the element stiffness matrix. Plan and Chen [16, 17] have shown that the assumed stress hybrid finite element formula-

tion can also be derived from generalized variational principles such as the Hu-Washizu principle or the Hellinger-Reissner principle.

The stiffness matrix g will be rank deficient if its rank is less than n - I, where n is the number of degrees of freedom and I is the number of rigid body modes. From the construction of the stiffness matrix K, it can be inferred that the necessary condition for it to have sufficient rank is m ~> n - l, where m is the number of independent I3-stress coefficients. A judicious choice of m is therefore necessary from an accuracy and efficiency point of view. Evaluation of K involves inversion of the (m × m) H matrix, and it is advantageous to use a minimum number of/3-parameters for improved efficiency. There are also indications that an increased number of ~.parameters leads to stiff elements. Although it is best to have m equal to n - l, this is not always possible from considerations of geometric isotropy.

Assuming variable degree polynomials for Airy's stress function in plane problems, the matrix P can be readily obtained. For example, stresses resulting from third order polynomials with no body forces can be delineated as

T~y - 0 y x O =Pl3.

0 0 0 1 7 (16)


In this case, there are 7/]-terms in the stress polynomial, i.e., m is equal to 7. Since the number of rigid body modes in the two-dimensional problem ! is equal to 3, (16) can be used for elements having up to 5 nodes. For elements with a higher number of nodes, it is necessary to increase the number of/]- terms to obtain rank sufficiency of the stiffness matrix. The matrix P corresponding to 12/]-terms, is used for elements for which (5 < 0.5n ~< 7), and is given as

i Y 0 0 0 X 0 y2 0 X 2 Xy 0 ] 0 1 x 0 0 y 0 X 2 y2 0 xy J 0 0 0 1 - y - x 0 0 - 2 x y - y ~ 2 - x ~ 2

and that corresponding to 18/] terms (7 < 0.5n ~< 10) is given as

i y 0 0 0 x 0 y2 0 x 2 x 3 y'~ "] xy 0 3x3y xy" 0 0 0 1 x 0 0 y 0 x 2 y2 0 xy 3xy 2 0 y3 0 x2y 0 0 0 1 - y - x 0 0 -2xy -y2/2 --X2/2 -3x"y 0 - 3 x y 2 -y3/3 -x3/3

Statistically, the likelihood of the occurrence of elements with more than 10 nodes is very small and hence higher order polynomials have been disregarded in this work. It should be noted that the assumed stresses are complete polynomials of the same degree and hence the resulting stiffness matrix is rotationally invariant. The present work differs from the polynomials that were proposed by Pian [6] that are discussed later in this section.

Continuing with the FEM formulation, the displacements on an element boundary are interpolated by using a linear function. For the ith side of an element

I"} [, ,,, o u : u, : L d = - 0 a l _ a/ l l

f d2i- | ) o/t, o lid', t

a/l~J d,1+1 ' o [.:,+,j (17)

where i~ is the length of side i, a is the distance of a boundary point from the ith node and d corresponds to the generalized displacements at nodal points. Traction on side i is given by a combination of (7) and (9) for the 7 0-terms as

"i2 X'/2 "il --Y/~il (--X'il +yni2)Jl~ 17

from which the matrix G~ is evaluated by integration along the boundary segments as G~ = fos~, Ri L ds for side i. The contribution to the element level matrix Gta for side i is then established from the connectivity of the nodes.

As opposed to the complete stress polynomials for the construction of P, Plan [6] assumed 5 /]-terms, in accordance with the requirement of rank sufficiency of four node quadrilateral elements (m = n - l). The corresponding polynomial matrix is given by the first five columns of the P matrix in (16). From inspection, it is apparent that the assumed stresses in the global coordinate system are not complete polynomials of the same degree and hence the element


stiffness matrix is not rotationally invariant. Pian and Chen [16] have extended the original hybrid stress formulation to one based on the Hellinger-Reissner principle. By introducing equilibrium as a constraint condition with internal displacements as Lagrange multipliers, the stress terms were expressed as complete polynomials of isoparametric coordinates. This accommodates elements of distorted geometries and preserves the invadance of the resulting stiffness matrix. Such an approach was taken by Pian and Sumihara [18] and Plan and Wu [19] for deriving the assumed stress in terms of natural coordinates and element geometry. However, such a formulation, based on the Hellinger-Reissner principle, necessitates interpolation of the displacement field in the interior of the element from the nodal values. This poses a problem for the Voronoi polygons in the present work and consequently has not been considered.

Cook [20] has suggested an alternative method to achieve element invariance without changing the stress polynomial. By calculating the element stiffness in a local element coordinate system, invariance in element response has been achieved. Upon establishment of a local coordinate system, calculation of the stiffness proceeds in the following manner.

(1) Nodal coordinates of an element are transferred to the local system and the stiffness matrix is calculated in the local system.

(2) The element stiffness matrix in the local system is then transformed to the global coordinate system and assembled in the global stiffness matrix. The transformation is accomplished by the operation

Kt~loha I = AIKIocalA , (19)

where A is the coordinate transformation matrix. (3) Stresses are computed in the local system as a',o~,,, -- P, oJ~o¢,,, and are then trans-

formed to the global coordinate system. In this paper, numerical results obtained from stress interpolation with 7, 12 and 18 ~-terms

have been compared with that suggested by Plan for 5, 10 and 18 E-terms. The transformation technique suggested by Cook has been implemented for rotational invarianee in the latter case. The choice of finding the angle the local system makes with the global system is not unique and alternative definitions of this angle are possible. Two choices are experimented with in this comparison analysis. In the first method, one of the axes of the local coordinate system is aligned with the longest edge of the element. In the second approach, the element is mapped onto a master plane by isoparametric transformation. The Jacobian matrix of the transformation is evaluated at the centroid and an [R][U] decomposition of [J] is performed. The global system is rotated by JR] to obtain the local coordinate axes.

3.2. Numerical examples

Computer codes are developed for analysis of plane strain problems with the assumed stress hybrid finite element method. Analytical results for some simple problems are given in this section and compared with those generated by the programs for complete stress polynomials (7, 12, 18/3-terms) and incomplete stress polynomials (5, 10, 18/3-terms) in conjunction with coordinate transformation for rotational invariance. Material properties for an isotropic elastic body are given as: Young's modulus (E), 2 kN/mm2; Poisson's ratio (u), 0.3.


I:

" J / \ " / . w " ~ ~

\ \ ,,

' / H

DX

Fig. 4. Square body subjected to tensile load.

3.2.1. Uniform tension problem A square body of size 40 mm 2 is discretized into 24 Voronoi polygons as shown in Fig. 4 and

is subjected to a uniformly applied tensile force of magnitude 10 kN. Both complete and incomplete polynomial, assumed stress hybrid methods yield exact end deflection of 4.55 mm in the X direction and a uniform normal stress o-~ of 0.25 kN/mm 2 in all the elements. A plot of the displacement of the nodal points in the X direction as a function of the position from the fixed end in Fig. 5 shows the expected linearity. To verify the effect of rotational variation, the body is now rotated by 45 ° to the global axes and is subjected to the, same tensile load. As

0.5

0,4

02

0.I.1

0.!

0.05

Nodal Imsiti~ in X dlmcti~

Fig. 5. Nodal displacement in X direction versus nodal position.

224 S. Ghosh, S.N. Mukhopadhyay, A material based finite element attalysis

expected, the results of deflection and stress in the direction of the load are again found to be the exact value for both methods due to the absence of shearing stresses in this problem.

3.2.2. Simple shear problem A simple shear deformation field is applied to the 40 x 40 mm 2 body discretized with 24

Voronoi polygons shown in Fig. 6. Both approaches of the assumed stress hybrid method yield an exact value of "rxy = 0.0385 kN/mm 2 and orx = or = 0 in every element. The shear stress distribution along a section A - A is plotted in Fig. 7.

3.2.3. Beam bending problem A 40 x 160 mm 2 cantilever beam, with one end fixed and the other end subjected to a pure

moment of 4 kN mm is analyzed. The beam is discretized into 28 Voronoi polygon elements as depicted in Fig. 8(a). The bending stress distribution or,. along a transverse section (Fig. 9), shows that both complete and incomplete polynomials yield reasonably good results. The discontinuities in the plots correspond to the stress jumps across the element boundary.

To study the convergence characteristics of the assumed stress model, various mesh refinements have been conducted. Comparison of the results for the same beam with 28 (Fig. 8(a)), 53 (Fig. 8(b)) and 152 (Fig. 8(c)) Voronoi polygon elements are conducted with the program for complete polynomials. Variation of bending stress ~, along the same transverse section is shown in Fig. l0 and the vertical displacements of the top surface are plotted in Fig. I I. These plots conclude the convergence of the results to the analytical solution with more refined meshes. The computed tip deflection at the free end with 152 elements is 4.35 mm, while the analytically calculated value is 4.368 mm corresponding to a difference of 0.412%. For 28 elements the tip deflection is 4.21 mm corresponding to an error of 3.42%.

To study the effect of rotational invariance, the beam with 53 elements is now rotated by 45 ° from the global axes system as illustrated in Fig. 12. A plot of the results obtained by various schemes is given in Fig. 13. The incomplete polynomial hybrid method in the global

¥

A

!

!

A

i i

I ! !

4 I !

! ! ' |

Fig. 6. Body subjected to simple shear (24 elements).

S. G h o s h , S . N . M , k h o p a d h y a y , A m a t e r i a l b a s e d f i n i t e e l e m e n t unalysis 225

1.00E~1 ~

L u . a ]

ca]

9 0 0 E ~ .

I k O 0 ~ . . . . I ' ' ' ' I . . . . I '

~ l ~ E t N &O~E~O0 1 .iaE*01 ?.40E*01

LEG~-N'D

n H y l ~ U h o ~

o ~ r e t k ~

_ _ :3

I I I [ I I I

3 . 2 0 E , 1 , 0 1

' I

4 .OQE, I .O ' I

Distan~ along Y direction (ram)

Fig. 7, Shear stress across section A-A (24 elements),

coordinate system seems to behave poorly especially near the edges. This is remedied by using the transformation methods using a local coordinate system. However, the global hybrid method with complete polynomials gives very good results even without any coordinate transformation.

To understand the sen~;itivity of numerical results to the shape of the elements, the beam is now discretized into 86 regular hexagonal elements as shown in Fig. 14. Figure 15 indicates that the results of both the complete and the incomplete global polynomial hybrid stress methods agree very well with the analytical values of bending stress. Figure 16 shows the distribution of the bending stress o" x in the entire beam. Except near the load application points, where the Saint Venant effect is predominant, the bending stress distribution conforms well with the theoretical stress distribution. Figure 17 shows the results when the beam is at 45 ° to the global axis system. The complete polynomial assumed stress method performs significantly better than the incomplete method. However, results in the latter case can be significantly improved by introducing a local coordinate system• These results establish a marked improvement in the quality of the solutions as the elements become more regular in shape.

As a final example in this section, the 40 x 160 mm 2 cantilever beam with re~3ular 86 elements, is subjected to a total downward end load of 0.05 kN distributed over the edge in a parabolic fashion as shown in Fig. 18. The shear across a transverse section is plotted in Fig.

226 S. Ghosh, S.N. Mukhopadhyay, A material based J~nite element analysis

¥

I Ca)

Cb)

Fig. 8, Beam subjected to pure end moment: (a) 28 elements, (b) 53 elements and (c) 152 elements.

19 and the bending stress ~, is plotted in Fig, 20. All methods are found to perform satisfactorily for this example.

It can be concluded from the above set of examples that the assumed stress hybrid finite element method with complete global stress polynomials or with incomplete local stress poiynomials have the capability to accurately model a domain that is discretized by tessellation techniques into Voronoi polygons,

S. Ghosh, S.N. l~L,!chopadhyay, A material based finite element analysis 227

1..50C--~ ..~ ".%

O.OOE,~

~ - , 5 .00B .M . |

o ~oommte rmlymm.

A ~m~e~Jvs~o

"1 ""

° ,

o ~ e . o o u o z , o o 1 . m u ~ . ~ 4 o e ~ u o a , o , 4 ,ooe .o ,

Distm~ aloali Y d~tioa (ram)

Fig. 9. Bending stress along a transverse section at x = 37 mm of the beam with 28 elements.

4. Modeling the effect of the second phase in Voronoi elements

In this section, the effect of the second phase materials in the matrix is incorporated into the element stiffness for determining the load response of heterogeneous media. The method implemented in this analysis is based on the introduction of a transformation strain in the regions of material discontinuity. The basic idea of transformation or eigen strain was first conceptualized by Eshelby [21]. Analytical formulations using this concept have resulted in methods like the self-consistent method and variational principles in composites. The present work focusses on the development of a numerical scheme based on this idea. It follows a finite element formulation for material discontinuities proposed by Accorsi [22, 23]. The formulation of Accorsi [22, 23] has been extended in this paper for domains of linear elastic materials, discretized into Voronoi elements.

4.1. Problem formula t ion

The proposed formulation subdivides the actual problem into two parts. They are (i) a homogeneous problem, corresponding to a single phase body constituted of the matrix


~ ~. C: 182 Elements

, . ~ o o w ° ~ ° ° . . . . o . . . ° . ~ . . ° . o ° . ° . ° ° ° . . ° P , o , ° ° ° - , - - - - ° ° ° , o °

- L O t . m

. l . s m . m . . . . i . . . . i . . . . ] ' ' ' ' i . . . . 1-

u o E , o o LOOE440 1 . a E , 0 1 & 4 O l ~ 0 1 ~ 1 0 1 1 ~ 1 4 .00E~41

Dilamm ~0.II Y dimcdoa (ram)

Fig. 10. Bending stress along a transverse section of the beam (complete polynomhls).

material, and (ii) a deviation problem, defined by the difference between the actual and the homogeneous field quantities. Let superscripts a, h and d denote quantities in the actual, homogeneous and deviation problems, respectively. The constitutive relation for the actual material is given as

o : -- ~ ( x ) : e ' , ( 2 0 )

where £(x) is the position dependent elasticity matrix of the material. Defining the trans-. formation strain by the equation

o'" = E(x ) : e" = Eh: (e" - e " ) , (21)

where E h is the elasticity matrix of homogeneous material and e* is the transformation strain in the discontinuity, one obtains

e" = [E h - E(x)]-t: gh: e* = C(x): Eh: e*, (22)


4.3"1E~00_

3.0E40: .

• 2.~E~O0.

&TaE-~ _

L ~ c 4 a T

(kmE~O0 ~ L4OE~M1 9 . M ~ 1 .aP .¢~ Distance along X din~doa (ram)

Fig. l i. Vertical deflection of top surface of the beam by complete polynomials.

/ ° - - -

J A I N

I

I . N

---I ' I

Fig. 12. Beam oriented at 45 ° and subjected to end moment (53 elements).


! |

i

I . S 4 E . ~ .

9 , ~ l E . m .

S . I l I l i ~ .

.F.mE.m

. I ~ , M .

LEGEND

o Glolml(complete polynom)

o I.oc~slde alUgned)

GlobalOncomplete polynom)

x Anoq4~l

v I.OC~RU deCOmlXm#lon)

. & . , . • . A

0.001~O0 &00g~40 1.110leVI L401E~I S~101h41 4.00|,01

lYamm ~ z uu~vmm dlm:doa (am)

Fig. 13, Bending stress along a transverse section (76 mm from fixed end) of the beam at 45 ~ using Ioeal and global coordinate methods with 53 elements,

b Y

Fig. 14. Beam with hexagonal elements subjected to pure end moment (86 elements).

S. Ghosh, S .N. Mukhopadhyay, A material based finite element analysis 231

| ~ .,q~E.O$ .

|

4 ~ g . m .

LEGEND

a e , , m p a ~ ~

o ~eemmm putmmamm

A 11~mmUem Valm

~ & ~ 1.aE¢~1 Ldmh41 ~

Dimam along Y dimct/Qa (ram)

Fig. 15. Bending stress along a transverse section at x ffi 19.32 mm of the beam with 86 elements.

where C(x)ffi [E h - £(x)] =1. Since the total strain is obtained by adding homogeneous and deviation strains as

E a ~ e h ~ e d ,

(22) takes the form

£h "l- e d -- C(X): E h : £ * . ( 2 3 )

Equation (23) is known as the consistency conuition. The constitutive relation for the deviation problem now becomes

Or d -- Or' -- Or h ---- E h : (£d -- £ , ) . ( 2 4 )

Each of the three problems consists of their respective equilibrium, kinematic and constitutive equations together with specified boundary conditions. Since the total problem is obtained by adding the homogeneous and deviation parts, it is assumed that the homogeneous problem


Max. -k_ ~ 3.636E+00

2.182E+(X)

7.272E-01

-7.272E-01

-2.182B+00

Mt.../*" .3,6s +0o

Fig. 16, Distribution of bending stress in the beam with 86 elements,

has the same displacement and traction boundary conditions as the total problem while these for the deviation problem are zero. The finite element implementation then takes place by first considering the homogeneous problem. The assembled equations for the homogeneous problem are written as

ghD h ~ F h , (25)

where K h, D h, F h are the global stiffness matrix, global displacement vector and global load vector, respectively. The homogeneous displacement vector is solved from the equation

D h h ~ l h = (K, , , ) F , , , , (26)

where the matrices K~, F~ are K h, F h modified for displacement boundary conditions. Next, the element equation for the discretized deviation problem is written as

Khd e= F*~ , whereF~* = fn~ B*E~e* d r , (27)


i .qoE--.m ._~. ~.

,l~E,m

o . o s , w 8 . ~ , o e l ~ t , m ~.4or,,m ~

O~tao~ ~ m~varm ¢ l ~ o a (w,)

Fig. 17. Bending stresses in the beam at 45 ° using local and global cooldinate methods with 86 hexagonal elements (x ~ 19.4 mm).

i

D% Fig. 18. Cantilever beam with distributed end load (86 elements).


.1,1(18~.

-1,g4E,4S.

.1.0U40,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X

/

o OIomKmmpiete polynom)

o I.mmsOncompkno polynom)

A OlomiOnmmpioto po~nom)

x A n e W !

/ A [ °.

r•X/I /

0,001~00 8,Q0l!4~) 1JlOl~t01 LdUNE¢,01 ~

Disuwe ~ Y dim:don (ram)

Fig. 19. Shear stress distribution along a transverse section at x ~ 45.3 mm of the beam with 86 elements.

B is the strain displacement matrix and n, refers to the element volume. Based on the development of the tessellated model, it is reasonable to assume that each elemeut consists of

~ingle inclusion occupying a region n, such that n, C ~¢. The elasticity tensor for the second phase is denoted by £r and the transformation strain is assumed to be non-zero only inside n, . The element load vector for the deviation problem may then be rewritten as

* -- fo B'£~ 8* dr. (28) Fe r

Assembly of (27) with the zero boundary conditions yields

h e * (29) K,,D = F,, .

The right-hand side of (29) contains the transformation strain e*, which is unknown at this stage. It is therefore imperative to assume a variation for the transformation strain and also to use the r: nsistency condition (23) in order to solve (29). In this work, two cases have been examineu, namely (a) Constant transformation strain in the second phase (b) Linear transformation strain in the second phase.


I 1.,q~-.4N. el ~ .°°* ,/

~ , , 4 d ~ . ~ . ""

, ~ o ~ n m m l m e p ~ / m m ) q

A A m l y ~ m l

. I u f f i m . m T ' ' ' ' I . . . . I ' ' ' ' I ' .v , , 'I . . . . I

e.om~o &ou~o i .~.+ol uw . ,m ~

Distance ,,tong Y dim:,;oa (ram)

Fig. 20, Fiexural stress along a transverse section at x = 45,3 mm of the beam with 86 elements.

4.1. I. Constant transformation strain If e* is constant within a discontinuity, the load vector in (28) reduces to

where

r t h , r * - ( R o ) B o e , F~ (30)

Rr - f n B d r . (31 ) r

Also, the consistency condition (23) for an element may be written in terms of generalized nodal displacements as

Bd h + Bd a = C(x)E~ e* . (32)

The consistency equation has been assumed to be satisfied only in an average sense within each element in this work. Thus, integrating (32) o v e r ~"~r yields the averaged consistency condition

, h , (33) Rre d h + Rred d = VrCeE~e**

236 S. Ghosh, S.N. Mukhopadhyay, A material based finite element analysts

where V r is the volume of the discontinuity and Cre = [E h - E r ] -l. This equation may then be substituted in (27) to solve for e*.

4.1.2. Polynomial transformation strain If e* is expressed as a linear function of the coordinates as

or

[ , ~x * = 0

~xy

• , 0 0 0 0 0 1 0 0 1 x y 0 0 i 0 0 0 0 0 1 x y J l e ; J

e* ffi P ' e* , (34)

the load vector in (28) becomes

where

F*~ = fa BtEhP*e * dv ffi ( s ; ) t e *r , r

(35)

R'° ffi fa P*tEhB dv . ( 3 6 ) g

In this case, the averaged consistency condition can be obtained by premuitiplying both sides of (32) by p . t£ , and integrating over/~ as

Rr~,d h + Rr,,d d = Q~e *r , w h e r e Q~ = f . P*tEhC(x)Ehp*e* du . g

(37)

An approach similar to that for constant strain is now followed to compute the coefficients e* of the strain polynomial, which are then used to evaluate the displacement of the deviation problem. From (31) and (36), it is obvious that e*', which is a function of R'~, is dependent upon the shape and size of the discontinuity since the integration is over/~ and the location of the discontinuity within the element is contained in the B matrix. In the numerical examples conducted, however, no significant differences were observed in the solutions for these two cases. The reason may be attributed to the otherwise low order interpolation functions that are chosen to construct the element stiffness equations.

4.1.3. Hybrid method with two.phase materials for Voronoi elements The method of analysis for two-phase materials (henceforth called the homogenization

method) is now coupled with the assumed stress hybrid finite element method with Voronoi elements for modeling the response of heterogeneous media. From the assumptions on stress interpolation in (8) and the relations of stress coefficients/3 in (13), the strain in an element can be expressed in terms of nodal displacements as

r = S:rr = SPH~tGd,

where S is the elastic compliance matrix. Hence, the strain-displacement matrix takes the form

8 = SPH-It ; . (38)


Once the strain-displacement matrix B is known, the R r matrix can be easily formed from (31) or (36) and the deviation problem can be solved for D d

4.2. The solution algorithm

The solution of the total heterogeneous problem is obtained by separately solving the homogeneous and deviation problems. (a) First, the homogeneous problem is solved by the assumed stress hybrid method. The steps are - The stress polynomial P for each element is chosen, depending upon the number of nodes in

each element. - Matrices H and G are formed and the element stiffness matrix is computed using (15). - The global stiffness matrix is formed and (26) is used to obtain the global displacement

vector of the homogeneous problem. (b) The presence of second phase material is now accounted for by solving the deviation problem. The subroutine for the calculation of D O , which is the global displacement vector of the deviation problem, is defined by the following steps: - Substituting (33) or (37) in (27) and performing a global assembly results in an equation of

the form A(Ehe *) = RD t', which is then solved to obtain Ehee *. h , r , r t h , r - The element load vectors are evaluated from values of Eee as Fe ffi (R~)E~e .

The global load vector F* is now assembled and (29) is solved to obtain D d. (c) The global displacement vector of the actual problem is obtained by adding the global displacement vectors of the homogeneous and deviation problems, i.e., D a = D h + D d. Stress in the matrix material is obtained as follows: - The array of stress parameters/3 for every element is calculated by using (13) from the

obtained values of nodal displacements. - The polynomial P is evaluated at any point within an element, depending upon the number

of nodes, and the stresses % are calculated at that point from (8). Stresses in the second phase are calculated as follows: - The polynomial P is evaluated at the point where the stresses are to be calculated. The

strain displacement matrix B at the corresponding point is calculated using (38). - The total strain e' is obtained as Bd, where d is the nodal displacement vector. - The stress o "t is then calculated from the relation o "t= E(x):e'.

4.3. Numerical examples

A computer code (HYBRID) is developed for plane strain problems based on the above algorithm. In this section, a performance analysis of HYBRID is conducted by comparing its output with the results of a displacement based conventional finite element program FEMDIS [24], for a few simple problems. Although tests have been performed with different stress polynomials in global and local coordinates, as mentioned in the previous section, the results presented in this section are conducted with the complete global stress polynomials only.

4.3.1. Tension test for a plate with a hole

A 70 x 70 mm 2 square plate with a circular cutout is subjected to a pure tension load of 7 kN per unit thickness. The material properties are the same as in the problems mentioned in

'238 S. Ghosh, S.N. Mukhopadhyay, A material based finite element analysis

the previous section. For FEMDIS, the material around the hole is discretized into 61 QUAD4 elements as shown in Fig. 21(a). For HYBRID with the proposed homogenization, two meshes are considered with the hole being completely contained in the central element in both -ases. In the first case, the domain is uniformly discretized into 49 Voronoi elements shown as squares in Fig. 21(b), and 169 Voronoi elements are introduced in the

Io

Ic

Ic

I:

I: i

t,

(L)

L

L

1 I I

I I N

=Y+

I I

I I

U,I! (b)

L

i

UL

I L

. |

n n n n n u l u u n i i n mnnuunnuuunnn u u n u u n n u n n n n n m n n n u u u u u n n n n u n n u n u m n n n n n n ~ u n u n n m u n l , , - n u m u n

mnnNnn nnnnnl u u n n u n l n u u u n n l n J U U U l U m m m m u n n n l n u N u n n u m u u m m n n n l n u u n n m n n m n n n l u u n n n m n u n n n n

Co) Fig. 21, Body under tensile load discrctized for (a) FEMDIS, (b) HYBRID with coarse mesh and (¢) HYBRID with fine mesh,


second case depicted in Fig. 21(c). The material of the hole is represented by a null elasticity matrix in the homogenization process.

The normal stress distribution o" x across a section through the hole by both methods is plotted in Fig. 22. The plot shows conclusively that the results are significantly improved through mesh refinement. With the finer mesh, stresses at a distance from the hole are in very good agreement with those computed by FEMDIS. Also, singularity near the hole is predicted much better with the finer mesh in HYBRID. It can be expected that better accuracy may be achieved with further refinement, although the averaged consistency condition used in this model may hinder exact representation.

4.3.2. Beam bending problem A 150 x 90 mm 2 cantilever beam having three square inclusions with Young's modulus

4 kN/mm 2 and Poisson's ratio 0.3, is subjected to a pure moment of 20.25 kN mm per unit thickness at the end. The problem is analyzed by four different methods, namely (a) conventional FEMDIS: domain discretized into 135 with QUAD4 elements shown in Fig.

23.

i L nE-01_ | ~ UlE-Cn

.I F l l J ~

o Hvesm ( ¢ . a ~ m ¢ ~

A I¢VI~B ( l~e m o ~

. ,¢'/

SQ °'; S S °;

1.f fJ l~'~ - / " ~ ' ..A #~ o,p°' ,.,~w

_, . . . . . . .

L10 |¢~ 7.NE¢,O0 1.~E¢41 1.88E4,01 2.A41[¢41 ~OOE,0t

Distance akm$ Y dix~lioa (m,)

Fig. 22. Normal stress distribution across a section through the hole.


¥

Fig. 23. Cantilever beam under end moment discretized for FEMDIS with 135 elements.

(b) FEMDIS with homogenization: domain discretized into 15 QUAD4 elements shown in Fig. 24.

(c) HYBRID: domain discretized into 15 square elements shown in Fig. 24. (d) HYBRID: domain discretized into 30 Voronoi elements shown in Fig. 25. The bending stress distribution ~ along a transverse section passing through the second phase materials is plotted in Fig. 26 for all the cases, In the matrix material, the stresses predicted by the homogenization algorithm with FEMDIS and HYBRID are found to agree well with those by FEMDIS alone. Within the inclusion, although the stresses predicted by homogenization in

j ,

/

n

I

~ ~ ' l ~ i D ~ l g m I

Fig, 24, Cantilever beam under end moment discretized for FEMDIS and HYBRID into 15 square elements.


• i

i

Fig. 25. Cantilever beam under end moment discretized for HYBRID into 30 Voronoi polygons.

1 . . M E , 4 8 .

) 4 . 1 . . q l . 4 8 . |

i

• 4.NE-Oe .

o I ~ D i 8

o PlSiOlS(Honmgenlmtlon)

A HYBfll0(mumre eiementm)

x HYIVU0(Voronol eiemenU~

kLiB~l~a&SlllllLLl~ll4141~4$1~4~ll~J~gll eelt144~&lll l l l l l~l1641151&65111llmlll l l l l tm14411

,.... % ".

i I ; I ~ I ' ' " I ' ' ' " I . . . . 't ' ' ' ' ; ' ' I

0.me,c0 1.aON~l s.eoe,ol s.4oe,ol v ~ e 4 ~ o . o u , m

D~wce ~ozg i d~ec~ (ram)

Fig. 26. Bending stresses in the cantilever beam across a section through the second phase inclusions.


Fig. 27. Composite beam subjected to pure end momc~lt (64 elements).

cases (b) and (c) compare rather well, they are higher than those produced by FEMDIS alone (case (a)). The discrepancy is because of the much larger size of elements in examples (b) and (c). With only a little mesh refinement in case (d) for 30Voronoi polygons, the results improved significantly even in the second phase. This example shows that even with relatively low resolution in the finite element mesh, the method developed can achieve a fairly good accuracy level for composite materials.

I .llllK.Oi

| 1.0M.Om, II

= mqgtto (win kwlmm.)

o l, Pmmo Omutt tmlmm)

&gab410 &gNu40 tdmh4~ Z.~N,4n ~

Disu=ce ~oas Y ~ (ram) Fig, 28, Bending stress across a transverse section (x = 22,3 mm) of the composite beam.


Finally a 40 × 160 mm ~ cantilever beam subjected to an end moment of 4 kN mm per unit thickness is analyzed by HYBRID. The beam has 64 dispersed inclusions with the same second phase properties as mentioned above. The finite element discretization into Voronoi cells is depicted in Fig. 27. The bending stress variation or along a transverse section passing through several second phase particulates is plotted in Fig. 28 and is compared with that in a homogeneous beam made of the matrix material. The bending stress distribution for the composite beam is shown in Fig. 29.

4.3.3. Analysis of composite gear The section of a gear tooth that was tessellated in the section of mesh generation is now

analyzed by HYBRID. A load of 1 kN is applied on a single tooth along the line of contact making an angle of 30 ° with the horizontal. Figure 30 shows the loading conditions on the individual tooth. For the sake of simplicity, the gear is analyzed in a static condition and the boundary conditions are chosen to simulate bending of the tooth only. The gear material (Young's modulus 2 kN/mm 2 and Poisson's ratio 0.3) is dispersed with 88 second phase particles (Young's modulus 2.5 kN/mm" and Poisson's ratio 0.3). The radii of the particles have a mean of 2.5 mm and standard deviation of 0.25 mm. The bending and the shear stress distribution along a section A - A through the gear tooth is shown is Figs. 31 and 32, respectively, and compared with those for a homogeneous material.

~ ' " ~ - ~ ~ ~ . ~ . o o

i~,i ~ i i~ ! :~

!:Z:!:i:!:i:i:i:i~::! ¸

i!iiii~ii~iiiiiiiiii

~ ; ~ : - : ~ . . ~ ::

r..:~..N.:.:.3?~

1.5598-01

-1.3S31~00

- 2 . ~ 1 ~ ) 0

1~ . ~ . :" - 5 . r / ~

Fig. 29. Distribution of bending stress in the composite cantilever beam.


1.111E-M,

1 4kgllll,il

_--.'l

Fig. 30. Gear tooth under concentrated load.

Fig. 31,

t

..'/ 2 ) ) ) ) ) , ) ) 1 ) 1 , ) 1 , )1~)I

!

, t ,¢'/W"

#0

•

# #

• l .?l l l .m ,,,~ Q I . ~ (Win

o 0

Dism~m alcgl X dinsctim ( ~ ) Normal stress along a section A-A of the gear tooth for homogeneous and heterogeneous materials,


- ~ . . 0 4 .

"~ .a.IBE.M .

! .1.11E-04.

t . . . . CA

sl

-1.Mg-g8.

Q

,4

o H~kmD (WU ~ )

o I.lYglglO (Wltlmut k l ~ / m ~

m;!

l! i l i l

t t p

• I I i t I J

I

.1J~E.m . . . . I . . . . I . . . . I ' '

-,1.~1E~11 - 1 ~ -6.d14F,.~O0 1 . M ~ ~t, mE~01

Di~tan~ along X dit~lioa (,.m) Fig. 32, Shear stress along a section A-A of the gear tooth for homogeneous and heterogeneous materials.

5. Conclusion

In this study, a new finite element method for the analysis of heterogeneous media with randomly dispersed second phase material is developed. It is based on the tessellation of a composite domain to yield a network of polygonal Voronoi elements, each containing one second phase material at most. The Voronoi cells are then used as elements in a two- dimensional finite element analysis. Element formulation of these polygons is carried out by an assumed stress hybrid method, with complete as well as incomplete global stress polynomials. The latter case lacks rotational invariance, and hence a formulation in a local element coordinate system is invoked. Both the complete global and incomplete local Voronoi cell finite element methods have produced satisfactory results for a wide variety of test problems. Furthermore, convergence is achieved through an increase in the degrees of freedom and incorporating more regularly shaped elements.

To account for the presence of second phase material in each element, effective properties are evaluated by a transformation or eigen strain formulation. Results of the assumed stress hybrid method coupled with a homogenization technique, when compared with those obtained by a displacement based finite element method show satisfactory agreement. Although the stresses are overestimated in the second phase material with a coarse mesh, refinement of the


finite element mesh significantly improves results. Nevertheless, alternatives to the averaged consistency condition in the second phase may be sought to enhance the accuracy even further.

The method presented in this paper would also be very effective in understanding microscopic behavior at critical locations of a component for which SEM micrographs are available. This study does not include the macro-micro scale effects and is therefore limited to materials with a rather low volume fraction of second phase materials. In materials with microscopic inclusions and high density, a more detailed two-scale analysis is warranted. Although the method proposed is effective for evaluating overall displacements and averaged stresses, true local stresses would require further discretization of the unit cell model.

Acknowledgment

The authors are grateful to Dr. Owen Richmond and Dr. R.L. Mallett of ALCOA Technical Center, PA for their invaluable suggestions and discussions on this work. Support of this work by the United States Army Research Office and National Science Foundation through grants DAAL03-91-G-0168 and MSS-9196137, respectively, are gratefully acknow- ledged.

References

[I] G. Duvaut, Analyse functionelle et m(~canique des mileaux continues, applications ~ I'~tude de materieux composites elustiques h structure periodiques. Homogeneisation, in: W,T. Koiter. ed., Theoretical and Applied Mechanics (North-Holland, Amsterdam, 1976).

[2] F, Devries, H. Dumontet, G, Duvaut and F. Lane, Homogenization and damage for composite structures, lnternat. J. Numer. Methods Engrg. 27 (1989) 285-298,

[3] P.J, Wray, O. Richmond and H,L, Morrison, Use of the Dirichlet tessellation for characterizing and modeling nonregular dispersions of second-phase particles, Metallography 16 (1983) 39-58.

[4] W.A. Spitzig, J.F. Kelly and O. Richmond, Quantitative characterization of second phase populations, Metallography 18 (1985) 235-261.

[5] S. Ghosh and S.N. Mukhopadhyay, A two dimensional automatic mesh generator for finite element analysis of random composites, Comput. & Structures 41 (1991) 245-256.

[6] T.H.H. Plan, Derivation of element stiffness matrices by assumed stress distribution, AIAA J. 2 (1964) 1333-1336.

[7] W.J. Sehroeder and M.S. Shephard, Geometry-based fully automatic mesh generation and the Delaunay triangulation, Internat. J. Numer. Methods Engrg. 26 (1988) 2530-2515.

[8] W.J. Sehroeder and M.S. Shephard, A combined octree/Delaunay method for fully automatic 3-d mesh generation, lnternat. J. Numer. Methods Engrg. 29 0990) 37-55.

[9] S.H. Lo, Delaunay triangulation of non-convex planar domains, lnternat. J. Numer. Methods Engrg. 28 (1989) 2695~2707.

[10J P.J. Green and R. Sibson, Computing Dirichlet tessellations in the plane, Comput. J. 21 (1978) 168-173. ({1] D. Rhynshurger, Analytic delineation of thiesson polygons, Oeog. Anal. 5 (1973) 133-144. [12] A. Bower, Computing dirichlet te~sellations, Comput. J. 24 (1981) 162-166. [13] D,F. Watson, Computing the n-dimensional Delaunay tesseilations with applications to Voronoi polytopes,

Comput. J. 24 (1981) 167-172. [14] B.N. Boots and D.J. Murdoch, The spatial arrangement of random Voronoi polygons, Comput. Geosci. 9

(1983) 351-365.


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A material based finite element analysis of heterogeneous media involving Dirichlet tessellations