Linear elasticity of planar delaunay networks - Semantic Scholar

ACTA MECHANICA

Acta Mechaniea 80, 61--80 (1989)

9 by Springer-Verlag 1989

Linear Elasticity of Planar Delaunay Networks: Random Field Characterization of Effective Moduli By M. Ostoja-Starzewski and C. Wang, West Lafayette, Indiana With 11 Figures

(Received December 16, 1988; revised March 2, 1989)

Summary A study is conducted of the influence of microscale geometric and physical randomness on effective moduli of a continuum approximation of disordered microstruetures. A particular class of microstructures investigated is that of planar Delaunay networks made up of linear elastic rods connected by joints. Three types of networks are considered: Delaunay networks with random geometry and random spring constants, modified Delaunay networks with random geometry and random spring constants, and regular triangular networks with random spring constants. Using a structural mechanics method, a numerical study is conducted of the first and second order characteristics of random fields of effective moduli. In view of duality of the Delaunay triangulations to the Voronoi tessellations, these results provide the basis for development of analytical models of various heterogeneous solids, e.g. granular, fibrous.

1. Introduction Scatter in constitutive coefficients and its dependence on the size of the body of a given solid are known to be present in elastic and inelastic response ranges of various materials. Those effects are caused by the heterogeneity of the material mierostructures, and thus are absent only in solids devoid of any internal defects and impurities, such as pure crystals grown under highly controlled laboratory conditions. B y the heterogeneity of the microstructure we understand the nonuniformity of its geometry and/or its physical properties. This nonuniformity m a y either be of a deterministic or stochastic type. I n this paper series we investigate the effects of stochastic nonuniformity -- that is randomness -- of microscale geometrical and physical properties of a specific type of planar elastic microstructures on their effective mechanical properties. The microstructures investigated are the so-called Delaunay graphs

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made up of linear elastic rods connected by universal joints; hereinafter they are caned linear elastic Delaunay networks. In view of both the usefulness of the graph representation in mechanics of heterogeneous solids [1] and the duality of Delaunay graphs and Voronoi tessenations, these networks may model various microstructures, e.g. granular, cellular, fibrous. As pointed out in Ref. [1], graph representation was implemented in the study of granular-type media since the seventies. While Oda, see e.g. [2], was concerned with the importance of fabric on meqhanical response of soils, Satake concentrated on the analogy between the governing relations of a discrete medium and ~ continuous medium [3], as well as on relating the anisotropy of microstructure with stress and strain [4]. The importance of fabric in the analysis of microstructural response was investigated with the help of statistical arguments in [5] and [6], and other works followed along the lines established there. A statistical approach was also employed in defining the stress tensor for particulate media [7] and hence in finding effective stiffnesses for assemblies of granules with linear elastic interactions [8]. The common characteristics of these and related other studies on effective continuum descriptions of mechanics of, granular media is the application of statistical arguments f~ volumes containing theoretically infinite numbers of grains. Thus, the effect of scale dependence -- that is, the ratio of scale L = A ~ of an approximating l~epresentative Volume Element (I~VE) A V to the average scale of grain d -- was not considered. Since for any finite Lid ratio, hereinafter denoted by 8, every RVE is somewhat different in its microstructure and its response, one has to account for the statistical scatter of its properties, and even, as we will show here, the scale dependence of the effective stiffnesses. Also, as pointed out in [1], the random discrete nature of materials plays an important role at the critical points of response, e.g. elastic-plastic transitions, damage evolution; see also [9]. The importance of probabilistic nature of the continuum approximation is coming now to be recognized in soil and solid mechanics (see e.g. [10]) and is one of main motivations for the development of random field models in one, two, and three dimensions [11]. However, these models lack a connection with the underlying material microstructure. It was this observation that provided the basic motivation for the research presented here, and 'which is also fully consistent with our basic philosophy: randomness in many media is the result o] their discrete nat~zre. X~'ollowing are the key objectives of our research: (i) characterization of stiffnesses of linear elastic Delaunay networks in terms of the first and second moment statistics; (ii) investigation of conventional statistical averaging methods using the paradigm of linear elastic Delaunay networks; (iii) investigation of effective moduli of two-phase and percolating networks.

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While a condensed account of some results on points (i) and (ii) has been given in [12], a detailed presentation of research concerning the first point is given in this paper.

2. Delaunay Network Models of Microstructure Representation of various heterogeneous solids (e.g. granular, fibrous) in terms of planar simple finite graphs has been discussed in [1]. Using the .terminology of t h a t reference, the dual graphs G and G' are chosen here as the Delaunay network and the Voro~oi tesselation, respectively: The Delaunay network (or De]aunay graph) G is a pair of two sets: V -- the set of vertices, and E - the set of edges connecting neighboring vertices, see [13]9 The notion of neighboring vertices is appreciated by noting that any Delaunay vertex is

Fig. 1. Delaunay network dual 5 A c t a Mech. 80/1--2

to

the Voronoi tessellation (2000 random Poisson points in a unit square)

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M. Ostoja-Starzewski and C. Wang:

an originating point of a Dirichlet region (i.e. grain) of the Voronoi tessellation, see Figs. 1 and 2. Clearly, G is a planar simple finite graph. The Delaunay vertices in Fig. 1 are generated with the Poisson point process of density equal to 2000 on the unit square in ]R2. As can be seen from Fig. 1 the resulting microstructure has a rather strong variability of edge lengths and angles between adjacent edges incident at a vertex. In order to investigate the effects of microscale geometric randomness, a so-called modi]ied Delaunay network, distinct from the (unmodi]ied) Delaunay network defined above, is introduced as another class of a G graph. The modification consists in requiring any two vertices to satisf~r a minimal distance condition, which is chosen here to equal 0.0182, while keeping the same Poisson process density, see Fig. 3. B y comparing Fig. 3 with Fig. 1, it is seen that this network is much less irregular than the unmodified one and, consequently, its dual graph has a much smaller variability

Fig. 2. The Voronoi tessellation of a unit square (2000 random Poisson points)

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in grain size and shape. Finally, for comparison purposes, we consider a regular triangular network m a d e up of equilateral triangles. This is seen as a limiting deterministic case of all irregular Delaunay graphs, and its dual graph is simply a hexagonal structure. The probability distributions of edge lengths, obtained from the computer simulations of the modified and unmodified networks, are shown in Fig. 4; the resulting average edge lengths for the unmodified, modified and regular networks are 0.02548, 0.02512 and 0.02420, respectively. All the I)elaunay networks are considered to represent discrete mechanical systems in the sense of each vertex being a joint, and each edge being a twoforce member acting as a linear elastic spring of a directly specified length l and of a deterministic or random spring constant k; the assignment of k to the edges is discussed in Sects. 5, 6 and 7. We assume all these networks to have on internal stress, that is all the edges are unstressed when the networks are not subjected

Fig. 3. The modified Delaunay network (2 000 points in a unit square) 5*

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to any external loads. Finally, we observe that the property of planarity of all G graphs (in the sense of graph theory terminology) renders these networks the spatially disordered mechanical systems with local (i.e. nearest-neighbor) interactions.

3. Effective Sfiffnesses We want to characterize the Delaunay networks defined above in terms of the effective moduli. That is, we want to find the moduli (~ in the relation

whereby we restrict possible deformations to infinitesimal strains. Equation (1) represents a continuum approximation of the network, in the sense that: (i) (3 is an effective modulus of any square window 1 L x L in ]R~ cut ou~ of the network, where L may not be smaller than the average cell size d; an example of such a window is shown in Fig. 5 a) ; (ii) a is the Cauchy stress meant in the sense of surface averaging, rather than volume averaging; (iii) e is the infinitesimal strain meant in the sense of volume averaging. Regarding point (i) we note that ~3 must depend on the location x of the center of the window and the actual realization ~ of the network therein. Thus, co C/2 -- the sample space of all possible realizations of t h e microstructure of a given kind of a Delaunay network; a regular Delaunay network with all edges having the same spring constants represents a deterministic case, i.e. ~2 is a singleton set. The choices listed in points (ii) and (iii) are made on the basis of conclusions reached in [14]. I t is important to point out that the interpretation of e in the sense of volume averaging coincides with the representation of strain of a discrete microstructure in a given window which is adopted in Section 4 and used throughout the ensuing analysis. The deformation prescribed on the boundaries of any window will be assumed uniform and hence the strain of the approximating continuum will be the same in both surface and volume averaging [15]. In view of the above comments, Eq. (1) should be rewritten as

(2) where 3 is the dimensionless ratio Lid. One may suppress here the explicit dependence on the window location x since the statistics of Delaunay vertices are spatially homogeneous in ]R2. 1 This window is ~he I~VE discussed in Section 2.

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I t follows t h a t C(x, 3) = {C(x, ~, 3) ; w C $2} is a random field. Finding C(x, ~) for a n y fixed 6 is analogous to local averaging of (he random field of spring stiffnesses k on the edge set E. Indeed, the determination of C(x, 8) can be done exactly via a local integration process in the one-dimensional case as demonstrated in the following steps : - - a Delaunay network reduces to a string of edges of random lengths and stiffnesses; - - consider a so-called averaging length L containing n edges; - - the effective compliance S~ of the string of length L is

{=1

where S~ = 1/#~; - - Eq. (3) defines a local integral process [11]. I t is well known t h a t Eq. (3) cannot be directly extended to two or three dimensions; t h a t is, no closed form expressions for C~i~'s can be derived. Hence, in order to reflect the actual physics (mechanics) of the network we resort to a computational structural mechanics method.

4. The Structural Mechanics Method We have found it convenient to adopt in the rest of analysis a vector rather t h a n a tensor formulation. Thus, the effective mechanical response of a single window is described, in place of Eq. (2), b y

(4) I t was stated in the foregoing section t h a t a window was cut out of the network as indicated b y a square with dashed lines in Fig. 5 a. This window is then treated as an independent mechanical structure (planar truss) as shown in Fig. 5 b. All the points of intersection of the edges of the original network b y the sides of the square window are pinned on these four sides. Next, we subject the window sides to a uniform deformation field with the strain components eu, e~ and e12, see Fig. 5c for an example of a deformed window. The actual mechanical response of this square-shaped truss is now solved b y a structural mechanics method. L e t {u} and [K] be the nodal displacements and the stiffness matrix of a given window {u} = [u(bij,

rr :J `''>, EKJ