Discrete vs continuum models for heterogeneous materials
Leonid Berlyand (Penn State)

The talk consists of two parts. In the first part we will compare continuum and discrete models of highly packed particle filled composites. Using an example of highly concentrated suspensions we start from a continuum PDE model and show how its discrete network approximation can be derived and justified. We call such an approximation the structural discretization (SD) as oppose to the numerical discretization (finite-differences, finite elements, etc). While in a numerical discretization its scale (mesh size) is adjustable depending on the desired precision, the scale of the structural discretization is determined by a given physical scale of the problem, such as particles sizes and distances between them (or lattice spacing in crystalline solids).

We show that the method of fictitious fluid domains, proposed in a recent joint work with Y. Gorb and A. Novikov (preprint, 2006), allows to determine rigorous asymptotic relation between the continuum and discrete (network) models. We demonstrate that the discrete model provides physical understanding of this problem which is not readily extractable from the corresponding continuum model and discuss computational advantages of the discrete model.

The second part of the talk is based on the joint work with M. Berezhnyy (J. Mech. Phys. Solids, 2006). In contrast with the first part, we start from a discrete model of a large number of concentrated masses (particles) connected by elastic spring and provide sufficient conditions on the geometry of the array of particles, when the network admits rigorous continuum limit. Our proof is based on the methods of mesocharacteristics and discrete Korns inequality. For generic non-periodic arrays of particles we describe the limiting continuum model in terms of local energy characteristic on the mezoscale (intermediate scale between the interparticle distances (small scale) and the domain sizes (large scale)), which represents local energy in the neighborhood of a point.

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