Mathematical modeling of composite materials leads to the solving PDEs with strongly oscillating coefficients. The problem of large number of equations can be solved using homogenization, that replaces heterogeneous material by an ‘equivalent‘ homogeneous one. This approach assumes periodic structure, which is not often true in reality. The first aim of the paper is to compare results obtained by solving the model problem describing the torsion of a bar applied to the random medium and the periodic one, respectively. The second aim is to present four algorithms generating samples of random structures of a two-component fibre composite material similar to the real one. and Obsahuje seznam literatury
The paper deals with a scalar diffusion equation c ut = (F[ux])x+f, where F is a Prandtl-Ishlinskii operator and c, f are given functions. In the diffusion or heat conduction equation the linear constitutive relation is replaced by a scalar Prandtl-Ishlinskii hysteresis spatially dependent operator. We prove existence, uniqueness and regularity of solution to the corresponding initial-boundary value problem. The problem is then homogenized by considering a sequence of equations of the above type with spatially periodic data cε and ηε when the spatial period ε tends to zero. The homogenized characteristics c∗ and η∗ are identified and the convergence of the corresponding solutions to the solution of the homogenized equation is proved.
We homogenize a class of nonlinear differential equations set in highly heterogeneous media. Contrary to the usual approach, the coefficients in the equation characterizing the material properties are supposed to be uncertain functions from a given set of admissible data. The problem with uncertainties is treated by means of the worst scenario method, when we look for a solution which is critical in some sense.
In the paper a Barenblatt-Biot consolidation model for flows in periodic porous elastic media is derived by means of the two-scale convergence technique. Starting with the fluid flow of a slightly compressible viscous fluid through a two-component poro-elastic medium separated by a periodic interfacial barrier, described by the Biot model of consolidation with the Deresiewicz-Skalak interface boundary condition and assuming that the period is too small compared with the size of the medium, the limiting behavior of the coupled deformation-pressure is studied.
Two-scale convergence is a powerful mathematical tool in periodic homogenization developed for modelling media with periodic structure. The contribution deals with the classical definition, its problems, the ''dua''' definition based on the so-called periodic unfolding. Since in the case of domains with boundary the unfolding operator introduced by D. Cioranescu, A. Damlamian, G. Griso does not satisfy the crucial integral preserving property, the contribution proposes a modified unfolding operator which satisfies the property and thus simplifies the theory. The properties of two-scale convergence are surveyed.