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2 Ostoja-Starzewski M.

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On the admissibility of an isotropic, smooth elastic continuum

Ostoja-Starzewski M.

Archives of Mechanics

2005

Vol. 57, nr 4

345--355

Many studies of elasticity of inhomogeneous materials - in both elastostatics and elastodynamics - assume the existence of locally isotropic, smooth stiffness tensor fields. We investigate the correctness of such a model in the simplest setup of anti-plane classical elasticity. We work with the concept of mesoscale (or apparent) moduli for a finite-size window placed in such a material, in accordance with the Hill condition for the Hooke law. The limit from mesoscale down to infinitesimal windows is admissible within the model of an assumed smooth, locally isotropic continuum. However, this limit is not admissible from the standpoint of a microstructure, and, in order to set up an inhomogeneous elastic medium, one must introduce its anisotropy. A separate argument against the local isotropy stems from the representation of a correlation function of a wide-sense stationary and isotropic random field, whose realizations are smooth stiffness tensor fields.

Random field models and scaling laws of heterogenous media

Ostoja-Starzewski M.

Archives of Mechanics

1998

Vol. 50, nr 3

549-558

In many problems of solid mechanics (e.g., stochastic finite elements, statistical fracture mechanics) there is a need for resolution of dependent fields over scales ont infinitely larger than the microscale. This task may be accomplished through a "meso-scale window" which becomes the classical Representative Volume Element (RVE) in the infinite limit relative to the microscale. It turns out that the material properties at such a mesoscale cannot be uniquely approximated by a random field of stiffness/compliance with locally isotropic realizations, but, rather, two random continuum fields with locally anisotropic relizations, corresponding respectively to Dirichlet and Neumann boundary conditions on the meso-scale, need to be introduced to bound the material response from above and from below. We discuss statistical characteristics of these two mesoscale random fields, including their spatial correlation structure, for anti-plane elastic response of random two-phase composites with Voronoi geometry at the percolation point. Particular attention is given to the scaling of effective responses obtained from both conditions, which sheds light on the minimum acceptable size of an RVE.