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Abstract
We determine the effective electric properties of a composite with high contrast. The energy density is given locally in terms of a convex function of the gradient of the potential. The permittivity may take very large values in a fairly general distribution of parallel fibers of tiny cross sections. For a critical size of the cross sections, we show that a concentration of electric energy may arise in a small region of space surrounding the fibers. This extra contribution is caused by the discrepancy between the behaviors of the potential in the matrix and in the fibers and is characterized by the density of the cross sections of the fibers with respect to the cross section of the body in terms of some suitable notion of capacity. Our results extend those established in [7] in the periodic case for the p-Laplacian to a general nonlinear framework and a nonperiodic distribution of fibers.
Highlights
We determine the effective electric properties of a composite with high contrast
The common feature of this body of work is the emergence of a concentration of energy in a small region of space surrounding the strong components
This extra contribution is characterized by a local density of the geometric perturbations in terms of an appropriate capacity depending on the type of equations
Summary
Composites comprising traces of materials with extreme physical properties have been investigated by several authors over the past decades in various contexts, such as diffusion equations [7, 11, 16, 26, 28], fluid mechanics [12], electromagnetic theory [9], linearized elasticity [6, 8]. The common feature of this body of work is the emergence of a concentration of energy in a small region of space surrounding the strong components This extra contribution is characterized by a local density of the geometric perturbations in terms of an appropriate capacity depending on the type of equations. We investigate the non periodic case and consider a more general non linear framework and fibers with arbitrarily shaped cross sections. This is worthwhile because fibers stem from draw plates and are likely to display anisotropic behaviors governed by general convex functions. The effective problem turns out to show the same general features as in the periodic case, provided the fibers are not too closely spaced (see (1.6)).
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