Stability analysis of the two-phase torsional rigidity near a radial\n configuration

2018/01/19 by Lorenzo Cavallina, Cavallina, Lorenzo
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Elasticity and Material Modeling #Composite Material Mechanics

paper · pdf · doi:10.48550/arxiv.1801.06298

Abstract

Let \Ω₀ denote the unit ball of \ℝ^N (N\≥ 2) centered at\nthe origin. We suppose that \Ω₀ contains a core, given by a smaller\nconcentric ball D₀, made of a (possibly) different material. We discover\nthat, depending on the relative hardness of the two materials, this radial\nconfiguration can either be a local maximizer for the torsional rigidity\nfunctional E or a saddle shape. In this paper we consider perturbations that\nsimultaneously act on the boundaries \∂ D₀ and \∂\Ω₀.\nThis gives rise to resonance effects that are not present when \∂ D₀\nor \∂\Ω₀ are perturbed in isolation. A detailed analysis of the\nsign of the second order shape derivative of E is then made possible by\nemploying the use of spherical harmonics.\n

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