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Register a new userSingular analysis of the stress concentration in the narrow regions between the inclusions and the matrix boundary
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We consider the Lam{e} system arising from high-contrast composite materials whose inclusions (fibers) are nearly touching the matrix boundary. The stress, which is the gradient of the solution, always concentrates highly in the narrow regions between the inclusions and the external boundary. This paper aims to provide a complete characterization in terms of the singularities of the stress concentration by accurately capturing all the blow-up factor matrices and making clear the dependence on the Lam{e} constants and the curvature parameters of geometry. Moreover, the precise asymptotic expansions of the stress concentration are also presented in the presence of a strictly convex inclusion close to touching the external boundary for the convenience of application.
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In the region between two closely located stiff inclusions, the stress, which is the gradient of a solution to the Lam{e} system with partially infinite coefficients, may become arbitrarily large as the distance between interfacial boundaries of inclusions tends to zero. The primary aim of this paper is to give a sharp description in terms of the asymptotic behavior of the stress concentration, as the distance between interfacial boundaries of inclusions goes to zero. For that purpose we capture all the blow-up factor matrices, whose elements comprise of some certain integrals of the solutions to the case when two inclusions are touching. Then we are able to establish the asymptotic formulas of the stress concentration in the presence of two close-to-touching $m$-convex inclusions in all dimensions. Furthermore, an example of curvilinear squares with rounded-off angles is also presented for future application in numerical computations and simulations.
In high-contrast composites, if an inclusion is in close proximity to the matrix boundary, then the stress, which is represented by the gradient of a solution to the Lam{e} systems of linear elasticity, may exhibits the singularities with respect to the distance $varepsilon$ between them. In this paper, we establish the asymptotic formulas of the stress concentration for core-shell geometry with $C^{1,alpha}$ boundaries in all dimensions by precisely capturing all the blow-up factor matrices, as the distance $varepsilon$ between interfacial boundaries of a core and a surrounding shell goes to zero. Further, a direct application of these blow-up factor matrices gives the optimal gradient estimates.
We study the stress concentration, which is the gradient of the solution, when two smooth inclusions are closely located in a possibly anisotropic medium. The governing equation may be degenerate of $p-$Laplace type, with $1<p leq N$. We prove optimal $L^infty$ estimates for the blow-up of the gradient of the solution as the distance between the inclusions tends to zero.
In the perfect conductivity problem of composites, the electric field may become arbitrarily large as $varepsilon$, the distance between the inclusions and the matrix boundary, tends to zero. The main contribution of this paper lies in developing a clear and concise procedure to establish a boundary asymptotic formula of the concentration for perfect conductors with arbitrary shape in all dimensions, which explicitly exhibits the singularities of the blow-up factor $Q[varphi]$ introduced in [29] by picking the boundary data $varphi$ of $k$-order growth. In particular, the smoothness of inclusions required for at least $C^{3,1}$ in [27] is weakened to $C^{2,alpha}$, $0<alpha<1$ here.
In high-contrast elastic composites, it is vitally important to investigate the stress concentration from an engineering point of view. The purpose of this paper is to show that the blowup rate of the stress depends not only on the shape of the inclusions, but also on the given boundary data, when hard inclusions are close to matrix boundary. First, when the boundary of inclusion is partially relatively parallel to that of matrix, we establish the gradient estimates for Lam{e} systems with partially infinite coefficients and find that they are bounded for some boundary data $varphi$ while some $varphi$ will increase the blow-up rate. In order to identify such novel blowup phenomenon, we further consider the general $m$-convex inclusion cases and uncover the dependence of blow-up rate on the inclusions convexity $m$ and the boundary datas order of growth $k$ in all dimensions. In particular, the sharpness of these blow-up rates is also presented for some prescribed boundary data.
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