Stress concentration factors for the Lamé system arising from composites

Abstract

For two neighbouring stiff inclusions, the stress, which is the gradient of a solution to the Lamé system of linear elasticity, may exhibit singular behavior as the distance between these two inclusions becomes arbitrarily small. In this paper, a family of stress concentration factors, which determine whether the stress will blow up or not, are accurately constructed in the presence of the generalized m-convex inclusions in all dimensions. We then use these stress concentration factors to establish the optimal upper and lower bounds on the stress blow-up rates in any dimension and meanwhile give a precise asymptotic expression of the stress concentration for interfacial boundaries of inclusions with different principal curvatures in dimension three. Finally, the corresponding results for the perfect conductivity problem are also presented. * d(d+1) 2 1 T for the elements of α-th column of the matrix B * , we obtain the new matrix B * α

Gradient asymptotics of solutions to the Lamé systems in the presence of two nearly touching $C^{1,γ}$-inclusions in all dimensions

“…A lower bound of the gradient was constructed by introducing a unified blow-up factor to prove the optimality of the blow-up rates in dimensions two and three in a subsequent work [35]. Miao and Zhao [44] further constructed the explicit stress concentration factors to establish the optimal gradient estimates in the presence of the generalized m-convex inclusions in all dimensions. The boundary case when the inclusions are nearly touching the matrix boundary was studied in [8,39,45].…”

Asymptotic analysis of the stress concentration between two adjacent stiff inclusions in all dimensions

“…) T by using systems of equations (4.5). This is different from that in [45], where Miao and Zhao utilized (4.5) to calculate…”

“…From the results in Theorems 2.1, 2.5, 2.6 and Corollary 2.7, we give a complete description for the singularities of the gradient and meanwhile answer all the optimality of the blow-up rates for two adjacent m-convex inclusions in all dimensions. This improves and makes complete the gradient estimates and asymptotics in the previous work [10,11,26,36,39,45]. By contrast with the gradient estimates, the main advantage of the precise asymptotic expansions lies in exhibiting all the higher and lower order singular terms in a unified expression.…”

“…In this paper, we make use of all the systems of equations in linear decomposition to capture all the blow-up factor matrices in any dimension and then establish the precise asymptotic expressions of the concentrated field for two adjacent m-convex inclusions in all dimensions. This is different from that in [36,39,45], which pursued to use a family of unified blow-up factors to characterize the singular behavior of the stress concentration.…”

“…However, the stress concentration factors, which are vital to the establishments of the lower bounds and asymptotic expansions on the gradients, are captured in [36,39] only when the domain and the boundary data satisfy some special symmetric conditions. Subsequently, Miao and Zhao [45] got ride of those strict symmetric conditions by accurately constructing the unified stress concentration factors for the generalized m-convex inclusions in all dimensions and then established the optimal upper and lower bounds on the gradient. Their results gave a perfect answer for the optimality of the stress blowup rate in any dimension.…”