Estimates and asymptotics for the stress concentration between closely spaced stiff C1, inclusions in linear elasticity

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

“…Remark 1.2. The asymptotic expansion in Theorem 1.1 improves the corresponding results in [16] in terms of the following two aspects: first, the gradient estimates in Theorems 1.1 and 1.3 of [16] are improved to be a precise asymptotic formula here; second, we get rid of the symmetric assumptions on the domain and boundary data added in Theorem 1.5 of [16] and then obtain the asymptotic expression in Theorem 1.1 for the more generalized C 1,γ -inclusions.…”

“…Note that the smoothness of inclusions require for at least C 2,γ in the elasticity problem considered above. Recently, by taking advantage of the Campanato's approach and W 1,p estimates for elliptic systems with right hand side in divergence form, Chen and Li [16] developed an adapted version of the iterate technique to establish the upper and lower bound estimates on the gradient of a solution to the Lamé systems with partially infinity coefficients in the presence of two adjacent C 1,γ -inclusions. The results obtained in [16] comprise of the following two parts: on one hand, the upper bounds on the blow-up rate of the gradient are established in two and three dimensions and a lower bound is constructed in dimension two; on the other hand, an asymptotic expansion of the gradient is only derived under the condition of the symmetric C 1,γ -inclusions and the boundary data of odd function type.…”

“…Thus we obtain the asymptotic formulas of the stress concentration in any dimension. Our idea is different from that in [16], where only partially systems of equations in linear decomposition were considered. In fact, our idea overcomes the difficulty faced in [16] that the blow-up factors in dimensions greater than two can not be captured to give an optimal information about the blow-up rate of the stress.…”

“…Our idea is different from that in [16], where only partially systems of equations in linear decomposition were considered. In fact, our idea overcomes the difficulty faced in [16] that the blow-up factors in dimensions greater than two can not be captured to give an optimal information about the blow-up rate of the stress. Moreover, we establish the asymptotic expansions of the stress concentration for the generalized C 1,γ -inclusions and boundary data, which means that we don't need to impose some special symmetric condition on the inclusions and the parity condition on the boundary data as in [16].…”

“…Remark 1.6. By contrast with the results in [16], the primary advantage of our idea lies in capturing the blow-up factor matrices in dimensions greater than two to obtain a unified asymptotic expansion in (1.18), which completely solves the optimality of the blow-up rate of the stress in higher dimensions.…”