Article

# Gradient Estimates for the Perfect and Insulated Conductivity Problems with Multiple Inclusions

• ##### Biao Yin
Communications in Partial Differential Equations (Impact Factor: 1.19). 09/2009; DOI: 10.1080/03605300903564000
Source: arXiv

ABSTRACT In this paper, we study the perfect and the insulated conductivity problems with multiple inclusions imbedded in a bounded domain in $\mathbb{R}^n, n\ge 2$. For these two extreme cases of the conductivity problems, the gradients of their solutions may blow up as two inclusions approach each other. We establish the gradient estimates for the perfect conductivity problems and an upper bound of the gradients for the insulated conductivity problems in terms of the distances between any two closely spaced inclusions.

0 Bookmarks
·
124 Views
• Source
##### Article: Optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions
[Hide abstract]
ABSTRACT: If stiff inclusions are closely located, then the stress, which is the gradient of the solution, may become arbitrarily large as the distance between two inclusions tends to zero. In this paper we investigate the asymptotic behavior of the stress concentration factor, which is the normalized magnitude of the stress concentration, as the distance between two inclusions tends to zero. For that purpose we show that the gradient of the solution to the case when two inclusions are touching decays exponentially fast near the touching point. We also prove a similar result when two inclusions are closely located and there is no potential difference on boundaries of two inclusions. We then use these facts to show that the stress concentration factor converges to a certain integral of the solution to the touching case as the distance between two inclusions tends to zero. We then present an efficient way to compute this integral.
12/2013;
• Source
##### Article: Singular Behavior of Electric Field of High Contrast Concentrated Composites
[Hide abstract]
ABSTRACT: A heterogeneous medium of constituents with vastly different mechanical properties, whose inhomogeneities are in close proximity to each other, is considered. The gradient of the solution to the corresponding problem exhibits singular behavior (blow up) with respect to the distance between inhomogeneities. This paper introduces a concise procedure for capturing the leading term of gradient's asymptotics precisely. This procedure is based on a thorough study of the system's energy. The developed methodology allows for straightforward generalization to heterogeneous media with a nonlinear constitutive description.
01/2014;
• Source
##### Article: Characterization of the gradient blow-up of the solution to the conductivity equation in the presence of adjacent circular inclusions
[Hide abstract]
ABSTRACT: We consider the conductivity problem in the presence of adjacent circular inclusions having arbitrary constant conductivity. When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution can be arbitrary large. We characterize the gradient blow-up by deriving an explicit formula for the singular term of the solution in terms of the Lerch transcendent function. This derivation is valid for inclusions having arbitrary constant conductivity. We illustrate our results with numerical calculations.
12/2013;