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If stiff inclusions are closely located, then the stress, which is thegradient of the solution, may become arbitrarily large as the distance betweentwo inclusions tends to zero. In this paper we investigate the asymptoticbehavior of the stress concentration factor, which is the normalized magnitudeof the stress concentration, as the distance between two inclusions tends tozero. For that purpose we show that the gradient of the solution to the casewhen two inclusions are touching decays exponentially fast near the touchingpoint. We also prove a similar result when two inclusions are closely locatedand there is no potential difference on boundaries of two inclusions. We thenuse these facts to show that the stress concentration factor converges to acertain integral of the solution to the touching case as the distance betweentwo inclusions tends to zero. We then present an efficient way to compute thisintegral.

Optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions