An asymptotic formula for the voltage potential in a perturbed ε-periodic composite medium containing misplaced inclusions of size ε

Abstract

We consider composite media made of homogeneous inclusions with C 1,α boundaries. Our goal is to compare the potential uε in a perfectly periodic composite with the potential u ε,d of a perturbed periodic medium, where the periodicity defects consist of misplaced inclusions. We give an asymptotic expansion of the difference u ε,d − uε away from the defects and show that, to first order, a misplaced inclusion manifests itself via a polarization tensor, which is characterized.

Asymptotics for the voltage potential in a periodic network with localized defects

“…We show that the first-order term in the expansion has the same structure as in (1). However, the smooth potential and Green's function that appear in the right-hand side are those of the limiting medium, obtained when the lattice size h tends to 0.In our setting, the scale of variations of the medium and the diameter of the defectuous zone are of the same order, much like the situation studied in [13]. In this work, the reference configuration is a continuous composite medium of smooth inclusions imbedded in a homogeneous matrix phase.…”

“…The perturbed medium is one where a few inclusions have been misplaced. The first term in the expansion of the difference between perturbed and reference potentials has the form (1), where the potential and Green's function on the right-hand side are those of the homogenized medium.Similar to [13], the main ingredients to prove convergence of the expansion are uniform W 1,∞ estimates on the potential u h of the reference domain and on the corresponding Green's function.In [13], such estimates were obtained as consequences of the '3-step compactness method' of Avellaneda and Lin [14] and as consequences of uniform W 1,∞ elliptic regularity results in composite media with piecewise Hölder coefficients [15,16]. In this paper, we will use uniform W 1,∞ estimates for the piecewise linear potential u h .…”

Asymptotics for the potential in a periodic network with localized defects

“…A similar problem was addressed in [4], where the influence of periodicity defects in a composite media made with inclusions was studied.…”

Artificial conditions for the linear elasticity equations

“…This problem enters the general framework of local perturbations for elliptic problems, which have been deeply studied. Among others, let us mention the following works using potential theory [38,5,3,4] and the reference monographs [30,31] for multiscale expansions. Following [10,36], the solution of (1.1) is given to first order by…”