Abstract
This paper is concerned with the scattering problems of a crack with Dirichlet or mixed impedance boundary conditions in two dimensional isotropic and linearized elasticity. The well posedness of the direct scattering problems for both situations are studied by the boundary integral equation method. The inverse scattering problems we are dealing with are the shape reconstruction of the crack from the knowledge of far field patterns due to the incident plane compressional and shear waves. We aim at extending the well known factorization method to crack determination in inverse elastic scattering, although it has been proved valid in acoustic and electromagnetic scattering, electrical impedance tomography and so on. The numerical examples are presented to illustrate the feasibility of this method.
Highlights
Consider the scattering of time harmonic elastic plane wave uin by a crack Γ in R2
We will denote by ∆∗ the Lame operator μ∆ + (μ + λ)∇(∇·) for brevity
The total displacement field ut is the superposition of the incident filed uin and the scattered field u, i.e., ut = uin + u
Summary
Consider the scattering of time harmonic elastic plane wave uin by a crack Γ in R2. Let the crack embedded in an isotropic and homogeneous elastic medium with constant density ρ = 1 and Lame constants μ and λ satisfying μ > 0, 2μ + λ > 0. The displacement field of the scattered wave u excited by the crack is governed by the Navier equation (1). For the scattering from hard crack the scattered field u has to satisfy the Dirichlet boundary condition (2). In the case of scattering from a coated crack with some material on one side, u satisfies mixed impedance boundary conditions (3). The scattered field u has to satisfy the Kupradze radiation condition [24]. Throughout this paper, the solution of Navier equation (1) satisfying the Kupradze radiation condition is called the radiating
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