Kazumi Tanuma

Complete Asymptotic Expansions of Solutions of the System of Elastostatics in the Presence of an Inclusion of Small Diameter and Detection of an Inclusion

We consider the system of elastostatics for an elastic medium consisting of an imperfection of small diameter, embedded in a homogeneous reference medium. The Lamé constants of the imperfection are different from those of the background medium. We establish a complete asymptotic formula for the displacement vector in terms of the reference Lamé constants, the location of the imperfection and its geometry. Our derivation is rigorous, and based on layer potential techniques. The asymptotic expansions in this paper are valid for an elastic imperfection with Lipschitz boundaries. In the course of derivation of the asymptotic formula, we introduce the concept of (generalized) elastic moment tensors (Pólya–Szegö tensor) and prove that the first order elastic moment tensor is symmetric and positive (negative)-definite. We also obtain estimation of its eigenvalue. We then apply these asymptotic formulas for the purpose of identifying with high precision the order of magnitude of the diameter of the elastic inclusion, its location, and its elastic moment tensors.

Stroh Formalism and Rayleigh Waves

The Stroh formalism is a powerful and elegant mathematical method developed for the analysis of the equations of anisotropic elasticity. The purpose of this exposition is to introduce the essence of this formalism and demonstrate its effectiveness in both static and dynamic elasticity. The equations of elasticity are complicated, because they constitute a system and, particularly for the anisotropic cases, inherit many parameters from the elasticity tensor. The Stroh formalism reveals simple structures hidden in the equations of anisotropic elasticity and provides a systematic approach to these equations. This exposition is divided into three chapters. Chapter 1 gives a succinct introduction to the Stroh formalism so that the reader could grasp the essentials as quickly as possible. In Chapter 2 several important topics in static elasticity, which include fundamental solutions, piezoelectricity, and inverse boundary value problems, are studied on the basis of the Stroh formalism. Chapter 3 is devoted to Rayleigh waves, for long a topic of utmost importance in nondestructive evaluation, seismology, and materials science. There we discuss existence, uniqueness, phase velocity, polarization, and perturbation of Rayleigh waves through the Stroh formalism.

Layer stripping for a transversely isotropic elastic medium

In this paper we develop a layer stripping algorithm for transversely isotropic elastic materials in three-dimensional space. We first prove that by making displacement and traction measurements at the boundary of these elastic materials one can determine the elastic tensor and all its derivatives at the boundary. The elastic measurements made at the boundary are encoded in the Dirichlet to Neumann (DN) map. Then we use the Riccati equation developed in [G. Nakamura and G. Uhlmann, A layer stripping algorithm in elastic impedance tomography, in Inverse Problems in Wave Propagation, IMA Vol. Math. Appl. 90, Springer-Verlag, New York, 1997, pp. 375-384] for the DN map in any anisotropic elastic medium to approximate the elastic tensor in the interior.

Local determination of conductivity at the boundary from the Dirichlet-to-Neumann map

We consider the problem of determining the conductivity of an isotropic, static conductive medium from the measurements of the electric potential on the boundary and the corresponding current flux across that boundary, that is, from the Dirichlet-to-Neumann map. Under some local regularity assumptions on the conductivity and on the boundary, we give a formula for reconstructing pointwisely the conductivity and its derivatives on the boundary from the localized Dirichlet-to-Neumann map.

A nonuniqueness theorem for an inverse boundary value problem in elasticity

The authors consider the inverse boundary value problem at the boundary for a two-dimensional anisotropic elastic medium. It is proved that the elastic tensor and its derivatives on the boundary cannot be identified from the full symbol of the Dirichlet to Neumann map.

Identification of the shape of the inclusion in the anisotropic elastic body

Consider an elastic body which occupies a domain Let D denote an open subset with Lipschitz boundary compactly contained in Ω. Suppose the elasticity tensor field of Ω has the form where Co is a homogeneous anisotropic elasticity tensor field. Denote by the traction on ∂Ω corresponding to a displacement field f on ∂Ω. We assume that (i) is connected; (ii) all component of H are just essentially bounded in D; (iii) H satisfies a jump condition in a relative neighbourhood of ∂D in D Under this assumption, we prove the unique determination of D by means of for infinitely many f. The proof is due to making use of a system of integral inequalities, the Runge approximation theorem and singular solutions.

Numerical Recovery of Conductivity at the Boundary from the Localized Dirichlet to Neumann Map

Numerical implementation of the reconstruction formulae of Nakamura and Tanuma [Recent Development in Theories and Numerics, International Conference on Inverse Problems 2003] is presented. With the formulae, the conductivity and its normal derivative can be recovered on the boundary of a planar domain from the localized Dirichlet to Neumann map. Such reconstruction method is needed as a preliminary step before full reconstruction of conductivity inside the domain from boundary measurements, as done in electrical impedance tomography. Properties of the method are illustrated with reconstructions from simulated data.

Angular Dependence of Rayleigh-Wave Velocity in Prestressed Polycrystalline Media with Monoclinic Texture

This article is concerned with Rayleigh waves propagating along the free surface of a macroscopically homogeneous, prestressed half-space. In the meso-scale, the half-space in question is taken to be a textured polycrystalline aggregate of cubic crystallites, which has the normal to its free surface being a 2-fold axis of monoclinic sample symmetry. Under the theoretical framework of linear elasticity with initial stress, an angular dependence formula, which shows explicitly how the phase velocity of Rayleigh waves depends on the propagation direction, the prestress, and the crystallographic texture, is derived from a constitutive equation motivated by Hartigs law. This velocity formula includes terms which describe the effects of texture on acoustoelastic coefficients, and it is correct to within terms linear in the initial stress and in the anisotropic part of the incremental elasticity tensor. Since its derivation makes no presumption on the origin of the initial stress, this velocity formula is meant to be applicable when the prestress is induced by plastic deformations such as those incurred during the surface enhancement treatment of low plasticity burnishing. The angular dependence formula assumes a simpler form when the texture of the prestressed half-space is orthorhombic.

Inverse problems for scalar conservation laws

The weak solution to the initial value problem for scalar conservation laws may develop a discontinuity called a shock. In this paper, we consider an inverse problem to determine the flux function entering the scalar conservation law by observing the shock developed by a single initial datum. We prove that the flux function on an interval can be uniquely determined by the shock. We also prove that this interval can be taken arbitrarily large by choosing an appropriate sequence of initial data. We then show that the derivatives of the flux function, and those of initial data, can be reconstructed in an explicit way from the coefficients of asymptotics of the shock wave.

Surface Impedance Tensors of Textured Polycrystals

A formula for the surface impedance tensors of orthorhombic aggregates of cubic crystallites is given explicitly in terms of the material constants and the texture coefficients. The surface impedance tensor is a Hermitian second-order tensor which, for a homogeneous elastic half-space, maps the displacements given at the surface to the tractions needed to sustain them. This tensor plays a fundamental role in Strohs formalism for anisotropic elasticity. In this paper we account for the effects of crystallographic texture only up to terms linear in the texture coefficients and give an explicit formula for the terms in the surface impedance tensor up to those linear in the texture coefficients.