Hiromichi Itou

Extracting the support function of a cavity in an isotropic elastic body from a single set of boundary data

In this paper a reconstruction problem of an unknown cavity in a homogeneous isotropic elastic body is considered. In both states of plane stress and plane strain, an extraction formula of the convex hull of the unknown polygonal cavity is established by using the enclosure method.

Reconstruction of a linear crack in an isotropic elastic body from a single set of measured data

An inverse problem related to a crack in elastostatics is considered. The problem is: extract information about the location and shape of an unknown crack from a single set of the surface displacement field and traction on the boundary of the elastic body. This is a typical problem from the nondestructive testing of materials. A version in a plane problem of elastostatics is considered. It is shown that, in a state of plane strain, the enclosure method which was introduced by Ikehata yields the extraction formula of an unknown crack provided: the crack is linear; one of the two end points of the crack is known and located on the boundary of the body; a well-controlled surface traction is given on the boundary of the body.

On reconstruction of a cavity in a linearized viscoelastic body from infinitely many transient boundary data

In this paper, a reconstruction problem of a cavity in a linearized viscoelastic body from dynamical boundary data is considered. As a result, we establish a formula extracting three kinds of information about the unknown cavity: the support function, the distance from an outer point of the body and the minimum radius of the open ball including the cavity centered at a point, with infinitely many sets of the boundary data. In the formula, observation time can be taken to be arbitrary.

An inverse problem for a linear crack in an anisotropic elastic body and the enclosure method

This paper considers the problem: extract information about the location and shape of an unknown crack from a single set of the surface displacement field and traction on the boundary of an arbitrary homogeneous and anisotropic elastic plate. In states of both plane stress and plane strain, an extraction formula of an unknown crack is given provided: the crack is linear; one of two end points of the crack is known and located on the boundary of the body; a well-controlled surface traction is given on the boundary of the body.

A Boundary Value Problem for an Infinite Elastic Strip with a Semi-infinite Crack

In this paper we study a boundary value problem for an infinite elastic strip with a semi-infinite crack. By using the single and double layer potentials this problem is reduced to a singular integral equation, which is uniquely solved in the Hölder spaces by the Fredholm alternative.

Nonlinear elasticity with limiting small strain for cracks subject to non-penetration:

A major drawback of the study of cracks within the context of the linearized theory of elasticity is the inconsistency that one obtains with regard to the strain at a crack tip, namely it becoming infinite. In this paper we consider the problem within the context of an elastic body that exhibits limiting small strain wherein we are not faced with such an inconsistency. We introduce the concept of a non-smooth viscosity solution which is described by generalized variational inequalities and coincides with the weak solution in the smooth case. The well-posedness is proved by the construction of an approximation problem using elliptic regularization and penalization techniques.

EXISTENCE OF A WEAK SOLUTION IN AN INFINITE VISCOELASTIC STRIP WITH A SEMI-INFINITE CRACK

We study an initial-boundary value problem in an infinite viscoelastic strip with a semi-infinite fixed crack. For this problem we prove the existence and uniqueness of a weak solution which is prescribed on each side of the extended crack in Sobolev-type spaces.

On reconstruction of an unknown polygonal cavity in a linearized elasticity with one measurement

In this paper we consider a reconstruction problem of an unknown polygonal cavity in a linearized elastic body [16]. For this problem, an extraction formula of the convex hull of the unknown polygonal cavity is established by means of the enclosure method introduced by Ikehata [2, 4]. The advantages of our method are that it needs only a single set of boundary data and we do not require any a priori assumptions for the unknown polygonal cavity and any constraints on boundary data. The theoretical formula may have possibility of application in nondestructive evaluation.

On the states of stress and strain adjacent to a crack in a strain-limiting viscoelastic body:

The viscoelastic Kelvin–Voigt model is considered within the context of quasi-static deformations and generalized with respect to a nonlinear constitutive response within the framework of limiting small strain. We consider a solid possessing a crack subject to stress-free faces. The corresponding class of problems for strain-limiting nonlinear viscoelastic bodies with cracks is considered within a generalized formulation stated as variational equations and inequalities. Its generalized solution, relying on the space of bounded measures, is proved rigorously with the help of an elliptic regularization and a fixed-point argument.

On Convergent Expansions of Solutions of the Linearized Elasticity Equation near Singular Points

In this paper, we introduce some convergent expansion formulae of solutions of two dimensional linearized elasticity equation (called Navier’s equation) around singular points such as a corner and a crack tip.