Application of invariant integrals to the problems of defect identification

Abstract

A problem of parameters identification for embedded defects in a linear elastic body using results of static tests is considered. A method, based on the use of invariant integrals is developed for solving this problem. A problem on identification the spherical inclusion parameters is considered as an example of the proposed approach application. It is shown that the radius, elastic moduli and coordinates of a spherical inclusion center are determined from one uniaxial tension (compression) test. The explicit formulae expressing the spherical inclusion parameters by means of the values of corresponding invariant integrals are obtained for the case when a spherical defect is located in an infinite elastic solid. If the defect is located in a bounded elastic body, the formulae can be considered as approximate ones. The values of the invariant integrals can be calculated from the experimental data if both applied loads and displacements are measured on the surface of the body in the static test. A numerical analysis of the obtained explicit formulae is fulfilled. It is shown that the formulae give a good approximation of the spherical inclusion parameters even in the case when the inclusion is located close enough to the surface of the body.

Shape derivative of the energy functional in a problem for a thin rigid inclusion in an elastic body

“…In spite of one-to-one correspondence between the spaces H(Ω 0 ) and H(Ω ε ), the set of admissible displacements K 0 (Ω 0 ) is not mapped into the set of admissible displacements K ε (Ω ε ) of perturbed problem under mapping (11). There is no such correspondence even for a rectilinear rigid inclusion, see [14,36].…”

“…At first, it is connected with the fact that the unit normal vector to γ 0 is not mapped into the unit normal vector to the perturbed crack γ ε . At second, the structure of functions from the set of rigid displacements R(γ − 0 ) is not preserved under mapping (11).…”

“…Apply the coordinate transformation (11) to functions and integrals in (13). We obtain the following variational inequality:…”

“…Remark 3. If perturbation (11) is such that the function V is compactly supported in Ω, then there is no need to consider the cut-off function θ. In this case, the proof of Theorem 1 goes through with θ = 1.…”

“…For these cases, the formula for the derivative of the energy functional with respect to the parameter of the domain perturbation can be presented as an invariant integral. Invariant integrals are widely used in the fracture mechanics (see [6,7]) and in the problems of identification of defects in elastic bodies (see [3,11]). …”