Application of invariant integrals to elastostatic inverse problems

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 for the spherical inclusion parameters identification is considered as an example of the proposed approach application. It is shown that a 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. The values of the 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.

Symmetry properties of the reciprocity gap functional in the linear elasticity

“…This property of the RGF was used by Andrieux et al (1999), Goldstein et al (2007Goldstein et al ( , 2008, Kaptsov and Shifrin (2008) for solving some elastostatic inverse problems.…”

“…In this case the values S kl are close to the values R 33kl corresponding to an infinite solid containing the crack G and subjected to the uniaxial tension in the direction of the axis x 3 . Calculations fulfilled by Goldstein et al (2007Goldstein et al ( , 2008 for spherical cavities and inclusions showed that a bounded body can be substituted by an infinite solid even if the spherical defect is located not so far from the surface of the body. Taking into account the suppositions, Eqs.…”

“…In particular, Stern et al (1976), Hong and Stern (1978) used it for the calculation of stress intensity factors in the crack problems. The functional was used also for identification of various types of defects (plane cracks, spherical cavities and inclusions) in elastic solids by means of static tests results (see, Andrieux et al 1999;Goldstein et al 2007Goldstein et al , 2008Kaptsov and Shifrin 2008). Application of the RGF to various inverse problems is discussed in the review by Bonnet and Constantinescu (2005 (2009) showed that all types of invariant integrals in isotropic, linear elastic bodies can be expressed by means of the RGF.…”

The relation between invariant integrals of the linear isotropic theory of elasticity and integrals defined by the duality principle

“…Note that, when the field u r is regular, all the integrals J r i , L r i and M r are zero; otherwise, they are generally non-zero. We will consider the invariant integrals for the sum of fields with superscripts f and r, which will be denoted by a superscript f + r. For the overall field we have (1.4) Integrals corresponding to the interaction between fields with superscripts f and r have the form (1.5) Integrals (1.5) are also invariant; they were introduced earlier 3 and used 12,13 to solve the problem of identifying a spherical cavity and an elastic spherical inclusion.…”

Identification of the geometry and elastic properties of rigid inclusions in thin plate

“…Для идентификации местоположения одиночных дефектов развит подход, основанный на применении функционала взаимности и инвариантных интегралов, так в работах [1,2] параметры дефекта (координаты центра, линейный размер дефекта) выражаются через отличные от нуля значения функционала взаимности в виде явных аналитических выражений. В работах [3,4] рассматривается задача идентификации конечного числа мелких эллипсоидальных дефектов в анизотропном линейно упругом теле.…”