Unique real-variable expressions of the integral kernels in the Somigliana stress identity covering all transversely isotropic elastic materials for 3D BEM

摘要

A formulation and computational implementation of the hypersingular stress boundary integral equation for the numerical solution of three-dimensional linear elastic problems in transversely isotropic solids is developed. The formulation is based on a new closed-form real variable expression of the integral kernel S_(ijk) giving tractions originated by an infinitesimal dislocation loop, the source of singularity work-conjugated to stress tensor. This expression is valid for any combination of material properties and for any orientation of the radius vector between the source and field points. The expression is based on compact expressions of U_(ik) in terms of the Stroh eigenvalues on the plane normal to the radius vector. Performing double differentiation of U_(ik) for deducing the second derivative kernel U_(ikjl) the stress influence function of an infinitesimal dislocation loop Σ_(ijkl)~(loop) are first obtained, obtaining then the integral kernel S_(ijk). The expressions of S_(ijk) and of the related kernels Σ_(ijkl)~(loop) and U_(ikjl) do not suffer from the difficulties of some previous expressions, obtained by other authors in different ways, with complex valued functions appearing for some combinations of material parameters and/or with division by zero for the radius vector at the rotational-symmetry axis. The expressions of the above mentioned kernels have been presented in a form suitable for an efficient computational implementation. The correctness of these expressions and of their implementation in a three-dimensional collocational BEM code has been tested numerically by solving problems with known analytic solutions for different classes of transversely isotropic materials. The obtained expressions will be useful in the development of BEM codes applied to composite materials, geomechanics and biomechanics. In particular, an application to biomechanics of the BEM code developed is shown. Additionally, these expressions can be employed in the distributed dislocation technique to solve crack problems.