Nonsingular traction BIEs for crack problems in elastodynamics

Abstract

The nonsingular traction BIEs are derived for the Laplace transforms in elastodynamic crack problems. Two different forms of the ®nal nonsingular traction BIEs are received with respect to the leading singularity of the integral kernels involved. In the ®rst one, the traction BIE is derived from the integral representation of stresses which involves hypersingular kernels. In the second way, the partially regularized integral representation of stresses with strongly singular kernels is used as a starting point. The singular integrals are transformed to nonsingular ones by making use of the Stokes theorem.

Wave propagation in the presence of empty cracks in an elastic medium

“…Another way is to use the Laplace and Fourier transforms of the displacement and traction integral equations, of the fundamental elastodynamics solutions, and of the displacements' and tractions' nodal values. The unknown displacements and tractions, as functions of time are then computed (by numerical inversion) from the unknown transformed variables, which are obtained for a series of Laplace or Fourier parameters [15,25,30,31,33]. The third alternative is the dual reciprocity method, where the equations of motion are expressed in a boundary integral form using the fundamental solutions of elastostatics [13,23].…”

“…When using this type of formulation, two main difficulties arise: the first is the continuity conditions that must be guaranteed by the density functions, and the second is the need to perform the integration of strongly singular and hypersingular kernels that come up after differentiating the classical ones. A number of approaches have been proposed to deal with both difficulties (among others, see [30] and [5]), in particular relating to the choice of the elements and collocation procedure to be used [10,12,21,24,32,46] and concerning the implementation of regularization techniques or non-singular formulations [9,12,33,42]. Other ways of dealing with the hypersingular integrals have been proposed in [19,29] and [43], while in [25] and [26] an indirect approach to evaluate these integrals for the plane-strain cases in 2D problems has been adopted.…”