The isogeometric boundary element algorithm for solving the plane strain problem of an elastic matrix containing an open material surface of arbitrary shape

TL;DR

This paper introduces an isogeometric boundary element method for solving plane strain elasticity problems involving complex open surfaces, utilizing NURBS for precise surface representation and validated through benchmark tests.

Contribution

The paper develops a novel IGABEM algorithm that accurately models elastic matrices with arbitrary open surfaces using NURBS and Gurtin-Murdoch surface models.

Findings

The algorithm achieves good convergence in benchmark problems.

Surface curvature significantly influences stress distribution.

Validated against problems with straight and curved surfaces.

Abstract

The paper presents the Isogeometric Boundary Element Method (IGABEM) algorithm for solving the plane strain problem of an isotropic linearly elastic matrix containing an open material surface of arbitrary shape. Theoretical developments are based on the use of the Gurtin-Murdoch model of material surfaces. The governing equations and the boundary conditions for the problem are reviewed, and analytical integral representations for the elastic fields everywhere in the material system are presented in terms of unknown traction jumps across the surface. To find the jumps, the problem is reduced to a system of singular boundary integral equations in terms of two unknown scalar components of the surface stress tensor. The system is solved numerically using the developed IGABEM algorithm in which NURBS are used to approximate the unknowns. The main steps of the algorithm are discussed and convergence studies are performed. The algorithm is validated using two benchmark problems involving the matrix subjected to a uniform far-field load and containing a surface along (i) a straight segment and (ii) a circular arc. Numerical examples are presented to illustrate the influence of governing parameters with a focus on the influence of curvature variation.