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 work develops an Isogeometric Boundary Element Method (IGABEM) to solve the plane-strain response of an infinite isotropic elastic matrix containing an open Gurtin–Murdoch surface of arbitrary smooth shape. The surface stresses $\sigma^{S}$ and $\omega^{S}$ are approximated with the same NURBS basis used for geometry, yielding a system of boundary-integral equations derived from the Kelvin fundamental solution $G_{kj}$ and traction jumps. The method is validated against benchmark problems for straight and circular-arc surfaces and extended to study curvature effects, including elliptical geometries, demonstrating accurate near-tip behavior without spectral filtering. The results quantify how surface curvature controls the distribution of surface stresses and the bulk stress field, with implications for modeling ultrathin reinforcements and graphene-like sheets in composites, and the framework opens avenues for multi-surface, homogenization, Steigmann–Ogden extensions, and 3D generalizations.

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.

Figures (16)

• Figure 1: Problem formulation.
• Figure 2: Illustration of the procedure for evaluation of singular integrals.
• Figure 3: Benchmark problem 1: Surface along the straight segment.
• Figure 4:
• Figure 5: