A symmetric Galerkin boundary/domain element method for finite elastic deformations

• Published in: Computer methods in applied mechanics and engineering 2000-09, Vol.189 (2), p.481-514
• Main Author: Polizzotto, Castrenze
• Format: Article
• Language: English
• Subjects: BEM; Boundary-integral methods; Computational techniques; Exact sciences and technology; Finite elasticity; Fundamental areas of phenomenology (including applications); Incompressibility; Mathematical methods in physics; Physics; Solid mechanics; Static elasticity; Static elasticity (thermoelasticity...); Structural and continuum mechanics; Variational principles
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/S0045-7825(99)00303-5

The Symmetric Galerkin Boundary Element Method (SGBEM) is reformulated for problems of finite elasticity with hyperelastic material and incompressibility, using fundamental solutions related to a (fictitious) homogeneous isotropic and compressible linear elastic material. The proposed formulation contains, besides the standard boundary integrals, domain integrals which account for the problem's nonlinearities through some (fictitious) initial strain and stress fields required to satisfy appropriate “consistency” equations. The boundary/domain integral equation problem so obtained is shown to admit a stationarity principle (a consequence of the Hu-Washizu one), which covers a number of particular cases, including the compressible case, as well as the fully linear case, either incompressible and compressible (so recovering known results of the literature). The continuum rate problem is also addressed and suitably discretized by boundary and domain elements. The related algebraic solving equation system is shown to exhibit symmetry and some sign definiteness features; it is formally solved in terms of the node values of the displacement gradient, the idrostatic pressure, the initial strain and stress fields, a long with the unknown boundary displacements and tractions. Details regarding the fundamental solutions and the stationarity principle are reported in three appendices at the end of the paper.