Multi-variable two-dimensional formulation using element-free basis functions for a new class of constitutive relations

Abstract

This article presents a numerical formulation using element-free basis functions to solve two-dimensional boundary value problems for constitutive models in which the linearized strain and stress are implicitly related. The formulation treats both the stress and displacement fields as primary unknowns. The stress field is computed from Airy’s function, thereby satisfying equilibrium equations, and needs to be \({\mathcal {C}}^{2}\) continuous to satisfy the traction continuity requirement. Since developing a higher-continuity basis function for an arbitrary-shaped quadrilateral element using mesh-based interpolation techniques is infeasible, element-free basis functions are used. The Airy’s function is assumed using a linear combination of Hermite-type Moving Least Squares approximation involving a complete fifth-degree polynomial function and a new fifth-degree spline weighting function. The displacement field is assumed to be a linear combination of Moving Least Squares approximation. The unknown nodal coefficients in the assumed linear combination of stress and displacement fields are computed such that the Frobenius norm of the constitutive relation is minimized, subject to the satisfaction of constraints arising from boundary conditions. The minimized value of the Frobenius norm of the constitutive relation reflects the accuracy of the solution. The viability, efficacy, and accuracy of the proposed numerical formulation are benchmarked using six boundary value problems for two constitutive relations: linear and non-linear elastic.