Multi-field formulations for solving plane problems involving viscoelastic constitutive relations

This article reports a multi-field numerical formulation for solving plane problems involving viscoelastic materials. Stress fields satisfying equilibrium equations are constructed using Airy’s potentials which are expressed as a linear combination of $C^2$ basis functions. The strain field is derived from a continuous displacement field obtained from a linear combination of $C^0$ basis functions. An appropriate linear combination of these stress and displacement basis functions is determined such that the resulting stress and strain fields satisfy the constitutive relation subjected to the satisfaction of the constraints arising from the boundary conditions. Since a viscoelastic constitutive relation involves stress, strain, and their rates, stress and displacement degrees of freedom or their rates can be considered as optimization variables for minimizing the error in satisfying the constitutive relation. Two Algorithms are proposed based on this choice of optimization variable. Accuracy and efficiency of the proposed algorithms are studied through five different boundary value problems involving four forms of the viscoelastic constitutive relations and for two loading histories. Using the developed rectangular element, viscoelastic beam bending problem is solved for the different constitutive relations studied.