Saravanan Umakanthan

We develop a methodology for solving truss problems involving viscoelastic materials where, of all the member forces that satisfy the nodal force equilibrium equation and nodal displacements that satisfy the displacement boundary conditions, those member forces and nodal displacements that satisfy the constitutive relation are sought. Since a rate type viscoelastic constitutive relation involves the rate of the stress or strain, this study explores the use of member forces, nodal displacements, and support reactions or their rates as independent variables. Assuming small deformations, the nodal force equilibrium and the displacement boundary condition results in a linear equality constraint between the independent variables. Then we find the unknown independent variables such that the root mean squared error in the constitutive relation of the members of the truss is minimized subject to the satisfaction of the linear constraint at selected times. The objective function is evaluated at selected times or integrated over subintervals of time. We explore six possible solution methods and benchmark them for their accuracy and efficiency. We study statically determinate and indeterminate truss whose members are modeled using rate and integral type viscoelastic constitutive relations for creep and oscillatory loading. For the standard linear solid model, we find that the proposed methods are more accurate than ABAQUS and, at times, require lesser computational wall time. We also demonstrate the applicability of the proposed methodology to fractional order and nonlinear 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 C2 basis functions. The strain field is derived from a continuous displacement field obtained from a linear combination of C0 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.