A finite element model for a coupled thermo-mechanical system: nonlinear strain-limiting thermoelastic body

TL;DR

This paper develops a finite element model for a nonlinear thermoelastic body with strain-limiting behavior, showing that near crack tips, strains grow slower than stresses, aligning with small strain theory.

Contribution

It introduces a coupled thermo-mechanical finite element model with nonlinear strain-limiting constitutive relations, suitable for analyzing crack-tip behavior.

Findings

Strain growth near crack tips is much slower than stress growth.

The model remains consistent with small strain assumptions.

Finite element solutions effectively capture thermoelastic responses.

Abstract

We investigate a specific finite element model to study the thermoelastic behavior of an elastic body within the context of nonlinear strain-limiting constitutive relation. As a special subclass of implicit relations, the thermoelastic response of our interest is such that stresses can be arbitrarily large, but strains remain small, especially in the neighborhood of crack-tips. Thus, the proposed model can be inherently consistent with the assumption of the small strain theory. In the present communication, we consider a two-dimensional coupled system-linear and quasilinear partial differential equations for temperature and displacements, respectively. Two distinct temperature distributions of the Dirichlet type are considered for boundary condition, and a standard finite element method of continuous Galerkin is employed to obtain the numerical solutions for the field variables. For a domain with an edge-crack, we find that the near-tip strain growth of our model is much slower than the growth of stress, which is the salient feature compared to the inconsistent results of the classical linearized description of the elastic body. Current study can provide a theoretical and computational framework to develop physically meaningful models and examine other coupled multi-physics such as an evolution of complex network of cracks induced by thermal shocks.