On a finite element model for two-dimensional thermoelastic problem

The study of elastic materials under thermal loading is an intense area of research from past several decades. The issues of expansion and contraction of materials when exposed to external mechanical forces and extreme temperatures are important in many engineering applications. Traditionally, the response of an elastic material is modeled within the framework of linearized theory of elasticity which is a first-order relationship between Cauchy stress and stretch tensors. However, the same relationship predicts singular strain when applied to model cracks and fractures. This clearly a violation of the basic assumption with which the theory is predicated upon. Additionally, when the linear theory applied to study the heat conduction, coupled with Fourier’s law, erroneously predict that the temperature propagates with infinite velocity. Thus, a captivating question is to develop the novel constitutive relations within the purview one can be content with large stress but most importantly infinitesimal strains at the crack-tip. Rajagopal [The elasticity of elasticity, ZAMP, 58(2):309–317, 2007] showed that one could arrive at a far-broader class of models by using the same linearization procedure as in the classical elasticity. A special sub-class of models in which linearized strain is a nonlinear function of Cauchy stress. More importantly, the strains are guaranteed to be bounded uniformly when such models are applied to study cracks and fractures. In this thesis, a main focus is on studying the numerical solution to a nonlinear thermoelastic model which is a coupled linear-quasilinear partial differential equation (PDEs) system. The field variables are displacement and temperature distribution in a linear, homogenous, and isotropic elastic solid. Due to the nonlinearity in the model, obtaining a closed form solution is a daunting task within the purview of the new framework developed in this thesis. Therefore, we seek a stable numerical solution to the partial differential equation system by using conforming bi-linear finite element method. And a weak formulation is derived based on the linearization at the PDE level using the Newton’s method. We consider two specific boundary value problems to study temperature distribution and the crack-tip stress-strain behavior. The numerical results depict a marked contrast to the classical linearized description of the material body. We observe a stress concentration at the crack-tip and more importantly, the near tip strain growth is much slower than the stress, which is consistent with the linearization procedure used in the derivation of the nonlinear model. Finally, our intention in this thesis is to provide a theoretical basis to develop physically meaningful models to study the evolution of complex network of cracks induced by thermal shocks.