Title:

A finite element method for the stress analysis of plane thermoviscoelastic bodies

The analysis presented in this thesis is an application of the finite element method to the determination of stresses in plane, linear viscoelastic bodies. In Chapter 1, the basic concepts of viscoelastic behaviour are introduced and some simple mathematical models are considered. Two classical forms for mathematical representation of viscoelastic behaviour are presented. These are the differential operator form and the hereditary integral form. An example problem is solved using the differential operator form by applying the elasticL viscoelastic analogy. The concept of timetemperature equivalence is introduced to take into account the affect of temperature on the viscoelastic material responses. The hereditary integral representation is adopted in this thesis owing to the ease with which the measured creep or relaxations can be incorporated and also to the superiority of integrals over the differential operator representation, in taking into account the affect of temperature on the responses. In Chapter 2, a stress analysis for plane problems in linear thermoviscoelasticity using the finite element formulation is presented. The method employed is based on the assumptions that: (1) the material is isotropic, homogeneous and linear, (2) the stressstrain laws are expressed in the hereditary integral form, (3) the material is thermorheologically simple, which implies the validity L of timetemperature equivalence hypothesis, and finally, (4) the viscoelastic stress response is independent of thermal response. The associated computer program includes both linear and quadratic isoparametric elements and the frontal method of Gaussian elimination is employed. The element matrices that result in the equilibrium equations involve hereditary integrals which are approximated by a stable and converging trapezoidal finite difference scheme for time marching. The solutions for two problems are compared with analytical results evaluated by the integral transform method. Since it is assumed that the thermal response is independent of the stress response, a nonlinear heat conduction analysis may be performed concurrently with the stress analysis. The basic equations and the finite element method of analysis for heat conduction are presented briefly. For approximate solutions which require less computer time alternative forms of the equilibrium equations utilizing an iterative technique is presented and an example problem is included. Finally, the affect of incompressibility is considered for an axisymmetric problem. The equlibrium equations based on the theorem of minimum potential energy are inaccurate for Poisson's ratios in the vicinity of 0.5, and degenerate completely for the case of incompressibility. Based on the assumptions made in Chapter 2, an alternative formulation of the viscoelastic stress analysis is presented in Chapter 3. The problem now has two displacement variables and a mean pressure variable. A variational principle is presented whose simultaneous minimisation with respect to the displacement variables and pressure variable leads to the equilibrium equations which are valid for all admissible values of Poisson's ratio. The hereditary integrals that result are approximated by a finite difference scheme for time marching and the finite element method is used for spatial discretisation and solution. An example problem is solved to examine the accuracy of the present formulation. Similar to the iterative technique developed in Chapter 4, an alternative form of the equilibrium equation is also presented here for approximate solutions that require less computer time. In Chapter 4 an orthotropic viscoelastic stress analysis is presented with particular reference to the modelling of drying stresses in timber. The heat and mass transfer taking place during drying are assumed to be independent of viscoelastic stress relaxation. Luikov's partial differential equations are used to model the heat and mass transfer and the finite element method is used for the solution. The shrinkage strains that develop during drying of timber are taken as input to the stress analysis problem. The effects of temperature and moisture content on the viscoelastic material responses are taken into account by postulating the validity of the timetemperature and moisture content equivalence hypothesis. The materials which can be characterised thus are called thermohygrorheologically simple models. Since all the material responses for wood which are required for stress analysis are yet to be evaluated experimentally, only a parametric study based on limited experimental results is presented in this thesis. A review of the literature relevant to each of the chapters is presented at the beginning of the chapters themselves and not as an independent entity.
