Numerical Analysis of the Behaviour of Rubber-Like Polymers with Hyperbolic Models of Nonlinear Thermoelasticity

Melnik, R.V.N., Strunin, D.V. and Roberts, A.J.

Proceedings of the XVI World Congress on Scientific Computation, Applied Mathematics and Simulation, Eds. Deville, M. and Owens, R., Paper 141-1, Lausanne, ISBN 3-9522075-1-9, pp. 1--6, 2000

Abstract:

In many industrial applications we have to deal with materials that exhibit typically nonlinear thermomechanical behaviour. Rubber-like polymers. Rubber-like polymers are one of the most cited examples of such materials. It is not a surprising fact, inasmuch as desirable thermomechanical properties of rubber-like polymers at an economical cost made these materials relatively cheap alternative to more traditional metallic materials in many consumer products, as well as in automotive and aircraft industries. The existing applications of polymers and the design of polymers with improved thermomechanical characteristics require the development of adequate mathematical tools to describethe coupled thermomechanical dynamics of these materials.

Although rubber-like polymers provide a classical example where nonlinear effects are essential in the adequate description of material dynamics, only a few recent papers have discussed numerical results obtained for these materialswith general nonlinear models of coupled thermoelasticity. In dealing with the nonlinear behaviour of such materials we must give a proper weight to the fact that nonlinearities in the dynamics of rubber-like materials are closely interwoven with the effect of thermal relaxation time. Mathematical models of hyperbolic thermoelasticity, known as models with relaxation times, are based on hyperbolic-type equations for temperature. Such models deserve more attention than has been accorded in the past. Despite an increasing attention that has been given in recent years to the numerical analysis of nonlinear effects in classical thermoelasticity, the results on the influence of the combined effects of nonlinearity and the thermal relaxation on thermomechanical properties of materials are virtually absent in the literature. However, in many cases these effects become key factors in the adequate description of the dynamical behaviour of thermoelastic systems. In particular, it is well-known that nonlinear terms in the equation of motion and the energy balance equation may lead to the formation of discontinuous profiles of displacement and temperature (shock profiles). In the regions of such discontinuities the combined effects of nonlinearity and the thermal relaxation become significant. From a practical point of view, these effects are especially important in such areas as manufacturing of electronic components and process engineering to name just a few.

In this paper we deal with rubber-like solid polymers that, in contrast to many other solid materials such as metals or ceramics, are energy elastic, and since they have relatively large thermal relaxation times, they demonstrate hyperbolic effects in a more pronounced fashion compared to "hard" solids. We study numerically a combined effect of hyperbolicity and nonlinearity on the dynamics of such materials. Modelling the thermomechanical behaviour of a ring-shaped body with the Lord-Shulman-type model, we demonstrate an intrinsic connection between nonlinear effects and hyperbolicity of the dynamics. In particular, we show that the vanishing relaxation time can lead to the superposition of nonlinearities.

Key words: polymer modelling; coupled thermomechanical dynamics; hyperbolic effects and thermomechanical interactions; nonlinear effects in thermoelasticity; Lord-Shulman-type models; rubber-like solid polymers; energy elastic materials; numerical analysis; shock profiles; thermal relaxation time.