Lord–Shulman Nonlinear Generalized Thermoviscoelasticity of a Strip

Abstract

In this paper, the response of a bounded one-dimensional medium (strip) subjected to a thermal shock is investigated. The strip is made of a linear visco-elastic material, of which the time-dependency of the elastic modulus is described by the simple Kelvin–Voigt model. To obtain the displacement, stress and temperature within the strip, the Lord and Shulman theory of generalized thermo-elasticity containing a single relaxation time is used. Three coupled equations, namely, the equation of motion, the modified Fourier law and the second law of thermodynamics, are established in terms of the displacement, temperature and heat flux. It is worth noting that the second law of thermodynamics is not linearized and is kept in the nonlinear form. The equations derived are first represented in a dimensionless presentation. Then, they are discretized using the well-known generalized differential quadrature. To obtain the response of the strip in time, the Newmark time marching scheme is implemented. It should be mentioned that due to the nonlinear nature of the governing equations, the successive Picard algorithm is used. The results of the present study are compared with those available in the literature for an elastic strip. Besides, numerical results are given for a strip made of Kelvin–Voigt visco-elastic material. The effects of visco-elastic parameter, coupling parameter, thermal relaxation time and nonlinearity are discussed in numerical examples.