Numerical solution of Lord-Shulman thermopiezoelectricity dynamical problem

Abstract

Abstract. Using Lord-Shulman hypothesis we formulate the initial boundary value problem and its corresponding variational problem of a generalized linear thermopiezoelectricity in terms of the displacement, electrical potential, temperature increment and heat flux, which describes the dynamic behavior of the coupled mechanic, electric and heat waves in pyroelectric materials. We construct the corresponding energy balance equation and determine input data regularity for the variational problem, which guarantees the existence, uniqueness and stability of its solution in the problem energy norm. Based on these results, we propose a numerical scheme for solving this problem, which includes spatial finite element semi-discretization and one-step recurrent time integration procedures and generalizes the similar one for classic thermopiezoelectricity problem. We give the sufficient conditions on the values of the scheme parameters which guarantee properties of conservatism and unconditional stability of the scheme. The rest of the article is devoted to the analysis of performed numerical experiments with 1D model problem and their results are then compared with the ones obtained by the other researchers.

Nonuniform laminated beam of Lord–Shulman type

“…In Ref. 23, the authors considered the numerical solution of the Lord–Shulman thermopiezoelectricity dynamical problem. A model of the equations of generalized thermoelasticity based on the Lord–Shulman theory in an isotropic elastic medium under the dependence of the modulus of elasticity on the reference temperature was established in Ref.…”