Galerkin-type solution for the theory of strain and temperature rate-dependent thermoelasticity

Abstract

Ultrafast heating has gained serious attention due to technological advancements in recent years. The thermoelastic process accounting for the finite speed of thermal signals is therefore becoming increasingly important. With the help of extended thermodynamics, Yu et al. (Meccanica 53(10):2543–2554, 2018) have established a thermoelastic model by including the strain rate and temperature rate in the constitutive equations. This model is also an attempt to remove the discontinuity in the displacement field under temperature rate-dependent thermoelasticity theory. The present work seeks to derive the representation of a Galerkin-type solution in the context of this recently proposed model in terms of elementary functions. The representation theorem of the Galerkin-type solution of the system of equations for steady oscillation is also proved. In accordance with this theorem, we finally establish a theorem which expresses the general solution of the system of homogeneous equations of steady oscillation in terms of metaharmonic functions.