Exact self similar solutions for nonlinear coupled heat elastic wave systems with temperature dependent material properties

Abstract

In this work, we study a coupled nonlinear thermoelastic system in which the heat capacity, thermal conductivity, and elastic modulus are all temperature-dependent. The thermal conduction equation is formulated in a generalized radial geometry governed by a parameter $\nu$, which encompasses planar, cylindrical, and spherical configurations as special cases. A complete Lie symmetry classification of the admissible constitutive functions is carried out, yielding three principal classes: exponential laws, power laws, and a unified class that contains both as limiting cases. For each class, the admitted symmetry algebras are determined, optimal systems of one-dimensional subalgebras are constructed, and the governing partial differential equations are reduced to coupled ordinary differential equation systems via similarity transformations. Exact closed-form solutions are obtained for each class using a combination of integrating factor methods, flux-variable substitutions, and power-law and logarithmic ans\"{a}tze. For the exponential class, the thermal field exhibits a logarithmic similarity profile while the displacement field decays algebraically, with the geometry parameter absorbed into the leading-order transformation. For the power-law class, the thermal field follows a self-similar power-law profile and the displacement field decomposes into distinct elastic, self-similar, and thermoelastically driven modes. Travelling-wave solutions are also constructed, with elementary forms recovered in limiting cases. The influence of geometry, particularly the structural distinction between cylindrical and other configurations, is identified and physically interpreted. All solutions are discussed in terms of the interplay between thermal diffusion, elastic stiffness, and thermoelastic coupling.