Basic Problems of Coupled Thermoelasticity with Thermal Relaxation and Pre-Stress: Aspects Observed in Exact and Asymptotic Solutions

Abstract

Some 1-and 2-D dynamic problems of coupled thermoelasticity in an unbounded solid, half-space and slab are treated. The problems are basic, and have been addressed in the literature. This treatment considers (possibly finite) pre-stress and both Fourier and thermal relaxation models, and generates analytical results from exact solutions or inversions of asymptotic expressions for exact transform solutions. The results illustrate the role of wave speeds in solution behavior, how pre-stress affects these speeds, and how the models themselves define solution behavior, especially near wave fronts.

Transient Plane Wave Solutions to Homogeneous Equations of Orthotropic Thermoelasticity With Thermal Relaxation: Illustration

“…Transient studies [ 1,2] suggest that the complementary (homogeneous) solutions for isotropic, coupled thermoelasticity with thermal relaxation allow plane wave propagation without attenuation or dispersion. A more recent transient study [3] considers orthotropic, coupled thermoelasticity with relaxation and finds complementary solutions that allow steps in temperature and stress to propagate as plane waves without attenuation or dispersion.…”

“…To facilitate study, its coefficients are expressed as functions of damping coefficient, thermal relaxation time, thermoelastic characteristic length, and four speeds. Two of the four arise in isothermal isotropic studies [12]; the other two are their analogs [2,3] for Fourier heat conduction [13]. It is these speeds (in dimensionless form) that introduce dependence on propagation direction and the remaining material constants.…”

Reflection of transient plane step-stress waves: Some considerations of orthotropy and thermoelasticity

“…Available data [Brock 2009] suggest that 1 < c I− < c D < c F < c I+ . Equation (35a) shows that the components of u x , u y corresponding to c 3I uncouple from θ , i.e., are shear waves defined by arbitrary functions of τ − x.…”

“…The inequalities in (31) are based on data [Ignaczak and Ostoja-Starzewski 2010;Brock 2009]. The roots of (29a), (29b) and (29c) give dimensionless speeds c = c F± , c = c 1I , c 2I , c 3I and c = c 1II , c 2II , c 3II .…”

“…Moreover (35) and (36) give the dimensionless speeds in transient two-dimensional studies that are valid for short times (τ/ h I 1 and τ/ h II 1 respectively); see [Brock 2009]. …”