Governing equations of coupled thermoelasticity Sample Clauses

Governing equations of coupled thermoelasticity. The equation of motion for a 3D elastic body in the physical coordinate system (x, y, z) is stated as follows [41, 49] σi j, j + Xi = ρu¨i + ζ u˙i (4.1) where σi j and ui are stress and displacement components, respectively. Xi denotes body forces per unit volume, ρ is mass density, and ζ stands for the damping coef- ficient of the material. Likewise, the superscript dot (·) and the subscript comma (,) indicate the derivatives with respect to the time (t) and the space variables (x, y, z), respectively. In addition, the strain-displacement relations within the linear context of small de- formation theory are expressed as [41] εi j = 2 (ui, j + u j,i) (4.2) and according to GL generalized theory, Xxxxx’x law for a nonhomogeneous anisotropic thermoelastic material can be written as [41] σi j = Ci jpqεpq − βi j(T + t1T˙) (4.3) where Ci jpq is a fourth-order tensor containing all the elastic coefficients of a general nonhomogeneous anisotropic material. βi j = Ci jpqαpq is the second order tensor of thermoelastic moduli where in αi j is the coefficient of thermal expansion tensor. In