ABSTRACT
Thermo hypo-elastic solids are a class of materials in which the ordered rate constitutive equations consist of constitutive theories in which the stress rates are functions of the strain rates. We consider derivations of general ordered rate constitutive theories for thermo hypo-elastic solids for which the mathematical models are in the Eulerian description. In the derivations of the constitutive theories we consider conjugate Cauchy stress and strain measures and their convected time derivatives in contravariant and covariant bases as well as Jaumann stress and strain rates. In these constitutive theories, the first convected time derivative of the chosen deviatoric Cauchy stress tensor and the heat vector are expressed as functions of the convected time derivatives up to any desired order of the conjugate strain tensor, density ρ¯, temperature θ¯ and temperature gradient g¯. We begin all developments with entropy inequality, an essential conservation law for the development of the constitutive theory. The chosen Cauchy stress tensor is decomposed into equilibrium stress and deviatoric stress as necessitated by the entropy inequality. The constitutive equation for the equilibrium stress for both compressible and incompressible cases is established using entropy inequality. For the deviatoric Cauchy stress tensor, the entropy inequality does not provide any mechanism for establishing the constitutive theory. In the present work we use the theory of generators and invariants to (i) establish a most general form of the rate constitutive theory in which the first convected time derivative of the chosen deviatoric Cauchy stress tensor can be a function of the convected time derivatives up to any desired order of the conjugate strain tensor (and other arguments), (ii) specialize the general theory presented in (i) to second order thermo hypo-elastic solids and (iii) further specialize the theory presented in (ii) to first order thermo
hypo-elastic solids and demonstrate that the general constitutive theory of ordered thermo hypo-elastic solids of order one reduces to the well known hypo-elasticity with further assumptions. All derivations and details for (i)– (iii) are presented using contravariant and covariant bases as well as Jaumann rates for incompressible and compressible thermo hypo-elastic solids. Discussion and arguments are presented for the validity and usefulness of the contravariant, covariant and well as the validity of Jaumann rate constitutive equations. See references [6-9, 11, 14, 15, 39, 68, 72-75,78-80,103-110] for published literature.
