Constitutive theories for thermoelastic solids in Lagrangian description using Gibbs potential
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This paper presents constitutive theories for finite deformation of homogeneous, isotropic thermoelastic solids in Lagrangian description using Gibbs potential. Since conservation of mass, balance of momenta and the energy equation are independent of the constitution of the matter, the second law of thermodynamics, that is, entropy inequality, must form the basis for all constitutive theories of the deforming matter to ensure thermodynamic equilibrium during the evolution (Surana and Reddy in Continuum mechanics, 2012; Eringen in Nonlinear theory of continuous media. McGraw-Hill, New York, 1962). The entropy inequality expressed in terms of Helmholtz free energy is recast in terms of Gibbs potential. The conditions resulting from the entropy inequality expressed in terms of Gibbs potential permit the derivation of the constitutive theory for the strain tensor in terms of the conjugate stress tensor and the constitutive theory for the heat vector. In the work presented here, it is shown that using the conditions resulting from the entropy inequality, the constitutive theory for the strain tensor can be derived using three different approaches: (i) assuming the Gibbs potential to be a function of the invariants of the conjugate stress tensor and then using the conditions resulting from the entropy inequality, (ii) using theory of generators and invariants and (iii) expanding Gibbs potential in the conjugate stress tensor using Taylor series about a known configuration and then using the conditions resulting from the entropy inequality. The constitutive theories resulting from these three approaches are compared for equivalence between them as well as their merits and shortcomings. The constitutive theory for the heat vector can also be derived either directly using the conditions resulting from the entropy inequality or using the theory of generators and invariants. The derivation of the constitutive theory for the heat vector using the theory of generators and invariants with the complete set of argument tensors yields a more comprehensive constitutive theory for the heat vector. In the work, we consider both approaches. Summaries of the constitutive theories using parallel approaches (as described above) resulting from the entropy inequality expressed in terms of Helmholtz free-energy density are also presented and compared for equivalence with the constitutive theories derived using Gibbs potential. 2013 Springer-Verlag Wien.