Formalized approach to construction of the state equations for complex media under finite deformations

Abstract

All kinematic and force quantities in a complex medium are defined by the history of thermo-elastic-inelastic processes occurring in it. For description of the process history, the procedure based on the kinematics of superposition of small deformations on the finite ones has proved to be more suitable. In the present paper, we use this procedure to construct both the kinematic relations of thermo-elastic-inelastic processes under finite strains and the constitutive equations. The obtained multiplicative decomposition of the total deformation gradient into elastic, inelastic, and temperature terms is free of the shortcomings inherent in the kinematics description of other authors. The constitutive equations are constructed based on the total and elastic deformation gradients and the elastic potential. These equations satisfy the thermodynamic principles and the objectivity law. They are derived relative to the intermediate configuration close to the current one and reduced to exact evolutionary relations with the objective derivative, which appears as a result of the limiting process. We introduce a functional, which in a purely elastic process is the elastic potential, and use it as a term of the expression for free energy. The thermodynamic validity of the constitutive equations has been justified. The heat-conduction equation for thermo-elastic-inelastic process is constructed and the constraints, imposed by thermodynamics and the objectivity principle on the inelastic process involving structural changes of the material and on the inelastic and temperature kinematics, are determined. The obtained theoretical results are tested for correctness and usefulness in the context of some simple thermo-elastic-inelastic problems.