Engineering Transactions, 50, 3, pp. 107–164, 2002

Thermodynamical Theory of Inelastic Single Crystals

Piotr Perzyna
Institute of Fundamental Technological Research, Polish Academy of Sciences
Poland

The paper aims at the development of the thermodynamic theory of elasto-viscoplasticity of single crystals which takes account of the evolution of the dislocation substructure. The next objective is the application of the theory developed for the investigation of the adiabatic shear-band formation in single crystals under dynamic loading processes. The description of the kinematics of finite elasto-viscoplastic deformations of single crystal is based on notions of the Riemannian space of manifolds and the tangent space. A multiplicative decomposition of the deformation gradient is adopted and the Lie derivative is used to define all objectives rates for the introduced vectors and tensors. A general constitutive model is developed within the thermodynamic framework of the rate-type covariance constitutive structure with finite set of the internal state variables, and takes account of the effects as follows: (i) thermomechanical coupling; (ii) influence of covariance terms, lattice deformations and rotations and plastic spin; (iii) evolution of the dislocation substructure; (iv) deviation from the Schmidt rule of a critical resolved shear stress for slip; (v) rate sensitivity (viscosity). A notion of covariance is understood in the sense of invariance under arbitrary spatial diffeomorphisms. The developed thermo-viscoplasticity theory of single crystals is based on the axioms as follows: (i) existence of the free energy function; (ii) invariance with respect to any diffeomorphism (any superposed motion); (iii) assumption of the entropy production inequality; (iv) assumption of the evolution equations for the internal state variables in the particular rate-dependent form. To describe the evolution of the dislocation substructure, a finite set of the internal state variables is interpreted as follows: the density of mobile dislocations, the density of obstacle dislocations and the concentration of the point defects. Physical foundations and experimental motivations are given. Two fundamental constitutive equations of the rate-type for the Kirchhoff stress tensor and temperature are formulated. To show that the thermodynamic theory of viscoplasticity of single crystals takes account of all the mentioned effects, an analysis of the thermomechanical couplings and internal dissipation is presented. Particular attention is focused on synergetic effects, generated by cooperative phenomena of thermomechanical couplings and the influence of the evolution of the dislocation substructure. The initial boundary value problem (the evolution problem) for rate-dependent elasto-plastic single crystal has been proved to be well posed. Criteria for adiabatic shear-band localization of plastic deformation are obtained by assuming that some eigenvalue of the instantaneous adiabatic acoustic tensor for rate-independent response is equal to zero. The formation of the adiabatic shear-band is investigated. It has been found that the synergetic effects generated by cooperative phenomena of thermomechanical couplings and the influence of the evolution of the dislocation substructure play a fundamental role in the inception of localization. The results obtained are compared with available experimental observations.
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