Institute of Fundamental Technological Research

Mathematical and physical aspects of the applicability of the Onsager, Prigogine as well as the Glansdorff and Ziegler thermodynamic extremal principles (TEPs) to non-equilibrium thermodynamics are examined for systems at fixed temperature with respect to their ability to provide kinetic equations approved in materials science. TEPs represent an alternative to the classical phenomenological equations approach. As TEPs are, more or less, a pure mathematical tool, they cannot significantly contribute to a deeper physical understanding. However, if a system can be described by discrete characteristic (thermodynamic) parameters, it is demonstrated that application of Onsager’s TEP or Ziegler’s TEP represents a systematic way to derive a set of explicit evolution equations for these parameters. This approach can significantly simplify the treatment of the problem and, thus, can also be applied to rather complex systems, for which the classical approach, involving application of phenomenological equations, fails. The application of TEPs is demonstrated on plasticity with respect to constitutive equations as well as on grain growth and coarsening with respect to evolution equations of discrete parameters. No exploitation of Prigogine’s TEP has been reported for applications in materials science. Contrarily, Prigogine’s TEP can be invalidated if the coefficients of the dissipation function depend on the evolution of state variables with time. This is demonstrated by a further practical example worked out for the solute drag phenomenon. Glansdorff’s and Prigogine’s evolution criterion, however, is always fulfilled near the equilibrium state of convex Gibbs energy. Extensions of TEPs to non-linear non-equilibrium thermodynamics are demonstrated for homogeneous and quasi-homogeneous dissipation functions.

Non-equilibrium, Thermodynamics, Entropy, Onsager’s principle, Thermodynamic extremal principles

The energy criterion of stability of equilibrium is used to predict the formation of discrete twin bands running across the grains of polycrystalline solids. An uncontrolled rapid development of a twin band is assumed to occur when the associated release of elastic energy compensates the interfacial energy and intrinsic dissipation due to twinning. A simplified analysis is performed that leads to compact analytic formulae for the thickness of a twin band and for the critical resolved stress. The creation of subsequent discrete twin bands in a finite solid can be separated by the periods of quasistatic growth of existing twins under varying loads. An application of the energy criterion to twinning in TiAl intermetallics and its verification by observed data for the thickness of a twin band are presented.