Lukas Schneider

Solid-Fluid Mixtures of Frictional Materials in Geophysical and Geotechnical Context

Mathematical Preliminaries and Notations.- to Mixture Theory.- Constitutive Assumptions.- Entropy Principle and Transformation of the Entropy Inequality.- Thermodynamic Analysis I Liu Identities, One-Forms and Integrability Conditions.- Thermodynamic Analysis II Residual Inequality, Thermodynamic Equilibrium, Isotropic Expansion.- Reduced Model.- Discussions and Conclusions.

Discussions and Conclusions

We review the essentials of the theory, discuss the highlights of the achievements and the limitations of the model equations, and give an outlook of some still unsolved problems.

Thermodynamic Analysis I Liu Identities, One-Forms and Integrability Conditions

The extended entropy inequality, derived and stated at the end of Chap. 5 is used to derive the LIU identities and the reduced entropy inequality in Sect. 6.1. Further reductions of the former are only possible if the implications of the symmetry group of the material are accounted for. This is done here for isotropy of the mixture, but also requires a number of ad-hoc assumptions. The most significant ones of these suppose (i) that the LAGRANGE multiplier of the mixture energy equation is a universal function of the (empirical) temperature and its time rate of change and (ii) that the LAGRANGE multiplier of the constituent momentum equation is proportional to the negative constituent diffusion velocity with the LAGRANGE multiplier of the energy as proportionality factor. On the basis of a number of Lemmas proved by LIU and an additional theorem motivated by him and a few technical assumptions restricting the functional dependence of two-forms arising in the LIU identities, we are then able to define the constituent thermodynamic pressures, the constituent configuration pressures and constituent free enthalpies as quantities that are derivable from a HELMHOLTZ-like free energy (whose number of independent variables is drastically reduced) and the saturation pressure. Moreover, all LAGRANGE multipliers can be expressed in terms of these variables; more specifically, they all have the LAGRANGE multiplier of the energy as a common factor. Some of the above mentioned ad-hoc assumptions are physically motivated, others are mathematically enforced, but all clearly state the conditions for which the validity of the theory ensues.

Mathematical Preliminaries and Notations

In the first part of this chapter we present the symbolic and the Cartesian tensor notations and show how these are applied in this book. Tensor calculus is presumed known to the reader; so, only specifics and peculiarities pertinent to the work are discussed. In the second part the elements of exterior calculus are explained, but only to the extent as they are used in the thermodynamic approach treated later on, in particular in Chap. 5.

Thermodynamic Analysis II Residual Inequality, Thermodynamic Equilibrium, Isotropic Expansion

After the full exploitation of the LIU identities in the preceding chapter, we draw in this chapter some (but not all) inferences which follow from the condition that the entropy production density assumes its minimum value in thermodynamic equilibrium. This requirement implies that ∂π ρη /∂n I | E = 0, where π ρη is the entropy production density and n I are those independent variables which vanish in equilibrium. The evaluation of this condition first requires π ρη to be expressed in an appropriate form. Choosing for n I , in turn, the variables v α , \(\dot \theta \), ∇ θ and D α , which are the constituent velocities, the time rates of change of the temperature, the temperature gradient and constituent stretchings, allows evaluation of the equilibrium representations of the constituent interaction forces, entropy, heat flux vector and constituent stress tensors, which exhibit a clear structure of their dependences on (i) a thermodynamic potential (HELMHOLTZ-like free energy) and thermodynamic, configuration and saturation pressures, (ii) extra entropy flux, (iii) frictional effects via their production terms and (iv) interaction rate densities of constituent mass and volume fractions. It becomes very clear how the various equilibrium terms are affected if simplifying assumptions are made about the functional dependencies of the above mentioned production terms.

Entropy Principle and Transformation of the Entropy Inequality

After a very brief introduction into the recent developments of modern rational thermodynamics with essentially two competing mathematical postulates for the exploitation of the Second Law of Thermodynamics we concentrate on the entropy principle of I. MULLER with its LAGRANGE multipliers technique of exploitation by LIU. We sketch LIU’s proof of how the entropy inequality, augmented by the LAGRANGE multiplied balance laws’ is reduced to the so-called LIU identities and the reduced entropy inequality. In this process the physical assumption [A10] that external source terms cannot affect the material behaviour is significant. Computations for the fluid-solid saturated mixture with an arbitrary number of constituents of which some may be compressible are rather involved and are partly deferred into Appendices. The chapter ends with inequality (5.33) from which the concrete thermodynamic analysis ensues.

Introduction to Mixture Theory

After a general description of mixtures and multi-phase systems and their difference, reasons are given why their distinctions are premature prior to a complete thermodynamic exploitation of postulated constitutive relations by the Second Law of Thermodynamics. Consequently, both systems are here denoted as mixtures. Kinematics is treated first. Then, the general balance laws and their specializations for constituent mass, momenta, energy and entropy are discussed in global and local forms as well as jump conditions across singular surfaces. Based on Truesdell’s metaphysical principles the sum relations define the corresponding mixture quantities which obey the physical balance laws for the mixture as a whole.