Continuum Thermomechanics

The general goal of this book is to deduce rigorously, from the first principles, the partial differential equations governing the thermodynamic processes undergone by continuum media under forces and heat. Solids and fluids are considered in a unified framework. Reacting mixtures of fluids are also included for which general notions of thermodynamics are recalled, such as the Gibbs equilibrium theory. Linear approximate models are mathematically obtained by calculating the derivatives of the constitutive response functions. They include the classical models for linear vibrations of thermoelastic solids and also for wave propagation in fluids (dissipative and non-dissipative acoustics and internal gravity waves).

Contributed by world-renowned specialists on the occasion of Paul Germain's 80th birthday, this unique book reflects the foundational works and the intellectual influence of this author. It presents the realm of modern thermomechanics with its extraordinary wealth of applications to the behaviour of materials, whether solid or fluid. The thirty-one contributions follow an easygoing autobiographical sketch by Paul Germain, and highlight the power and richness of a methodological approach to the phenomenology of many materials. This approach combines harmoniously thermodynamics and continuum theory in order to provide exploitable, thermodynamically admissible models of a large variety of behaviours and phenomena, including those of diffusion, thermoelasticity, viscoplasticity, relaxation, hysteresis, wetting, shape-memory effects, growth, phase transitions, stability, fracture, shocks, machining of materials, microstructured solids, complex fluids, etc. Especially aimed at graduate students, researchers, and engineers in mechanical engineering and materials science, this book also presents the state of the art in an active field of research and opens new horizons in other scientific fields, such as applied mathematics and applied physics, because of the intellectual satisfaction and remarkable efficiency provided by the advocated approach.

This book deals with an important topic in rational continuum physics, thermodynamics.Although slim, it is fairly well self-contained; some basic notions in continuum mechanics, which a well-intentioned reader should but may not be familiar with, are collected in a final appendix. Modern continuum thermodynamics is a field theory devised to handle a large class of processes that typically are neither spatially homogeneous nor sequences of equilibrium states. The most basic chapter addresses the continuum theory of heat conduction, in which the constitutive laws furnish a mathematical characterization of the macroscopic manifestations of those fluctuations in position and velocity of the microscopic matter constituents that statistical thermodynamics considers collectively. In addition to a nonstandard exposition of the conceptual steps leading to the classical heat equation, the crucial assumption that energy and entropy inflows should be proportional is discussed and a hyperbolic version of that prototypical parabolic PDE is presented. Thermomechanics comes next, a slightly more complex paradigmatic example of a field theory where microscopic and macroscopic manifestations of motion become intertwined. Finally, a virtual power format for thermomechanics is proposed, whose formulation requires that temperature is regarded formally as the time derivative of thermal displacement. It is shown that this format permits an alternative formulation of the theory of heat conduction, and a physical interpretation of the notion of thermal displacement is given. It is addressed to mathematical modelers – or mathematical modelers to be – of continuous material bodies, be they mathematicians, physicists, or mathematically versed engineers.

This book is a unique presentation of thermodynamic methods of construction of continuous models. It is based on a uniform approach following from the entropy inequality and using Lagrange multipliers as auxiliary quantities in its evaluation. It covers a wide range of models — ideal gases, thermoviscoelastic fluids, thermoelastic and thermoviscoelastic solids, plastic polycrystals, miscible and immiscible mixtures, and many others. The structure of phenomenological thermodynamics is justified by a systematic derivation from the Liouville equation, through the BBGKY-hierarchy-derived Boltzmann equation, to an extended thermodynamics. In order to simplify the reading, an extensive introduction to classical continuum mechanics and thermostatics is included. As a complementary volume to Part II, which will contain applications and examples, and to Part III, which will cover numerical methods, only a few simple examples are presented in this first Part. One exception is an extensive example of a linear poroelastic material because it will not appear in future Parts.The book is the first presentation of continuum thermodynamics in which foundations of continuum mechanics, microscopic foundations and transition to extended thermodynamics, applications of extended thermodynamics beyond ideal gases, and thermodynamic foundations of various material theories are exposed in a uniform and rational way. The book may serve both as a support for advanced courses as well as a desk reference.

This book is a unique presentation of thermodynamic methods of construction of continuous models. It is based on a uniform approach following from the entropy inequality and using Lagrange multipliers as auxiliary quantities in its evaluation. It covers a wide range of models OCo ideal gases, thermoviscoelastic fluids, thermoelastic and thermoviscoelastic solids, plastic polycrystals, miscible and immiscible mixtures, and many others. The structure of phenomenological thermodynamics is justified by a systematic derivation from the Liouville equation, through the BBGKY-hierarchy-derived Boltzmann equation, to an extended thermodynamics. In order to simplify the reading, an extensive introduction to classical continuum mechanics and thermostatics is included. As a complementary volume to Part II, which will contain applications and examples, and to Part III, which will cover numerical methods, only a few simple examples are presented in this first Part. One exception is an extensive example of a linear poroelastic material because it will not appear in future Parts. The book is the first presentation of continuum thermodynamics in which foundations of continuum mechanics, microscopic foundations and transition to extended thermodynamics, applications of extended thermodynamics beyond ideal gases, and thermodynamic foundations of various material theories are exposed in a uniform and rational way. The book may serve both as a support for advanced courses as well as a desk reference.

This book presents different thermodynamic approaches in the area of constitutive theory: thermodynamics of irreversible processes, rational thermodynamics, and extended thermodynamics. These different approaches are analyzed with respect to their presuppositions, as well as to their results, and each method is applied to several important examples. In many cases these examples are archetypes for numerous technologically important materials; i.e. complex materials having an internal structure. Some of the examples dealt with in this book are liquid crystals, colloid suspensions, ans fiber suspensions. The book well serves students and researchers who have basic knowledge in continuum mechanics and thermodynamics. It provides a systematic overview of the vast field of thermodynamic constitutive theory, beginning from a historical perspective and concluding with outstanding questions in recent research.

A complete treatment of continuum thermodynamics with applications to material modelling, packed with examples and illustrations.