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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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Heat Conduction

Abstract
The central purpose of this chapter is to derive the Heat Equation, that is, the equation that describes temperature evolution in rigid conductors; this is done in Sect. 1.6, after the indispensable preliminaries are covered in Sects. 1.1–1.5. This derivation, which is of interest by itself, best exemplifies how constitutive issues are dealt with in modern continuum mechanics.
Paolo Podio-Guidugli

Chapter 2. Thermomechanics

Abstract
In this chapter, after we put together the balance and imbalance laws of thermomechanics in Sections 2.2 and 2.3, we further exemplify how a systematic use of the Coleman-Noll sieve leads to thermodynamically consistent constitutive prescriptions of increasing complexity, elastic in Section 2.4 and viscoelastic in Section 2.5.
Paolo Podio-Guidugli

Chapter 3. The Principle of Virtual Powers

Abstract
Hodiernal continuum mechanics is multiscale and multiphysics, in that it deals with situations where interdependent phenomena take place at different scales in material bodies that often must be regarded as an interactive composition of more than one of the basic structures of phenomenological physics, that is to say, the mechanical, thermal, electrical, and magnetic structures. Plurality of scales and physical structures calls for various adjustments of the standard modeling format.
Paolo Podio-Guidugli

Chapter 4. A Virtual Power Format for Thermomechanics

Abstract
When regarded as the composition of two material body structures, the one mechanical the other thermal, thermomechanics epitomizes multiphysics theories.
Paolo Podio-Guidugli

Chapter 5. A Physical Interpretation of Thermal Displacement

Abstract
Until recently, the use of thermal displacement—and of its gradient—in modern narrations of continuum thermodynamics (see e.g. [4, 5, 23–25, 54]) has been interesting but at bottom formal: a physical interpretation was wanted, a discrete, and hence microscopic, scale-bridging interpretation being both unequivocal and explicit, and not, as the one put forward in [23], just allusively inferred from the well-known statistical interpretation of temperature. Such an interpretation may be found in [56], a note whose contents are faithfully reproduced in this chapter.
Paolo Podio-Guidugli

Backmatter

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