Any theory of continuum thermomechanics presupposes, for the formulation of conservation laws and constitutive equations, some basic geometry, whose choice has deep consequences for the effectiveness of the model to describe the phenomena. In continua with a distribution of singularities representing local discontinuities of scalar fields (temperature, pressure, temperature) or vectorial fields (dislocations, disclinations, microcracks, integranular decohesions, vortices) this choice is especially relevant, as these singularities represent changes not only in the metric but also in the topology of the system.
This book gives a detailed theoretical basis for the thermomechanics of materials in the presence of field singularities. These features often appear in continuum mechanics, mainly in solids with microdefects or large deformations, or in granular systems, or in fluids with vorticity, to mention only a few examples. The creation of singularity distribution, and its evolution, involves a topological change (connectivity) besides a classical geometrical change (shape and size), in such a way that not only the metric tensor must be used, but also an affine connection which may be in turn related to nonvanishing torsion and curvature tensors as additional independent variables of the system. Furthermore, these changes in topology represent an irreversible contribution to the dynamics of the system and, accordingly, they must be suitably incorporated in the entropy production. Differential geometry, mechanical and thermodynamical laws converge in this book in the description of the mentioned systems and relate the irreversible part of the deformation to the change in the affine structure of the continuum, by extending and updating early ideas by Cartan in 1923. The book uses the general ideas of rational thermodynamics of Coleman, Truesdell and Noll, but with suitable changes to take into account his purposes.
The author takes as independent variables the metric tensor, the volume form, the torsion and curvature tensors, temperature and temperature gradient, and it incorporates, besides the usual thermal and viscous dissipation, the dissipation due to singularity, induced by the change in the affine structure of the continuum related to the change of topology. There follow the principles of determinism, local action, frame indifference and Clausius-Duhem inequality, but with some subtle modifications. For instance, in defining the times derivatives, the author outlines that their definition with respect to the continuum coordinates is more appropriate for the global laws and localization than with respect to an external referencial system; furthermore, the author refines the idea of frame indifference which was based on a metric tensor, to take into account affine connections, and it is shown how to redefine the divergence and rotational operators to take into account the extension of the geometrical internal structure of the system.
The book is organized in six chapters and five appendices, and it has a wide list of more than two hundred references. The three first chapters are devoted to general ideas: geometry and kinematics, conservation laws, and a general overview of continuum with singularity. The three other ones deal with three specific domains of application, namely thermoviscous fluids, thermoviscous solids, and solids with dry microcracks. The chapter on fluids pays special attention to turbulence, and to the fact that turbulent flows are not only chaotic, but they also imply changes in local topology due to creation and destruction of vortices, and emphasizes that the presence of the torsion field captures the vortices source. The chapter on thermoviscous solids deals, among other topics, with the analysis of ultrasonic signals through defected materials, which is of practical importance for exploration and diagnosis. Finally, in the chapter on solids with dry microcracks the author compares the micromechanics approach and the continuum approach in two situations. The appendices give a useful summary of mathematical preliminaries (vectors, tensors, topological spaces, manifolds, invariance groups and physical laws, affinely connected manifolds, Bianchi identities, and the theorem of Cauchy-Weil).
In my opinion, the book is excellent: it is well structured, well focused on an interesting topic, and it is clearly developed. It combines mathematical rigor with deeply physical motivations, many of them of much current interest in material sciences or in the fundamentals of thermodynamics. The proposals of the author seem to be a worthwhile contribution to a mathematically sound and physically fruitful description of many interesting phenomena.