Eulerian theory of deformable environments. Problems of dislocation and straightening in solids (Q2863026)

This bulky book intends to present an innovative theory concerning the deformations of fluids and solids observed in Eulerian coordinates. It is divided into two parts: the first one covers a unified Eulerian approach to fluids and solids, which is composed of 17 chapters; the second part presents distortions, charges and plastic contortions in solids and contains 9 chapters. Six appendices complete the book, which either recall some useful tools of mathematics or physics, or give some further complements to different notions presented in the book. NEWLINENEWLINENEWLINE Chapter 1 describes the deformations of solids in Lagrangian coordinates. The author defines the notion of continua, of displacements and of velocity fields in a Lagrangian frame. Of the distortion tensor, he gives a general definition of deformation, which can be simplified in the case of small deformations. He also defines a vector associated to rotations and explains how this vector is related to local rotations. The chapter ends with considerations on compatibility conditions which have to be imposed on the distortion field. NEWLINENEWLINENEWLINE In Chapter 2, the author moves to the description of fluid and solid deformations in an Eulerian system of coordinates. He defines the notions of local mean velocity, of local density and of particulate derivative. He points out the non-commutativity of the derivatives with respect to time and to space in Eulerian coordinates. Then the author again defines a global rotation linked to local rotations. Finally, the possible deformations of the material are analyzed, computed and illustrated with appropriate figures. NEWLINENEWLINENEWLINE In Chapter 3, the author describes the principles of Newtonian mechanics in Eulerian coordinates. He starts from the kinetic energy and draws computations which lead to the continuity of the inertial mass. Then, using the first (resp. second) principle of thermodynamics, he writes down the continuity of the energy (resp. entropy). In the case of a non-deformable body, the author derives the conservation equations for energy and momentum. NEWLINENEWLINENEWLINE Chapter 4 is devoted to the description of the motion of viscoelastic fluids. Again, starting from the thermokinetic balance and from the conservation of energy and entropy, the author derives the equations of motion. NEWLINENEWLINENEWLINE Chapter 5 goes on with the description of the motion of viscoelastic fluids, but now considering a phenomenological approach. The author here starts from the free energy \(f(\tau ,T)\) of a particle, where \(\tau \) (resp. \(T\)) is the volumic expansion (resp. temperature), and he writes \(f(\tau ,T)=\varepsilon _{0}+f_{0}(T)-k_{0}(T)\tau +k_{1}(T)\tau ^{2}-k_{4}(T)\tau ^{3}\), \(\varepsilon _{0}\) (resp. \(f_{0}(T)\)) being the potential energy of interaction of a particle with its neighbors at \(\tau =0=T\) (resp. pure thermic component of the free energy of the fluid). He derives from the state equations the cases of a perfect gas, of an ideal liquid and of a perfect fluid, taking the first (resp. second, third) order of the asymptotic expansion of the free energy. The author then describes the transition liquid-gas and derives Clapeyron's formula. He finally gives the Navier-Stokes equations for viscoelastic perfect fluids and concludes with the description of wave propagation in viscoelastic fluids.NEWLINENEWLINEChapter 6 describes the evolution equations for chemical fluids, composed of \(N\) chemical species. In this case, the author writes down the continuity equation for each species, which links the local density \(n_{x}\), the speed \( \overrightarrow{\phi }_{x}\) and the volumic source \(S_{x}\), for the species \( x\) through \(\frac{dn_{x}}{dt}=S_{x}-\text{div}(n_{x}\overrightarrow{\phi } _{x})\). He then writes down the diffusion equation for each species and the global balance equation from which he derives the evolution equation for viscoelastic chemical fluids. The chapter again ends with the phenomenological relations, here for a chemical fluid, by introducing the state functions and equations. NEWLINENEWLINENEWLINE Chapter 7 goes deeper into the analysis of chemical fluids. The author first specializes the state equations for a perfect chemical gas and for a perfect chemical liquid. Then the case of binary fluid mixtures is considered. The author here introduces the collision frequency and Boltzmann probability. The chapter ends with the description of a special situation: the precipitation transition in a non-miscible liquid solute. NEWLINENEWLINENEWLINE In Chapter 8, the author introduces and studies the distortion and contortion fields for solids in Eulerian coordinates. He especially intends to prove that Eulerian coordinates are also helpful for the study of the motion of solids. The chapter starts with a description of solids at a macroscopic level, showing the crystalline structure of solids, introducing local frames attached to the crystals, and finally computing the projections of different quantities and resulting equations in these local frames. After some considerations of compatibility equations, the author introduces the contortion and distortion tensors. The chapter ends with the description of two cases: pure flexion and pure torsion. NEWLINENEWLINENEWLINE Chapter 9 moves to the description of elastic solids. Starting from the distortion and contortion tensors, as exposed in the preceding chapter, the author introduces the deformation, the stress tensor and a rotation vector. He then writes down the principle of Newtonian dynamics for a solid and the evolution equations for an elastic solid. Finally, he computes the phenomenological relationships for an elastic solid. NEWLINENEWLINENEWLINE Chapter 10 further analyzes the motion of elastic solids from a phenomenological point of view. In the case of a perfect isotropic solid, the author introduces the Lamé coefficients and computes Young's modulus and Poisson's ratio in a pure traction case. He also gives some considerations of anisotropic elastic solids and thermoconduction coefficients. In the case of perfect isotropic solids, the author writes down the evolution thermodynamics equations and studies the effects of pure shear perturbations and the wave propagation in such solids. NEWLINENEWLINENEWLINE Chapter 11 is devoted to the description of self-diffusive materials obtained when a hole or an extra crystal occurs in the grid. The author here gives again the continuity equations. He expresses Newton principles, which lead to the evolution equations. The chapter again ends with the phenomenological relationships in this case. NEWLINENEWLINENEWLINE Chapter 12 starts with the state equations for self-diffusive materials considering a phenomenological approach. Then the author writes down the transport equation and the matrix of kinetic coefficients. He describes the phenomena when pairs of holes or extra crystals are being created or destroyed in the grid. The evolution to a thermodynamical equilibrium and relaxation processes for pointwise defects are considered. Finally, the author examines possible analogies between pure shear effects in isotropic self-diffusive solids and magnetic effects, and studies the wave propagation in such isotropic self-diffusive solids. NEWLINENEWLINENEWLINE The purpose of Chapter 13 is to derive the evolution equations for solid solutions. Starting from the description of lacunar solid solutions, the author derives the evolution equations and presents the phenomenological relationships. NEWLINENEWLINENEWLINE Chapter 14 presents the precipitation and phase transition processes in solid solutions, starting with the description of the free interaction energy in a binary solid solution. NEWLINENEWLINENEWLINE Chapter 15 intends to derive the evolution equations for anelastic and plastic solids. The author first describes the two notions, then the Newtonian principles of dynamics which lead to the evolution equations, and once more the chapter ends with the phenomenological relationships for such solids. NEWLINENEWLINENEWLINE Chapters 16 and 17 are respectively devoted to the phenomenological description of anelastic and plastic solids. In Chapter 16, relaxation and hysteresis phenomena are studied. The author then describes the case of pure shears and the wave propagation in isotropic and anelastic solids. The chapter ends with the study of some structural transformations such as the martensitic ones. Chapter 17 describes the plasticity of solids, then the creep, or traction, or fatigue tests. This chapter 17 ends the first part of the book. NEWLINENEWLINENEWLINE The second part of the book starts with Chapter 18, which deals with the description of dislocation charge densities and fluxes and charge densities of disinclinations in solids. It first describes the plastic charges. As an illustration of this notion, the author describes the Volterra pipes. He then defines different types of charge densities and fluxes which lead to the continuity equations. NEWLINENEWLINENEWLINE The long Chapter 19 studies topological singularities associated with dislocation or disinclination charges. With appropriate figures, the author illustrates these two kinds of deformations that may occur. Throughout the whole chapter, these deformations are analyzed in different contexts. NEWLINENEWLINENEWLINE In Chapter 20, the author describes the dislocation charge fluxes and Orovan's relationships. He considers the continuity equations for dislocation charges and writes down the relationships between the dislocation fluxes and charges. The chapter ends with the properties of the Orovan relationships. NEWLINENEWLINENEWLINE Chapter 21 moves to the description of charged solids and Pearl and Kohler forces. The author establishes the way to write down the evolution equations for such solids from the corresponding ones for elastic and anelastic solids. The expression of the Pearl and Kohler force is then deduced and discussed in detail in the case of a dislocation line. NEWLINENEWLINENEWLINE Chapter 22 extends the results of Chapter 21, giving a phenomenological description of perfect charged solids. The author starts with a perfect charged solid and writes down the Maxwellian equations, first in the case of constant volumic expansion and then for usual solids with low variations of the volumic expansion of the solid. NEWLINENEWLINENEWLINE Chapter 23 is devoted to further study dislocation fields, energies and interactions between charges for a perfect solid. The author here considers different examples where he computes these physical quantities. NEWLINENEWLINENEWLINE In Chapter 24, the author introduces the Lorentz transformation in order to study the dynamics of a dislocation. He gives examples of such Lorentz transformations. He then defines the relativistic dynamics of a dislocation in different contexts, and finally establishes a link between the Pearl and Kohler force and the Lorentz relativistic force. NEWLINENEWLINENEWLINE In Chapter 25, the author considers a non-straight dislocation in a perfect solid, and he presents the rope model. He computes an expression for the Pearl and Kohler force in this context, and presents non-relativistic dynamical equations for rope models. Several examples are described. NEWLINENEWLINENEWLINE The final Chapter 26 presents the localized curvature singularities which may occur in a perfect solid. NEWLINENEWLINENEWLINE Throughout the whole book, the author takes care to illustrate the notions he presents with appropriate figures, and many chapters end with a table gathering the different equations they contain. Even if the book is written in French, it will surely be useful to students or reasearchers who want to learn more on solid or fluid mechanics.