Angelo Morro - Böcker

For graduate students and researchers, this self contained text provides a carefully structured, coherent, and comprehensive treatment of the mathematical modelling in electromagnetism of continuous media. The authors provide a systematic review of known subjects along with many original results.Part I reviews basic notions and approaches in electromagnetism (Maxwell's equations, Green's functions, harmonic fields, dispersive effects) and emphasizes the physical motivation for the modelling of non-conventional materials. The frequency-dependent properties (such as conductivity, polarizability, and magnetizability), which enter wave diffraction and dispersion, are shown, and these lead to a discussion of models of materials with fading memory in the time domain. Part II develops the thermodynamics of electromagnetic and thermoelectromagnetic materials with memory and provides a systematic account of thermodynamic restrictions. Existence, uniqueness and stability problems are investigated. Also, variational formulations and wave propagation solution are established.Part III is devoted to more involved models which are motivated by the interest in materials and structures with non-conventional properties. The mathematical modelling deals with non-linearity, non-locality and hysteresis. In non-linear materials attention is focussed on the generation of harmonics and in discontinuity waves. Non-locality is examined in a general way and hence is applied to superconductivity. Hysteresis is developed for magnetism. A review of known schemes is given along with new results about the modelling of hysteresis loops. The wide application of technologies in new mechanical, electronic and biomedical systems calls for materials and structures with non-conventional properties (e.g materials with 'memory'). Of equal importance is the understanding of the physical behaviour of these materials and consequently developing mathematical modelling techniques for prediction.Includes appendices that include some properties of Bessel functions, Fourier transforms and Sobolev spaces, compact operators and eigenfunctions, differential operators in curvilinear coordinates, and finite formulation of electromagnetism.

A crucial stability condition in linear viscoelasticity is that the Fourier cosine transform of the stress relaxation modulus be positive definite. The subject of this book is the derivation of this condition from thermodynamics and its implications for the mathematical analysis of the equations of linear viscoelasticity.The authors investigate the connection between thermodynamic restrictions and well-posedness of initial and boundary value problems. A thorough thermodynamic analysis of linear viscoelasticity is included. New results are established and previous ones are shown to follow as particular cases from the general scheme. The authors demonstrate that significant improvements can be obtained in existence, uniqueness, and asymptotic stability theorems by starting from the thermodynamic restrictions as mathematical hypotheses for the initial boundary value problems.Describes general mathematical modeling of viscoelastic materials as systems with fading memory.Discusses the interrelation between topics such as existence, uniqueness, and stability of initial boundary value problems, variational and extremum principles, and wave propagation.Demonstrates the deep connection between the properties of the solution to initial boundary value problems and the requirements of the general physical principles.Discusses special techniques and new methods, including Fourier and Laplace transforms, extremum principles via weight functions, and singular surfaces and discontinuity waves.Royalties from the sale of this book are contributed to the SIAM Student Travel fund.

This monograph provides a comprehensive and self-contained treatment of continuum physics, illustrating a systematic approach to the constitutive equations for wide-ranging classes of materials.  Derivations of results are detailed through careful proofs, and the contents have been developed to ensure a self-contained and consistent presentation.Part I reviews the kinematics of continuous bodies and illustrates the general setting of balance laws.  Essential preliminaries to continuum physics – such as reference and current configurations, transport relations, singular surfaces, objectivity, and objective time derivatives – are covered in detail.  A chapter on balance equations then develops the balance laws of mass, linear momentum, angular momentum, energy, and entropy, as well as the balance laws in electromagnetism.Part II is devoted to the general requirements on constitutive models, emphasizing the application of objectivity and consistency with the second law of thermodynamics.  Common models of simple materials are then reviewed, and in this framework, detailed descriptions are given of solids (thermoelastic, elastic, and dissipative) and fluids (elastic, thermoelastic, viscous, and Newtonian).A wide of variety of constitutive models are investigated in Part III, which consists of separate chapters focused on several types of non-simple materials: materials with memory, aging and higher-order grade materials, mixtures, micropolar media, and porous materials.  The interaction of the electromagnetic field with deformation is also examined within electroelasticity, magnetoelasticity, and plasma theory.Hysteretic effects and phase transitions are considered in Part IV.  A new approach is established by treating entropy production as a constitutive function in itself, as is the case for entropy and entropy flux. This proves to be conceptually and practically advantageous in the modelling of nonlinear phenomena, such as those occurring in hysteretic continua (e.g., plasticity, electromagnetism, and the physics of shape memory alloys).Mathematical Modelling of Continuum Physics will be an important reference for mathematicians, engineers, physicists, and other scientists interested in research or applications of continuum mechanics.

This monograph provides a comprehensive and self-contained treatment of continuum physics, illustrating a systematic approach to the constitutive equations for wide-ranging classes of materials.  Derivations of results are detailed through careful proofs, and the contents have been developed to ensure a self-contained and consistent presentation.Part I reviews the kinematics of continuous bodies and illustrates the general setting of balance laws.  Essential preliminaries to continuum physics – such as reference and current configurations, transport relations, singular surfaces, objectivity, and objective time derivatives – are covered in detail.  A chapter on balance equations then develops the balance laws of mass, linear momentum, angular momentum, energy, and entropy, as well as the balance laws in electromagnetism.Part II is devoted to the general requirements on constitutive models, emphasizing the application of objectivity and consistency with the second law of thermodynamics.  Common models of simple materials are then reviewed, and in this framework, detailed descriptions are given of solids (thermoelastic, elastic, and dissipative) and fluids (elastic, thermoelastic, viscous, and Newtonian).A wide of variety of constitutive models are investigated in Part III, which consists of separate chapters focused on several types of non-simple materials: materials with memory, aging and higher-order grade materials, mixtures, micropolar media, and porous materials.  The interaction of the electromagnetic field with deformation is also examined within electroelasticity, magnetoelasticity, and plasma theory.Hysteretic effects and phase transitions are considered in Part IV.  A new approach is established by treating entropy production as a constitutive function in itself, as is the case for entropy and entropy flux. This proves to be conceptually and practically advantageous in the modelling of nonlinear phenomena, such as those occurring in hysteretic continua (e.g., plasticity, electromagnetism, and the physics of shape memory alloys).Mathematical Modelling of Continuum Physics will be an important reference for mathematicians, engineers, physicists, and other scientists interested in research or applications of continuum mechanics.

The book may be viewed as an introduction to time-harmonic waves in dissipative bodies, notably viscoelastic solids and fluids. The inhomogeneity of the waves, which is due to the fact that planes of constant phase are not parallel to planes of constant amplitude, is shown to be strictly related to the dissipativity of the medium. A preliminary analysis is performed on the propagation of inhomogeneous waves in unbounded media and of reflection and refraction at plane interfaces. Then emphasis is given to those features that are of significance for applications. In essence, they regard surface waves, scattering by (curved) obstacles, wave propagation in layered heterogeneous media, and ray methods. The pertinent mathematical techniques are discussed so as to make the book reasonably self-contained.

This book deals with aspects of thermodynamic restrictions in modern continuum mechanics and with particular problems of the kinetic theory and statistical mechanics. It stresses the interplay between statistical and phenomenological modelling of physical phenomena including homogenization techniques for media with microstructure. Diverse approaches to either derivation or justification of macroscopic models by microscopic theories are presented. From the kinetic theory, the problem of existence of solutions to the Boltzmann equation and particular solutions to the discrete velocity models are also considered. The book includes papers concerning viscoelasticity treated within the framework of both rational and extended thermodynamics. Phenomenological theories of hyperbolic heat conduction in solids and fluids are also discussed.

This book contains the papers presented at the conference on “Mathematical Models and Methods for Smart Materials”, held in Italy in 2001. The papers are divided into four parts:”Methods in Materials Science” deals mainly with mathematical techniques for the investigation of physical systems, such as liquid crystals, materials with internal variables, amorphous materials, and thermoelastic materials. Also, techniques are exhibited for the analysis of stability and controllability of classical models of continuum mechanics and of dynamical systems.”Modelling of Smart Materials” is devoted to models of superfluids, superconductors, materials with memory, nonlinear elastic solids, and damaged materials. In the elaboration of the models, thermodynamic aspects play a central role in the characterization of the constitutive properties.”Well-Posedness in Materials with Memory” deals with existence, uniqueness and stability for the solution of problems, most often expressed by integrodifferential equations, which involve materials with fading memory. Also, attention is given to exponential decay in viscoelasticity, inverse problems in heat conduction with memory, and automatic control for parabolic equations.”Analytic Problems in Phase Transitions” discusses nonlinear partial differential equations associated with phase transitions, and hysteresis, possibly involving fading memory effects. Particular applications are developed for the phase-field model with memory, the Stefan problem with a Cattaneo-type equation, the hysteresis in thermo-visco-plasticity, and the solid-solid phase transition.