Mauro Fabrizio - 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 is a work in four parts, dealing with the mechanics and thermodynamics of materials with memory, including properties of the dynamical equations which describe their evolution in time under varying loads. The first part is an introduction to Continuum Mechanics with sections dealing with classical Fluid Mechanics and Elasticity, linear and non-linear. The second part is devoted to Continuum Thermodynamics, which is used to derive constitutive equations of materials with memory, including viscoelastic solids, fluids, heat conductors and some examples of non-simple materials. In part three, free energies for materials with linear memory constitutive relations are comprehensively explored. The new concept of a minimal state is also introduced. Formulae derived over the last decade for the minimum and related free energies are discussed in depth. Also, a new single integral free energy which is a functional of the minimal state is analyzed in detail. Finally, free energies for examples of non-simple materials are considered. In the final part, existence, uniqueness and stability results are presented for the integrodifferential equations describing the dynamical evolution of viscoelastic materials. A new approach to these topics, based on the use of minimal states rather than histories, is discussed in detail. There are also chapters on the controllability of thermoelastic systems with memory, the Saint-Venant problem for viscoelastic materials and on the theory of inverse problems.

This monograph deals with the mechanics and thermodynamics of materials with memory, including properties of the dynamical equations that describe their evolution in time under varying loads.

This monograph deals with the mechanics and thermodynamics of materials with memory, including properties of the dynamical equations that describe their evolution in time under varying loads.

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.