As any human activity needs goals, mathematical research needs problems. -David Hilbert Mechanics is the paradise of mathematical sciences. -Leonardo da Vinci Mechanics and mathematics have been complementary partners since N- ton's time, and the history of science shows much evidence of the bene?cial in?uence of these disciplines on each other. Driven by increasingly ela- rate modern technological applications, the symbiotic relationship between mathematics and mechanics is continually growing. However, the increasingly large number of specialist journals has generated a duality gap between the partners, and this gap is growing wider. Advances in Mechanics and Mathematics (AMMA) is intended to bridge the gap by providing multidisciplinary publications that fall into the two f- lowing complementary categories: 1. An annual book dedicated to the latest developments in mechanics and mathematics; 2. Monographs, advanced textbooks, handbooks, edited volumes, and selected conference proceedings. The AMMA annual book publishes invited and contributed compreh- sive research and survey articles within the broad area of modern mechanics and applied mathematics. The discipline of mechanics, for this series, includes relevantphysicalandbiologicalphenomenasuchas:electromagnetic,thermal, viii Series Preface and quantum e?ects, biomechanics, nanomechanics, multiscale modeling, - namical systems, optimization and control, and computation methods. - pecially encouraged are articles on mathematical and computational models and methods based on mechanics and their interactions with other ?elds. All contributions will be reviewed so as to guarantee the highest possible sci- ti?c standards. Each chapter will re?ect the most recent achievements in the area.

This book contains geometrical and thermodynamical issues indispensable for development of a rational theory of thermoviscoplasticity. Geometrical picture of coupled thermomagnetomechanical histories of damaged solids is built both by means of Kroener's incompatibility approach as well by Eshelbian implanting eigenstrains. Duality of Euclidean anholonomic and non-Euclidean natural state space is also outlined in this book.

Damaged inelastic materials of differential type, discrete and infinitesimal memory are obtained from principle of thermo-inelastic memory. Issue of plastic spin is considered. Postulate of minimal plastic work and corresponding non-associativity 4-tensor are then used to show whether associativity of flow rule holds. Postulates of Drucker, Iliushin and Hill are discussed. Thermodynamics of inelasticity is extensively discussed in classical, rational, extended and endochronic version with account to statistical thermodynamics. A non-steady aging is used in endochronic thermodynamics to cover creep-pasticity coupled inelastic histories. Multiaxial dynamic experiments with cylindrical, ``bichierino'' and cruciform specimen from austenitic stainless steels are analyzed. Quasi-rate independence and Rabotnov's plastic delay is combined with tensor representation. Inelastic ferromagnetics are treated by means of extended as well endochronic thermodynamics. For low cycle fatigue the experimentally observed displacement of magnetic induction history with respect to stress history is analyzed.

Self consistent method applied to inelastic polycrystals is based on constrained micro-rotations and free meso-rotations. A special attention is devoted to slight disorder of polycrystal grains. The theory is confronted with classical J2-theory. Different inelastic multiaxial stress histories are analyzed and corresponding active slip systems determined. For numerical results micro quasi rate independence and relaxed Taylor's model are used. The theory of inelastic micromorphic polycrystals with couple stresses needs a very small number of necessary material constants. Nonproportionality of strain history as well as intergranular continuity are related to antisymmetry of stress tensor.

Key topics:

* Includes a detailed description of the geometry of thermo-deformation with local evolving natural state configuration

* Provides a comparative review of various models of thermodynamics (classical, rational, endochronic, statistical) with special approach to inelastic high speed histories

* Introduces quasi-rate independence and its application to plastic waves, ratcheting, and diffuse localization

* Explores the sensor representation approach to thermo-inelastic coupled fields connected to a generalized associativity of flow rule as well as a comparison with the J2-approach

* Examines micromechanics based on micro grain approach leading to reduced number of material constants

* Provides biaxial cruciform specimen Hopkinson bar results

* Reexamines the Hill's yield function for nonproportional stress-thermo-strain histories

This book is intended for material science experts and professionals interested in impact experiments, continuum mechanics researchers, engineers in research institutes and graduate and Phd students aiming to apply FEM to calculate strength of structures at time varying thermo-mechanical excitations.

From the reviews: "This book exposes in a concise manner the essentials of the modern theory of viscoplasticity in relation with its basic thermomechanical foundations ? . In that sense the book represents a true ? contribution to the field. ? addresses professionals or younger readers (graduate students, PhDs, research engineers) who want to apprehend a difficult subject matter that pertains to both continuum mechanics ? and its applications in materials science. ? It is generally well and concisely written and ? generously documented." (Gérard A. Maugin, Mathematical Reviews, Issue 2010 c)