Roger Fosdick - Böcker

This IMA Volume in Mathematics and its Applications SHOCK INDUCED TRANSITIONS AND PHASE STRUCTURES IN GENERAL MEDIA is based on the proceedings of a workshop that was an integral part of the 1990-91 IMA program on "Phase Transitions and Free Boundaries." The workshop focused on the thermodynamics and mechanics of dynamic phase transitions that are mainly inertially driven and brought together physicists, metallurgists, mathematicians, engineers, and molecular dynamicists with interests in these problems. Financial support of the National Science Foundation made the meeting pos­ sible. We are grateful to J .E. Dunn, Roger Fosdick, and Marshall Slemrod for organizing the meeting and editing the proceedings. A vner Friedman Willard Miller, .Jr. PREFACE When a body is subjected to a strong shock the material may suffer severe local structural changes. Rapid solidification, liquification, or vaporization can oc­ cur, and, moreover, complex structural heterogeneity is often left in the wake of the passing wave. Thus, inertially driven shock waves raise fundamental questions involving experiment, theory, and mathematics which bear on phase stability and metastability, as well as on reaction kinetics and appropriate measures of phase structure.

The paper of Admal & Tadmor, "A Uni ed Interpretation of Stress in Molecular S- tems," takes up the various existing microscopic de nitions of the Cauchy stress tensor. Here the ambition is to establish a unifying framework in which all of these molecular surfacial interactions can be derived and the connections between them made evident. Developments in this paper draw upon the non-equilibrium statistical mechanics of Irving & Kirkwood and Noll, together with spatial averaging techniques. Extensions of the early work of Irving & Kirkwood to include multibody potentials and a generalization of the lemmas of Noll to include non-straight bonds are incorporated. Connections to the direct spatial averaging - proach of Murdoch and Hardy are exposed and the troublesome sources of non-uniqueness of the stress tensor are identi ed. Finally, numerical experiments based on molecular - namics and lattice statics are reported. These contrast the various de nitions of stress, - cluding convergence questions related to the size of the domain over which spatial averaging is performed.It is natural to wonder about the connection between works focused on the microscopic foundation of stress and more kinematically-focused works, such as those of Ericksen, P- teri, and Zanzotto, which emphasize the utility of and explore the validity of the Cauchy- Born rule. Podio-Guidugli's paper, "On (Andersen-)Parrinello-Rahman Molecular Dyn- ics, the Related Metadynamics, and the Use of the Cauchy-Born Rule," discusses scale bridging between molecular dynamics and continuum mechanics for Parrinello-Rahman molecular dynamics.

This book is dedicated to the memory of Donald E. Carlson, who enjoyed a long distinguished career as a Professor of Theoretical and Applied Mechanics at the University of Illinois at Urbana-Champaign; he influenced the mechanics community through his teachings, publications and interactions. The book disseminates unique research investigations which give breadth and depth to the field of continuum mechanics and provide a vision for future advancements. The common bond of Don's publications was his sound theoretical developments and fundamental insight. He carefully delineated the principal roles of kinematics, conservation laws (including the second law of thermodynamics) and constitutive assumptions not only in his discussions and writings about fundamentals in mechanics but also in his work on the formulation of initial-boundary value problems that arise in modeling the behavior of elastic and thermoelastic bodies. This book expands this lucid practice by applying these roles to model a plethora of physical phenomena on the foundations and applications of modern continuum mechanics.This is a hardbound spinoff edition previously published in the Journal of Elasticity, volumes 104 and 105, 2011.

The paper of Admal & Tadmor, "A Uni ed Interpretation of Stress in Molecular S- tems," takes up the various existing microscopic de nitions of the Cauchy stress tensor. Here the ambition is to establish a unifying framework in which all of these molecular surfacial interactions can be derived and the connections between them made evident. Developments in this paper draw upon the non-equilibrium statistical mechanics of Irving & Kirkwood and Noll, together with spatial averaging techniques. Extensions of the early work of Irving & Kirkwood to include multibody potentials and a generalization of the lemmas of Noll to include non-straight bonds are incorporated. Connections to the direct spatial averaging - proach of Murdoch and Hardy are exposed and the troublesome sources of non-uniqueness of the stress tensor are identi ed. Finally, numerical experiments based on molecular - namics and lattice statics are reported. These contrast the various de nitions of stress, - cluding convergence questions related to the size of the domain over which spatial averaging is performed.It is natural to wonder about the connection between works focused on the microscopic foundation of stress and more kinematically-focused works, such as those of Ericksen, P- teri, and Zanzotto, which emphasize the utility of and explore the validity of the Cauchy- Born rule. Podio-Guidugli's paper, "On (Andersen-)Parrinello-Rahman Molecular Dyn- ics, the Related Metadynamics, and the Use of the Cauchy-Born Rule," discusses scale bridging between molecular dynamics and continuum mechanics for Parrinello-Rahman molecular dynamics.

Recent developments in biology and nanotechnology have stimulated a rapidly growing interest in the mechanics of thin, flexible ribbons and Mobius bands.This edited volume contains English translations of four seminal papers on this topic, all originally written in German; of these, Michael A. Sadowsky published the first in 1929, followed by two others in 1930, and Walter Wunderlich published the last in 1962.The volume also contains invited, peer-reviewed, original research articles on related topics.Previously published in the Journal of Elasticity, Volume 119, Issue 1-2, 2015.

This book is dedicated to the memory of Donald E. Carlson, who enjoyed a long distinguished career as a Professor of Theoretical and Applied Mechanics at the University of Illinois at Urbana-Champaign; he influenced the mechanics community through his teachings, publications and interactions. The book disseminates unique research investigations which give breadth and depth to the field of continuum mechanics and provide a vision for future advancements. The common bond of Don's publications was his sound theoretical developments and fundamental insight. He carefully delineated the principal roles of kinematics, conservation laws (including the second law of thermodynamics) and constitutive assumptions not only in his discussions and writings about fundamentals in mechanics but also in his work on the formulation of initial-boundary value problems that arise in modeling the behavior of elastic and thermoelastic bodies. This book expands this lucid practice by applying these roles to model a plethora of physical phenomena on the foundations and applications of modern continuum mechanics.This is a hardbound spinoff edition previously published in the Journal of Elasticity, volumes 104 and 105, 2011.

Recent developments in biology and nanotechnology have stimulated a rapidly growing interest in the mechanics of thin, flexible ribbons and Mobius bands. This edited volume contains English translations of four seminal papers on this topic, all originally written in German;

Walter Noll's leadership was central to recent progress in setting the mathematical foundations of continuum physics. In this volume his life and contributions are revisited in an extensive essay that includes a complete list of his research contributions, and the conceptual tools that he provided are put in evidence in original articles pointing to the continuing vitality of his legacy. The volume will appeal to readers with an interest in the many-faceted role that mathematics can play in the advancement of continuum theories of material behavior.This is a reprint of the Journal of Elasticity, Volume 135, Numbers 1-2, April 2019