Statistical Mechanics, Molecular Modeling, and the Notion of Stress

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 hard bound spinoff from a special issue of the Journal of Elasticity (volume 100: 1-2) features an English translation of an important 1955 paper by Walter Noll, “Die Herteitung der Grundgleichungen der Thermomechanik der Kontinua aus der statistischen Mechanik.” In this paper, Noll addresses and analyses the seminal paper of Irving and Kirkwood, published five years earlier, on “The Statistical Mechanical Theory of Transport Processes. IV, The Equations of Hydrodynamics.” Noll gives new interpretations and provides a firm setting for ideas advanced by Irving & Kirkwood that clearly and directly relate to the basic principles of continuum mechanics. However, the original German paper of Noll seems not to have gained the attention that it deserved as the field of statistical mechanics grew both fundamentally and in applications. By providing an English translation of Noll’s paper, Lehoucq & Von Lilienfeld-Toal have provided a great service to the scientific community. The Noll translation is presented here to expose fundamental ideas of statistical mechanics that are of major importance in the modeling of small-scale behaviour and its link to macroscopic observations. In recent years there has been a rapidly increasing reliance upon and interest in multiscale methods in computation. This has accentuated the need to establish meaningful connections between atomistic and continuum descriptions of contact interactions such as stress and heat flux. In recognition of Noll’s contribution, the translation is accompanied by four relevant and invited papers, including one, entitled “Thoughts on the Concept of Stress,” by Noll himself.