Statistical Mechanics, Molecular Modeling, and the Notion of Stress

Here is an English translation of a 1955 paper by Walter Noll exposing fundamentals of statistical mechanics that are important in the modeling of small-scale behaviour and its link to macroscopic observations. Includes four invited papers on related topics.

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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.