Limits to Poisson's ratio in isotropic materials

A long-standing question is why Poisson's ratio v nearly always exceeds 0.2 for isotropic materials, whereas classical elasticity predicts v to be between -1 to 1/2. We show that the roots of quadratic relations from classical elasticity divide v into three possible ranges: -1 < v <= 0, 0 <= v <= 1/5, and 1/5 <= v < 1/2. Since elastic properties are unique there can be only one valid set of roots, which must be 1/5 <= v < 1/2 for consistency with the behavior of real materials. Materials with Poisson's ratio outside of this range are rare, and tend to be either very hard (e.g., diamond, beryllium) or porous (e.g., auxetic foams); such substances have more complex behavior than can be described by classical elasticity. Thus, classical elasticity is inapplicable whenever v < 1/5, and the use of the equations from classical elasticity for such materials is inappropriate.Comment: Physical Review B, in pres

Tributes to Family Law Scholars Who Helped Us Find Our Path

At some point after the virus struck, I had the idea that it would be appropriate and interesting to ask a number of experienced family law teachers to write a tribute about a more senior family law scholar whose work inspired them when they were beginning their careers. I mentioned this idea to some other long-term members of the professoriate, and they agreed that this could be a good project. So I reached out to some colleagues and asked them to participate. Many agreed to join the team. Some suggested other potential contributors, and some of these suggested faculty members also agreed to submit a tribute. The authors have written about a diverse group of distinguished scholars in the area of family law. We have included 12 scholars who have contributed substantially to the field, and they have also influenced those who have written about them here. The honored scholars and the tribute authors are as follows (organized alphabetically by the honoree): Homer H. Clark Jr. (1918-2015), by Ann Laquer Estin Cooper Davis, by Melissa MurrayPeggy Mary Ann Glendon, by June Carbone Herma Hill Kay (1934-2017), by Barbara A. Atwood Robert Levy, by Paul M. Kurtz Marygold (Margo) Shire Melli (1926-2018), by J. Thomas Oldham & Bruce M. Smyth Martha Minow, by Brian H. Bix Robert Mnookin, by Elizabeth S. Scott Twila Perry, by R.A. Lenhardt Dorothy E. Roberts, by Jessica Dixon Weaver Carol Sanger, by Solangel Maldonado Barbara Bennett Woodhouse, by Sacha M. Coupet Each colleague who participated in this project chose the scholar whose work he or she would celebrate. So, the list of those honored here is subjective and, to a certain extent, serendipitous. This Article is part of a Family Law Quarterly issue that also honors other pioneering contributors to the family law field. We hope to make this a continuing project and to have future opportunities to recognize the many scholars who have had a profound impact on their students – and on all of us – in addition to having an important impact on the development of the law. I trust the reader will find these tributes of interest

Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups

International audienceWhen performing statistics on elements of sets that possess a particular geometric structure, it is desirable to respect this structure. For instance in a Lie group, it would be judicious to have a notion of a mean which is stable by the group operations (composition and inversion). Such a property is ensured for Riemannian center of mass in Lie groups endowed with a bi-invariant Riemannian metric, like compact Lie groups (e.g. rotations). However, bi-invariant Riemannian metrics do not exist for most non compact and non-commutative Lie groups. This is the case in particular for rigid-body transformations in any dimension greater than one, which form the most simple Lie group involved in biomedical image registration. In this paper, we propose to replace the Riemannian metric by an affine connection structure on the group. We show that the canonical Cartan connections of a connected Lie group provides group geodesics which are completely consistent with the composition and inversion. With such a non-metric structure, the mean cannot be defined by minimizing the variance as in Riemannian Manifolds. However, the characterization of the mean as an exponential barycenter gives us an implicit definition of the mean using a general barycentric equation. Thanks to the properties of the canonical Cartan connection, this mean is naturally bi-invariant. We show the local existence and uniqueness of the invariant mean when the dispersion of the data is small enough. We also propose an iterative fixed point algorithm and demonstrate that the convergence to the invariant mean is at least linear. In the case of rigid-body transformations, we give a simple criterion for the global existence and uniqueness of the bi-invariant mean, which happens to be the same as for rotations. We also give closed forms for the bi-invariant mean in a number of simple but instructive cases, including 2D rigid transformations. For general linear transformations, we show that the bi-invariant mean is a generalization of the (scalar) geometric mean, since the determinant of the bi-invariant mean is the geometric mean of the determinants of the data. Finally, we extend the theory to higher order moments, in particular with the covariance which can be used to define a local bi-invariant Mahalanobis distance