Related papers: A Riemannian approach to strain measures in nonlin…
We consider various notions of strains; quantitative measures for the deviation of a linear transformation from an isometry. The main approach, which is motivated by physical applications and follows the work of Patrizio Neff and co-workers…
We provide an easy approach to the geodesic distance on the general linear group GL(n) for left-invariant Riemannian metrics which are also right-O(n)-invariant. The parametrization of geodesic curves and the global existence of length…
We consider the two logarithmic strain measures\[\omega_{\rm iso}=\|\mathrm{dev}_n\log U\|=\|\mathrm{dev}_n\log \sqrt{F^TF}\|\quad\text{ and }\quad \omega_{\rm vol}=|\mathrm{tr}(\log U)|=|\mathrm{tr}(\log\sqrt{F^TF})|\,,\]which are…
The logarithmic strain measures $\lVert\log U\rVert^2$, where $\log U$ is the principal matrix logarithm of the stretch tensor $U=\sqrt{F^TF}$ corresponding to the deformation gradient $F$ and $\lVert\,.\,\rVert$ denotes the Frobenius…
In this paper we propose an anisotropic extension of the isotropic exponentiated Hencky energy, based on logarithmic strain invariants. Unlike other elastic formulations, the isotropic exponentiated Hencky elastic energy has been derived…
The introduction of the quadratic Hencky strain energy based on the logarithmic strain tensor log V is a milestone in the development of nonlinear elasticity theory in the first half of the 20th century. Since the original manuscripts are…
To begin with, we identify the equations of elastostatics in a Riemannian manifold, which generalize those of classical elasticity in the three-dimensional Euclidean space. Our approach relies on the principle of least energy, which asserts…
With a f-left-invariant Riemannian metric on a Lie group $G$, we mean a Riemannian metric which is conformally equivalent to a left-invariant Riemannian metric, with the conformal factor $f$. In this article, we study the geometry of such…
This paper is the third and final component of a three-part effort on observers contracting a Riemannian distance between the state of the system and its estimate. In Part I, we showed that such a contraction property holds if the system…
In [1], it is established that a convergent observer with an infinite gain margin can be designed for a given nonlinear system when a Riemannian metric showing that the system is differentially detectable (i.e., the Lie derivative of the…
Various properties of isoperimetric, functional, Transport-Entropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of…
In this work, we show that a class of metric-affine gravities can be reduced to a Riemann-Cartan one. The reduction is based on the cancelation of the nonmetricity against the symmetric components of the spin connection. A heuristic proof,…
The geodesic orbit property is useful and interesting in itself, and it plays a key role in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly…
We consider a family of isotropic volumetric-isochoric decoupled strain energies $$ F\mapsto W_{\rm eH}(F):=\widehat{W}_{\rm eH}(U):=\left\{\begin{array}{lll} \frac{\mu}{k}\,e^{k\,\|{\rm dev}_n\log…
Let (N,g) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their `almost' versions). We define a left invariant Riemannian metric on N compatible with g to be minimal,…
The problem of recovering the configuration of points from their partial pairwise distances, referred to as the Euclidean Distance Matrix Completion (EDMC) problem, arises in a broad range of applications, including sensor network…
Many phenomena are naturally characterized by measuring continuous transformations such as shape changes in medicine or articulated systems in robotics. Modeling the variability in such datasets requires performing statistics on Lie groups,…
The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and…
Let N be a nilpotent Lie group and let S be an invariant geometric structure on N (cf. symplectic, complex or hypercomplex). We define a left invariant Riemannian metric on N compatible with S to be "minimal", if it minimizes the norm of…
Stochastic contraction analysis is a recently developed tool for studying the global stability properties of nonlinear stochastic systems, based on a differential analysis of convergence in an appropriate metric. To date, stochastic…