Search

This work is concerned with the study of the extreme rays of the convex cone of 3 × 3 quasiconvex quadratic forms (denoted by C 3 ). We characterize quadratic forms f ∈ C 3 , the determinant of the acoustic tensor of which is an extremal polynomial, and conjecture/discuss about other cases. We prove that in the case when the determinant of the acoustic tensor of a form f ∈ C 3 is an extremal polynomial other than a perfect square, then the form must itself be an extreme ray of C 3 ; when the determinant is a perfect square, then the form is either an extreme ray of C 3 or polyconvex; finally, when the determinant is identically zero, then the form f must be polyconvex. The zero determinant case plays an important role in the proofs of the other two cases. We also make a conjecture on the extreme rays of C 3 , and discuss about weak and strong extremals of C d for d ≥ 3 , where it turns out that several properties of C 3 do not hold for C d for d > 3 , and thus case d = 3 is special. These results recover all previously known results (to our best knowledge) on examples of extreme points of C 3 that were proved to be such. Our results also improve the ones proven by Harutyunyan and Milton (Commun Pure Appl Math 70(11):2164–2190, 2017) on weak extremals in C 3 (or extremals in the sense of Milton) introduced in (Commun Pure Appl Math XLIII:63–125, 1990). In the language of positive biquadratic forms, quasiconvex quadratic forms correspond to nonnegative biquadratic forms and the results read as follows: if the determinant of the y (or x ) matrix of a 3 × 3 nonnegative biquadratic form in x , y ∈ R 3 is an extremal polynomial that is not a perfect square, then the form must be an extreme ray of the convex cone of 3 × 3 nonnegative biquadratic forms (C 3) ; if the determinant is identically zero, then the form must be a sum of squares; if the determinant is a nonzero perfect square, then the form is either an extreme ray of C 3 , or is a sum of squares. The proofs are all established by means of several classical results from linear algebra, convex analysis (geometry), real algebraic geometry, and the calculus of variations. [ABSTRACT FROM AUTHOR]

In the paper we deal with shells with non-zero Gaussian curvature. We derive sharp Korn's first (linear geometric rigidity estimate) and second inequalities on that kind of shell for zero or periodic Dirichlet, Neumann, and Robin type boundary conditions. We prove that if the Gaussian curvature is positive, then the optimal constant in the first Korn inequality scales like h, and if the Gaussian curvature is negative, then the Korn constant scales like h , where h is the thickness of the shell. These results have a classical flavour in continuum mechanics, in particular shell theory. The Korn first inequalities are the linear version of the famous geometric rigidity estimate by Friesecke et al. for plates in Arch Ration Mech Anal 180(2):183-236, 2006 (where they show that the Korn constant in the nonlinear Korn's first inequality scales like h ), extended to shells with nonzero curvature. We also recover the uniform Korn-Poincaré inequality proven for 'boundary-less' shells by Lewicka and Müller in Annales de l'Institute Henri Poincare (C) Non Linear Anal 28(3):443-469, 2011 in the setting of our problem. The new estimates can also be applied to find the scaling law for the critical buckling load of the shell under in-plane loads as well as to derive energy scaling laws in the pre-buckled regime. The exponents 1 and 4/3 in the present work appear for the first time in any sharp geometric rigidity estimate. [ABSTRACT FROM AUTHOR]

We prove that an elasticity tensor with orthotropic symmetry is extremal if the determinant of its acoustic tensor is an extremal polynomial that is not a perfect square. [ABSTRACT FROM AUTHOR]