Lamination exact relations and their stability under homogenization

Relations between components of the effective tensors of composites that hold regardless of composite's microstructure are called exact relations. Relations between components of the effective tensors of all laminates are called lamination exact relations. The question of existence of sets of effective tensors of composites that are stable under lamination, but not homogenization was settled by Milton with an example in 3D elasticity. In this paper we discuss an analogous question for exact relations, where in a wide variety of physical contexts it is known (a posteriori) that all lamination exact relations are stable under homogenization. In this paper we consider 2D polycrystalline multi-field response materials and give an example of an exact relation that is stable under lamination, but not homogenization. We also shed some light on the surprising absence of such examples in most other physical contexts (including 3D polycrystalline multi-field response materials). The methods of our analysis are algebraic and lead to an explicit description (up to orthogonal conjugation equivalence) of all representations of formally real Jordan algebras as symmetric $n\times n$ matrices. For each representation we examine the validity of the 4-chain relation|a 4th degree polynomial identity, playing an important role in the theory of special Jordan algebras

Scaling instability of the buckling load in axially compressed circular cylindrical shells

In this paper we initiate a program of rigorous analytical investigation of the paradoxical buckling behavior of circular cylindrical shells under axial compression. This is done by the development and systematic application of general theory of "near-flip" buckling of 3D slender bodies to cylindrical shells. The theory predicts scaling instability of the buckling load due to imperfections of load. It also suggests a more dramatic scaling instability caused by shape imperfections. The experimentally determined scaling exponent 1.5 of the critical stress as a function of shell thickness appears in our analysis as the scaling of the lower bound on safe loads given by the Korn constant. While the results of this paper fall short of a definitive explanation of the buckling behavior of cylindrical shells, we believe that our approach is capable of providing reliable estimates of the buckling loads of axially compressed cylindrical shells.Comment: 29 page

Normality condition in elasticity

Strong local minimizers with surfaces of gradient discontinuity appear in variational problems when the energy density function is not rank-one convex. In this paper we show that stability of such surfaces is related to stability outside the surface via a single jump relation that can be regarded as interchange stability condition. Although this relation appears in the setting of equilibrium elasticity theory, it is remarkably similar to the well known normality condition which plays a central role in the classical plasticity theory

Exact scaling exponents in Korn and Korn-type inequalities for cylindrical shells

Understanding asymptotics of gradient components in relation to the symmetrized gradient is im- portant for the analysis of buckling of slender structures. For circular cylindrical shells we obtain the exact scaling exponent of the Korn constant as a function of shell's thickness. Equally sharp results are obtained for individual components of the gradient in cylindrical coordinates. We also derive an ana- logue of the Kirchho? ansatz, whose most prominent feature is its singular dependence on the slenderness parameter, in marked contrast with the classical case of plates and rods

Rigorous derivation of the formula for the buckling load in axially compressed circular cylindrical shells

The goal of this paper is to apply the recently developed theory of buckling of arbitrary slender bodies to a tractable yet non-trivial example of buckling in axially compressed circular cylindrical shells, regarded as three-dimensional hyperelastic bodies. The theory is based on a mathematically rigorous asymptotic analysis of the second variation of 3D, fully nonlinear elastic energy, as the shell's slenderness parameter goes to zero. Our main results are a rigorous proof of the classical formula for buckling load and the explicit expressions for the relative amplitudes of displacement components in single Fourier harmonics buckling modes, whose wave numbers are described by Koiter's circle. This work is also a part of an effort to quantify the sensitivity of the buckling load of axially compressed cylindrical shells to imperfections of load and shape