The set GU f of possible effective elastic tensors of composites built from two materials with elasticity tensors C1 > 0 and C2 = 0 comprising the set U = {C1, C2} and mixed in proportions f and 1−f is partly characterized. The material with tensor C2 = 0 corresponds to a material which is void. (For technical reasons C2 is actually taken to be nonzero and we take the limit C2 → 0). Specifically, recalling that GU f is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate p-mode material, that is easily compliant to p ≤ 5 independent applied strains, yet supports any stress in the orthogonal space. Thus the material can easily slip in certain directions along the walls. The region outside the walls contains "complementary Avellaneda material" which is a hierarchical laminate which minimizes the sum of complementary energies.
Understanding asymptotics of gradient components in relation to the symmetrized gradient is important 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 analogue of the Kirchhoff ansatz, whose most prominent feature is its singular dependence on the slenderness parameter, in marked contrast with the classical case of plates and rods.
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. IntroductionA circular cylindrical shell loaded by an axial compressive stress will buckle producing a variety of buckling patterns [4,20,6], including the single-dimple buckle [32,15], shown in Figure 1. In the soda can experiments [14] this dimple consistently appeared with an audible Figure 1: Single-dimple buckling pattern in buckled soda cans [14]. 1 click, corresponding to the drop in load in Figure 1 and disappears (also with a click) upon unloading. This suggests that the local material response is still linearly elastic, while the global non-linearity is purely geometric. The abrupt nature of the observed buckling suggests that the trivial branch, whose stress and strain are well-approximated by linear elasticity, becomes unstable with respect to the observed buckling variation.The classical shell theory supplies the following formula for the critical stress [24,27] (see also [28]):where E and ν are the Young modulus and the Poisson ratio, respectively, and h = t/R is the ratio of the wall thickness to the radius of the cylinder. A large body of experimental results summarized in [20,32] show that not only the theoretical value of the buckling load is about 4 to 5 times higher than the one observed in experiments, but the critical stress σ cr scales like h 3/2 with h, in stark contradiction to (1.1). Such paradoxical behavior is generally attributed to the sensitivity of the buckling load to imperfections of load and shape [1,26,29,10,30], due to the subcritical nature of the bifurcation [18,19,23,16] in the von-Kármán-Donnell equations. Yet, such an interpretation of the experimental results does not give a quantification of sensitivity to imperfections, and does little to explain the paradoxical h 1.5 scaling of the critical stress. These questions have been raised in [5,32,15], where a combination of heuristic arguments and numerical simulations were used to address the problem. In situations where the classical shell theory gives predictions inconsistent with experiment, one can question whether "sensitivity to imperfections" is the true source of the inconsiste...
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 shells 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 4/3 , where h is the thickness of the shell. These results have 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, James and Müller for plates [14] (where they show that the Korn constant in the nonlinear Korn's first inequality scales like h 2 ), extended to shells with nonzero curvature. We also recover the uniform Korn-Poincaré inequality proven for "boundary-less" shells by Lewicka and Müller in [37] 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.
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.
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