Some experiments on the elastic stability of some highly elastic bodies

Beatty, Millard F.; Dadras, P.

doi:10.1016/0020-7225(76)90092-6

Cited by 18 publications

(18 citation statements)

References 6 publications

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“…At first sight this would seem to agree well with the experiments performed by Beatty and Hook [8] and Beatty and Dadras [7], They investigated the compression of columns of various aspect ratios between greased parallel plates and reported that whilst thin columns buckled under a sufficiently large applied load those of larger aspect ratio started deforming inhomogeneously by bulging in the middle. However, as is pointed out by Simpson and Spector [19], the theoretical values of Abar obtained for the material considered in Sec.…”

supporting

confidence: 77%

“…This can be overcome by assuming for example that a and d<& JdX are of the same order of magnitude when X is small (which is to be expected in practice). Instead of assuming specific technical hypotheses on <X> of this type we will just suppose that there is a solution in (0, 1] to = 0 (4)(5)(6)(7)(8)(9)(10)(11)(12)(13) and denote the largest such solution by X . The first step is to show that u, r, e Hl (3?…”

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confidence: 99%

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Buckling and barrelling instabilities on non-linearly elastic columns

Davies¹

1991

Quart. Appl. Math.

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1. Introduction. The question of predicting what happens to an elastic column when it is compressed uniaxially has long been a subject of investigation (see [9,17] for a bibliography). We address here the compression problem for a circular cylinder of radius h of compressible hyperelastic material which satisfies appropriate ellipticity and growth hypotheses. We study the mixed displacement-traction boundary value problem for the deformation x of the column in which the compression ratio (ratio of deformed to undeformed length) is prescribed, the surface traction at its ends has no tangential component, and the lateral surface is specified to be stress free.Results on the compression of a circular cylinder of general compressible material have previously been obtained by Simpson and Spector. They gave a necessary and sufficient condition for the equilibrium equations of elasticity linearized about xA to have an axisymmetric or barrelling solution [17], and showed that such a solution does exist for materials with certain specific stored energy functions [17,19]. Analogous conditions have been obtained by Ogden [ 16] for the existence of planar solutions to the linearized equations for a rectangular column, and by Davies [9] who considered the two-dimensional case of an elastic rectangle and showed that this would always have a solution for a general class of materials.As in previous treatments of this and similar problems we first give conditions which ensure that for each compression ratio A 6 (0, 1) there is precisely one homogeneous solution xA to the nonlinear problem with diagonal gradient. These are stated in Sec. 2 and apply to columns of general constant cross-section. The stability of x^ (in a sense made precise in Sec. 3) is then investigated, and the Complementing Condition (cf. Simpson and Spector [20]) for the linearized problem is discussed.In Sec. 5, results of Ball [5] are used to rigorously obtain an equation for Abar, the compression ratio X at which x; can first cease to be a weak local minimum of the deformation energy with respect to barrelling perturbations. Section 6 contains a comparison of Abar with ABUC , the compression ratio corresponding to the onset of planar buckling of a column of square cross-section of side 2h . (Calculation

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confidence: 77%

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confidence: 99%

Buckling and barrelling instabilities on non-linearly elastic columns

Davies¹

1991

Quart. Appl. Math.

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“…Negrón-Marrero (1999) studied the bifurcation of the axisymmetric hyperelastic cylinders subject to nonlinear mixed boundary conditions and found that the eigenfunctions can be classified into those that are symmetric about the mid-plane, representing either necked or barrelled configurations of the cylinder, and those that break this symmetry. Finite axisymmetric deformations of thick-walled carbon-black filled rubber tubes were also studied experimentally by Beatty and Dadras (1976). They found that for aspect ratios less than 5 tubes exhibit radially or axially symmetric bulging modes of deformation, distinct from the familiar Euler buckling that occurs for longer tubes.…”

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confidence: 99%

Nonlinear axisymmetric deformations of an elastic tube under external pressure

Zhu

Luo

Ogden

2010

European Journal of Mechanics - A/Solids

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International audienceThe problem of the finite axisymmetric deformation of a thick-walled circular cylindrical elastic tube subject to pressure on its external lateral boundaries and zero displacement on its ends is formulated for an incompressible isotropic neo-Hookean material. The formulation is fully nonlinear and can accommodate large strains and large displacements. The governing system of nonlinear partial differential equations is derived and then solved numerically using the Cþþ based object-oriented finite element library Libmesh. The weighted residual-Galerkin method and the Newton-Krylov nonlinear solver are adopted for solving the governing equations. Since the nonlinear problem is highly sensitive to small changes in the numerical scheme, convergence was obtained only when the analytical Jacobian matrix was used. A Lagrangian mesh is used to discretize the governing partial differential equations. Results are presented for different parameters, such as wall thickness and aspect ratio, and comparison is made with the corresponding linear elasticity formulation of the problem, the results of which agree with those of the nonlinear formulation only for small external pressure. Not surprisingly, the nonlinear results depart significantly from the linear ones for larger values of the pressure and when the strains in the tube wall become large. Typical nonlinear characteristics exhibited are the ''corner bulging'' of short tubes, and multiple modes of deformation for longer tubes

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“…The Lagrangian strain tensor is given by 2 = ( − )where is the identity matrix. A material is said to be pathindependent or hyperelastic if the work done by the stresses during the deformation process depends only on its initial configuration at time t = 0 and the final configuration at time t [17]. As a consequence of the path-independence:…”

Section: Hyperelasticitymentioning

confidence: 99%

“…Heil [15] and Marzo [16] performed a numerical simulation of the post-buckling behaviour of tubes under external pressure. Finite axisymmetric deformations of thick-walled carbon-black filled rubber tubes were also studied experimentally by Beatty and Dadras [17]. They found that for aspect ratios less than 5 tubes exhibit radially or axially symmetric bulging modes of deformation, distinct from the familiar Euler buckling that occurs for longer tubes.…”

Section: Introductionmentioning

confidence: 98%