On axisymmetric solutions for compressible nonlinearly elastic solids

Horgan, Cornelius O.

doi:10.1007/978-3-0348-9229-2_6

Cited by 16 publications

(21 citation statements)

References 23 publications

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“…We first consider the material model (10). The cavitation problem for this material model has been solved explicitly by Shang and Cheng [44] for the case when δ = 0.…”

Section: Asymptotic Results For the Cauchy Elastic Materials Modelmentioning

confidence: 99%

“…When the material is incompressible, the radially axisymmetric deformation can be determined to within an arbitrary constant irrespective of the form of the strain-energy function, and determination of the critical tension and post-buckling deformation is then reduced to the evaluation of an integral. When the material is compressible, the two-point boundary value problem can be solved by a shooting procedure in the most general case (see, e.g., [8]), but many studies have focused on finding closed-form solutions for specific material models (see, e.g., [9][10][11][12]). There also exists a large body of literature concerned with the effects of anisotropy, material inhomogeneity, surface tension, and plastic behavior; see, e.g., [13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

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On the near-critical behavior of cavitation in elastic plane membranes

Geng

Cai

2017

International Journal of Non-Linear Mechanics

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This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractMaterial cavitation under tensile loading is often studied by assuming the pre-existence of a small void. In this case the void would initially grow but without significant change in its size, and cavitation is said to take place if this slow growth is followed by rapid growth at higher load values. In the limit when the original void radius δ tends to zero, there will be no growth until a load or stretch measure, λ say, reaches a well-defined critical value λ cr at which a cavity appears suddenly. In this paper we study the near-critical asymptotic behavior of cavitation in plane membranes when δ is not zero but small, and show that the near-critical behavior is governed by a scaling law in the form λ − λ cr = C(δ/L) m , where L is the undeformed outer radius of the plane membrane, and C and m are non-dimensional constants. The positive power m in general depends on the material model used, but for the three classes of material models considered, it happens to be equal to 2(1 + ν)/(3 + ν) in each case, where ν is Poisson's ratio for infinitesimal deformations. If a pre-existing void is viewed as an imperfection, then this scaling law describes the imperfection sensitivity of cavitation: it states that in the presence of imperfections significant void growth would occur when λ were increased to within an order (δ/L) m interval around λ cr .

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“…We first consider the material model (10). The cavitation problem for this material model has been solved explicitly by Shang and Cheng [44] for the case when δ = 0.…”

Section: Asymptotic Results For the Cauchy Elastic Materials Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the near-critical behavior of cavitation in elastic plane membranes

Geng

Cai

2017

International Journal of Non-Linear Mechanics

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“…It is worth mentioning that by taking µ = 0 in Equation (11), a hyperelastic material whose strainenergy function depends on J , so that it is of the Varga type [Horgan 1995;Horgan 2001], is obtained. This class of materials admits, under the present axisymmetry conditions, an exact solution of the equilibrium Equation (12) of the form r (R) = ξ R 2 + η, where ξ and η are arbitrary constants.…”

Section: Verification Of the Micromechanical Predictionmentioning

confidence: 99%

Finite strain micromechanical analysis for thermoelastoplastic multiphase materials

Aboudi

2008

JOMMS

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A micromechanical model that is based on the homogenization technique for periodic composites is developed for the prediction of the response of multiphase materials undergoing large deformations. Every one of the constituents is supposed to be either a rate-independent thermoelastoplastic material or a thermoelastic one, both of which are formulated in the framework of finite strains. Hyperelastic constituents are obtained as a special case. The resulting macroscopic (global) constitutive equations of the composite involve the instantaneous mechanical and thermal tangent tensors. The reliability of the prediction is examined by comparisons with the composite cylinder assemblage model, which is formulated for a finite strain rate-independent thermoplasticity and is valid under axisymmetric loading. Applications are given for a system of a rubber-like matrix reinforced by metallic fibers. In addition, the behavior of rate-independent elastoplastic laminated materials undergoing large deformations and subjected to in-plane loading is investigated. Finally, the response of an elastoplastic auxetic metallic material, which is capable of generating a negative Poisson's ratio at any stage of a finite strain loading is examined by employing the proposed micromechanical model.

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“…A number of other explicit solutions to this problem for a range of different strain-energy functions have been collected together and augmented by Horgan [25,26]. In section 8 of [26] Hadamard materials of the form…”

Section: Vol 59 (2008)mentioning

confidence: 99%

Using null strain energy functions in compressible finite elasticity to generate exact solutions

Haughton

2007

Z. angew. Math. Phys.

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In this paper we characterize those strain energy functions in unconstrained nonlinear elasticity that satisfy the equations of equilibrium identically. The idea is to construct a useful, physically reasonable strain-energy function containing one or more components which are null, in such a way that exact solutions may be obtained from the resulting equilibrium equations. We show that the dilatation is a universal null energy while there may be others that depend on the actual problem. To obtain the null energies for a given problem it is often convenient to formulate the variational problem and look at the Euler-Lagrange equations. Specific examples are used to illustrate some of the potential uses of the method in finding exact solutions for physically meaningful constitutive models.

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On axisymmetric solutions for compressible nonlinearly elastic solids

Cited by 16 publications

References 23 publications

On the near-critical behavior of cavitation in elastic plane membranes

On the near-critical behavior of cavitation in elastic plane membranes

Finite strain micromechanical analysis for thermoelastoplastic multiphase materials

Using null strain energy functions in compressible finite elasticity to generate exact solutions

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