A bifurcation problem for a compressible nonlinearly elastic medium: growth of a micro-void

Horgan, Cornelius O.; Abeyaratne, Rohan

doi:10.1007/bf00043585

Cited by 160 publications

(132 citation statements)

References 7 publications

(14 reference statements)

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“…Adapting the results given by Horgan and Abeyaratne [6], we first note that in this case the equilibrium equation takes the form…”

Section: Asymptotic Results For the Blatz-ko Materials Modelmentioning

confidence: 95%

“…The critical tension can be obtained by taking two limits one after the other: firstly the undeformed cavity radius tending to zero, and then the deformed cavity radius tending to zero. This has been demonstrated by Horgan and Abeyaratne [6], and is justified by the rigorous results of Sivaloganathan et al [7]. 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.…”

Section: Introductionmentioning

confidence: 85%

“…In this spirit, the explicit (plane-strain) cavitation solution found by Horgan and Abeyaratne [6] for…”

Section: Problem Descriptionmentioning

confidence: 97%

“…Despite this difference, the bifurcation diagram, showing the dependence of cavitation size on the applied tensile pressure, is nonetheless of the same shape as that for more traditional buckling problems such as the pitch-fork bifurcation associated with Euler buckling. A pre-existing void can be viewed as an imperfection, and the associated imperfect bifurcation diagram can be viewed as an "unfolding" of the perfect bifurcation diagram, in exactly the same manner as in Euler buckling; see, e.g., Figure 1 in [6], or the three figures in the current paper. In the case of Euler buckling, a simple near-critical analysis would yield an amplitude equation of the form…”

Section: Introductionmentioning

confidence: 99%

“…Horgan and Abeyaratne [6] considered cavitation associated with a Blatz-Ko material, and derived two asymptotic expressions for the deformed void radius r(δ) valid for λ < λ cr and λ > λ cr , respectively, where δ is the undeformed void radius, λ is the azimuthal stretch imposed at the outer surface and λ cr its critical value. It was shown that when λ < λ cr , the deformed radius r(δ) is of order δ, whereas when λ > λ cr , the leading order term in r(δ) is independent of δ, indicating that in this parameter regime the azimuthal stretch r(δ)/δ would tend to infinity as δ → 0.…”

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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“…Adapting the results given by Horgan and Abeyaratne [6], we first note that in this case the equilibrium equation takes the form…”

Section: Asymptotic Results For the Blatz-ko Materials Modelmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 85%

“…In this spirit, the explicit (plane-strain) cavitation solution found by Horgan and Abeyaratne [6] for…”

Section: Problem Descriptionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Geng

Cai

2017

International Journal of Non-Linear Mechanics

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Energy Estimates and Cavity Interaction for a Critical‐Exponent Cavitation Model

Henao

Serfaty

2012

Comm Pure Appl Math

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We consider the minimization of $\int_{\Omega _\varepsilon } {|D{\bf u}|^p } d{\bf x}$ in a perforated domain $\Omega _\varepsilon : = \Omega \backslash \cup _{i = 1}^M B_\varepsilon ({\bf a}_i )$ of $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^n$ among maps $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf u} \in W^{1,p} (\Omega _\varepsilon ,\R^n )$ that are incompressible (det $D{\bf u} \equiv 1$) and invertible, and satisfy a Dirichlet boundary condition u = g on ∂Ω. If the volume enclosed by g(∂Ω) is greater than |Ω|, any such deformation u is forced to map the small holes Bε(ai) onto macroscopically visible cavities (which do not disappear as ε → 0). We restrict our attention to the critical exponent p = n, where the energy required for cavitation is of the order of $\sum\nolimits_{i = 1}^M {v_i |\log \varepsilon |}$ and the model is suited, therefore, for an asymptotic analysis (v1,…, vM denote the volumes of the cavities). In the spirit of the analysis of vortices in Ginzburg‐Landau theory, we obtain estimates for the “renormalized” energy showing its dependence on the size and the shape of the cavities, on the initial distance between the cavitation points a1,…,aM, and on the distance from these points to the outer boundary ∂Ω. Based on those estimates we conclude, for the case of two cavities, that either the cavities prefer to be spherical in shape and well separated, or to be very close to each other and appear as a single equivalent round cavity. This is in agreement with existing numerical simulations and is reminiscent of the interaction between cavities in the mechanism of ductile fracture by void growth and coalescence. © 2012 Wiley Periodicals, Inc.

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A three‐scale compressible microsphere model for hyperelastic materials

Dal

Cansız

Miehé

2018

Numerical Meth Engineering

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This paper presents a seminumerical homogenization framework for porous hyperelastic materials that is open for any hyperelastic microresponse. The conventional analytical homogenization schemes do apply to a limited number of elementary hyperelastic constitutive models. Within this context, we propose a general numerical scheme based on the homogenization of a spherical cavity in an incompressible unit hyperelastic solid sphere, which is denoted as the mesoscopic representative volume element (mRVE). The approach is applicable to any hyperelastic micromechanical response. The deformation field in the sphere is approximated via nonaffine kinematics proposed by Hou and Abeyaratne (JMPS 40:571-592,1992). Symmetric displacement boundary conditions driven by the principal stretches of the deformation gradient are applied on the outer boundary of the mRVE. The macroscopic quantities, eg, stress and moduli expressions, are obtained by analytically derived pointwise geometric transformations. The macroscopic expressions are then computed numerically through quadrature rules applied in the radial and surface directions of the sphere. A three-scale compressible microsphere model is derived from the developed seminumerical homogenization framework where the micro-meso transition is based on the nonaffine microsphere model at every point of the mRVE. The numerical scheme developed for the derivation of macroscopic homogenized stresses and moduli terms as well as the modeling capability of the three-scale microsphere model is investigated through representative boundary value problems. KEYWORDS compressible hyperelasticity, homogenization methods, microsphere model, rubberlike materials INTRODUCTIONPorous polymeric materials exhibiting compressible hyperelastic behavior are common in engineering practice due to their properties, such as higher energy absorption and noise reduction. Despite the abundance of conventional hyperelastic constitutive models treating rubberlike materials incompressible, we lack a rigorous yet relatively simple treatment of this class of materials. The compressible mechanical response of porous elastomers results from preexisting voids introduced deliberately or unintentionally during the production process. The mechanical behavior of porous elastomers is intrinsically related to cavitation, which is attributed to the sudden growth of these preexisting voids under high triaxial stresses. The study of cavitation is particularly interesting since the initiation, growth, and coalescence of voids lead to 412

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A bifurcation problem for a compressible nonlinearly elastic medium: growth of a micro-void

Cited by 160 publications

References 7 publications

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

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

Energy Estimates and Cavity Interaction for a Critical‐Exponent Cavitation Model

A three‐scale compressible microsphere model for hyperelastic materials

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