Bifurcation and Stability of an Incompressible Elastic Body Under Homogeneous Dead Loads With Symmetry

Sithigh, G. P. Mac; Chen, Yi-Chao

doi:10.1093/qjmam/45.2.293

Cited by 8 publications

(3 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The TK instability was first noted by Kearsley (1986) based on the experimental results of Treloar (1949), with its plane-strain counterpart analysed slightly earlier by Ogden (1985). It has subsequently been studied by MacSithigh (1986), Ogden (1987), Chen (1987), MacSithigh & Chen (1992, Batra et al (2005), andSteigmann (2007). However, our focus in the current study will be on a third instability, that associated with axisymmetric necking.…”

Section: Introductionmentioning

confidence: 87%

Axisymmetric necking versus Treloar-Kearsley instability in a hyperelastic sheet under equibiaxial stretching

Wang¹,

Jin²,

Fu³

2021

Preprint

View full text Add to dashboard Cite

We consider bifurcations from the homogeneous solution of a circular or square hyperelastic sheet that is subjected to equibiaxial stretching under either force-or displacement-controlled edge conditions. We derive the condition for axisymmetric necking and show, for the class of strain-energy functions considered, that the critical stretch for necking is greater than the critical stretch for the Treloar-Kearsley (TK) instability and less than the critical stretch for the limiting-point instability. An amplitude equation for the bifurcated necking solution is derived through a weakly nonlinear analysis and is used to show that necking initiation is generally sub-critical. Abaqus simulations are conducted to verify the bifurcation conditions and the expectation that the TK instability should occur first under force control, but when the edge displacement is controlled the TK instability is suppressed, and it is the necking instability that will be observed. It is also demonstrated that axisymmetric necking follows a growth/propagation process typical of all such localization problems.

show abstract

Section: Introductionmentioning

confidence: 87%

Axisymmetric necking versus Treloar-Kearsley instability in a hyperelastic sheet under equibiaxial stretching

Wang¹,

Jin²,

Fu³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The TK instability was first noted by Kearsley [60] based on the experimental results of Treloar [61], with its plane-strain counterpart analyzed slightly earlier by Ogden [62]. It has subsequently been studied by MacSithigh [63], Ogden [64], Chen [65], MacSithigh and Chen [66], Batra et al [67], and Steigmann [68]. However, our focus in this study will be on a third instability, that associated with axisymmetric necking.…”

Section: Introductionmentioning

confidence: 99%

Axisymmetric necking versus Treloar–Kearsley instability in a hyperelastic sheet under equibiaxial stretching

Wang

Jin

2022

Mathematics and Mechanics of Solids

View full text Add to dashboard Cite

We consider bifurcations from the homogeneous solution of a circular or square hyperelastic sheet that is subjected to equibiaxial stretching under either force- or displacement-controlled edge conditions. We derive the condition for axisymmetric necking and show, for the class of strain-energy functions considered, that the critical stretch for necking is greater than the critical stretch for the Treloar–Kearsley (TK) instability and less than the critical stretch for the limiting-point instability. An amplitude equation for the bifurcated necking solution is derived through a weakly nonlinear analysis and is used to show that necking initiation is generally sub-critical. Abaqus simulations are conducted to verify the bifurcation conditions and the expectation that the TK instability should occur first under force control, but when the edge displacement is controlled, the TK instability is suppressed, and it is the necking instability that will be observed. It is also demonstrated that axisymmetric necking follows a growth/propagation process typical of all such localization problems.

show abstract

“…However, requiring the material to satisfy the Baker-Ericksen inequalities, we find that one of these two classes does not allow the construction of multiphase homogeneous deformations, and so it is disregarded. For the remaining class, we use an analysis analogous to those developed in different contexts [18][19][20][21] and find that generally the equilibrium states are not uniquely determined for a given load. In this case, we determine equilibrium solutions with two equal principal stretches, which we name symmetric, and equilibrium solutions with all different principal stretches, which we name asymmetric.…”

mentioning

confidence: 99%

Stress-Induced Phase Transformations of an Isotropic Incompressible Elastic Body Subject to a Triaxial Stress State

Foti

2006

J Elasticity

View full text Add to dashboard Cite

The present paper concerns the stable multiphase isochoric deformations for an isotropic elastic body subject to a surface traction of uniform Piola stress with two equal principal forces which are opposite to the third. To model the occurrence of such deformations, we consider a strain energy density function which depends on the first principal invariant of deformation through a non-convex function and which has an added linear dependence on the second invariant. We establish existence conditions for equilibrium multiphase deformations which give restrictions on the morphology of the connecting phases as well as on the orientation of the flat interfaces between the phases. Finally, by considering a special, but representative, form for the non-convex strain energy function, we show that there exists a Bcritical'' value of the external load which allows for the emergence of stable coexistent deformation fields.

show abstract

Bifurcation and Stability of an Incompressible Elastic Body Under Homogeneous Dead Loads With Symmetry

Cited by 8 publications

References 0 publications

Axisymmetric necking versus Treloar-Kearsley instability in a hyperelastic sheet under equibiaxial stretching

Axisymmetric necking versus Treloar-Kearsley instability in a hyperelastic sheet under equibiaxial stretching

Axisymmetric necking versus Treloar–Kearsley instability in a hyperelastic sheet under equibiaxial stretching

Stress-Induced Phase Transformations of an Isotropic Incompressible Elastic Body Subject to a Triaxial Stress State

Contact Info

Product

Resources

About