Wang, M, Jin, L and Fu, Y (2022) Axisymmetric necking versus Treloar–Kearsley instability in a hyperelastic sheet under equibiaxial stretching. Mathematics and Mechanics of Solids. ISSN 1081-2865

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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.

Item Type: Article
Additional Information: article is distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 License ( which permits non-commercial use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (
Uncontrolled Keywords: Necking; Treloar-Kearsley instability; equibiaxial tension; elastic localization; nonlinear elasticity
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Q Science > QA Mathematics > QA76 Computer software
T Technology > T Technology (General)
Divisions: Faculty of Natural Sciences > School of Computing and Mathematics
Depositing User: Symplectic
Date Deposited: 24 Feb 2022 09:21
Last Modified: 13 Apr 2022 10:24

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