Stability of an internally constrained, hyperelastic slab

• Published in: International journal of non-linear mechanics 1998-09, Vol.33 (5), p.867-906
• Main Authors: Beatty, M.F., Pan, F.
• Format: Article
• Language: English
• Subjects: Buckling; constrained materials – Bell models – neo-Hookean model – small deformations on large – non-linear elasticity; elastic stability; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Physics; Solid mechanics; Static buckling and instability; Structural and continuum mechanics
• ISSN: 0020-7462; 1878-5638
• DOI: 10.1016/S0020-7462(97)00059-0

The elastic stability of an internally constrained thick plate is investigated on the basis of the well-known Euler dead load criterion of existence of adjacent equilibrium states. Two classes of internally constrained materials are examined—the Bell constrained material characterized by tr V=3, and the incompressible material for which det V=1, V being the left-stretch tensor. Although this study focuses mainly on instability of a Bell constrained slab, some new results are presented as well for an incompressible neo-Hookean slab. Graphical illustrations of the critical stretch as a function of the slenderness measure of the slab are provided for two classes of Bell materials—the simple hyperelastic Bell material model and an empirical model first described in experiments by Bell. The oscillatory nature of the stability results for Bell’s empirical model differ from the monotone solutions found for both the simple hyperelastic Bell material and the neo-Hookean material models. Estimates for buckling loads of thin plates deduced from the thick-plate analysis are compared with classical results. The critical load found for the simple hyperelastic Bell material coincides with the result deduced from classical thin plate theory, while the solution found for Bell’s law, which is parabolic in the engineering strain, does not. The half-space stability problem is examined as an extension of results deduced for a thick plate based on Bell’s law.