New results in the theory of plane strain flexure of incompressible isotropic hyperelastic materials

Abstract

New results on the classical problem of bending by end moments for incompressible isotropic hyperelastic materials within the framework of nonlinear elasticity are investigated and presented in this paper. The particular results of concern here include (i) the adaptation of Rivlin's standard analysis to the case where one end of the beam is fixed and the other end is subjected to a bending moment; and (ii) results on the finite bending of (infinitesimally) thin isotropic hyperelastic plates which are valid for large deformations , extending the classical results from the linear elasticity theory which are restricted to small deformations. An interesting feature observed in this context is that a flexed thin plate develops an oscillatory surface along the circular arc near the free end, due to local (small) deviations of the radius of curvature. A potential application to the bending of a biological soft tissue, namely the aortic valve leaflet, is briefly described by way of an example. Finally, some new results are obtained for finite bending of hyperelastic materials that exhibit limiting chain extensibility at the molecular level and involve constraints on the deformation. The amount of bending that such materials can sustain is limited by the constraint. On using a limiting chain extensibility model, closed-form solutions for the Cauchy stress components, the bending moment and the normal out-of-plane force required to sustain the bending deformation are derived.

Large Isotropic Elastic Deformations: On a Comprehensive Model to Correlate the Theory and Experiments for Incompressible Rubber-Like Materials

“…However, in comparison, the more accurate predictions of this model over the Gent model have been demonstrated (see, e.g., [24,28]). This model was then successively adopted for application to the mechanical behaviour of rubber-like materials for various boundary value problems including the investigation of elastic instabilities [16,[28][29][30][31][32].…”