Asymptotic results for bifurcations in pure bending of rubber blocks

Abstract

The bifurcation of an incompressible neo-Hookean thick hyperelastic plate with a ratio of thickness to length η and subject to pure bending is considered within a plane-strain framework. The two incremental equilibrium equations corresponding to a nonlinear prebuckling state of strain are reduced to a fourth-order linear eigenproblem that displays a multiple turning point. It is found that for 0 < η < ∞ the plate experience an Euler-type buckling instability which in the limit η → ∞ degenerates into a surface instability. Singular perturbation methods enable us to capture this transition, while direct numerical simulations corroborate the analytical results.

Buckling condensation in constrained growth

“…Consequently, the wavelength of the instability becomes independent of the elasticity ratio m I =m II . This condensation phenomenon is confirmed by plotting the relative components of the displacement field (since the absolute value is not given by a linear analysis, Coman and Destrade, 2008). Fig.…”

“…A boundary-layer analysis is well suited to describe this limiting case (Coman and Destrade, 2008). Let us mention that calculations in this section are done mostly with the help of a symbolic calculator and start with the growing case:…”

Bending instabilities of soft biological tissues

“…whereṡ is the incremental nominal stress and u is the incremental mechanical displacement, and by the satisfaction of appropriate boundary conditions. In the present context of pure bending in nonlinear elasticity, these equations have been written down several times (see (Coman and Destrade, 2008) and references therein), usually as a second-order system of coupled differential equations for the components of the displacement. Instead, we present them here as a first-order system for the displacement and the traction: this is the so-called Stroh formulation, which proves to be optimal for the subsequent numerical resolution of the boundary value problem.…”

Bifurcation of Elastic Multilayers

“…bifurcation problem. These findings relied on a generalization of previous results for plane-strain bending of an elastic block given by Rivlin [23] and on analyses of incremental bifurcations [24][25][26][27][28][29][30].…”