Elastic nonlinearities in a one-dimensional model of fracture

Abstract

The dynamics of rapid brittle cracks is commonly studied in the framework of linear elastic fracture mechanics where nonlinearities are neglected. However, recent experimental and theoretical work demonstrated explicitly the importance of elastic nonlinearities in fracture dynamics. We study two simple one-dimensional models of fracture in order to gain insights about the role of elastic nonlinearities and the implications of their exclusion in the common linear elastic approximation. In one model we consider the decohesion of a nonlinear elastic membrane from a substrate. In a second model we follow the philosophy of linear elastic fracture mechanics and study a linearized version of the nonlinear model. By analyzing the steady state solutions, the velocity-load relations and the response to perturbations of the two models we show that the linear approximation fails at finite crack tip velocities. We highlight certain features of the breakdown of the linear theory and discuss possible implications of our results to higher dimensional systems.

The 1/r singularity in weakly nonlinear fracture mechanics

“…Note that these parameters cannot be determined by the asymptotic analysis as they depend on the global crack problem. Xðy; vÞ is a known universal function (Freund, 1998;Bouchbinder et al, 2008). in Eq.…”

“…In this section we derive, in detail, the weakly nonlinear theory, which was previously outlined in the results of Bouchbinder et al (2008). Consider the deformation field /, which is assumed to be a continuous, differentiable and invertible mapping between a reference configuration x and a deformed configuration x 0 such that…”

“…This solution is r-independent and results from solving a vectorial linear ordinary differential equation in y that depends on gðy; vÞ and thus on the second order elastic moduli. For the particular case of a neo-Hookean material, !ðy; vÞ (for a general v) was obtained in Bouchbinder et al (2008 …”

“…This physical situation motivates a controlled expansion of a general nonlinear elastic material constitutive law in powers of displacement-gradients, beyond the common small displacement-gradient approximation of LEFM. Such an approach was taken recently, when a weakly nonlinear fracture mechanics theory was developed by deriving the leading nonlinear elastic correction, up to second order displacementgradients, to the 1= ffiffi ffi r p singular fields of LEFM Bouchbinder et al, 2008). It is important to note that this approach is entirely general as any material exhibits nonlinear elastic response to some degree.…”

Weakly nonlinear fracture mechanics: experiments and theory

“…We will show the weakly nonlinear theory to be applicable at scales within a dynamic lengthscale nl from a crack's tip. nl denotes the scale where second order displacement-gradients become non-negligible compared to the first order ones, and deformation-dependent material behavior (Gao 1996;Buehler et al 2003;Buehler and Gao 2006;Bouchbinder and Lo 2008) is initiated.…”