A Strain-Gradient Virtual-Internal-Bond Model

Abstract

Nonlinear continuum models with nonconvex elastic energies result in equilibrium equations that lose ellipticity at some critical level of deformation. Beyond this critical strain, discontinuous-strain solutions emerge, and due to the absence of an internal length scale, the equilibria computed using finite element methods strongly depend on the selected mesh size. In particular, this problem presents itself in nonlinear models of fracture. One such example is the Virtual Internal Bond (VIB) model (Gao and Klein, 1998; Klein and Gao, 1998, 2000; Zhang et al., 2001), where the constitutive law is found by averaging over a random network of cohesive bonds with nonconvex Lennard-Jones-type potentials. The model successfully predicts critical stress level and direction of the deformation zone for nucleation of fracture that appears as a localized zone of high strain. However, without an internal length scale it cannot predict the size of the localized deformation zone. One way to introduce a length scale into the VIB model is to employ a straingradient nonlinear elasticity theory. Phenomenological models incorporating higher-order gradient term in the constitutive law have been used to study localization phenomena in materials (Coleman; 1983; Aifantis, 1984; Coleman and Hodgdon, 1985; Fleck and Hutchinson, 1998; Shi et al., 2000). Recently, a derivation of such theories from periodic lattice microstructures has been presented by Triantafyllidis and Bardenhagen (1993) and Bardenhagen and Triantafyllidis (1994). This derivation introduces a natural length scale into the continuum model: a characteristic lattice spacing . The derivation of the strain-gradient theory in the one-dimensional case given by Triantafyllidis and Bardenhagen (1993) is reviewed and discussed in this paper. It is shown that the positive sign of the strain-gradient coefficient is crucial for the existence of strain localization zones that replace strain discontinuities resulting from the local approximation. On the other hand, the positive sign of strain-gradient term requires a rather special and not always realistic choice of the long-range interaction potentials. It also leads to a wrong qualitative behavior of the dispersion relation.