Stability of an expanding circular cavity and the failure of amorphous solids

Abstract

Recently, the existence and properties of unbounded cavity modes, resulting in extensive plastic deformation failure of two-dimensional sheets of amorphous media, were discussed in the context of the athermal Shear-Transformation-Zones (STZ) theory. These modes pertain to perfect circular symmetry of the cavity and the stress conditions. In this paper we study the shape stability of the expanding circular cavity against perturbations, in both the unbounded and the bounded growth regimes (for the latter the unperturbed theory predicts no catastrophic failure). Since the unperturbed reference state is time dependent, the linear stability theory cannot be cast into standard time-independent eigenvalue analysis. The main results of our study are: (i) sufficiently small perturbations are stable, (ii) larger perturbations within the formal linear decomposition may lead to an instability; this dependence on the magnitude of the perturbations in the linear analysis is a result of the non-stationarity of the growth, (iii) the stability of the circular cavity is particularly sensitive to perturbations in the effective disorder temperature; in this context we highlight the role of the rate sensitivity of the limiting value of this effective temperature. Finally we point to the consequences of the form of the stress-dependence of the rate of STZ transitions. The present analysis indicates the importance of nonlinear effects that were not taken into account yet. Furthermore, the analysis suggests that details of the constitutive relations appearing in the theory can be constrained by the modes of macroscopic failure in these amorphous systems.

Simulations of a stretching bar using a plasticity model from the shear transformation zone theory

“…The simulation method presented can be used as a basis for further study. It can be readily adapted to a wide variety of geometries, and we have written a preliminary version in polar coordinates for computing STZ elastoplasticity problems involving circular and elliptical holes in an infinite medium, for which a variety of exact and asymptotic solutions have been examined [21,23]. We also aim to study some newer versions of the STZ theory [24,25,26], and hope to incorporate orientational effects when a non-isotropic STZ density is employed.…”

Cavitation in Amorphous Solids

“…These studies observed evidence of the curvature dependence of the surface energy on the measured cavitation rate, an effect that has itself been the subject of a significant amount of study [19][20][21][22]. One particularly notable contribution from simulation was the mapping out of the point at which the gas-liquid spinodal dips below the glass line and the glass must become unstable to cavitation [23].Continuum mechanics approaches to modeling the kinetics of cavitation have been developed over many years by numerous researchers [6,[24][25][26][27][28][29][30]. Often preexisting cavities are assumed to exist due to voids, inclusions, or intersections of grains, and the effect of surface energy is neglected.…”

“…Continuum mechanics approaches to modeling the kinetics of cavitation have been developed over many years by numerous researchers [6,[24][25][26][27][28][29][30]. Often preexisting cavities are assumed to exist due to voids, inclusions, or intersections of grains, and the effect of surface energy is neglected.…”

“…Often preexisting cavities are assumed to exist due to voids, inclusions, or intersections of grains, and the effect of surface energy is neglected. Various assumptions have been considered regarding the plastic constitutive behavior around the cavity [6,[24][25][26][27][28][29][30]. In these cases, because plastic and elastic energy both scale with the volume of the void, above a critical stress cavity growth becomes unbounded.…”

Notch Fracture Toughness of Glasses: Dependence on Rate, Age, and Geometry

“…The fracture toughness is the critical value of the stress-intensity-factor, K Ic , at which crack propagation initiates and global failure occurs. Recent work [33,[38][39][40][41][42][43][44][45][46][47] suggests that this onset (and in fact also the subsequent propagation [33]) is controlled by a local cavitation instability occurring when the hydrostatic tension 1 3 trσ surpasses a threshold level σ c . We adopt this failure criterion here.…”