Dynamic failure in amorphous solids via a cavitation instability

Abstract

The understanding of dynamic failure in amorphous materials via the propagation of free boundaries like cracks and voids must go beyond elasticity theory, since plasticity intervenes in a crucial and poorly understood manner near the moving free boundary. In this Letter we focus on failure via a cavitation instability in a radially-symmetric stressed material, set up the free boundary dynamics taking both elasticity and visco-plasticity into account, using the recently proposed athermal Shear Transformation Zone theory. We demonstrate the existence (in amorphous systems) of fast cavitation modes accompanied by extensive plastic deformations and discuss the revealed physics.The principal difficulty in describing the mechanical failure of materials using elasticity theory is that this approach does not provide a law of motion for the free boundary that naturally occurs when a crack or a void is propagating. Recently, phenomenological phase-field models [1] were proposed in order to overcome this fundamental problem. In reality the high stress concentration near the moving boundary must be associated with some form of plastic deformations that are rather poorly understood in the context of amorphous materials; the consequences of these plastic deformations are usually referred to as the "process zone" whose properties one usually does not dare to probe too closely. The aim of this Letter is to do just the opposite [2]. We set up the simplest (from the point of view of symmetries) example of free boundary dynamics, i.e. that of a circular cavity responding to radially symmetric stresses at infinity (or equivalently high pressure inside the cavity). This problem has a rather long history [3,4]; the main contribution of our study is the elucidation of the role of amorphous plasticity and the detailed interaction between elasticity and plasticity, throughout the system and in particular near the free boundary where plastic dynamics is explicitly described in terms of the athermal Shear Transformation Zone theory (STZ) [5,6]. This theory automatically includes hardening and rate-dependent effects in addition to a proper Eulerian description of the equations of motion which allows the discussion of inertial effects and of large deformations including accelerating and catastrophic cavitation instability.Consider then an infinite 2D medium with a circular hole around the origin of radius R(t), loaded by a radially symmetric stress σ ∞ at infinity. The symmetry dictates a radial velocity field v that is independent of the azimuthal angle θ, i.e. v r (r, t) = v(r, t) and v θ (r, t) = 0. This velocity field implies the components of the total rate of deformation tensorIn this Letter we restrict the elastic part of the deformation to be small, allowing us to decompose the total rate of deformation tensor into its elastic and plastic parts.Denoting the material time derivative as D t = ∂ t +v∂ r ,where the plastic part D pl ij will be discussed below. The (linear) elastic part ǫ el ij in the present symmetry is determined by th...

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

“…This is understood as significantly higher rate of deformation is developed above the cavitation threshold, where unbounded growth takes place [14], compared to below the threshold where the rate of deformation vanishes at a finite time. We note that the dependence of χ ∞ onD pl affects both the zeroth and first order solutions such that R (0) and R (1) increase.…”

“…The application of the full fledged theory of STZ to crack propagation is still daunting (although not impossible) due to the tensorial nature of the theory and the need to deal with an extremely stiff set of partial differential equations with a wide range of time-scales and length-scales involved. For that reason it seemed advantageous to apply the full theory to a situation in which the symmetries reduce the problem to an effectively scalar theory; this is the problem of a circular cavity developing under circular symmetric stress boundary conditions [13,14]. While this problem does not reach the extreme conditions of stress concentration that characterizes a running slender crack, it still raises many physical issues that appear also in cracks, in particular the give-and-take between elasticity and plasticity, the way stresses are transmitted to moving boundaries (in apparent excess of the material yield stress) and most importantly for this paper, the possible existence of dynamical instabilities of the moving free boundary.…”

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.…”