Free-boundary dynamics in elastoplastic amorphous solids: The circular hole problem

Abstract

We develop an athermal shear-transformation-zone (STZ) theory of plastic deformation in spatially inhomogeneous, amorphous solids. Our ultimate goal is to describe the dynamics of the boundaries of voids or cracks in such systems when they are subjected to remote, time-dependent tractions. The theory is illustrated here for the case of a circular hole in an infinite two-dimensional plate, a highly symmetric situation that allows us to solve much of the problem analytically. In spite of its special symmetry, this example contains many general features of systems in which stress is concentrated near free boundaries and deforms them irreversibly. We depart from conventional treatments of such problems in two ways. First, the STZ analysis allows us to keep track of spatially heterogeneous, internal state variables such as the effective disorder temperature, which determines plastic response to subsequent loading. Second, we subject the system to stress pulses of finite duration, and therefore are able to observe elasto-plastic response during both loading and unloading. We compute the final deformations and residual stresses produced by these stress pulses. Looking toward more general applications of these results, we examine the possibility of constructing a boundary-layer theory that might be useful in less symmetric situations.

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.…”

“…While the shear response of amorphous solids has received a significant amount of attention in the theoretical physics and molecular simulation literature over the past decade [1][2][3][4][5][6][7][8], significantly less attention has been devoted to hydrostatic loading in such systems [9,10]. This omission appears significant since experimental studies in metallic glass (MG) and other amorphous solids reveal nanocavities [11,12] that form during or subsequent to deformation and strongly implicate cavitation in the physics of the fracture process zone, even when the fracture behavior is relatively brittle [13][14][15].…”

“…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.…”

“…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.…”

Dynamic failure in amorphous solids via a cavitation instability

“…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.…”