Universal Breakdown of Elasticity at the Onset of Material Failure

Abstract

We show that, in the athermal quasi-static deformation of amorphous materials, the onset of failure is accompanied by universal scalings associated with a divergence of elastic constants. A normal mode analysis of the non-affine elastic displacement field allows us to clarify its relation to the zero-frequency mode at the onset of failure and to the crack-like pattern which results from the subsequent relaxation of energy.Experiments on nanoindentation of metallic glasses [1], on granular materials [2] and on foams [3], demonstrate that at very low temperature and strain rates, the microstructural mechanisms of deformation involve highly intermittent stress fluctuations. These fluctuations can be accessed in molecular dynamics simulations, but are best characterized numerically via "exact" implementation of a-thermal quasi-static deformation: alternating elementary steps of affine deformation with energy relaxation [4] permits one to constrain the system to reside in a local energy minimum (inherent structure) at all times. As illustrated in figure 1, macroscopic stress fluctuations arise from a series of reversible (elastic) branches corresponding to deformation-induced changes of local minima. These branches are interrupted by sudden irreversible (plastic) events which occur when the inherent structure annihilates during a collision with a saddle point.[5] These transitions constitute the most elementary mechanism of deformation and failure for disordered materials at low temperature.Using this quasi-static protocol, recent studies of both elasticity [6] and plasticity [5], could identify important properties of elasto-plastic behavior which arise solely from the geometrical structure of the potential energy landscape. Tanguy et al [6] have observed that, following reversible (elastic) changes of the inherent structures, molecules undergo large scale non-affine displacements. They have shown these non-affine displacements to be related to the breakdown of classical elasticity at small scales and to quantitative differences between measured Lamé constants and their Born approximation. Malandro and Lacks [5] have shown that the destabilization of a minimum occurs through shear-induced collision with a saddle. At the collision, a single normal mode sees its eigenvalue going to zero. Building on this work, we studied the irreversible (plastic) event following the disappearance of an inherent structure: subsequent material deformation in search of a new minimum involves non-local displacement fields -in the likeness of nascent cracks-controlled by long-range elastic interactions. [7] Several molecular displacement fields thus appear to be closely related to the geometrical structure of the potential energy landscape: (i) non-affine displacements along elastic branches, (ii) the single normal mode controlling the annihilation of an inherent structure, and (iii) the overall deformation occurring during an irreversible event. In order to piece together a complete picture of elasto-plasticity at the nanoscale, we need ...

Continuum limit of amorphous elastic bodies. III. Three-dimensional systems

“…In the recent work [15] de Pablo and coworkers pointed out the possibility of regions of negative shear modulus in quenched amorphous systems -such regions being stabilized by the "normal" material in which they are embedded. The typical size of these regions is much smaller than the size for elastic inhomogeneities discussed in this work, implying they are more likely to be linked to elementary rearrangements taking place at the onset of plastic deformation, which usually imply small numbers of particles [6,7,8,9,24], or even localization along a shear band [10,11]. Such a difference in elastic and plastic deformations was also observed for 2D systems in Ref.…”

“…Plotting the rescaled and averaged frequencies y = (ωL/2πc T ) 2 as a function of mode number p, 6 it is clear that the degeneracies and the associated step-like behavior begin to show up only for large systems, containing at least 32000 particles (lateral size 32). For the largest systems however (lateral size 64) the discreteness of the low frequency spectrum is well apparent, typically up to the 4th eigenfrequencies.…”

“…They are simply the Born estimates, that can be, in a system with pairwise interactions, computed from the reference configuration by carrying out a simple summation over all pairs of interacting particles (see for example ref. [6])…”

“…In order to make the comparison as direct as possible, the same type of potential was chosen, i.e. a slightly polydisperse Lennard-Jones potential U ij (r) = 4ǫ (σ ij /r) 12 − (σ ij /r) 6 where the σ ij are taken uniformly distributed between 0.8σ and 1.2σ, corresponding to a polydispersity index of 0.12. This is expected to be enough to destabilise a polydisperse crystal [21], and indeed no sign of crystallisation or demixing was observed in our simulations.…”

“…Moreover, the field of plastic deformation of glassy materials, which has attracted considerable attention recently [6,7,8,9,10,11] may be expected to be related to elastic heterogeneities. Other points of interest include the experimental evidence for dynamical heterogeneities in deeply supercooled systems [12], which again could be expected to give rise to "frozen in" heterogeneities in low temperature systems.…”