Strain localization in a shear transformation zone model for amorphous solids

Abstract

We model a sheared disordered solid using the theory of Shear Transformation Zones (STZs). In this mean-field continuum model the density of zones is governed by an effective temperature that approaches a steady state value as energy is dissipated. We compare the STZ model to simulations by Shi, et al.[Y. Shi et al. PRL 98, 185505 (2007)], finding that the model generates solutions that fit the data, exhibit strain localization, and capture important features of the localization process. We show that perturbations to the effective temperature grow due to an instability in the transient dynamics, but unstable systems do not always develop shear bands. Nonlinear energy dissipation processes interact with perturbation growth to determine whether a material exhibits strain localization. By estimating the effects of these interactions, we derive a criterion that determines which materials exhibit shear bands based on the initial conditions alone. We also show that the shear band width is not set by an inherent diffusion length scale but instead by a dynamical scale that depends on the imposed strain rate.

Spontaneous formation of permanent shear bands in a mesoscopic model of flowing disordered matter

“…in the experimental work of Goyon et al 6,7 , who showed that a single flow curve was not able to account for the flow profile of a colloidal paste in a microfluidic channel. Such a size dependence of the flow profile hints to the existence of length scales that are not part of the simple rheological description, and several models have recently been put forward to account for the existence of such material specific length scales, either phenomenologically 8,9 or by an approximate coarse graining of a mesoscopic description of plastic flow 10 . The general structure couples a local rheological model with a diffusion equation that governs the behaviour of a "fluidity" parameter.…”

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

“…Manning et al [9,10] have simulated a variety of models to investigate shear-localization in a one-dimensional model, compared with analytic results. Pechenik [11] has derived a multi-dimensional elastoplastic model, and a two-dimensional model has been developed to study the necking instability in a stretched bar [12].…”

Criteria for Shear Banding in Time-Dependent Flows of Complex Fluids

“…However most practical flows involve a strong time-dependence, whether perpetually or during a startup process in which a steady flow is established from an initial rest-state. Data in polymers [8][9][10][11][12][13][14][15], surfactants [16][17][18], soft glasses [19][20][21][22], and simulations [23][24][25][26][27][28][29][30] reveals that shear bands often also arise during these time-dependent flows, and can be sufficiently long lived to represent the ultimate flow response of the material for practical purposes, even if the constitutive curve is monotonic, dΣ/dγ > 0.In view of these widespread observations, crucially lacking is any known criterion for the onset of banding in time-dependent flows. This Letter provides such criteria, with the same fluid-universal status as the criterion given above in steady state: independent of the internal constitutive properties of the particular fluid in question, and depending only on the shape of the experimentally measured rheological response function.…”