Mean-Field Description of Plastic Flow in Amorphous Solids

Abstract

Failure and flow of amorphous materials are central to various phenomena including earthquakes and landslides. There is accumulating evidence that the yielding transition between a flowing and an arrested phase is a critical phenomenon, but the associated exponents are not understood, even at a mean-field level where the validity of popular models is debated. Here, we solve a mean-field model that captures the broad distribution of the mechanical noise generated by plasticity, whose behavior is related to biased Lévy flights near an absorbing boundary. We compute the exponent θ characterizing the density of shear transformation PðxÞ ∼ x θ , where x is the stress increment beyond which they yield. We find that after an isotropic thermal quench, θ ¼ 1=2. However, θ depends continuously on the applied shear stress; this dependence is not monotonic, and its value at the yield stress is not universal. The model rationalizes previously unexplained observations and captures reasonably well the value of exponents in three dimensions. Values of exponents in four dimensions are accurately predicted. These results support the fact that it is the true mean-field model that applies in large dimensions, and they raise fundamental questions about the nature of the yielding transition.

Nonlinear plastic modes in disordered solids

“…This finding supports the robustness of our definition of plastic modes and the usefulness of our framework. It also supports the picture proposed by a number of recent studies [19][20][21][22][23][24] that assumes that the (reversible) destabilization process of a soft spot in a deformed glass is predominantly coupled to the external load and not to other coexisting (reversible) destabilization processes.…”

“…Shear transformations are known to self-organize in spatially correlated patterns [10][11][12][13][14][15][16] in solids subjected to large stresses and low deformation rates. Their densities and other statistical properties, and mechanical consequences, are a subject of much recent debate [17][18][19][20][21][22][23][24][25][26]. Two questions, central to theoretical descriptions of elastoplasticity, that we address in this work are whether shear transformations can be predicted a priori and if so, how.…”

Non-trivial rheological exponents in sheared yield stress fluids

“…In this study we discussed how a well-defined HB scaling can be modified by taking into account an effective shear-rate dependence of the elastic modulus, and/or the local relaxation processes, and we explained why we can observe a HB behaviour with a 1/2 exponent in many sheared athermal amorphous materials in the framework of ALYS models. Moreover, we discussed the validity range of this HB behaviour, corresponding to an intermediate shear-rate regime which is bounded (i) at low shear rates by the development of non-trivial stress fluctuations close to the critical point of zero shear rate 19,32,37 , and (ii) at high shear rates by the crossover towards intermediate scaling regimes depending on the relaxation dynamics and succeded by a completely fluidized, Newtonian regime. Highlighting furthermore the artefacts that may correspondingly arise when fitting numerical or experimental flow curves, we have thus provided alternative scenarios for measuring non-trivial HB exponents apart from its mean-field predicted value of 1/2.…”

“…More refined features of the rheology such as the exponent θ characterizing the density of shear transformations, discussed for instance in Ref. 37 , are known not to be correctly captured by diffusive ALYS models such as the HL model, but they are beyond the scope of the present work centered on the averaged mean stress of the flow curve.…”

“…As we have argued in Ref. 31 in a toymodel picture, such ALYS models should be suited for describing athermal systems, and they encompass in fact several existing models 10,31,[33][34][35][36][37][38] . Such a stress-threshold-based dynamics has to be distinguished from thermally activated process for which some energy barrier has to be overcome 35,39,40 ; here local barriers disappear when the stress threshold is reached.…”