Distributed thresholds vs distributed slip amplitudes

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Scaling properties and the yielding transition

Upon stress, materials exhibit deformation. Up to a stress value they show linear elastic properties. Passed that value, their deformation is still reversible, but nonlinear. Further load leads to irreversible deformation, and, eventually a plastic flow. When driven slowly, one may encounter sudden stress drops discussed previously. Finally, the material is likely to break (Figure 1.11). While the phenomenon presented here is not particular to amorphous solids as similar behavior is observed for crystalline materials as well, the processes behind the macroscopic behavior are different. While in crystals the stress drops in the flow curve are caused by the pinning of dislocations on defects, in amorphous materials they are related to the particle rearrangements.

Stress redistribution: shear transformations as inclusions

We have not told much yet about the way stress is redistributed upon rearrangements in mesomodels. Upon coarse graining, in mesomodels particle rearrangements are handled as material inclusions embedded into the continuous elastic bulk. These inclusions may undergo plastic deformation, but are squeezed by the surrounding elastic material, thus will wind up in a stressed state while inducing stress in the rest of the bulk as well. These inclusions are known as Eshelby inclusions and the stress field induced by them is known for various inclusion shapes [57, 20, 40, 71, 85, 178, 193, 194]. Mesomodels use such inclusions as a continuum counterpart of the shear transformations. When undergoing a plastic deformation, the stress-free reference frame of the Eshelby inclusions changes: a new shape corresponds to the zero stress state. The strain between the initial and new stress-free configurations is called the eigenstrain. Inclusions undergoing an eigenstrain induce a long-range displacement and stress field in the material, which in the far-field limit are strikingly similar to those of a shear transformation [48, 10, 174, 183] (Figure 2.7).
In particular, as shown on Figure 2.7, the elastic fields induced by such an inclusion are highly anisotropic. For instance, for an ellipsoidal inclusion at the origin, undergoing a pure plastic shear deformation with the principal axis oriented along ±π/4, the shear stress in the far-field approximation has the form G(r, θ) ∝ − cos 4θ r2 (2.15) Mesomodels then have at their basis a set of interacting Eshelby inclusions via 2.15 (Figure 2.8). Most of these models use a lattice arrangement for the inclusions [30, 29, 31, 128, 127, 126, 171, 190, 16, 28] but there have been several off-lattice attempts put into practice as well [79, 77, 78].
When working on a lattice, similarly to depinning (as we deal with elastic interactions after all), the elastic Eshelby interactions can be defined by an elastic kernel Gij = G(ri, rj) which gives the stress on site j when a plastic event happened on site i. The Eshelby kernel in two dimensions decays as 1/r2 which makes it a long range interaction. While working with long range interactions has its pitfalls, similar 1/x2 interactions occur in crack front propagation or triple contact line propagation (1 + 1 depinning, i.e. line in a plane) and there are available methods to deal with them. In 2 + 1 depinning (2d manifold in a 3d space) it was found that the 1/r2 kernel falls in the mean field universality class. This observation led to the conclusion that, since the Eshelby kernel is long-ranged, it should exhibit mean field universality [47, 73].

Discretization of the Eshelby fields

We have discussed that in the coarse grained model, blocks of the material are substituted with Eshelby inclusions containing several shear transformations. The details regarding the calculation of the Eshelby elastic fields are given in the Appendix A. In this rather technical section let us discuss two delicate, but important issues, as follows:
1. How to impose periodic boundary conditions on a finite system?
2. How to discretize the solution so that it can be used as a translation invariant elastic propagator? In order to perform numerical simulations, the elastic fields have to be discretized somehow. Moreover, in order to avoid unwanted boundary effects, periodic boundary conditions have to be imposed. Since we intend to work on a lattice, our model is particularly sensitive to discretization as various schemes may increase unphysical lattice effects. The main question of this section is thus: how to obtain a stable and physical elastic propagator on a finite, but periodic square lattice? The answer is not obvious given that what we have so far is the continuous solution of the Eshelby problem in an infinite system. Note that the (discretized) elastic propagator is the operator that translates the (discretized) plastic strain field into the internal stress field: |σeli = ˆG |ǫpi.

Are fluctuations kernel-dependent?

When talking about universality, one would expect properties called “universal” to be independent of the details of the particular model used. In this spirit, instead of the Eshelby kernel, mean field approaches were used to model amorphous systems [73, 47]. Mean field approaches very well reproduce properties as avalanche-like behavior or hardening, but it only recently started to become clear that a mean field approach is unable to capture all the generic properties, in particular those related to localization [107, 106, 172]. Here we show that the nature of the kernel and even the details of its building has important effects on the scaling of the fluctuations.
To that end, we focus on the fluctuation of the strain field and compare these fluctuations between the mean field and quadrupolar kernels.

Strain and displacement fluctuations in the finite element kernels

We saw the emergence of a diffusive regime for the quadrupolar Fourier kernel and we would expect the same diffusive long-time behavior for the finite element kernels. After all, they originate from the similar Eshelby inclusion problem. For the finite element kernels we have access to the displacement fields, therefore we can measure the displacement fluctuations as well. As we can see on Figure 3.7, there is no much difference between the fluctuations at low strains: both strain and displacement start out diffusively, for both of the kernels. On the other hand, the long time behavior is very different: while after an intermediate superdiffusive regime fluctuations for the m3 kernel converge to a second diffusive behavior, fluctuations in the m2 kernel saturate. Despite the fact that both kernels were meant to solve the same problem of the Eshelby inclusion (but in different geometries) and both of them have very similar symmetries, one of them reflects the same diffusive behavior as the Fourier quadrupolar kernel, while fluctuations in the other one saturate just as a mean field kernel would.
Note that although strain fluctuations in the m2 kernel indeed saturate, they do so at a much longer time than the mean field kernel. This observation reflects the extreme sensitivity on the particular discretization, and, at the same time raises the question whether the mean field kernel is sufficient to capture the physics behind amorphous plasticity? If the m2 kernel that exhibits a very similar quadrupolar symmetry to the Fourier kernel or the m3 kernel shows saturation at long times just as the mean field kernel, why would not a mean field approach work? To answer these questions, in the next section we investigate the effect of mean field on the strain fluctuations and correlations by fine-tuning between a mean field kernel and the Fourier quadrupolar kernel.

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Soft modes control fluctuations

We have seen that the Fourier quadrupolar kernel leads to a qualitatively different behavior than the mean field kernel. Furthermore, the m2 finite element kernel shows mean field-like saturation of the fluctuations at long times while the m3 kernel reproduces diffusion, in agreement with the Fourier quadrupolar kernel. Moreover, we have seen that the slightest mean field contribution into the kernel leads to the saturation of strain fluctuations. We could therefore conclude, that the inner structure of the kernel has a considerable impact on the properties the system exhibits. Let us therefore take a closer look to the details of the kernel.
In the following, we conduct an analysis based on the eigenvalues and eigenvectors of the elastic kernel showing how these properties affect the localization, in particular, the formation of shear bands. Moreover, we show that shear bands are soft modes of the kernel.

Scaling properties in the lattice model

In order to better comply with the MD conditions, here we will be using the quasistatic dynamics with synchronous flips introduced in chapter 2. While extremal dynamics was suitable to extract universal properties, it cannot be applied to MD, where a constant strain or stress dynamics is the most common. We therefore chose a constant strain load dynamics with vanishing strain rate. This dynamics involves synchronous pruning of all the unstable sites, until no more unstable sites are present, and only then is the loading increased. To ensure quasistatic loading, in between avalanches the strain is always adjusted such that only the weakest site yields. The strain is then held constant until the avalanche triggered by the first event is over. It was found in particular that this dynamics gives more accurate avalanche statistics [106], but it is not expected to affect the universal properties in any way.
Moreover, we will be using the two finite element kernels m2 and m3 depending on the loading geometry. Their Fourier-discretized counterparts will be referred to as m2F and m3F. In the previous chapters m3F was referred to as the Fourier quadrupolar kernel. As discussed previously, we have displacements available only for m2 and m3. Please refer to chapter 3 for details regarding the kernels. Recall that m3, m2F and m3F feature soft modes, however m2 does not.

Table of contents :

Preface
I Amorphous plasticity 
1 Introduction 
1.1 Phenomenology of amorphous plasticity
1.2 Modeling strategies
1.3 Scaling properties and the yielding transition
1.4 Analogy between depinning and plasticity mesomodels
1.5 Overview of Part I
1.6 Conclusions
2 Mesoscopic models 
2.1 A tour of mesomodels
2.2 Ingredients
2.2.1 Threshold dynamics
2.2.2 Loading
2.2.3 Stress redistribution: shear transformations as inclusions
2.3 Our models
2.4 Conclusions
3 Building elastic kernels: all about Eshelby 
3.1 Building elastic kernels
3.1.1 The Eshelby inclusion
3.1.2 Discretization of the Eshelby fields
3.1.3 Fourier discretization
3.1.4 A finite element method
3.2 Are fluctuations kernel-dependent?
3.2.1 Mean field depinning vs plasticity
3.2.2 Strain and displacement fluctuations in the finite element kernels .
3.2.3 Fluctuations of the strain field
3.3 Soft modes control fluctuations
3.3.1 Eigenvalues and eigenmodes of the elastic kernel
3.4 Conclusions
4 Scaling properties and finite size effects 
4.1 Review of MD results
4.1.1 Avalanches
4.1.2 Localization and diffusion
4.1.3 Density of shear transformation zones
4.2 Scaling properties in the lattice model
4.2.1 Distributed thresholds vs distributed slip amplitudes
4.2.2 Avalanches
4.2.3 Density of shear transformation zones
4.3 Diffusion
4.3.1 Fluctuations of the plastic strain field
4.3.2 Fluctuations of the displacement field
4.3.3 Trajectories and soft modes
4.4 Scaling properties of a minimal kernel
4.5 Summary of scaling relations
4.6 Conclusions
5 Application to amorphous composites 
5.1 Inclusions in an amorphous bulk
5.2 Introducing inhomogeneities
5.3 Size dependent flow stress
5.3.1 Amorphous matrix
5.3.2 Amorphous composites
5.4 Hardening and localization
5.4.1 Statistical hardening of the amorphous matrix
5.4.2 Inclusion hardening
5.4.3 Localization and shear band percolation
5.4.4 The weakest band
5.5 An analytical model
5.5.1 Percolation
5.5.2 Effective plastic behavior is defined by the weakest band
5.6 Conclusions
Appendix A Eshelby inclusions 
Appendix B Eigenvalues and eigenvectors of translation invariant 
Appendix C Evolution equation of soft modes 
Appendix D Finite element calculations 
D.1 Discretization of the displacement field and strains
D.2 Solving the equilibrium equations
D.3 Point force
D.4 Stresses
D.5 Eigenstrain on a plaquette – mode 2
D.6 Eigenstrain on a plaquette – mode 3
II Dewetting of thin liquid layers 
6 Soft line in quenched disorder 
6.1 Introduction and motivation
6.2 Basic concepts
6.3 The simulation method
6.4 Inhomogeneities
6.4.1 Extended inhomogeneities
6.4.2 Point-like inhomogeneities
6.5 Soft line in quenched disorder
6.6 Conclusions
Conclusions and perspectives
Publications

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