A tour of mesomodels
Mesoscale models represent a coarse grained description of amorphous plasticity. In this context, they follow a semi-continuous approach: the material is divided into elastoplastic blocks, and the interaction between these blocks is considered. These blocks may contain several shear transformations, however, since the precise boundaries of shear transforma-tions are still unclear, there is no exact definition of the proper discretization scale. The size of the block has to be large enough for continuum elasticity to hold, but at the same time small enough to account for the local heterogeneities of the plastic behavior related to the structure.
As sketched on Figure 2.1, mesomodels developed so far are built up of several main ingredients, and the combination of these ingredients gives the cooking details of the model. Some of the models involve thermal activation [30, 78], but in the mechanical response we are mainly concerned with athermal models. It can be readily seen, that the zoo of mesomodels has many dimensions and the complexity of this “model space” just keeps increasing over time. The extent to which the particular details of the individual models aﬀect the outcome is a matter of debate and establishing a common framework would be crucial to properly carry out an organized, apples-to-apples comparison.
To give an impression about the various models, here we briefly review some of them. In the subsequent sections of this chapter, we discuss the main ingredients of this models clarifying what aspects are relevant to the generic phenomenology of amorphous plasticity.
It all started with the work of Bulatov and Argon [30, 29, 31] where the plane was tes-sellated into hexagonal elements forming a hexagonal lattice. One element corresponded to one inclusion prone to undergo an eigenstrain (plastic deformation) and hence induc-ing a residual stress in the rest of the material. An exact calculation was carried out to account for the stress field due to an eigenstrain and an Arrhenius-like activation mechanism was proposed to overcome the barrier between the two configurations.
To study the inhomogeneous flow in yield stress fluids, Picard et al. [137, 138, 139] developed a model based on the far field interactions of the elastoplastic blocks. Disorder was considered through the rates at which the blocks switched between an elastic and a plastic regime. It was found that at low shear rate, the flow is governed by the cooperative bursts of plastic events.
Similarly to Picard, Nicolas et al. [128, 127, 126] proposed a viscoplastic evolution to model the inhomogeneous flow of soft amorphous solids. The discretization here was performed on a square lattice, but stresses were resolved on a finer mesh. The building tiles could alter between an elastic and a plastic state and the states could be interchanged with predefined rates. The model successfully recovered the rheology of silicon oil droplets in a water-glycerin mixture in a microchannel flow, as well as the Herschel-Bulkley exponent of the flow.
In another model, Homer et al. [78, 79] proposed an oﬀ-lattice discretization. In this model, the solid was discretized on an irregular triangular mesh and clusters composed of several triangles would correspond to a single shear transformation. A kinetic Monte Carlo method was used to drive the system and the spatial correlation between the activated zones was studied.
Lin et. al proposed a time-delayed activation mechanism in their mesoscopic model . They worked on a square grid, and inclusions could suﬀer a plastic deformation with a given probability per unit time, thus fixed rate. Unstable zones could be restabi-lized in case the stress on the unstable site dropped before it yielded. With this model, the Herschel-Bulkley flow was recovered and the density of shear transformation was investigated.
In all the models discussed so far, the disorder from the amorphous structure is represented by disorder in the dynamics. Along another line of mesomodels however, quench disorder is considered, i.e. plasticity is related to threshold instead of rates. Baret et al.  proposed a finite diﬀerence discretization and, taken from depinning studies, an extremal dynamics for the onset of a plastic event was considered. This method was later on refined [171, 190] where an analytic expression of the kernel was used, for plane loading conditions. A similar approach was used by Budrikis and Zapperi , but with strain controlled quasistatic load. In what follows, our methods resemble the most to these latter approaches.
The main idea behind mesomodels is to capture the essential phenomenology of amorphous plasticity. In this view, they are similar to the Ising model used as a model system for magnetic materials. As of today however, the consensus behind mesomodels are more fragmented: many diﬀerent methods and implementations are available. In order that they can indeed serve as the Ising model of amorphous plasticity, it has to be clarified first which details of these models are important and which are not to the phenomenology.
There have been many diﬀerent attempts to build coarse grained models based on inter-acting shear transformations. There are however several common concepts these models share. The key ingredients behind any mesomodel are the elastic interactions between individual rearrangements and the disorder associated to the amorphous structure. The dynamics of the amorphous systems is then led by the competition between elasticity and disorder : in the depinning analogy, elasticity tries to smoothen the manifold, while disorder roughens it by repeated pinning. The resulting stationary statistics is thus defined by the interplay between elasticity and disorder . In the mesomodels developed so far, disorder can be of two flavors: it can either enter into the dynamics [130, 83, 84, 126, 107], or into the landscape [171, 16, 28, 105]. Examples of stochastic dynamics include time-delayed models [130, 107] where the yield of unstable zones does not happen instantaneously, but with a given probability per unit time. This time delay is intended to model the coupling between neighboring zones during the rearrangements. The impact of the particular choice of the stochastic dynamics on the universal properties, in particular, avalanche scaling is currently a matter of debate [130, 83] and is beyond the scope of present work. In what follows, in the models we use we are going to introduce disorder through the landscape which would correspond to a quenched disorder.
From now on, we will be only focusing on models with a disorder in the landscape. Across this chapter we therefore review the origins of this static disorder. As the eﬀect of the particular form of the disorder on the phenomenology is not clarified, in later chapters we are testing various forms.
For completeness of the model presentation, we discuss several driving protocols, which are strongly related to the quenched disorder nature of the model. As it was revealed within separate works [171, 28, 105] and confirmed in subsequent chapters how-ever, the use of a particular driving protocol is irrelevant to the universal properties.
On the other hand, the use of the particular elastic interaction can have a dramatic impact on the phenomenology. This issue is going to be addressed in the next chapter, here however we provide a brief overview on the origin of the elastic kernel used in lattice models to capture the elastic interactions.
The onset of plastic events can be understood via a deformation in the potential energy landscape of the amorphous solid , as shown schematically on Figure 2.2. Initially, the system is in a stable equilibrium position, i.e. in an energy minimum. External loading stress however remodels the potential energy landscape. If the loading is small, the solid experiences reversible, elastic deformations. At this stage the depth of the energy minimum changes, but during the process, the minimum remains a minimum. As the loading increases however, the minimum eventually transforms into a saddle. At this point the system is no longer in a stable position, thus irreversible deformations take place in order to reach a new stable configuration. Plastic deformations thus happen at a threshold value of the loading.
All of the mesomodels use some sort of threshold criterion for the onset of a plastic event. If the stress in a region exceeds a certain threshold value, the zone yields. Here we show that such a criterion is in fact the direct consequence of the multistability of the rearranging zones, that is the possibility of multiple stable equilibrium configurations sub-ject to the same macroscopic conditions. We separate the eﬀects of elasticity and disorder and we argue here that the threshold dynamics is a direct outcome of the competition between elasticity and disorder upon coarse-graining in the direction of propagation.
Disorder leads to multistability
In this section we show the natural emergence of a threshold dynamics and the related stick-slip events from the multistability of the rearranging zones [148, 171]. In order to simplify the picture, let us consider a simple one dimensional problem as an example, early developed in the close context of solid friction [175, 34, 18] and rate independent plasticity [142, 141], namely the motion of a single point in a random potential. The point is connected to an elastic spring and we control the position of the loading end of the spring. Let us denote by x the position of the point and by y the position of the loading end of the spring. Let us further denote by V (x) the random potential, defined Such a system exhibits multistability when disorder overcomes elasticity. Figure 2.3 presents the graphical solution of the equilibrium equation ∂tx = 0. Equilibrium positions for a fixed position yi of the loading end of the spring are given by the intersections V ′(x) = −k(y − yi). Among these equilibrium positions the ones with V ′′(x) > −k are stable. It is clear that if the spring is stiﬀ enough, the intersection points follow continuously the shape of V ′(x) meaning that for each y∗ corresponds one and only one x∗ at which the point is in stable equilibrium. The dynamics is thus smooth and this is what we call weak pinning conditions  (Figure 2.3 (a)).
On the other hand, if the spring constant k is small with respect to the gradient of the force landscape, there are multiple stable x∗ positions for a fixed y∗ position of the loading, so the system exhibits multistability (Figure 2.3 (b)). The actual x∗ positions that are visited depend on the history of the loading, i.e. the previous positions that have been visited. This situation is called strong pinning . In this case, as the loading increases, the motion of the point is governed by sudden jumps between subsequent stable positions.
We can thus see that the nature of the dynamics is given by the interplay between the disordered potential and elasticity. If the potential traps are deep and narrow enough compared to the spring stiﬀness, the system exhibits a stick-slip dynamics. Note that the very same potential could result in a smooth dynamics when loading with a stiﬀer spring. In strain controlled experiments the stiﬀness is set by the material’s elastic properties, while the random landscape by its inherent disordered structure. The type of dynamics is thus entirely defined by the material properties.
Multistability leads to slips
Although in the strong pinning case multiple stable positions are available, one can define an eﬀective potential Wef f (y) = W [x∗i, y] associated to each stable position (x∗i, y) for fixed y. As shown on Figure 2.4, the eﬀective potential is composed of a set of truncated parabola-like curves. Upon driving, the system jumps from one local minimum to another one as soon as the elastic force exceeds the threshold value V ′(x) of the local maxima of the random force. We recover thus here the phenomenology of the instability inducing local rearrangements at the atomic scale in amorphous materials .
We represented on Figure 2.4 such a history-dependent sample trajectory and we see that a threshold dynamics in this simple case of an isolated point is a direct consequence of the multistability of the system. In particular, when coarse-graining at scale b, the dynamics of jumps between basins of V ′(x) is entirely controlled by a series of threshold forces (Figure 2.4).
The phenomenology remains unchanged when dealing with higher dimensional man-ifolds. In this case however, the disorder has to be compared to the internal elasticity of the manifold (rather than the stiﬀness of a loading spring). We see thus in this simple model system that thresholds can be thought of as the eﬀect of the strong pinning potential. When coarse graining in the propagation direction at a scale superior to b, the depinning equation 1.6 can be rewritten as
∂h (r, t) = P f ext(t) + G ∗ hr,( t ) − f cr,[ h r,( t )] (2.3)
where the P () function accounts for the positive part of its argument, that is P (x) = x if x > 0 and P (x) = 0 if x ≤ 0. f c is the threshold force resulting from the coarse graining. In these depinning-like models thus the disorder of the landscape enters into the model through the threshold forces.
Analogously to depinning, a similar equation of motion can be written for the evolu-tion of the plastic strain in amorphous systems: ∂ǫp (r, t) = P (Σload + G ∗ ǫpr,( t ) − σc[(r, ǫ pr,( t ))]) (2.4)
Recall that the internal stress induced by all the former plastic slips is given by the convolution G ∗ ǫpr,( t ) . The heterogeneity of the yield stress at mesoscopic scale is represented by the quenched random variable σc defined by its average σc and correla-tion σcr,( ǫ p)σcr( + r,δ ǫ p + δǫp) ∝ f (rδ )g(δǫp). Above the mesoscopic scale l at which the coarse graining is performed, we consider short range correlations, that is f (rδ ) → 0 if |rδ | ≫ l. Moreover, thresholds are uncorrelated in between successive events: g(δǫp) → 0 if δǫp ≫ e0 where e0 is the typical plastic strain associated to one elementary event.
Disordered landscape in mesomodels
One way to represent structural disorder in mesomodels is thus through thresholds. As they reflect the underlying disorder, thresholds should be considered random. As any random variable, thresholds are sensitive to their distribution and correlation. While distributed thresholds lead to the statistical hardening of the material  by the sys-tematic elimination of the weak zones, constant thresholds do not allow for such a phe-nomenology. Transient nevertheless is an important aspect of amorphous plasticity, be-cause in many cases materials fail before reaching a stationary plastic flow. An important question to address therefore is: how sensitive mesomodels are to the particular choice of the threshold distribution?
Another aspect of the disordered potential is its correlation along the direction of propagation, or, in the plasticity picture, the correlation over accumulating plastic strain. We have chosen to coarse grain at larger strain amplitudes, i.e. for each of our rearrange-ments δǫp ≫ e0. Still, the strain amplitude of each rearrangement could be random as it comes from a random landscape. The slip amplitude would then correspond to the distance between the minima of the asperities on Figure 2.4 and is related to the corre-lation of thresholds upon plastic deformation. A priori, the actual distribution of these slip amplitudes may be of importance.
Figure 2.5 shows two possible archetypes of the random landscape. In the first one, the potential barriers fluctuate resulting in distributed σc thresholds, but at the same time they are highly correlated in space, resulting in a narrow distribution of the slip amplitudes δǫp. In the second one, the barriers have a narrow distribution, but they do not show spatial correlation meaning that the slip amplitude distribution is wide. To investigate the impact of the threshold and slip amplitude distribution on the scaling properties, in chapter 4 we test these two extreme cases: one with distributed thresholds and constant slip amplitudes and the other one with constant thresholds and distributed slip amplitudes, but, theoretically, any combination of the two distributions is possible. While there is still debate whether the thresholds and the slip increments are independent or correlated, in the view of a quenched underlying potential the former sets the height of the barrier and the latter the distance between barriers, which, theoretically are two independent features of the potential.
Similarly to experiments, there are various protocols that come into account when driv-ing the system, but we expect universal properties to be invariant as long as quasistatic loading conditions are fulfilled. Nevertheless, diﬀerent protocols are appropriate for sam-pling diﬀerent properties. For example, avalanches are better defined in strain controlled load , while finite size eﬀects may be easier to investigate with extremal dynamics . In the upcoming sections we review some of the possible loading protocols, all of which are quasistatic loading applied to an athermal system. Finite strain rate , constant stress  and kinetic Monte Carlo methods  are available, but we are not going to use them. Most of the mesomodels consider homogeneous elasticity, loading is then considered homogeneous throughout the system.
Table of contents :
I Amorphous plasticity
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
2 Mesoscopic models
2.1 A tour of mesomodels
2.2.1 Threshold dynamics
2.2.3 Stress redistribution: shear transformations as inclusions
2.3 Our models
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
4 Scaling properties and finite size effects
4.1 Review of MD results
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
4.2.3 Density of shear transformation zones
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
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
5.5.2 Effective plastic behavior is defined by the weakest band
Appendix A Eshelby inclusions
Appendix B Eigenvalues and eigenvectors of translation invariant
Appendix C Evolution equation of soft modes