We take the lowest eigenvalue of this local elastic tensor, i.e. an eective local shear modulus, as the relevant local
coarse-grained order parameter to relate the structural disorder of the glass to its observed
heterogeneous mechanical response. We will show that in contrast to other local structural
indicators previously analysed the local shear modulus presents an important degree of
correlation with the heterogeneous dynamics in the sheared glasses. Moreover the use of
the coarse-grained technique introduced in the previous chapter will anable us to estimate
the length scale at which this correlation is maximal.
Our approach is reminiscent of similar studies attempting to relate structure and dynamics
in static supercooled liquids. As illustrated1 in various numerical [172, 16, 173, 174,
175, 176, 177] and experimental studies (for excellent reviews on models and experiments we refer the reader to [178]), over (essentially) the last two decades, supercooled liquids near
the glass transition show peculiar dynamical features that have been associated to spatially
heterogeneous dynamical relaxation. Yet, their structure, as measured by two-point correlation
functions, appears homogeneous and unspectacular. There are various theoretical
pictures that try to incorporate connections between static and dynamic properties and it is
tempting (and maybe fruitful) to try and test some of the ideas developed in the framework
of supercooled liquids to achieve a better understanding of the structure/dynamics relation
in the related problem of driven glasses. Let us mention some of these approaches. In a
picture based on dynamical facilitation [179, 180] one postulates the existence of mobile
and immobile regions, with the implicit assumption that these regions have a structural
origin. Frustration-based theories [181] infer dynamical behavior by assuming the existence
of domains with a preferred local order. Alternatively, one can attempt to connect static
and dynamical properties through the congurational entropy [182, 183] through two-point
density correlations [184], through elastic properties [185, 186], or through the idea of a
rough energy landscape [187, 188].

It is an almost impossible task to try and summarize the ensemble of references that are
dealing with the question of a structural signature of the dynamics of glassy materials. In
fact what arouses the curiosity is the apparent paradox with glassy materials that can be
formulated as follows, how can a system with the apparent structure of a liquid behave
so dierently than the liquid ? It is to answer this crucial question that so many papers
have been devoted to try and extract some hidden structural order parameter in glassy
materials. Here we present some (the list is by no means exhaustive) of these local order
parameters that have been considered in glasses. Most of the discussion below concerns
results obtained through the extensive use of numerical simulations, as experimentally it
remains dicult to measure local quantities and even more to correlate local structural
and local dynamical information.