  In this work we calculate the local elastic moduli in a weakly polydisperse
2DLennard-Jones glass undergoing a quasistatic shear deformation at zero
temperature. The numerical method uses coarse grained microscopic expressions
for the strain, displacement and stress fields. This method allows us to
calculate the local elasticity tensor and to quantify the deviation from linear
elasticity (local Hooke's law) at different coarse-graining scales. From the
results a clear picture emerges of an amorphous material with strongly
spatially heterogeneous elastic moduli that simultaneously satisfies Hooke's
law at scales larger than a characteristic length scale of the order of five
interatomic distances. At this scale the glass appears as a composite material
composed of a rigid scaffoldingand of soft zones. Only recently calculated in
non homogeneous materials, the local elastic structure plays a crucial role in
the elasto-plastic response of the amorphous material. For a small macroscopic
shear strain the structures associated with the non-affine displacement field
appear directly related to the spatial structure of the elastic moduli.
Moreover for a larger macroscopic shear strain we show that zones of low shear
modulus concentrate most of the strain in form of plastic rearrangements. The
spatio-temporal evolution of this local elasticity map and its connection with
long term dynamical heterogeneity as well as with the plasticity in the
material is quantified. The possibility to use this local parameter as a
predictor of subsequent local plastic activity is also discussed.
