  We numerically and theoretically study macroscopic properties of dense,
sheared granular materials. In this process we first introduce an invariance in
Newton's equations, explain how it leads to Bagnold's scaling, and discuss how
it relates to the dynamics of granular temperature. Next we implement numerical
simulations of granular materials in two different geometries-- simple shear
and flow down an incline-- and show that measurements can be extrapolated from
one geometry to the other. Then we observe non-affine rearrangements of
clusters of grains in response to shear strain and show that fundamental
observations, which served as a basis for the Shear Transformation Zone (STZ)
theory of amorphous solids, can be reproduced in granular materials. Finally we
present constitutive equations for granular materials, based on the dynamics of
granular temperature and STZ theory, and show that they match remarkably well
with our numerical data from both geometries.
