Consideration of boundary-value problems in mechanics of materials with dis
ordered microstructures leads to the introduction of an intermediate scale,
a mesoscale, which specifies the resolution of a finite-element mesh relat
ive to the microscale. The effective elastic mesoscale response is bounded
by the Dirichlet and Neumann boundary-value problems. The two estimates, se
parately, provide inputs to two finite-element schemes-based on minimum pot
ential and complementary energy principles, respectively-for bounding the g
lobal response. While in the classical case of a homogeneous material, thes
e bounds are convergent with the finite elements becoming infinitesimal, th
e presence of a disordered non-periodic microstructure prevents such a conv
ergence and leads to a possibility of an optimal mesoscale. The method is d
emonstrated through an example of torsion of a bar having a percolating two
-phase microstructure of over 100 000 grains. By passing to an ensemble set
ting, we arrive at a hierarchy of two random continuum fields! which provid
es inputs to a stochastic finite-element method.