  This study addresses the question of the quantitative reconstruction of
heterogeneous distributions of isotropic elastic moduli from full strain field
data. This parameter identification problem exposes the need for a local
reconstruction procedure that is investigated here in the case of materials
with small contrast. To begin with the integral formulation framework for the
periodic linear elasticity problem, first- and second-order asymptotics are
retained for the strain field solution and the effective elasticity tensor.
Properties of the featured Green's tensor are investigated to characterize its
decomposition into an isotropic term and an orthogonal part. The former is then
shown to define a local contribution to the volume integral equations
considered. Based on this property, then the combination of multiple strain
field solutions corresponding to well-chosen applied macroscopic strains is
shown to lead to a set of local and uncoupled identities relating,
respectively, the bulk and shear moduli to the spherical and deviatoric
components of the strain fields. Valid at the first-order in the weak contrast
limit, such relations permit point-wise conversions of strain maps into
elasticity maps. Furthermore, it is also shown that for macroscopically
isotropic material configurations a single strain field solution is actually
sufficient to reconstruct either the bulk or the shear modulus distribution.
Those results are then revisited in the case of bounded media. Finally, some
sets of analytical and numerical examples are provided for comparison and to
illustrate the relevance of the obtained strain-modulus local equations for a
parameter identification method based on full-field data.
