# Converting strain maps into elasticity maps for materials with small contrast

## TL;DR

This paper develops a local method to convert strain maps into elasticity maps for materials with small contrast, enabling point-wise reconstruction of elastic moduli from full strain field data.

## Contribution

It introduces a new local reconstruction procedure using asymptotic analysis and Green's tensor properties for materials with small contrast, allowing direct conversion of strain to elasticity maps.

## Key findings

- Point-wise conversion formulas for strain to elasticity maps.
- Single strain field suffices for isotropic materials to reconstruct bulk or shear modulus.
- Validated with analytical and numerical examples.

## Abstract

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.