It is well known that the overall properties are a function of grain properties. Thus, each grain is assumed to be embedded in a homogenous equivalent medium exhibiting a response equal to the average response of all grains. In this context, the overall elastic and inelastic strain tensors are obtainable through the micro–macro methodology, which means that there are no state variables at this level. Hence, the macroscopic Cauchy stress tensor $\Sigma \xaf\xaf$ is also deduced from the same concept. In fact, after determining the granular inelastic strain rate as the sum of the contributions from all activated slip systems, the transition from the single to polycrystal response is performed by following the well-known averaging procedures. For homogenous elastic media, the overall stress $\Sigma \xaf\xaf$ (the overall strain $E\u02d9\xaf\xafe$) is simply the volume average of the granular stresses $\sigma \xaf\xaf$ (granular strains $\epsilon \xaf\xafe$) [58,59]. For the overall inelastic strain, the averaging procedure is usually not straightforward since it involves localization tensors [60]. In the special case of a single-phase polycrystal with isotropic elasticity, as in the actual case of elasto-inelastic behavior with homogeneous elasticity, the overall elastic and inelastic strain rates ($E\u02d9\xaf\xafe$ and $E\u02d9\xaf\xafin$) are equal to the average of granular rates (i.e., $\epsilon \u02d9\xaf\xafe$ and $\epsilon \u02d9\xaf\xafin$) [60]. The rates of change of overall elastic and inelastic strains are therefore computed, respectively, as follows:
Display Formula

(46)$E\u02d9\xaf\xafe=\u2211g=1Ngvg\epsilon \u02d9\xaf\xafe$

Display Formula(47)$E\u02d9\xaf\xafin=\u2211g=1Ngvg\epsilon \u02d9\xaf\xafin$