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Error propagation in multiscale approaches to the elasticity of polycrystals
The error propagation properties of the Voigt, Reuss, Voigt-Reuss-Hill-Gilvarry, and Hershey schemes for the determination of the integral elastic response of texture free polycrystalline aggregates with cubic structure were studied. The sensitivity of the homogenized polycrystalline shear modulus was tested (i) analytically on the partial derivatives of the shear modulus with respect to the individual elastic constants within extremal Voigt and Reuss schemes, and (ii) numerically for all four methods. The sensitivity of the Hershey shear modulus on the input parameters, the single crystalline elastic constants B, C′, C44, is shown to be within the limiting values found for the Voigt and Reuss schemes. This conclusion is illustrated numerically on a set of five cubic materials with very different physical properties. The influence of the bulk modulus was found to be approximately two orders of magnitude smaller than that of C' and C44. The Hershey modulus was also found to be non-linear, asymmetric, and strongly dependent on the level of the elastic anisotropy of the studied system.