Abstract.
We show that the averaged response of random isotropic Cauchy elastic material can be described analytically. It leads to a higher gradient model with explicit expressions for the dependence on the second derivatives of the mean field. A subsequent penalty formulation coincides with a linear elastic micro-stretch model with specific choice of constitutive parameters, depending only on the average cut-off length (the internal characteristic length scale L c > 0). Thus the microstretch displacement field can be interpreted as an approximated mean field response for these parameter ranges. The mean field free energy in this micro-stretch formulation is not uniformly pointwise positive, nevertheless, the model is well posed.
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Neff, P., Jeong, J., Münch, I. et al. Mean field modeling of isotropic random Cauchy elasticity versus microstretch elasticity. Z. angew. Math. Phys. 60, 479–497 (2009). https://doi.org/10.1007/s00033-008-8090-2
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DOI: https://doi.org/10.1007/s00033-008-8090-2