Maximum-Entropy Based Estimates of Stress and Strain in Thermoelastic Random Heterogeneous Materials
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Maximum-Entropy Based Estimates of Stress and Strain in Thermoelastic Random Heterogeneous Materials Maximilian Krause1 · Thomas Böhlke1
Received: 23 September 2019 © The Author(s) 2020
Abstract Mean-field methods are a common procedure for characterizing random heterogeneous materials. However, they typically provide only mean stresses and strains, which do not always allow predictions of failure in the phases since exact localization of these stresses and strains requires exact microscopic knowledge of the microstructures involved, which is generally not available. In this work, the maximum entropy method pioneered by Kreher and Pompe (Internal Stresses in Heterogeneous Solids, Physical Research, vol. 9, 1989) is used for estimating one-point probability distributions of local stresses and strains for various classes of materials without requiring microstructural information beyond the volume fractions. This approach yields analytical formulae for mean values and variances of stresses or strains of general heterogeneous linear thermoelastic materials as well as various special cases of this material class. Of these, the formulae for discrete-phase materials and the formulae for polycrystals in terms of their orientation distribution functions are novel. To illustrate the theory, a parametric study based on Al-Al2 O3 composites is performed. Polycrystalline copper is considered as an additional example. Through comparison with full-field simulations, the method is found to be particularly suited for polycrystals and materials with elastic contrasts of up to 5. We see that, for increasing contrast, the dependence of our estimates on the particular microstructures is increasing, as well. Keywords Maximum entropy method · Linear thermoelasticity · Heterogeneous materials · Homogenisation · Statistical second moments Mathematics Subject Classification 74B10 · 74M25
B T. Böhlke
[email protected] M. Krause [email protected]
1
Chair for Continuum Mechanics, Institute of Engineering Mechanics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
M. Krause, T. Böhlke
1 Introduction Heterogeneous materials, owing to their fabrication process, generally possess random microstructures, allowing for the application of statistical continuum theories to mechanical problems, as described by, e.g., Beran [3]. By using this framework to project the random heterogeneous material properties onto homogeneous effective properties, the “mean field” problem is obtained out of the more complex “full field” problem. This projection is achieved via homogenization methods, which can be numerical in nature, such as those used in FE [9], FFT [17] or NTFA [8] approaches. These numerical methods depend on fullfield calculations of representative volumes as an intermediate step, which requires detailed microstructure knowledge. Full-field calculations are generally more numerically expensive than analytical solutions, which offer explicit or implicit solutions in terms of straightforward formulas. Exact (i.e., precise
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