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-Al2O3 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.
Highlights
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], fast Fourier transform (FFT) [17] or NTFA [8] approaches
Exact analytical homogenizations are known for special microstructures [2], while analytical estimates such as self-consistent estimates, the Mori-Tanaka method [16] and differential schemes [19] suffice in many common applications
Kreher and Pompe [14] detail an approach to produce analytical approximations of firstand second-order statistical moments of stresses and strains in heterogeneous thermoelastic materials under consideration of effective material values obtained by homogenization methods or experiments
Summary
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]. Kreher and Pompe [14] detail an approach to produce analytical approximations of firstand second-order statistical moments of stresses and strains in heterogeneous thermoelastic materials under consideration of effective material values obtained by homogenization methods or experiments. This approach employs the maximum entropy methods commonly seen in statistical thermodynamics and physics (see Jaynes [11], [12]), and is often abbreviated as MEM in the following. Kreher and Pompe [13] mention an exact analytical solution obtainable by differentiating the effective strain energy, if its dependence on the phase stiffnesses is known This approach was used most prominently in Ponte Castaneda’s works on second-order homogenization methods [20]. Effective quantities are denoted by a bar, e.g., ψ
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