Homogenization of the Linear and Non-linear Mechanical Behavior of Polycrystals

This work is dedicated to the numerically efficient simulation of the material response of polycrystalline aggregates. Therefore, crystal plasticity is combined with a new non-linear homogenization scheme, which is based on piecewise constant stress polarizations with respect to a homogeneous reference medium and corresponds to a generalization of the Hashin-Shtrikman scheme. This mean field approach accounts for the one- and two-point statistics of the microstructure

Homogenization of the Linear and Non-linear Mechanical Behavior of Polycrystals

This work is dedicated to the numerically efficient simulation of the material response of polycrystalline aggregates. Therefore, crystal plasticity is combined with a new non-linear homogenization scheme, which is based on piecewise constant stress polarizations with respect to a homogeneous reference medium and corresponds to a generalization of the Hashin-Shtrikman scheme. This mean field approach accounts for the one- and two-point statistics of the microstructure

On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with elasticity tensors \BC_1>0 and \BC_2=0 comprising the set U=\{\BC_1,\BC_2\} and mixed in proportions $f$ and $1-f$ is partly characterized. The material with tensor \BC_2=0 corresponds to a material which is void. (For technical reasons \BC_2 is actually taken to be nonzero and we take the limit \BC_2\to 0). Specifically, recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is easily compliant to $p\leq 5$ independent applied strains, yet supports any stress in the orthogonal space. Thus the material can easily slip in certain directions along the walls. The region outside the walls contains "complementary Avellaneda material" which is a hierarchical laminate which minimizes the sum of complementary energies.Comment: 39 pages, 11 figure

Effective elastic properties of 3D stochastic bicontinuous composites

We study effective elastic properties of 3D bicontinuous random composites (such as, e.g., nanoporous gold filled with polymer) considering linear and infinitesimal elasticity and using asymptotic homogenization along with the finite element method. For the generation of the microstructures, a leveled-wave model based on the works of J. W. Cahn [J.W. Cahn. Phase separation by spinodal decomposition in isotropic systems. The Journal of Chemical Physics, 42(1):93-99, 1965.] and Soyarslan et al. [C. Soyarslan, S. Bargmann, M. Pradas, and J. Weissmüller. 3D stochastic bicontinuous microstructures: generation, topology and elasticity. Acta Materialia, 149: 326-340, 2018.] is used. The influences of volume element size, phase contrast, relative volume fraction of phases and applied boundary conditions on computed apparent elastic moduli are investigated. The nanocomposite behaves distinctly different than its nanoporous counterpart without any filling as determined by scrutinized macroscopic responses of gold-epoxy nanocomposites of various phase volume fractions. This is due to the fact that, in the space-filling nanocomposite the force transmission is possible in all directions whereas in the nanoporous gold the load is transmitted along ligaments, which hinges upon the phase topology through network connectivity. As a consequence, we observe a distinct elastic scaling law for bicontinuous metal-polymer composites. A comparison of our findings with the Hashin-Shtrikman, the three-point Beran-Molyneux and the Milton-Phan-Tien analytical bounds for the same composites show that computational homogenization using periodic boundary conditions is justified to be the only tool in accurate and efficient determination of the effective properties of 3D bicontinuous random composites with high contrast and volume fraction bias towards the weaker phase

On the Analytical Estimation for Isotropic Approximation of Elastic Properties applied to Polycrystalline Cubic Silicon used at Solar Cells

The present contribution is concerned with the effective mechanical parameters of polycristalline silicon used for solarcells. Thereby, an analytical scheme for the prediction of elastic properties of polycrystals is reviewed, applied and verified.Emphasis is on first-order bounds and derived estimates. Based on cubic symmetry of a single crystal the projector representationis exploited for the description of the constitutive equations. The elasticity tensor is developed for several assumptions whichstem from rather classical homogenization schemes. This results in a rational representation and determination of linear elasticmaterial parameters of a homogeneous, macroscopically isotropic comparison material. The analytically determined parametersare compared to experimental data. It is found that the present procedure reproduces the experimental results in very closeproximity

Elasticity of plagioclase feldspars

AbstractElastic properties are reported for eight plagioclase feldspars that span compositions from albite (NaSi3AlO8) to anorthite (CaSi2Al2O8). Surface acoustic wave velocities measured using Impulsive Stimulated Light Scattering and compliance sums from high‐pressure X‐ray compression studies accurately determine all 21 components of the elasticity tensor for these triclinic minerals. The overall pattern of elasticity and the changes in individual elastic components with composition can be rationalized on the basis of the evolution of crystal structures and chemistry across this solid‐solution join. All plagioclase feldspars have high elastic anisotropy; a* (the direction perpendicular to the b and c axes) is the softest direction by a factor of 3 in albite. From albite to anorthite the stiffness of this direction undergoes the greatest change, increasing twofold. Small discontinuities in the elastic components, inferred to occur between the three plagioclase phases with distinct symmetry ( , , and ), appear consistent with the nature of the underlying conformation of the framework‐linked tetrahedra and the associated structural changes. Measured body wave velocities of plagioclase‐rich rocks, reported over the last five decades, are consistent with calculated Hill‐averaged velocities using the current moduli. This confirms long‐standing speculation that previously reported elastic moduli for plagioclase feldspars are systematically in error. The current results provide greater assurance that the seismic structure of the middle and lower crusts can be accurately estimated on the basis of specified mineral modes, chemistry, and fabric