Convolutional Neural Networks for the Localization of Plastic Velocity Gradient Tensor in Polycrystalline Microstructures

Recent work has demonstrated the potential of convolutional neural networks (CNNs) in producing low-computational cost surrogate models for the localization of mechanical fields in two-phase microstructures. The extension of the same CNNs to polycrystalline microstructures is hindered by the lack of an efficient formalism for the representation of the crystal lattice orientation in the input channels of the CNNs. In this paper, we demonstrate the benefits of using generalized spherical harmonics (GSH) for addressing this challenge. A CNN model was successfully trained to predict the local plastic velocity gradient fields in polycrystalline microstructures subjected to a macroscopically imposed loading condition. Specifically, it is demonstrated that the proposed approach improves significantly the accuracy of the CNN models, when compared with the direct use of Bunge-Euler angles to represent the crystal orientations in the input channels. Since the proposed approach implicitly satisfies the expected crystal symmetries in the specification of the input microstructure to the CNN, it opens new research directions for the adoption of CNNs in addressing a broad range of polycrystalline microstructure design and optimization problems.

Attenuation and Phase Velocity of Elastic Wave in Textured Polycrystals with Ellipsoidal Grains of Arbitrary Crystal Symmetry

This study extends the second-order attenuation (SOA) model for elastic waves in texture-free inhomogeneous cubic polycrystalline materials with equiaxed grains to textured polycrystals with ellipsoidal grains of arbitrary crystal symmetry. In term of this work, one can predict both the scattering-induced attenuation and phase velocity from Rayleigh region (wavelength >> scatter size) to geometric region (wavelength << scatter size) for an arbitrary incident wave mode (quasi-longitudinal, quasi-transverse fast or quasi-transverse slow mode) in a textured polycrystal and examine the impact of crystallographic texture on attenuation and phase velocity dispersion in the whole frequency range. The predicted attenuation results of this work also agree well with the literature on a textured stainless steel polycrystal. Furthermore, an analytical expression for quasi-static phase velocity at an arbitrary wave propagation direction in a textured polycrystal is derived from the SOA model, which can provide an alternative homogenization method for textured polycrystals based on scattering theory. Computational results using triclinic titanium polycrystals with Gaussian orientation distribution function (ODF) are also presented to demonstrate the texture effect on attenuation and phase velocity behaviors and evaluate the applicability and limitation of an existing analytical model based on the Born approximation for textured polycrystals. Finally, quasi-static phase velocities predicted by this work for a textured polycrystalline copper with generalized spherical harmonics form ODF are compared to available velocity bounds in the literature including Hashin–Shtrikman bounds, and a reasonable agreement is found between this work and the literature.

Addition Formula and Related Integral Equations for Heine—Stieltjes Polynomials

It is shown that symmetric products of Heine–Stieltjes quasi-polynomials satisfy an addition formula. The formula follows from the relationship between Heine–Stieltjes quasi-polynomials and spaces of generalized spherical harmonics, and from the known explicit form of the reproducing kernel of these spaces. In special cases, the addition formula is written out explicitly and verified. As an application, integral equations for Heine–Stieltjes quasi-polynomials are found.

Spherical harmonics analysis based on the Reuss model in elastic macro strain and stress determination by powder diffraction

In this paper a new approach to macro strain/stress analysis by generalized spherical harmonics is presented. It consists of expanding the stress tensor weighted by texture in a series of generalized spherical harmonics with the ground state of expansion specific to the classical Reuss model of an isotropic polycrystal. Like previously reported models having a ground state of hydrostatic type [Popa & Balzar (2001).J Appl Cryst.34, 187–195] and of Voigt type [Popaet al.(2014).J Appl Cryst.34, 154–159], the actual model is appropriate for use with Rietveld refinement.

Elastic macro strain and stress determination by powder diffraction: spherical harmonics analysis starting from the Voigt model

A new approach for the determination of the elastic macro strain and stress in textured polycrystals by diffraction is presented. It consists of expanding the strain tensor weighted by texture in a series of generalized spherical harmonics where the ground state is defined by the strain/stress state in an isotropic sample in the Voigt model. In contrast to similar expansions already reported by other authors, this new approach provides expressions valid for any sample and crystal symmetries and can easily be implemented in whole powder pattern fitting, including Rietveld refinement. An earlier article [Popa & Balzar (2001).J. Appl. Cryst.34, 187–195] reported a similar model, but with a spherical harmonics expansion around the hydrostatic strain/stress state of the isotropic polycrystal. The availability of several different models is beneficial in order to allow one to select the representation in which the ground state is the closest to the actual stress state in the sample.