Anisotropic Polycrystals Research Articles

A machine learning model to predict yield surfaces from crystal plasticity simulations

We introduce a microstructurally informed machine learning model for predicting the anisotropic yield surfaces of polycrystalline materials. A full-field, spatially resolved crystal plasticity model is employed to generate a data set describing the yield response of an aluminum alloy, enabling the training of a neural network yield function and the calibration of 3D yield criteria of plastically anisotropic polycrystals. This novel formulation explores the flexibility of neural networks to describe complex-shaped yield loci and avoids common problems associated with conventional 3D yield functions, such as the non-trivial parameter identification and non-uniqueness of the anisotropy coefficients. Here, Bayesian optimization is applied to obtain an optimal neural network architecture and allows for an automated model design. The neural network yield function is able to learn intrinsic properties such as the convexity of the yield hull and tension–compression symmetry from a relatively small number of data points. The fully data-driven yield criterion can accurately reproduce multiaxial flow response and planar anisotropy despite of its material blind initial state. Stress gradients can also be computed from the neural network through automatic differentiation as derived quantities with good fidelity. This allows the calculation of r-values and provides a pathway for implementing the neural network yield model into finite element codes.

Elastic wave propagation in anisotropic polycrystals: inferring physical properties of glacier ice

An optimization problem is proposed for inferring physical properties of polycrystals given ultrasonic (elastic) wave velocity measurements, made across multiple sample orientations. The feasibility of the method is demonstrated by inferring both the effective grain elastic parameters and the grain c -axis orientation distribution function (ODF) of ice-core samples from Priestley glacier, Antarctica. The method relies on expanding the ODF in terms of a spherical harmonic series, which allows for a non-parametric estimation of the sample ODF. Moreover, any linear combination of the Voigt (strain) and Reuss (stress) homogenization scheme is allowed, although for glacier ice, the exact choice is found to matter little for bulk elastic behaviour, and thus for inferred physical properties, too. Finally, the accuracy of the inferred grain elastic parameters is discussed, including the well-posedness and shortcomings of the inverse problem, relevant for future adoptions in glaciology, geology and elsewhere.

Constitutive Relations of Anisotropic Polycrystals: Self-Consistent Estimates.

In this paper, the elastic constitutive relation of polycrystals contains the effect of the mesostucture coefficients. We consider a general case and derive the average elastic constitutive relation pertaining to polycrystals of cubic crystals with any symmetry of crystalline orientation in their statistical distribution. Following Budiansky and Wu, we used self-consistent estimates of eigenstrain to obtain the effective elastic constitutive relation of polycrystals in an explicit form. For the Voigt assumption and the Reuss assumption, the effective elastic constitutive relation of polycrystals on cubic crystals contains the the mesostructure coefficients up to linear terms. In general, the linear term expression works well for materials such as aluminum, the single crystal of which has weak anisotropy. However the same expression (which allows the anisotropic part of the effective elastic constitutive relation to depend only linearly on the mesostructure coefficients) does not suffice for materials such as copper, in which the single crystal is strongly anisotropic. Per the Taylor theorem, we expand the expression based on the self-consistent estimates with respect to the mesostructure coefficients up to quadratic terms for anisotropic polycrystals of cubic crystals. While our numerical data are very close to those of Morris, our expression is much simpler.

Computational thermomechanics for crystalline rock. Part II: Chemo-damage-plasticity and healing in strongly anisotropic polycrystals

We present a thermal–mechanical–chemical-phase field model that captures the multi-physical coupling effects of precipitation creeping, crystal plasticity, anisotropic fracture, and crack healing in polycrystalline rock at various temperature and strain-rate regimes. This model is solved via a fast Fourier transfer solver with an operator-split algorithm to update displacement, temperature and phase field, and chemical concentration incrementally. In nuclear waste disposal in salt formation, brine inside the crystal salt may migrate along the grain boundary and cracks due to the gradient of interfacial energy and pressure. This migration has a significant implication on the permeability evolution, creep deformation, and crack healing within rock salt but is difficult to incorporate implicitly via effective medium theories compared with computational homogenization. As such, we introduce a thermodynamic framework and a corresponding computational implementation that explicitly captures the brine diffusion along the grain boundary and crack at the grain scale. Meanwhile, the anisotropic fracture and healing are captured via a high-order phase field that represents the regularized crack region in which a newly derived non-monotonic driving force is used to capture the fracture and healing due to the solution–precipitation. Numerical examples are presented to demonstrate the capacity of the thermodynamic framework to capture the multiphysics material behaviors of rock salt.

Relationship between the thermal stability of coercivity and the aspect ratio of grains in Nd-Fe-B magnets: Experimental and numerical approaches

The influence of the aspect ratio of grains on the thermal stability of coercivity of Nd-Fe-B hot-deformed magnets was systematically investigated by experimental and numerical approaches. With increasing amount of Nb doping from 0.2 at.% to 0.6 at.% in the hot-deformed magnets, the aspect ratio of grains, defined as the ratio of the width to the height of a Nd2Fe14B grain (Dc/Dab), decreased while the average grain size was preserved. No change of temperature coefficient of coercivity (β) was observed in the as hot-deformed samples with different grain aspect ratios. However, the reduction of aspect ratio (increase of the height and reduction of the width of platelet shaped Nd2Fe14B grains) improves β for the sample infiltrated with a Nd-Cu eutectic alloy; i.e., from β = −0.42 %/°C to −0.40 %/°C for the aspect ratio reduction from 4.7 to 3.1. Micromagnetic simulations indicated that the grain aspect ratio is not a dominant factor to the thermal stability of coercivity for exchange-coupled anisotropic polycrystals. While smaller aspect ratio reduces demagnetizing field caused by magnetically isolated grains, resulting in the improvement of the thermal stability of the coercivity for Nd-Cu infiltrated hot-deformed magnets.

The role of elastic anisotropy on the macroscopic constitutive response and yield onset of cubic oligo- and polycrystals

The single crystal elastic anisotropy is one of the most important properties affecting the macroscopic response of polycrystalline materials, both in the elastic regime and the early stages of plastic deformation. In this work, the impact of monocrystalline parameters on the mechanical behaviour of cubic polycrystals is studied in detail. The analysis is conducted with an RVE-based computational homogenisation framework which includes several original strategies and criteria developed to enable the efficient statistical analysis of virtual micro-structures. In particular, these developments include a stress-driven adaptation to the usually employed strain-driven homogenisation approach, used to study the plastic polycrystalline response in stress space. The effects of number of grains and realisations considered, finite element mesh discretisation level, and algorithms used to generate both the grain morphology and orientation distribution are also assessed. Considering a number of materials with cubic symmetry, expressions for the number of grains required for an elastically isotropic response are deduced, along with bounds for the homogenised elastic stiffness components of oligocrystals, in both cases as a function of the monocrystalline elastic anisotropy. In the plastic domain, multiple microscopic and macroscopic criteria to describe the yield onset of such polycrystals are evaluated and compared. This is done for both elastically isotropic and anisotropic polycrystals, where once again the effect of different levels of monocrystalline elastic anisotropy are evaluated. The numerically derived yield surface of isotropic polycrystals is fitted to both Tresca and von Mises yield functions, and for anisotropic oligocrystals, the statistical bounds for the expected distribution of micro-yield stresses are deduced, in terms of both the number of grains and of the level of single crystal elastic anisotropy. The results obtained are thus a step towards clarifying the effect of important micro-structural parameters in establishing the minimum representative volume element (RVE) size for metallic alloys in elasto-plastic applications.

Progressive modelling and experimentation of hydrogen diffusion and precipitation in anisotropic polycrystals

Hydrogen embrittlement of zirconium alloys is a major concern in nuclear industry. It is known that diffusion of hydrogen atoms in zirconium lattice depends on the localized state of stresses, the patterning of which is significantly affected by the elastic and plastic anisotropy of zirconium crystals. This study focuses on developing a model to understand the effects of such localized stress fields on hydrogen diffusion and precipitation. For this purpose, a crystal plasticity finite element code is updated and coupled with a series of newly developed subroutines for simulating stress-assisted-diffusion of hydrogen atoms in zirconium polycrystals. An electron backscatter diffraction experiment is also conducted on a hydrided Zircaloy-2 sample to compare the measured distribution of the hydride phase with the calculated one. It is shown that even in the absence of macroscopic loads, thermal residual stresses within zirconium grains can disturb the uniform distribution of hydrogen atoms by inducing large hydrostatic stresses at the vicinity of grain boundaries. Such stress concentrations can further affect the sequence of hydrides nucleation. Also, it is shown that by alternating local stress fields, hydride induced transformation strain can change the distribution of the hydrogen atoms in zirconium matrix.

Fourier Spectral Phase Field Simulations of Elastically Anisotropic Heterogeneous Polycrystals

Abstract In this study, we developed a multi-order, phase field model to compute the stress distributions in anisotropically elastic, inhomogeneous polycrystals and study stress-driven grain boundary migration. In particular, we included elastic contributions to the total free energy density and solved the multicomponent, nonconserved Allen–Cahn equations via the semi-implicit Fourier spectral method. Our analysis included specific cases related to bicrystalline planar and curved systems as well as polycrystalline systems with grain orientation and applied strain conditions. The evolution of the grain boundary confirmed the strong dependencies between grain orientation and applied strain conditions and the localized stresses were found to be maximum within grain boundary triple junctions.

Impact of grain shape on the micromechanics-based extraction of single-crystalline elastic constants from polycrystalline samples with crystallographic texture

A micromechanics-based method (“inverse self-consistent approximation”) that can extract all the independent elastic constants of single crystals from those of polycrystals with crystallographic texture was newly developed. In the developed method, all the elastic constants in an anisotropic polycrystal were measured by resonant ultrasound spectroscopy, and the crystallographic orientations and shapes of the grains were analyzed. Then, the elastic constants of a single crystal, which reproduce those of the polycrystal, were determined on the basis of Eshelby's inclusion theory and the effective-medium approximation, taking into account the elastic interaction between the grains, which reflects their shapes and orientations. The developed method determined the elastic constants of pure Cu and pure Mg single crystals from those of polycrystals quite precisely. The differences between the elastic constants obtained using our method and the values measured using single crystals were only ∼ 1%. In contrast, the application of an inverse Voigt–Reuss–Hill approximation, which cannot consider the effect of the grain shape on the elastic interaction, resulted in a relatively poor evaluation for pure single-crystalline Cu which exhibits strong elastic anisotropy. This indicates that the grain shape clearly affects the elastic interaction between the grains exhibiting high elastic anisotropy and influences the extracted single-crystalline elastic constants. In terms of the actual application of the inverse self-consistent approximation, the single-crystalline elastic constants of AZ31 Mg alloy, whose single crystals cannot be prepared easily, were clarified.

Displacement field of doubly periodic array of dislocation dipoles in elastically anisotropic media

The displacement field for dislocation dipoles periodically arranged along both x- and y-directions is found to be conditionally convergent. That is, different displacement fields are obtained depending on the order of the summation to be adopted. From the two summations, one can be performed analytically; however, the other one has to be performed numerically. We first derive analytic expressions for the displacement field of periodic array of dipoles along one (either x or y) direction considering anisotropic elasticity; they are then applied for the numerical summation (practically truncated) along the other direction. The resulting displacement field needs to be corrected by subtracting the spurious displacement field, whose expressions are analytically derived. As a first application, we employ the displacement and corresponding stress fields in a 2D discrete dislocation plasticity (DDP) model of a fine-grained polycrystal under shear loading. To this end, anisotropic plane-strain DDP method is utilised to solve the underlying boundary value problem. Subsequently, predictions of size-dependent plastic behaviour in anisotropic polycrystals with grain sizes in the range are presented.

Bounds and an isotropically self‐consistent singular approximation of the linear elastic properties of cubic crystal aggregates for application in materials design

AbstractFor crystal aggregates, the orientation distribution of single crystals affects the anisotropic linear elastic properties. In the singular approximation for cubic materials, this influence is reflected by a fourth‐order texture coefficient. From this approximation, the statistical bounds of Voigt, Reuss and Hashin‐Shtrikman, and an isotropically self‐consistent singular approximation can be obtained. Here, an approximation is called isotropically self‐consistent, if, for a vanishing texture, it results in the isotropic self‐consistent approximation. The isotropically self‐consistent singular approximation has the following advantages: i) it lies between the bounds of Voigt, Reuss and Hashin‐Shtrikman, ii) it offers a useful approximation of the effective material behavior of textured anisotropic polycrystals, and iii) it can be used for material design purposes tailoring anisotropic properties mainly depending on the crystallographic texture. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Phase field modeling of directional fracture in anisotropic polycrystals

A phase field theory for modeling deformation and fracture of single crystals, polycrystals, and grain boundaries is developed. Anisotropies of elastic coefficients and fracture surface energy are addressed, the latter enabling favorable cleavage on intrinsically weak planes in crystals. An order parameter increases in value as damage accumulates in an element of material. The shear elastic coefficients deteriorate with cumulative damage regardless of local strain state, while the effective bulk modulus degrades only under tensile volumetric deformation. Governing equations and boundary conditions are derived using variational methods. An incremental energy minimization approach is used to predict equilibrium crack morphologies in finite element simulations of deforming polycrystals. Thin layers of material, representative of glassy second phases near grain boundaries, are assigned possibly different properties than surrounding crystals. Results of simulations of polycrystals subjected to tensile loading are reported, with base properties representative of silicon carbide or zinc. Key findings include (i) a tendency for intergranular over transgranular fracture as the grain boundary surface energy is reduced or as cleavage anisotropy is increased and (ii) an increase in overall ductility and strength, the latter similar to Hall–Petch scaling, as the absolute size of the polycrystal is reduced while holding the ratio of phase field regularization length to grain size fixed.

Elastic mechanical grain interactions in polycrystalline materials; analysis by diffraction-line broadening

Experimental investigations have revealed that the Neerfeld–Hill and Eshelby–Kröner models, for grain interactions in massive, bulk (in particular, macroscopically isotropic) polycrystals, and a recently proposed effective grain-interaction model for macroscopically anisotropic polycrystals, as thin films, provide good estimates for the macroscopic (mechanical and) X-ray elastic constants and stress factors of such polycrystalline aggregates. These models can also be used to calculate the strain variation among the diffracting crystallites, i.e. the diffraction-line broadening induced by elastic grain interactions can thus be predicted. This work provides an assessment of diffraction-line broadening induced by elastic loading of polycrystalline specimens according to the various grain-interaction models. It is shown that the variety of environment, and thus the heterogeneity of the stress–strain states experienced by each of the individual grains exhibiting the same crystallographic orientation in a real polycrystal, cannot be accounted for by traditional grain-interaction models, where all grains of the same crystallographic orientation in the specimen frame of reference are considered to experience the same stress–strain state. A significant degree of broadening which is induced by the heterogeneity of the environments of the individual crystallites is calculated on the basis of a finite element algorithm. The obtained results have vast implication for diffraction-line broadening analysis and modelling of the elastic behaviour of massive polycrystals.

K вопросу ползучести анизотропных поликристаллов.

K вопросу ползучести анизотропных поликристаллов.

Influence of structure disorder on the lattice thermal conductivity of polycrystals: A frequency-dependent phonon-transport study

It is widely accepted that the lattice thermal conductivity of a polycrystal mainly depends on its grain sizes, phonon mean free paths, and grain-boundary thermal resistance. However, uncertainties always exist on how much grain misalignment and a wide grain size distribution in a real polycrystal could affect the thermal analysis. Considering frequency-dependent phonon mean free paths, the influence of these factors is carefully examined by phonon Monte Carlo simulations for a series of disordered silicon polycrystals with grain sizes ranging from 1 to 400 nm. More generally, simulations are also performed on thermally anisotropic polycrystals. Despite all structure variation, this work suggests that the “direction-averaged” lattice thermal conductivity of a polycrystal is always close to that of an aligned polycrystal, with an effective grain size matching the interface density of the studied polycrystal.

Micromechanically based stress and strain‐rate flow potentials for anisotropic polycrystals

AbstractThe dissipation of rigid‐viscoplastic material behaviour is related to the stress flow potential and its Legendre‐Fenchel conjugated strain‐rate flow potential. In this paper micromechanically based approximation of these flow potentials, expressed in terms of the deviatoric stresses and strain‐rates, is derived from texture measurements. The dual potentials for single face‐centered cubic crystals are approximated in terms of Fourier expansion with tensorial texture coefficients. The macroscopic flow stress and strain‐rate potentials for an aggregate of polycrystals can be deduced from those of single crystal under the assumption of either homogeneous strain‐rate, or homogeneous stress field from the codf measurements. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerically determined enrichment functions for the extended finite element method and applications to bi‐material anisotropic fracture and polycrystals

AbstractStrain singularities appear in many linear elasticity problems. A very fine mesh has to be used in the vicinity of the singularity in order to obtain acceptable numerical solutions with the finite element method (FEM). Special enrichment functions describing this singular behavior can be used in the extended finite element method (X‐FEM) to circumvent this problem. These functions have to be known in advance, but their analytical form is unknown in many cases. Li et al. described a method to calculate singular strain fields at the tip of a notch numerically. A slight modification of this approach makes it possible to calculate singular fields also in the interior of the structural domain. We will show in numerical experiments that convergence rates can be significantly enhanced by using these approximations in the X‐FEM. The convergence rates have been compared with the ones obtained by the FEM. This was done for a series of problems including a polycrystalline structure. Copyright © 2010 John Wiley & Sons, Ltd.

Prediction on Nominal Stress-Strain Curve of Anisotropic Polycrystal with Texture by FE Analysis

In order to predict nominal stress-strain (S-S) curve in tensile test for a commercial purity titanium (hereafter, CP-Ti) with texture, three boundary conditions are elucidated as well as plastic anisotropy in three-dimensional finite element (FE) analysis. Three boundary conditions are a free case, a full constraint case at both ends of specimen, and a simulated case based on empirical data. In the simulated case, the shape change and the nominal S-S curve can be well predicted with deviation smaller than 10%. The origin of localized necking is concluded not due to the plastic instability but due to the constraint at both ends of specimen. Deviation point of the nominal S-S curve at the simulated case from that of the free case has been suggested as the onset of localized necking of CP-Ti, which is determined as 0.25 in transverse direction and 0.50 in rolling direction.

Elasticity tensor and ultrasonic velocities for anisotropic cubic polycrystal

The orientation distribution of crystallites in a polycrystal can be described by the orientation distribution function (ODF). The ODF can be expanded under the Wigner D-bases. The expanded coefficients in the ODF are called the texture coefficients. In this paper, we use the Clebsch-Gordan expression to derive an explicit expression of the elasticity tensor for an anisotropic cubic polycrystal. The elasticity tensor contains three material constants and nine texture coefficients. In order to measure the nine texture coefficients by ultrasonic wave, we give relations between the nine texture coefficients and ultrasonic propagation velocities. We also give a numerical example to check the relations.

A Note on the Effective Magnetic Permeability of Polycrystals

Many classical estimates for the effective behavior of heterogeneous materials can be reinterpreted in terms of inclusion problems. However, in the case of cubic polycrystals, a cubic permeability tensor for single crystals has to be written. In the framework of linear behavior, the description of the cubic symmetry reduces to isotropy. The heterogeneity of polycrystals, which results from single crystal anisotropy, cannot be described, and the classical estimates for the overall behavior of heterogeneous materials cannot be used. In this paper, we propose a particular description of the cubic symmetry for the magnetic permeability. We then derive estimates for the effective permeability of polycrystals from the solution of the basic inclusion problem, for both macroscopically isotropic and anisotropic polycrystals.