Anisotropic Polycrystalline Materials Research Articles

Crack growth in anisotropic brittle and polycrystalline materials by adaptive phase field model using variable-node elements

Crack growth in anisotropic brittle and polycrystalline materials by adaptive phase field model using variable-node elements

Implementation and experimental validation of nonlocal damage in a large-strain elasto-viscoplastic FFT-based framework for predicting ductile fracture in 3D polycrystalline materials

Ductile materials, such as metal alloys, can undergo substantial deformation before failure. Additionally, these materials are usually of polycrystalline composition and exhibit strongly anisotropic behavior at small length scales. Previously developed fast Fourier transform (FFT)-based models can model ductile fracture of isotropic materials or the elastic–plastic behavior of anisotropic polycrystalline materials; however, there remains a need to couple both capabilities. This work extends a large-strain FFT-based crystal plasticity model to simulate ductile fracture of polycrystalline materials. A triaxiality-based continuum damage mechanics (CDM) formulation is incorporated into a large-strain elasto-viscoplastic FFT (LS-EVPFFT) framework. The CDM formulation is augmented with an integral-based nonlocal regularization approach that correctly handles gas-phase material necessary to model unconstrained surfaces. To validate the damage-enabled LS-EVPFFT framework, mesoscale copper tensile coupons were machined using microwire electrical discharge machining and experimentally characterized using electron backscatter diffraction. In-situ optical digital image correlation was performed during uniaxial testing to provide a side-by-side comparison of the experimental and computational strain fields and stress–strain responses. The damage-enabled LS-EVPFFT framework can simulate the complete macroscopic stress–strain response of ductile polycrystals to failure. The model reproduces necking behavior that qualitatively agrees with experimental observations. By leveraging the relatively low computational cost of the damage-enabled LS-EVPFFT framework, the framework presented here allows the ductile fracture response of 3D polycrystalline materials to be tractably predicted.

Representation of Hashin–Shtrikman Bounds in Terms of Texture Coefficients for Arbitrarily Anisotropic Polycrystalline Materials

The present work generalizes the results of Bohlke and Lobos (Acta Mater. 67:324–334, 2014) by giving an explicit representation of the Hashin–Shtrikman (HS) bounds of linear elastic properties in terms of tensorial Fourier texture coefficients not only for cubic materials but for arbitrarily anisotropic linear elastic polycrystalline materials. Based on the HS bounds as given by Walpole (J. Mech. Phys. Solids 14(3):151–162, 1966) and tensor functions for the representation of the crystallite orientation distribution function, it is shown that the HS bounds are represented in terms of the exact same second- and fourth-order texture coefficients which appear in the consideration of the Voigt and Reuss bounds. The derived representations in terms of tensorial texture coefficients are valid for all symmetry classes in elasticity and present expressions highly attractive for the description of physical quantities in terms of tensorial variables. In order to make these results also accessible for the community of quantitative texture analysis, transformation relations between experimentally obtained Bunge’s or Roe’s coefficients and the tensorial texture coefficients are given. The representation of the present work offers a finite and low dimensional parametrization of the fully anisotropic Hashin–Shtrikman bounds, which can be used in inverse materials design problems in order to explore the set of possible materials properties or for the determination of optimal microstructural influence with respect to prescribed material properties. Examples for orthotropic polycrystals of cubic materials and transversely isotropic polycrystals of hexagonal materials (showing the connection and applicability of the results also to fiber orientation distributions) are discussed. Finally, an implementation in Mathematica® 11 of the HS bounds for arbitrarily anisotropic materials is offered, such that readers can reproduce all the results of this work and use them for their own purposes.

Fracture of solar-grade anisotropic polycrystalline Silicon: A combined phase field–cohesive zone model approach

This work presents a novel computational framework to simulate fracture events in brittle anisotropic polycrystalline materials at the microscopical level, with application to solar-grade polycrystalline Silicon. Quasi-static failure is modeled by combining the phase field approach of brittle fracture (for transgranular fracture) with the cohesive zone model for the grain boundaries (for intergranular fracture) through the generalization of the recent FE-based technique published in [M. Paggi, J. Reinoso, Comput. Methods Appl. Mech. Engrg., 31 (2017) 145–172] to deal with anisotropic polycrystalline microstructures. The proposed model, which accounts for any anisotropic constitutive tensor for the grains depending on their preferential orientation, as well as an orientation-dependent fracture toughness, allows to simulate intergranular and transgranular crack growths in an efficient manner, with or without initial defects. One of the advantages of the current variational method is the fact that complex crack patterns in such materials are triggered without any user-intervention, being possible to account for the competition between both dissipative phenomena. In addition, further aspects with regard to the model parameters identification are discussed in reference to solar cells images obtained from transmitted light source. A series of representative numerical simulations is carried out to highlight the interplay between the different types of fracture occurring in solar-grade polycrystalline Silicon, and to assess the role of anisotropy on the crack path and on the apparent tensile strength of the material.

Propagation of material and surface profile uncertainties on MEMS micro‐resonators using a stochastic second‐order computational multi‐scale approach

SummaryThis paper aims at accounting for the uncertainties because of material structure and surface topology of micro‐beams in a stochastic multi‐scale model. For micro‐resonators made of anisotropic polycrystalline materials, micro‐scale uncertainties exist because of the grain size, grain orientation, and the surface profile. First, micro‐scale realizations of stochastic volume elements are obtained based on experimental measurements. To account for the surface roughness, the stochastic volume elements are defined as a volume element having the same thickness as the microelectromechanical system (MEMS), with a view to the use of a plate model at the structural scale. The uncertainties are then propagated up to an intermediate scale, the meso‐scale, through a second‐order homogenization procedure. From the meso‐scale plate‐resultant material property realizations, a spatially correlated random field of the in‐plane, out‐of‐plane, and cross‐resultant material tensors can be characterized. Owing to this characterized random field, realizations of MEMS‐scale problems can be defined on a plate finite element model. Samples of the macro‐scale quantity of interest can then be computed by relying on a Monte Carlo simulation procedure. As a case study, the resonance frequency of MEMS micro‐beams is investigated for different uncertainty cases, such as grain‐preferred orientations and surface roughness effects. Copyright © 2016 John Wiley & Sons, Ltd.

Semi-analytical modeling of acoustic beam divergence in homogeneous anisotropic half-spaces

Beam divergences of acoustical fields in semi-infinite homogeneous anisotropic media are calculated based on a semi-analytical model. The model for a plane source in a semi-infinite homogeneous anisotropic medium is proposed as an extended model for a point source in an infinite medium. Beam divergences propagating along crystallographic axes 〈100〉, 〈110〉, and 〈111〉 in a cubic crystal, a single crystalline Ni-based alloy, are measured and compared to calculation results for verifying the model. The contribution of beam divergence attenuation to the total attenuation for propagating in anisotropic polycrystalline materials is quantitatively evaluated in isolation from scattering attenuation effects.

Materials design of elastic properties of multiphase polycrystalline composites using model functions

AbstractFor a chosen combination of materials for a multiphase polycrystalline composite, a crystallite orientation distribution function for each phase is formulated as a superposition of analytic central model von Mises‐Fisher functions. The chosen central model functions allow the analytic integration of orientation averages of arbitrarily anisotropic fourth‐order tensors. Based on this result, the first‐order bounds of Voigt and Reuss and the geometric average of the stiffness can be expressed explicitly in closed forms for arbitrarily anisotropic polycrystalline materials and number of phases depending on the volume fractions and the influence of the crystallographic texture of each constituent. These expressions can then be used for the materials design aiming the determination of volume fractions and orientation distribution of selected materials for prescribed effective properties. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Homogenization estimates for texture evolution in halite

In this work, the recently developed “second-order” self-consistent method [Liu, Y., Ponte Castañeda, P., 2004a. Second-order estimates for the effective behavior and field fluctuations in viscoplastic polycrystals. J. Mech. Phys. Solids 52 467–495] is used to simulate texture evolution in halite polycrystals. This method makes use of a suitably optimized linear comparison polycrystal and has the distinguishing property of being exact to second order in the heterogeneity contrast. The second-order model takes into consideration the effects of hardening and of the evolution of both crystallographic and morphological texture to yield reliable predictions for the macroscopic behavior of the polycrystal. Comparisons of these predictions with full-field numerical simulations [Lebensohn, R.A., Dawson, P.R., Kern, H.M., Wenk, H.R., 2003. Heterogeneous deformation and texture development in halite polycrystals: comparison of different modeling approaches and experimental data. Tectonophysics 370 287–311], as well as with predictions resulting from the earlier “variational” and “tangent” self-consistent models, included here for comparison purposes, provide insight into how the underlying assumptions of the various models affect slip in the grains, and therefore the texture predictions in highly anisotropic and nonlinear polycrystalline materials. The “second-order” self-consistent method, while giving a softer stress-strain response than the corresponding full-field results, predicts a pattern of texture evolution that is not captured by the other homogenization models and that agrees reasonably well with the full-field predictions and with the experimental measures.

Convex plastic potentials of fourth and sixth rank for anisotropic materials

It is briefly reminded how the theory of dual plastic potentials has been used in the past to generate analytical expressions for plastic potentials of anisotropic polycrystalline materials with a known crystallographic texture. Such constitutive models are fairly general, and the identification of their parameters can readily be done on the basis of data obtained from a texture measurement. As a result, they are suitable for engineering applications such as elastic–plastic finite element models for forming processes. However, the yield loci generated in this way are not automatically convex. Therefore, a new variant of the method has now been developed, which preserves the advantages of the old method, but for which convexity can at least been tested by means of a mathematical criterion. In addition, it has turned out to be possible to slightly modify plastic potentials which do not satisfy the criterion, in order to achieve convexity. An example of a plastic potential modified in this way is discussed. After modification, it was still a good analytical approximation of the plastic potential directly derived from the Taylor–Bishop–Hill theory on the basis of the crystallographic texture of the material.

Heterogeneous deformation and texture development in halite polycrystals: comparison of different modeling approaches and experimental data

Modeling the plastic deformation and texture evolution in halite is challenging due to its high plastic anisotropy at the single crystal level and to the influence this exerts on the heterogeneity of deformation over halite polycrystals. Three different assumptions for averaging the single crystal responses over the polycrystal were used: a Taylor hypothesis, a self-consistent viscoplastic model, and a finite element methodology. The three modeling approaches employ the same single crystal relations, but construct the polycrystal response differently. The results are compared with experimental data for extension at two temperatures: 20 and 100 °C. These comparisons provide new insights of how the interplay of compatibility and local equilibrium affects the overall plastic behavior and the texture development in highly anisotropic polycrystalline materials. Neither formulation is able to completely simulate the texture development of halite polycrystals while, at the same time, giving sound predictions of microstructural evolution. Results obtained using the finite element methodology are promising, although they point to the need for greater resolution of the individual crystals to capture the full impact of deformation heterogeneities.

Numerical correlation of material structure weaknesses in anisotropic polycrystalline materials

Repeated computer experiments were implemented to investigate material structure weaknesses in polycrystalline materials. The computational investigation relies on behaviors of mesoscopic stress responses in a simulated polycrystalline aggregate containing a fairly large number of constituent grains. A Kroner-Kneer structure-based model was adopted and refined to provide an efficient numerical solution to local mesoscopic stresses, calculated in terms of grain-averaged fields at a material scale of grain size, in arbitrarily polygon-shaped grains. Three criteria have been proposed for classifying material structure weaknesses in the simulated polycrystalline materials. It is found that material structure weakness of three types can be well correlated by a defined “Orientation-Geometry Factor” O and “Relevance Parameter”. Every correlated relation, incorporating effects of grain orientation and geometry, provides a base for discerning material structure weaknesses. The homogenization of an anisotropic material is also discussed.

Computer Identification of Structural Weaknesses in Locally Anisotropic Polycrystalline Materials

Computer experiments were performed to investigate behavior of mesoscopic stress responses in a simulated polycrystalline material sample containing a fairly large number of constituent grains for a number of polycrystalline materials. Kro¨ner-Kneer structure-based model was adopted and refined to provide an efficacious numerical approach to local mesoscopic stresses. The approach is developed on a concept of average fields of grains for arbitrarily polygon-shaped grains. Three criteria were proposed for classifying speculated material structure weaknesses in all simulated material samples. It is found that material structure weaknesses can be well correlated by defined “Orientation-Geometry Factor” and “Relevance Parameter.” Not only grain-orientation but also grain geometry exerts strong influences on mesoscopic stress distribution, hence the distribution of material structure weaknesses in simulated polycrystalline material samples. Computer experiments lead to correlated relationships that links material structure weaknesses with local microstructure, and a database for discrimination of material structure weaknesses in the material samples. The homogenization of materials with locally anisotropic microstructure is also discussed.

Anisotropic Thermal Expansion of Barium Hexaferrite Using Dynamic High-temperature X-ray Diffraction

Barium hexaferrite is a well-known ceramic permanent magnet and due to its high coercivity, remanence, and large uniaxial magnetic anisotropy, finds applications that compete with metallic magnets. Even though a number of the high-temperature properties of barium hexaferrite have been studied extensively, its anisotropic thermal expansion has not been reported so far. Dynamic high-temperature x-ray diffraction (HT-XRD) is one powerful method to obtain thermal expansion data for anisotropic polycrystalline materials in a very short period of time. In this paper the anisotropic nature of the thermal expansion coefficient of the barium hexaferrite phase is reported with the use of a dynamic HT-XRD setup. The thermal expansion coefficient (linear fit) was determined to be 8.36 × 10−6 K−1 along the a–b plane to 1.4 × 10−5 K−1 along the c axis between the temperature range of 293 to 1343 K.

Effective transport coefficients of textured materials

A method is proposed for calculating effective transport coefficients of two-phase anisotropic polycrystalline materials with an arbitrary distribution of anisotropic ellipsoidal particles of the first phase.

Critical Grain Size/Preexisting Flaw Size Relations in Anisotropic Brittle Solids

A known solution for the critical condition of thermoelastic microcrack initiation is adopted to evaluate the extent of crack propagation after initiation and the arrest condition. It is shown that the critical temperature for crack initiation and the strength of thermoelastically anisotropic polycrystalline materials is not determined by the grain size only but also by the ratio of grain‐to‐flaw size. The variation of slopes in the strength‐grain size plot is interpreted in terms of preexisting annular flaw‐size change. The major implications of the analysis are discussed in relation to experimental results on a number of polycrystalline ceramics.

A hardening model for anisotropic materials

Results from proportionate-loading experiments under plane-stress states illustrate the existence of a field of uniform-hardening potentials for the yield and creep deformation behavior of isotropic, slightly anisotropic, aelotropic and orthotropic polycrystalline materials in the initially strain-free condition. For two different plane-stress states, it is shown that a linear functional relationship holds between the plastic-strain increment ratio and the stress ratio in these materials and that, consequently, the field is adequately modeled by a uniform-hardening anisotropic function that is quadratic in the components of deviatoric stress. Anisotropic plane-stress yield functions are formulated for any stage in the deformation process by combining the uniform-hardening function with the kinematic-hardening rule. The resulting functions, which correspond to rigid translations of initial yield loci according to Ziegler's rule, provide good agreement with experimental observations on a marked Bauschinger effect and an absence of cross hardening.

Internal strain energy and the strength of brittle materials

The theory of Clarke for the formation of grain boundary cracks in anisotropic polycrystalline materials, is re-examined in the light of recent experimental data. The theory predicts correctly the conditions for the formation of grain boundary cracks of length similar to a grain dimension. However, the theory cannot be used to explain the experimentally observed strength/grain size and strength/irradiation dose relationships, for example for BeO. The theory supposes that the process controlling catastrophic fracture is the growth of a crack from a grain boundary pore with an energy absorption rate corresponding to the grain boundary surface energy of ∼103 erg/cm2. In practice, the process controlling catastrophic fracture is the subsequent growth of a crack from a grain dimension, with a higher energy absorption rate corresponding to an effective surface energy of ∼104 erg/cm2.

The dimensional changes produced in graphite by absorption of bromine

It is well known that, in general, polycrystalline graphites have substantially lower volume coefficients of thermal expansion than the crystallites of which they are composed. It has previously been suggested that this is because microporosity, orientated parallel to the layer planes, absorbs part of the c-axis expansion of the crystallites. If this is true then the volume expansion of polycrystalline graphite when it absorbs bromine should also be lower than that of the crystallites and there should be a correlation between linear thermal expansion and initial linear growth per unit bromine concentration. The dimensional changes of a wide variety of polycrystalline graphites, when they absorb bromine, have been investigated and the correlation between initial growth per unit concentration of bromine and thermal expansion coefficient has been demonstrated. It has also been observed that when the strain of the crystallites in the c-axis direction becomes large the orientated microporosity fills up. When this happens the volume growth of the polycrystalline graphites per unit concentration of bromine approaches that of the crystallites. At even higher crystallite strains in the c-axis direction, anisotropic polycrystalline materials tend to expand faster than the crystallites and this is considered to be caused by the generation of additional porosity. This effect is less marked in the isotropic polycrystalline graphites. The relation between growth due to bromination and dimensional changes produced by fast neutron irradiation is discussed. Data on the effect of graphitization temperature on the subsequent dimensional changes produced by bromination are also described.