Single Crystal Stiffness Research Articles

Estimation of directional single crystal elastic properties from nano-indentation by correlation with EBSD and first-principle calculations

In this study, a two-step optimisation process for the estimation of the single crystal stiffness tensor from the indentation modulus is presented. This was accomplished by using a 2-dimensional data correlation method for nano-indentation, EBSD and ab initio data, as well as EDS for phase separation in a multi-phase material. Here, a single-phase Fe-24Ni-0.4C austenitic steel and a two-phase Fe-9.5Ni-0.5Co Seymchan meteorite were used as example materials.In a first step, the combination of high-speed nano-indentation mapping data with elemental and orientation distribution maps allowed the estimation of indentation moduli along specific crystal surface planes normal directions, (001), (011) and (111) using a least squares optimisation based on starting values from DFT or experimentally determined stiffness tensors. A second global optimisation step to estimate the single crystal stiffness tensor using pre-solved solution parameters of the Vlassak-Nix equations yielded reasonable correspondence between the experimentally determined stiffness tensors from correlative nano-indentation data and other methods. The presented method demonstrates generally the possibility to derive directionally sensitive elastic properties from high-speed nano-indentation for cubic materials statistically.

Elastic Anisotropy of 1,3,5‐Triamino‐2,4,6‐Trinitrobenzene as a Function of Temperature and Pressure: A Molecular Dynamics Study

AbstractThe equation of state of the triclinic compound 1,3,5‐triamino‐2,4,6‐trinitrobenzene (TATB) as well as its second‐order isothermal elastic tensor were computed through classical molecular dynamics simulations under various temperature and pressure conditions. Hydrostatic pressures similar to previous diamond anvil cell experiments were imposed up to 60 GPa and temperatures chosen between 100 and 900 K in conjunction with the most recent version of an all‐atom fully flexible molecule force field. The isothermal elastic constants were computed using the generalized Hooke's law by fitting Cauchy stress vs. linear strain curves for pressures below 50 GPa. Along isobaric loadings, TATB single crystal stiffnesses are found to undergo softening, less pronounced at high pressure, while maintaining its elastic anisotropy. On the other hand, along an isothermal loading, a non‐linear increase is observed in the elastic constants with respect to pressure with a significant decrease in anisotropy. Towards a precise mesoscopic modelling of TATB single crystal mechanical behavior, we provide “ready to plug‐in” analytical formulations of the P, V, T equation of state and pressure‐temperature dependent non‐linear elasticity.

Quantitative texture analysis of slates: An insight into slate’s elastic properties and their contribution to the understanding of seismic wave reflections in anisotropic materials

Slates are low-grade metamorphic rocks characterized by an alignment of the phyllosilicates, resulting in the development of a marked foliation. A consequence of this is a strong directional dependence of their elastic properties, which are tightly bound to their texture. The Rietveld analysis applied to synchrotron x-ray diffraction images of slates from the Truchas Syncline (NW Iberian Peninsula) will provide us with the orientation distribution function (ODF) of the principal minerals that constitute the rock. This textural analysis shows an exceptional alignment of phyllosilicates (muscovite and chlorite) in contrast with a poor orientation of other minerals such as quartz. In a later treatment of the data, the elastic properties of the slates, as well the variations of the seismic wave velocities, have been calculated from the ODF and the single crystal stiffness tensors. The results of both the elastic properties and the wave velocities can be used for the construction of models that allow to understand the reflections of seismic waves in markedly anisotropic materials, which is a matter of a great interest in the studies of the Earth’s lithosphere with seismic methods.

Numerical study on the effective stiffness of topology-optimized lattice structures made of orthotropic crystal grains with optimal orientation

Controlling the orientation of crystal grains in metal additive manufacturing is an active field of research. Assuming that three-dimensional control of orthotropic metal crystal grain can be achieved, we numerically studied the effective performance of a topology-optimized lattice made of crystal grains with optimal directions in orientation. The objective function is maximized for an effective isotropic stiffness. The effective properties of the lattice are calculated using the homogenization method. Each discretized finite element is regarded as a crystal grain and its artificial density and orientation are optimized. Through several numerical studies using a single orthotropic crystal of Ni exhibiting large anisotropy in its single crystal stiffness, the optimization and the usefulness of the lattice structure composed of optimal orientation crystal grains is confirmed. In the design of a lattice with effective isotropic stiffness, an improvement in stiffness was observed compared with that using equivalent isotropic Ni as a material.

Homogenization and Materials Design of Anisotropic Multiphase Linear Elastic Materials Using Central Model Functions

It is shown in the present work that for linear elastic multiphase materials the orientation average of a stiffness tensor can be given in a closed form if central model functions are taken into consideration for the description of the material orientation distribution. This holds for arbitrary numbers of material constituents and for arbitrary degree of anisotropy of any of the constituents. The first-order bounds of Voigt and Reuss, the geometric average of Aleksandrov and Aisenberg and, specifically, the Hashin–Shtrikman bounds are given in closed forms depending on the single crystal stiffnesses of the material constituents, on their volume fractions and, for elasticity, on a central orientation and two scalar parameters per central model function. The central orientation and the two scalar parameters, referred to as texture eigenvalues in the present work, reflect the complete influence of the corresponding central model function on the elastic properties. The set of all admissible texture eigenvalues is derived. The closed form expressions can be used, e.g., for efficient and low dimensional homogenization procedures taking the crystallographic texture into account or, e.g., for the determination of estimates for unknown single crystal behavior. It is further shown, that central crystallite orientation distributions exist which conserve the anisotropy of the single crystal behavior but invert the direction of the anisotropy. The closed form expressions documented in the present work are used for an exemplary materials design problem in order to determine favorable volume fractions and texture eigenvalues for the orientation distributions of considered constituents for given prescribed anisotropic properties.

Calculations of single crystal elastic constants for yttria partially stabilised zirconia from powder diffraction data

Yttria Stabilised Zirconia (YSZ) is a tough, phase-transforming ceramic that finds use in a wide range of commercial applications from dental prostheses to thermal barrier coatings. Micromechanical modelling of phase transformation can deliver reliable predictions in terms of the influence of temperature and stress. However, models must rely on the accurate knowledge of single crystal elastic stiffness constants. Some techniques for elastic stiffness determination are well-established. The most popular of these involve exploiting frequency shifts and phase velocities of acoustic waves. However, the application of these techniques to YSZ can be problematic due to the micro-twinning observed in larger crystals. Here, we propose an alternative approach based on selective elastic strain sampling (e.g., by diffraction) of grain ensembles sharing certain orientation, and the prediction of the same quantities by polycrystalline modelling, for example, the Reuss or Voigt average. The inverse problem arises consisting of adjusting the single crystal stiffness matrix to match the polycrystal predictions to observations. In the present model-matching study, we sought to determine the single crystal stiffness matrix of tetragonal YSZ using the results of time-of-flight neutron diffraction obtained from an in situ compression experiment and Finite Element modelling of the deformation of polycrystalline tetragonal YSZ. The best match between the model predictions and observations was obtained for the optimized stiffness values of C11 = 451, C33 = 302, C44 = 39, C66 = 82, C12 = 240, and C13 = 50 (units: GPa). Considering the significant amount of scatter in the published literature data, our result appears reasonably consistent.

Structural and elastic properties of ternary metal nitrides TixTa1 − xN alloys: First-principles calculations versus experiments

First-principles pseudopotential calculations of the lattice constants and of the single-crystal elastic constants for TixTa1−xN (0≤x≤1) alloys with B1-rocksalt structure were first carried out. These calculations were performed using density functional perturbation theory (DFPT) within the virtual crystal approximation (VCA) for the disordered alloys and the supercell method (SC) for the ordered alloys, with the ABINIT program. For disordered structures, partial comparisons of the lattice constants and of the bulk modulus are provided with calculations that used the coherent potential approximation (CPA) with the exact muffin-tin orbitals (EMTO). For the exchange-correlation potential we used the generalized gradient methods (GGA). The calculated equilibrium lattice parameters by VCA are in good agreement with the stress-free lattice parameters a0 and exhibit a positive deviation from Vegard's rule corresponding to a positive bowing parameter while the calculated single-crystal stiffness c12 and c44, gradually increase when c11 decreases from TaN to TiN. In a second stage, we have estimated by homogenization methods the averaged stiffnesses <cij>, the Young modulus and Poisson ratio of polycrystalline TixTa1−xN (0≤x≤1) alloys considering a random orientation of crystallites. Finally, comparisons are made with the experimental effective out-of-plane shear elastic modulus C44 and the effective out-of-plane longitudinal elastic constant C33 measured by Brillouin light scattering and picosecond ultrasonics, respectively, on thin films elaborated by magnetron sputtering.

Structural and elastic properties of single-crystal and polycrystalline Ti1−xZrxN alloys: A computational study

First-principles pseudopotential calculations of the lattice constants and of the single-crystal elastic constants of Ti1−xZrxN (0≤x≤1) alloys were carried out. These calculations were performed using density functional perturbation theory (DFPT) within the virtual crystal approximation (VCA) for the disordered alloys and the supercell method (SC) for the ordered alloys. For the exchange-correlation potential we used both the local density (LDA) and the generalized gradient methods (GGA). The calculated equilibrium lattice parameters exhibit a positive deviation from Vegard's rule corresponding to a positive bowing parameter, while the calculated single-crystal stiffnesses, namely C11, C12 and C44, gradually decrease from TiN to ZrN. In a second stage, in the frame of anisotropic elasticity, we have estimated by homogenization methods the averaged stiffnesses 〈Cij〉, direction dependent Young's moduli and Poisson's ratios of polycrystalline Ti1−xZrxN (0≤x≤1) alloys considering a {111}-fiber texture.

Effects of crystallographic texture on ultrasonic wave velocities in steels

Studies of ultrasonic wave velocities in commercial steel sheet products have been made and the results have been interpreted in terms of crystallographic textures measured by X-ray diffraction. Elasticity tensors were calculated as weighted averages of the Voigt and Reuss bounds using texture coefficients together with single crystal stiffness data, and these were then employed to predict the ultrasonic wave velocities. For guided bulk waves of types So and SHo excellent agreement was obtained between measured and predicted values. The application of these types of measurements to process control in metallurgical annealing lines has been briefly discussed.

The elastic properties of crystalline syndiotactic polypropylene

The single crystal stiffness constants of different structures of syndiotactic polypropylene have been calculated using a molecular modelling package. These have been used to predict some mechanical properties of a uniaxially aligned aggregate of crystals. The extensional modulus of such an aggregate is compared with measurements of the crystal modulus determined from shifts in the (002) diffraction peak of an oriented sample subjected to stress. Measurements of the crystalline strain have also been performed on off-meridional reflections with a view to obtaining estimates of Poisson's ratios. It is found that the theoretical chain modulus of the 2 l helical conformation is very low (~1.5GPa). Incorporation of this conformation into a crystalline environment gives a crystal modulus about 10 times greater, with the value varying slightly between each of the three crystalline structures considered. Although the theoretical values are greater than the experimental value, ~8 GPa, all these results support the contention that syndiotactic polypropylene has a low crystal modulus. Experimental measurements of lateral crystal strain, arising from extensional loading, are considerably greater than predicted from realistic values for Poisson's ratios. This discrepancy is interpreted in terms of the additional lateral forces generated when a two phase system is strained extensionally.

Seismic anisotropy and shear-wave splitting in lower-crustal and upper-mantle rocks from the Ivrea Zone—experimental and calculated data

To quantify the relationships between anisotropy, S-wave splitting and tectonics, we determined the seismic properties of lower-crustal and upper-mantle rocks outcropping in the Ivrea Zone (Northern Italy). We obtained P- and S-wave seismic velocities by laboratory direct velocity measurements and/or by calculations based on the modal compositions of the rocks, the lattice preferred orientations (LPOs), and the single crystal stiffness coefficients. Measured P- and S-wave velocities (6.0–7.5 km s −1 and 3.6–4.2 km s −1) are typical of the lower crust. The P-wave anisotropy is in the range 0–10%. Shear-wave birefringence is in the range 0.0–0.6 km s −1, with typical values between 0.0 and 0.2 km s −1. In many cases, the birefringence is clearly related to fabric elements (foliation, lineation). Mafic rocks such as anorthosite or pyroxene-bearing gabbros exhibit low P-wave anisotropies ( < 5%) and low shear-wave birefringences (less than 0.1 km s −1). In contrast, the seismic properties of felsic rocks such as biotite-bearing gneisses and mafic rocks such as amphibolites display high V p anisotropy (10%) and strong birefringence (0.3 km s −1). Biotite and amphibole preferred orientations clearly control seismic anisotropy and particularly shear-wave splitting. In these rocks, maximum splitting is observed in directions parallel to the foliation with the fast split shear wave polarized parallel to the foliation plane. To have an overview of the seismic properties of this lower-crustal section at a broader scale, we calculated from our data the anisotropic seismic properties of several hypothetical samples that are perhaps more representative of the regional anisotropy than each sample individually. For instance, the average lower-crustal sample displays an anisotropy of 5.5% for P waves and a birefringence around 0.14 km s −1 for S waves propagating parallel to the foliation. We observe little splitting for waves propagating at high angle to the foliation.

Fabric-related seismic anisotropy in upper-mantle xenoliths: evidence from measurements and calculations

The seismic properties and their directional dependence have been measured in four mantle peridotites from picritic tuffs (Vitim, Baikal region, Russia). Geothermobarometric data suggest a source depth for the xenoliths corresponding to the spinel-garnet transition zone (40–90km). The experimental data include the measurement of P- and S-wave velocities and densities at confining pressures to 600 MPa and temperatures to 600°C, and the determination of velocity anisotropy and shear-wave splitting. The measured velocities are compared with velocity calculations based on lattice preferred orientation (LPO) of olivine (orthopyroxene), the modal composition of the rocks and the single crystal stiffness coefficients. Macroscopic velocity anisotropy is strongly controlled by the LPO of olivine. Measured (600 MPa, room temperature) and calculated P-wave anisotropies compare fairly well, although the experimentally determined velocities are markedly lower than the calculated velocities. The correspondence between measured and calculated shear-wave splitting data is less pronounced. This is attributed to grain boundary effects (retrogressive reactions, interstitial glass, incomplete crack closure) and to the fact that the LPO of the cpx fraction has not been included in the calculation. The magnitude of P-wave anisotropy corresponds to that observed in mantle peridotites exposed at the surface (Ivrea zone, northern Italy: Ronda, southern Spain).

3-D seismic velocities calculated from lattice-preferred orientation and reflectivity of a lower crustal section: examples of the Val Sesia section (Ivrea zone, northern Italy)

SUMMARY To quantify the seismic properties of lower crustal rocks and to better constrain the origin of the lower crustal seismic reflectivity, we determined the complete 3-D seismic properties of a lower crustal section. Eight representative samples of the main lithologic and structural units outcropping in the Val Sesia (Ivrea zone) were studied in detail. The seismic velocities were calculated using the single crystal stiffness coefficients and the lattice preferred orientation (LPO) of each mineral in all samples. The 21 stiffness coefficients characterizing the elastic behaviour of each rock are determined. Mafic and ultramafic rocks such as pyroxenite and pyroxenebearing gabbros display complex shear wave properties. These rocks are weakly birefringent (maximum 0.1 km s-') and it is difficult to find consistent relationships between the seismic properties and the rock structure. On the other hand, seismic properties of deformed felsic rocks are essentially controlled by mica. They display strong S-wave birefringence (0.3 km s-') and relatively high Vp anisotropy (7.6 per cent). Amphibole also strongly influences the rock birefringence patterns. For both kind of rocks, the foliation is highly birefringent and the fast polarized shear wave is systematically oriented parallel to the foliation. We show that the number of mineral phases in the rock strongly controls the anisotropy. The seismic anisotropy has a complex role in the P-wave reflectivity. Compared to the isotropic case, anisotropy enhances the reflection coefficient for about 60 per cent of the possible lithological interfaces. For 40 per cent of the interfaces, the reflection coefficient is much lower when one considers the medium as anisotropic.

P- and S-wave anisotropy and shear-wave splitting at pressure and temperature in possible mantle rocks and their relation to the rock fabric

The seismic properties and their directional dependence have been measured in three potentially mantle-derived rocks (peridotite, Ivrea zone, Italy; lherzolitic xenolith, Massif Central, France; serpentinite). The experimental data include the measurement of elastic wave velocities and densities at confining pressures up to 600 MPa and temperatures up to 600°C (600 MPa), and the determination of velocity anisotropy and shear-wave splitting. The petrophysical data can be related to mineral composition and to microstructural characteristics of the rocks. Macroscopic anisotropy of P- and S-wave velocities is strongly controlled in the peridodites by the olivine and in the serpentinite by the serpentine (antigorite) lattice preferred orientation (LPO), respectively. Shear-wave splitting was found to be an important characteristic of the anisotropic rocks. The study shows that for the interpretation of shear-wave splitting, the direction of wave propagation in relation to the LPOs of the major minerals and to the symmetry of their lattice fabric is of great importance. In the Ivrea peridotite exhibiting rhombic symmetry, there is practically no shear-wave splitting to be observed for S-waves propagating parallel to the olivine [100]- axes, and the maximum shear-wave splitting is not coupled with the direction of highest P-wave velocity. In the transversely isotropic Massif Central xenolith shear waves do not split when propagating parallel to the olivine [010] maximum coinciding with the direction of the radial symmetry axis of the olivine fabric. The experimental data are principally in accordance with velocity calculations based on the measured olivine LPO and the olivine single crystal stiffness coefficients. Implications for the interpretation of seismic data are discussed.

An experimental investigation of the Hashin-Shtrikman bounds on two-phase aggregate elastic properties

Existing data supporting or disputing the validity of the Hashin-Shtrikman bounds on the elastic properties of multiphase aggregates often do not consider porosity, elastic anisotropy, or experimental errors. In this experiment, two-phase aggregates of KCl + (NH 4Br, TlBr, CsCl, NaCl, Cu, and LiF) at every 20% volume fraction were vacuum hot-pressed and the compressional and shear velocities were measured with a computer-controlled ultrasonic interferometer to ±0.2%. The ratio of the shear moduli, μ, (phase 2/KCl) varied from about 1 to 5, producing a range of separations between the theoretical two-phase Hashin-Shtrikman bounds for the composites. Samples were generally 99% or better of the theoretical density, with less than 1% velocity anisotropy. Porosity corrections were applied assuming spherical pores, based on the observed velocity-pressure behaviour. Velocities agreed with the HS bounds calculated from the end-member single-crystal stiffnesses when anisotropy was taken into account. The velocity data were also used to estimate the bulk modulus, K, and shear modulus of the second phase by means of the matrix method — taking the K and μ of KCl as known and calculating the moduli of the other phase assuming that the measured velocities were the two-phase Hashin-Shtrikman bounds or the Voigt-Reuss-Hill average. A narrow range of moduli estimates results only if the μ's of both phases are fairly closely matched. For μ's mismatched by a factor of 5, the theoretical uncertainty in the estimates can be 10 times larger than the experimental uncertainty. Estimates using the VRH average can lie outside the HS-based results.

Theoretical bounds for thermal expansion, specific heat, and strain energy due to internal stress

Theoretical bounds on the strain energy due to internal stress in polycrystalline materials are found using energy principles. These bounds are used to derive bounds on bulk thermal expansion; the expressions depend only upon the single-crystal stiffness and compliance tensors, the thermal expansion tensors, and the partial volumes of the mineral components. A description of the geometric configuration of the phases is not required in the analysis. Calculations based on published data for corundum, rutile, calcite, and quartz show that bounds on thermal expansion for polycrystalline calcite, which is the most anisotropic of the four minerals, differ by only 6%. The specific heat of the polycrystalline forms of these minerals is found to be very nearly equal to the single-crystal values. Theory is used to find hot-pressing pressure and temperature required for minimizing internal stresses at room conditions and to estimate internal stresses in surface rock that is free of internal stress at depth.