Elastic Tensor Research Articles

Resonant ultrasound spectroscopy for textured materials

Resonant ultrasound spectroscopy (RUS) is a method in which the least-squares fitting of a sample's measured resonance frequencies is used to determine the sample material's elastic constants. Difficulties arise when RUS is applied to textured materials, which are composed of crystallites that are neither completely aligned nor randomly oriented; such materials would have effective elastic tensors with 21 independent elastic constants, and this would overburden the conventional RUS analysis process. In any case, the elastic nature of some textured materials may be represented as the weighted average of the elastic tensors of the constituent crystalline material rotated in the directions of the individual crystallites. The collection of weights is the orientation distribution function (ODF). Now the least-squares fitting of a sample's measured resonance frequencies may be used to determine not the sample's elastic constants but instead its ODF. This is not trivial since the weights must be positive and sum to one, but it may be done with a simple modification of the conventional RUS software.

Ultrasonic scattering from polycrystals with transversely isotropic material texture

During laser-based metal additive manufacturing (AM), the melted powder is subjected to rapid cooling and the resulting microstructures often have material texture. In many cases, the texture is uniaxial (i.e., transversely isotropic) such that one symmetry axis defines the material response. Therefore, ultrasonic inspection methods that exploit the scattering from the microstructure are complicated by the resulting texture which affects the coherent propagation and scattering. In this presentation, a generalized approach is described in which the covariance of the elastic modulus tensor for a uniaxial ensemble of cubic crystals is expressed in terms of a fundamental set of constants. These constants are determined for an arbitrary texture from synthetic polycrystals created using DREAM.3D. With this information, calculations for wave velocity, attenuation, and diffuse scattering can be made efficiently for any wave type and propagation direction relative to the material symmetry axis. Results are compared with analytical expressions for simplified cases and then more generalized textures are examined. Finally, prospects for characterization of components created using metal AM are discussed within the context of input data from electron backscatter diffraction measurements. These results are expected to provide insight regarding the inversion of measurement data for texture characterization.

Enabling resonant ultrasound spectroscopy in extreme environments

Elastic moduli are fundamental thermodynamic susceptibilities that connect to thermodynamics, electronic structure, and mechanic properties. Thus, determining the changes of elastic moduli as a function of time or temperature is a powerful tool to study several thermodynamic and materials phenomena. Resonant Ultrasound Spectroscopy (RUS) determines elastic constants with high accuracy and precision from a single measurement of the mechanical resonances of a sample. Conventionally, the quantitative extraction of elastic moduli with RUS assumes free boundary conditions but lead to unstable positioning of the sample making it incompatible with extreme environments like high magnetic fields. In practice, even holding the sample produces contact forces that deviate from free-boundary conditions shifting resonant frequencies. In this talk, we show that, we can reduce the free-boundary conditions (by gluing/adhering the sample to the transducer), while still being able to obtain the full elastic tensor by a simple model of the sample-transducer interaction. This means elastic constants can be determined to within the uncertainty of conventional RUS, but with significant improvements in sample stability and control of sample orientation. We show the results of studying of several magnetic and structural phase-transitions using this new method.

Odd elasticity and topological waves in active surfaces.

Odd elasticity describes active elastic systems whose stress-strain relationship is not compatible with a potential energy. As the requirement of energy conservation is lifted from linear elasticity, new antisymmetric (odd) components appear in the elastic tensor. In this work we study the odd elasticity and non-Hermitian wave dynamics of active surfaces, specifically plates of moderate thickness. These odd moduli can endow the vibrational modes of the plate with a nonzero topological invariant known as the first Chern number. Within continuum elastic theory, we show that the Chern number is related to the presence of unidirectional shearing waves that are hosted at the plate's boundary. We show that the existence of these chiral edge waves hinges on a distinctive two-step mechanism. Unlike electronic Chern insulators where the magnetic field at the same time gaps the spectrum and imparts chirality, here the finite thickness of the sample gaps the shear modes, and the odd elasticity makes them chiral.

Mechanical and dynamical stability of major oxide constituents of Portland cement clinker: a density functional theory study

AbstractQuantum‐mechanical calculations based on the density functional theory (DFT) enable an effective characterization of a variety of properties of materials at the atomistic level, although at a high computational cost. Here, we fully exploit parallel computing to apply such methods to the study of lattice dynamics and mechanical response of the major oxide constituents of Portland cement clinker: , , , and . Raman spectra and the evolution of the elastic tensor (and associated mechanical properties) with pressure are predicted for all oxide systems. We devote much attention to the assessment of dynamical and mechanical stability of the many previously proposed polymorphic forms of the most abundant oxide: . Specifically, five different crystalline models of are analyzed: only two turn out to be dynamically stable. The mechanical response of is further analyzed as a function of temperature through a quasi‐harmonic description of its lattice dynamics.

PYXIS: an integrated software package for synchrotron micro/nanodiffraction data analysis

PYXIS, a software package based on the MATLAB platform, is suitable for the in-depth analysis of synchrotron X-ray Laue micro/nanodiffraction and energy scan data. The main objective of the software is to analyze the following three aspects of the data. First, in a single Laue diffraction pattern, it can remove the background, search and index the peaks, and obtain the crystal orientation and the elastic strain tensor. A novel zone-axis assisted indexing algorithm has been developed to speed up this process. Second, a batch of Laue patterns obtained from a scan can be analyzed simultaneously to map the orientation and strain distributions, which is achieved efficiently by applying a peak position comparison strategy. An analysis of a Laue microdiffraction scan of the cross section of Li7La3Zr2O12 solid-state electrolyte has been taken as an example in this article to illustrate these functions. The third unique function in PYXIS involves data mining of energy scans. Considering a Ni-based superalloy as an example, the energy scan data sets are processed to reconstruct the diffraction peaks in three dimensions, allowing extraction of the lattice constants, lattice misfit and dislocation characteristics from the peak profiles.

Crystal structure and elastic and phonon properties of realgar versus pressure

Realgar, α-As4S4 (space group P21/n), is one of the best-known arsenic sulfide minerals because of its extended use in the past as a red pigment and its employment in modern times for advanced optical and electronic technological applications. From a geological perspective, the main realgar deposits are hydrothermal and epithermal, but it is also a relevant phase found between the upper mantle and Earth's crust and therefore one of the main sources of arsenic. Despite this widespread use and interest, few experimental and theoretical studies have been focused on the characterization of the structural, elastic and vibrational properties of realgar, especially their variation with pressure. Some quantities, such as the cohesive energy between the As4S4 units and the elastic moduli, have never been reported in the scientific literature. The present work deals with a density functional theory investigation of the cited properties of realgar using the recently proposed PBEh-3c method, which was devised in particular to deal with crystalline solids characterized by weak van der Waals interactions. This approach is validated against the available crystal-chemical, mechanical and spectroscopic data from previous studies, finding a generally good agreement. The equation-of-state parameters of the energy versus unit-cell volume data were V 0 = 767.13 (9) Å3, B 0 = 15.73 (8) GPa and B′ = 9.1 (2), with the bulk modulus value (B 0) in good agreement with the value obtained from the elastic tensor analysis (B = 16.1 GPa). The cohesive energy was found to be about 146.1 kJ mol–1, a value that follows the typical ones of organic crystals. The present work provides new insights into this peculiar mineral that, from a mineralogical point of view, could be considered a prototype of a heterodesmic structure made by inorganic molecular clusters.

Internally Balanced Elasticity Tensor in Terms of Principal Stretches

Internally Balanced Elasticity Tensor in Terms of Principal Stretches

Data-Driven Based Characterization of Anisotropic Mechanical Properties for Cancellous Bone From Low-Resolution CT Images.

Exploring the anisotropic mechanical behavior of cancellous bone is crucial for in-vivo bone biomechanical analysis. However, it is challenging to characterize anisotropic mechanical behaviors under low-resolution (LR) clinical CT images due to a lack of microstructural information. The data-driven method proposed in this article accurately characterizes the anisotropic mechanical properties of cancellous bone from LR clinical CT images. The trabecular bone cubes of sheep are used to obtain a high-resolution (HR) micro-CT and an LR clinical CT image dataset. First, an auto-encoder model is trained using HR image data. Microstructural features are extracted by the encoder. A fast super-resolution (FSR) model is trained to map LR bone cubes to the features extracted from corresponding HR samples. The pretrained FSR model is used to convert LR clinical CT images to encoded microstructural features. The features are later used to predict target histomorphological parameters, anisotropic elastic tensors, and fabric tensors based on a fully connected neural network. The data-driven model accurately predicts the elastic tensor and fabric tensor of trabecular bones with LR CT images with 0.6 mm/pixel spatial resolution. It was verified that LR clinical CT images could generate microstructural information using a generative deep-learning model and an up-sampling operation. This study proves that clinical medical images of cancellous bone can be used for analysis of complex mechanical properties using a data-driven method, which is useful for real-time bone defect diagnosis and personalized bone prosthesis design in clinical application.

Piezoelectricity of lead-free hybrid perovskite monolayer (ATHP)2XY4 (X = Ge, Sn; Y = Cl, Br, I): Cross-scale calculation and simulation

Two-dimensional (2D) piezoelectric material is the core part of all kinds of micro-piezoelectric devices (piezoelectric energy harvesters, piezoelectric sensors, etc.). Based on the first principle, this paper theoretically predicts a kind of organic–inorganic lead-free hybrid perovskite monolayer (ATHP)2XY4 (X = Ge, Sn; Y = Cl, Br, I) with a large out-of-plane piezoelectric coefficient (d31). Combined with the finite element simulation, it shows excellent performance in the piezoelectric energy harvester. Through the first-principles theoretical calculation, the density, elastic tensor, piezoelectric stress tensor, and dielectric coefficient of the (ATHP)2XY4 monolayer can be obtained. The thermodynamic stability and mechanical stability of the materials are judged by ab initio molecular dynamics and stress–strain relationship analysis, respectively. The piezoelectric coefficient of the material is calculated, in which the d31 of (ATHP)2SnBr4 is the largest (d31 = 35.04 pm/V), which is still 7 times that of the bulk AlN and at least one order of magnitude larger than that of other 2D materials. The finite element simulation results of a simple piezoelectric beam show that the voltage output of the piezoelectric beam reaches 27.79 V. Our research shows that lead-free hybrid perovskite monolayer (ATHP)2XY4 has strong competitiveness in the application of environment-friendly and bio-friendly micro-piezoelectric devices. In addition, the cross-scale simulation method from the first principles to the finite element method is of great significance for the optimization of micro-piezoelectric devices.

Effect of collagen fibril orientation on the anisotropic properties of peri-implant bone.

In orthopedic and dental surgery, the implantation of biomaterials within the bone to restore the integrity of the treated organ has become a standard procedure. Their long-term stability relies on the osseointegration phenomena, where bone grows onto and around metallic implants, creating a bone-implant interface. Bone is a highly hierarchical material that evolves spatially and temporally during this healing phase. A deeper understanding of its biomechanical characteristics is needed, as they are determinants for surgical success. In this context, we propose a multiscale homogenization model to evaluate the effective elastic properties of bone as a function of the distance from the implant, based on the tissue's structure and composition at lower scales. The model considers three scales: hydroxyapatite foam (nanoscale), ultrastructure (microscale), and tissue (mesoscale). The elastic properties and the volume fraction of the elementary constituents of bone matrix (mineral, collagen, and water), the orientation of the collagen fibril relative to the implant surface, and the mesoscale porosity constitute the input data of the model. The effect of a spatiotemporal variation in the collagen fibrils' orientation on the bone anisotropic properties in the proximity of the implant was investigated. The findings revealed a strong variation of the components of the effective elasticity tensor of the bone as a function of the distance from the implant. The effective elasticity appears to be primarily sensitive to the porosity (mesoscale) rather than to the collagen fibrils' orientation (sub-micro scale). However, the orientation of the fibrils has a significant influence on the isotropy of the bone. When analyzing the symmetry properties of the effective elasticity tensor, the ratio between the isotropic and hexagonal components is determined by a combination of the porosity and the fibrils' orientation. A decrease in porosity leads to a decrease in bone isotropy and, in turn, an increase in the impact of the fibrils' orientation. These results demonstrate that the collagen fibril orientation should be taken into account to properly describe the effective elastic anisotropy of bone at the organ scale.

An adaptive FEM for the elastic transmission eigenvalue problem with different elastic tensors and different mass densities

An adaptive FEM for the elastic transmission eigenvalue problem with different elastic tensors and different mass densities

Concurrent multiscale and multi‐material optimization method for natural vibration design of porous structures

AbstractThis paper proposes a concurrent multiscale optimization method of macrostructure topology and microstructure shapes for porous structures, aimed at maximizing a specified natural frequency. The multi‐material distribution of the macrostructure and the shape of the microstructures are optimized by topology and shape optimization, respectively. The homogenized properties of the porous materials are calculated using the homogenization method, and the homogenized elastic tensor and density are applied to the macrostructure. The optimum distribution of the porous material in the macrostructure is determined by a multi‐material topology optimization using the generalized solid isotropic material with penalization (GSIMP) method, which is a method to obtain multi‐material topology by implementing the successive binarization. The KS (Kreisselmeier–Steinhauser) function is introduced to solve the repeated natural frequencies issue that lies in the maximization of a specified natural frequency. An area‐constrained multiscale optimization problem is formulated as a distributed‐parameter optimization problem, and the sensitivity functions are derived using the Lagrange multiplier method and the adjoint variable method. Based on the obtained sensitivity functions, the design variables are updated using the H1 gradient method (i.e., a nonparametric shape/topology optimization method). The effectiveness of the proposed method for maximizing a specified natural frequency is confirmed through 2D and 3D numerical examples, in which the influences of sub‐regions of the macrostructure and anisotropic material of the microstructures are also investigated and the results are discussed.

Exploring the role of Type-II residual stresses in a laser powder bed fusion nickel-based superalloy using measurement and modeling

Far-field high-energy X-ray diffraction microscopy (ff-HEDM) and the crystal plasticity finite element method (CPFEM) are used to investigate the role of grain-scale (Type-II) residual stresses on the fatigue life of additively manufactured (AM) Inconel alloy 625 (IN-625). Grain-averaged orientations, centroids, and residual elastic strain tensors from ff-HEDM data are used to instantiate a crystal plasticity model to simulate the effect of residual stresses at the grain scale. Simulation results indicate that the presence of tensile residual strains increase stress localization and heterogeneity within grains, triggering an earlier onset of plasticity. A microscale fatigue indicator parameter (FIP) is computed to model the impact of these residual strains on the cycles to fatigue crack nucleation. The crack nucleation model, based on the computed FIPs, predicts a significant reduction in the number of cycles for fatigue crack nucleation for mid- and high-cycle fatigue due to the residual strain induced localization, while the residual strains have minimal impact on low-cycle fatigue life.

High temperature decomposition and age hardening of single-phase wurtzite Ti1−xAlxN thin films grown by cathodic arc deposition

Wurtzite TmAlN (Tm=transition metal) themselves are of interest as semiconductors with tunable band gap, insulating motifs to superconductors, and piezoelectric crystals. Characterization of wurtzite TmAlN is challenging because of the difficulty to synthesize them as single-phase solid solution and such thermodynamic, elastic properties, and high temperature behavior of wurtzite Ti1−xAlxN is unknown. Here, we investigated the high temperature decomposition behavior of wurtzite Ti1−xAlxN films using experimental methods combined with first-principles calculations. We have developed a method to grow single-phase metastable wurtzite Ti1−xAlxN (x=0.65, 0.75, 085, and 0.95) solid-solution films by cathodic arc deposition using low duty-cycle pulsed substrate-bias voltage. We report the full elasticity tensor for wurtzite Ti1−xAlxN as a function of Al content and predict a phase diagram including a miscibility gap and spinodals for both cubic and wurtzite Ti1−xAlxN. Complementary high-resolution scanning transmission electron microscopy and chemical mapping demonstrate decomposition of the films after high temperature annealing (950∘C), which resulted in nanoscale chemical compositional modulations containing Ti-rich and Al-rich regions with coherent or semicoherent interfaces. This spinodal decomposition of the wurtzite film causes age hardening of 1–2 GPa. Published by the American Physical Society 2024

Origin of brittle behavior of doped PbTe-based thermoelectric materials

Finding new efficient thermoelectric materials is a significant challenge for materials science. It is crucial to have a comprehensive understanding of material-property relationships to develop new materials successfully, given that minor structural or compositional changes can result in significant property changes. This paper extensively utilizes advanced theoretical approaches and investigates the impact of n- and p-type impurities on the mechanical characteristics of PbTe thermoelectric materials. n- and p-type doping of PbTe were studied using various techniques, including elastic tensor calculations, crystal orbital Hamilton population method, and local vibrational theory. Our findings reveal the specific ways in which doping type affects the material's mechanical properties. This information can aid researchers in optimizing PbTe doping strategies.

Anisotropic structural, vibrational, electronic, optical, and elastic properties of single-layer hafnium pentatelluride: an ab initio study.

Motivated by the highly anisotropic nature of bulk hafnium pentatelluride (HfTe5), the structural, vibrational, electronic, optical, and elastic properties of single-layer two-dimensional (2D) HfTe5 were investigated by performing density functional theory (DFT)-based first-principles calculations. Total energy and geometry optimizations reveal that the 2D single-layer form of HfTe5 exhibits in-plane anisotropy. The phonon band structure shows dynamic stability of the free-standing layer and the predicted Raman spectrum displays seven characteristic Raman-active phonon peaks. In addition to its dynamic stability, HfTe5 is shown to exhibit thermal stability at room temperature, as confirmed by quantum molecular dynamics simulations. Moreover, the obtained elastic stiffness tensor elements indicate the mechanical stability of HfTe5 with its orientation-dependent soft nature. The electronic band structure calculations show the indirect-gap semiconducting behavior of HfTe5 with a narrow electronic band gap energy. The optical properties of HfTe5, in terms of its imaginary dielectric function, absorption coefficient, reflectance, and transmittance, are shown to exhibit strong in-plane anisotropy. Furthermore, structural analysis of several point defects and their oxidized structures was performed by means of simulated STM images. Among the considered vacancy defects, namely , , VTeout, VTein, , and VHf, the formation of VTeout is revealed to be the most favorable defect. While and VHf defects lead to local magnetism, only the oxygen-substituted VHf structure possesses magnetism among the oxidized defects. Moreover, it is found that all the bare and oxidized vacant sites can be distinguished from each other through the STM images. Overall, our study indicates not only the fundamental anisotropic features of single-layer HfTe5, but also shows the signatures of feasible point defects and their oxidized structures, which may be useful for future experiments on 2D HfTe5.

First principles study on polarization and piezoelectric properties of group substitution regulated lead-free organic perovskite ferroelectrics

Organic ferroelectrics are desirable for the applications in the field of wearable electronics due to their eco-friendly process-ability, mechanical flexibility, low processing temperatures, and lightweight. In this work, we use five organic groups as substitution for organic cation and study the effects of organic cations on the structural stability, electronic structure, mechanical properties and spontaneous polarization of metal-free perovskite <i>A</i>-NH<sub>4</sub>-(PF<sub>6</sub>)<sub>3</sub> (<i>A</i> = MDABCO, CNDABCO, ODABCO, NODABCO, SHDABCO) through first-principles calculations. Firstly, the stabilities of the five materials are calculated by molecular dynamics simulations, and the energy values of all systems are negative and stable after 500 fs, which demonstrates the stabilities of the five materials at 300 K. The electronic structure calculation shows that the organic perovskite materials have wide band gap with a value of about 7.05 eV. The valence band maximum (VBM) and Cconduction band minimum (CBM) are occupied by different elements, which is conductive to the separation of electrons and holes. We find that organic cations have an important contribution to the spontaneous polarization of materials, with a contribution rate over 50%. The presence of hydrogen atoms in the substituting groups (MDABCO, ODABCO) enhances the hydrogen bond interaction between the organic cations and <inline-formula><tex-math id="Z-20240616143151">\begin{document}${\rm PF}_6^- $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20240385_Z-20240616143151.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20240385_Z-20240616143151.png"/></alternatives></inline-formula> and increases the displacement of the organic cation, resulting in an increase in the contribution of the polarization of the organic cation to the total polarization. In addition, we observe large piezoelectric strain components, the calculated value of <i>d</i><sub>33</sub> is 36.5 pC/N for CNDABCO-NH<sub>4</sub>-(PF<sub>6</sub>)<sub>3</sub>, 32.3 pC/N for SHNDABCO-NH<sub>4</sub>-(PF<sub>6</sub>)<sub>3</sub>, which is larger than the known value of <i>d</i><sub>33</sub> of MDABCO-NH<sub>4</sub>-I<sub>3</sub>(14pC/N). The calculated value of <i>d</i><sub>14</sub> is 57.5 pC/N for ODABCO-NH<sub>4</sub>-(PF<sub>6</sub>)<sub>3</sub>, 27.5 pC/N for NODABCO-NH<sub>4</sub>-(PF<sub>6</sub>)<sub>3</sub>. These components are at a high level among known organic perovskite materials and comparable to many known inorganic crystals. The large value of <i>d</i><sub>14</sub> is found to be closely related to the large value of elastic compliance tensor <i>s</i><sub>44</sub>. The analysis of Young’s modulus and bulk’s modulus shows that these organic perovskite materials have good ductility. These results indicate that these organic materials are excellent candidates for future environmentally friendly piezoelectric materials.

An equivariant graph neural network for the elasticity tensors of all seven crystal systems

An equivariant graph neural network model enables the rapid and accurate prediction of complete fourth-rank elasticity tensors of inorganic materials, facilitating the discovery of materials with exceptional mechanical properties.

Vibration of solid and thin-walled slender structures made of soft materials by high-order beam finite elements

In this work, high-order beam (1D) finite element models for the modal analysis of structures made of compressible and nearly-incompressible hyperelastic soft materials are presented in the well-established framework of Carrera Unified Formulation (CUF). In this investigation, the modal behavior of soft structures subjected to progressively increasing loads can be correctly predicted by higher-order structural theories, and the influence of pre-stress conditions applied on the modal response of structures is investigated. The mathematical formulation of hyperelastic isotropic materials is presented in terms of invariants of the right Cauchy–Green strain tensor, obtaining the most general expression of the Piola–Kirchoff 2 stress tensor and tangent elasticity tensor, both independent of the model adopted for the material strain energy function. Governing equations in matrix forms for the static nonlinear analysis and subsequent vibration problem around non-trivial equilibrium states are derived through the Principle of Virtual Displacements (PVD) under a total Lagrangian formulation, defining the fundamental nuclei of stiffness matrices and internal and external forces vector, all independent of expansion theories and kinematic models adopted in the mathematical modeling of finite elements. Actual numerical results are obtained by an iterative Newton–Raphson linearized scheme coupled with line-search algorithms, and they are compared with results obtained by the commercial code ABAQUS. Our proposed models are tested with large strain problems involving hyperelastic slender and thin-walled structures, for which mode aberration such as crossing or bearing are observed.