Elastic Tensor Research Articles

Temperature-dependent properties of wurtzite and rock salt ScxAl1-xN for applications in BAW resonators

Abstract Scandium aluminum nitride (AlxSc1-xN) is known for its tremendous enhancement of piezoelectric properties. This material can improve the energy efficiency of a wide range of applications, for example bulk acoustic wave (BAW) resonators. Such devices heat up during operation and are thus sensitive to temperature-dependent material properties. As a result, device characteristics change with temperature, causing undesirable shifts in achievable frequency and performance. These shifts are well known for AlN-based resonators, where the frequency shifts are compensated without increasing the bandwidth of the frequency filter. For the ternary alloy AlxSc1-xN, many temperature-dependent structural, mechanical and piezoelectric properties have not or have been insufficiently investigated. This review provides an overview of relevant temperature dependent properties of hexagonal wurtzite (x < 0.5) and cubic rock salt AlxSc1-xN (x > 0.5) crystals up to 2000 K. The properties studied are the thermal expansion coefficient (TEC) α, the elastic tensor C ij, the piezoelectric coefficient d 33, the piezoelectric stress coefficient e 33 and the permittivity ε33. Based on these material properties, the derived characteristics for BAW resonators are determined from 0 to 1400 K, which are the resonance frequency f R and the electromechanical coupling coefficient k t 2.

Pressure-induced band gap enhancement and temperature-dependent thermoelectric characterization of semiconducting Transition Metal Chalcogenides LiMS (M = Cu, Ag)

The first principles study of Transition Metal Chalcogenides LiMS (M = Cu, Ag) has been conducted utilizing the concepts of Density Functional Theory (DFT). Both the compounds possess a cubic structure with space group F-43m. The mechanical stability of LiCuS and LiAgS has been predicted based on the values of elastic tensor. These Chalcogenides possess an interesting electronic band structure which reveals their semiconducting nature with a direct band gap. The electronic structure is further subjected to high pressure and the changes in band gap are deeply observed. For these materials, the changes in band gap in response to applied pressure suggest the possibility of their use in band gap engineering. The phonon curves are investigated over the Brillouin zone's high symmetry directions which reveal the lattice dynamical stability of the presently studied compounds. Moreover, the atoms present at different positions inside a crystal lattice exhibit vibrations of different intensities and in different directions which have been studied through the Eigenvector representations. The thermoelectric properties of LiCuS and LiAgS have been characterized using Boltzmann's theory over the range of temperature between 100 K and 1600 K. The calculated thermoelectric parameters reveal the high thermoelectric efficiency of these materials with excellent values of the figure of merit. The interesting properties of ternary Chalcogenides LiMS (M = Cu, Ag) make them promising candidates for thermoelectric applications.

Anisotropic odd elasticity with Hamiltonian curl forces

Abstract A host of elastic systems consisting of active components exhibit path-dependent elastic behaviors not found in classical elasticity, which is known as odd elasticity. Odd elasticity is characterized by antisymmetric (odd) elastic modulus tensor. Here, from the perspective of geometry, we construct the Hamiltonian formalism to show the origin of the antisymmetry of the elastic modulus that is intrinsically anisotropic. Furthermore, both non-conservative stress and the associated nonlinear constitutive relation naturally arise. This work also opens the promising possibility of exploring the physics of odd elasticity in dynamical regime by Hamiltonian formalism.

A trust region algorithm for finite-strain plasticity with strongly coupled hardening

As extensions to our Lagrangian finite-strain plasticity framework based on the approximate exponential integrator, we introduce two new algorithms: (i) a fixed-radius trust region root finder for the nonlinear system involving the elastic right Cauchy Green tensor and plastic multiplier (ii) a partitioned approach for the hardening variables, which are determined in a staggered form using the strongly-coupled concept typically adopted for multiphysics problems. This allows the use of intricate hyperelastic laws combined with recent yield functions, which otherwise would involve a laborious treatment, and the use of corresponding work-hardening. Work-hardening would introduce significant nonlinearities in the constitutive system if used in a fully-coupled form. For the partitioned approach, a dynamic relaxation algorithm is adopted. This allows the efficient solution of the two nonlinear equations without significant drifting. Results show that robustness and efficiency are significantly improved. Herein, algorithms are described in detail. Significant testing is performed with imposed strains up to 100 for a carbon steel. This contributes to the robustness of equilibrium iterations. Drifting is also assessed as a function of number of steps in the dynamic relaxation algorithm. Numerical experimentation is performed in the 3D tension test for an anisotropic yield function.

Fourier Transform Approach to Boundary Domain Integral Equations for Elastic Composites With Domain Decomposition and Multi Reference Parameters

ABSTRACTIn this article, displacement and strain periodic boundary domain integral equations for homogenization problems of elastic composites are derived in the context of FFT homogenization methods. The resolution methods based on regular grids and discrete Green's tensors are presented. The displacement based equations can be used to solve problems in arbitrary domains under periodic and non‐periodic boundary conditions. The strain based integral equation is obtained from the combination of the displacement based equations for different domains, each one having its own reference elasticity tensor. In the latter, the strain values inside every phases are connected to material mismatch parameters on the phase boundary. It was shown that by decomposing suitably domains by stiffness and using adapted reference parameters, the iteration schemes converge faster.

Conditions for strong ellipticity of 4th-order elasticity tensors

Conditions for strong ellipticity of 4th-order elasticity tensors

Multi-scale topology optimization of periodic structures with orthotropic multiple materials using the element-free Galerkin method

ABSTRACT A multi-scale topology optimization model is proposed for orthotropic multi-material periodic structures using the element-free Galerkin method. The macro multi-material distribution is optimized with the alternating active-phase algorithm, and the effective elastic tensor of multi-type microstructures is determined by energy-based homogenization. The feasibility of the model is verified and the effects of the quantities of material types and design subdomains, the Poisson's ratio factor and the multi-material off-angle on topological configuration and compliance are studied. The results indicate that the quantity of material types affects the mechanical properties of the periodic structure, and different microstructure configurations can be obtained under different numbers of design subdomains. The increase in the Poisson's ratio factor and decrease in the off-angle can enhance the effective elastic properties of the microstructure and reduce the compliance. Reasonable values of the Poisson's ratio factor and multi-material off-angle are suggested to be 2–4 and 0–45°.

Planar structured materials with extreme elastic anisotropy

Designing anisotropic structured materials by reducing symmetry results in unique behaviors, such as shearing under uniaxial compression. While rank-deficient materials such as hierarchical laminates have been shown to exhibit extreme elastic anisotropy, there is limited knowledge about the fully anisotropic elasticity tensors achievable with single-scale fabrication techniques. No established upper and lower bounds on anisotropic moduli exist. In this paper, we estimate the range of anisotropic stiffness tensors achieved by single-scale two-dimensional structured materials. We first develop a database of periodic anisotropic single-scale unit cell geometries using linear combinations of periodic cosine functions. The database covers a wide range of anisotropic elasticity tensors, which are then compared with the elasticity tensors of hierarchical laminates. We identify the regions in the property space where hierarchical design is necessary to achieve extremal properties. We demonstrate a method to construct various 2D functionally graded structures using this cosine function representation. These graded structures seamlessly interpolate between unit cells with distinct patterns, allowing for independent control of several functional gradients, such as porosity, anisotropic moduli, and symmetry. When designed with unit cells positioned at extreme parts of the property space, these graded structures exhibit unique mechanical behaviors such as selective strain energy localization, compressive strains under tension, and localized rotations.

Nonlinear Reduced Order Modeling of Heated Structures with Temperature-Dependent Properties

This paper focuses on extending coupled structural–thermal reduced order modeling approaches for the prediction of the nonlinear geometric response of heated structures to efficiently account for temperature-dependent structural and thermal properties. Structurally, a linear dependence of the elasticity tensor and thermal expansion coefficient is assumed consistently with typical material behavior. The resulting structural reduced order model (ROM) equations exhibit polynomials of the temperature generalized coordinates, the coefficients of which are readily identified. A different strategy is proposed for conductance and capacitance properties which typically depend nonlinearly on temperature. It relies on splitting the temperature generalized coordinates into a small set of dominant ones having a nonlinear effect on the ROM conductances and capacitances and a much larger set affecting them linearly. The inclusion of uncertainty in the structural ROM is also addressed extending earlier work limited to temperature independent properties. These strategies are exemplified on a representative hypersonic vehicle panel undergoing a full mission profile and with aero-thermal–structural coupling. The ROM predictions are observed to be very close to the full finite element ones for the mean model. Moreover, an uncertainty analysis demonstrates the strong sensitivity of the response to the structural model in the strongly nonlinear regions of the trajectory.

Fracture modeling in composite structures containing orthotropic materials by incorporating strain orthogonal decompositions into phase-field method of interfacial damage

PurposeThe phase-field method of interfacial damage is used to simulate the damage in composite structures containing the brittle orthotropic materials and their interface.Design/methodology/approachIn the brittle fracture modeling, the strain tensor is decomposed into positive and negative parts characterizing tension and compression behaviors. By requiring an elastic energy preserving transformation involving the elastic stiffness tensor, these two strain parts must satisfy the orthogonality condition in the sense that the elastic stiffness tensor responds as a metric. However, most of the recent phase-field methods for brittle fracture do not verify this orthogonality condition. Additionally, to describe the damage in structures with anisotropic phases, recent studies have used multiple phase-field variables, with each preferential orientation represented by a phase-field variable to describe the bulk damage of component materials. This approach increases the complexity of simulation procedure. These disadvantages motivate the present study aimed at enhancing the simulation method.FindingsThe present study improves the phase-field method of interfacial damage by (1) incorporating the strain orthogonality condition into the phase-field method; (2) using only one phase-field variable instead of multiple phase-field variables to simulate damage in component orthotropic phases; and (3) investigating the interaction between interfacial damage and bulk damage as well as the effect of orientation tensor of preferential orientation in each orthotropic phase and the interfacial parameters on crack branching in composite structures.Originality/valueThrough several simulation examples, the present simulation method is proven to be accurate, effective, and helps the simulation process simpler than previous relevant methods.

Surprises in the theory of elasticity: pairwise atomic interactions, central forces, and Cauchy relations

Although the theory of elasticity has been well known for many years there are still some commonly accepted misconceptions that continue to be taught in solid-state physics courses. In this paper we review some of these misconceptions through a detailed study of the relationship between macroscopic and microscopic theories of elasticity in a simple-cubic crystal with one atom per primitive cell. The analysis is performed at different stages of complexity. In the first-neighbor approximation we find that the crystal is simply unstable. If pairwise interaction up to second neighbors is considered the crystal is stable but the elastic constants present additional symmetries known as Cauchy relations which are not observed experimentally. The reasons for this behavior are analyzed, pointing out that the invariance of the potential energy with respect to rotations implies necessarily that pairwise interactions must be central. Only if the atomic interactions involve the participation of 3-bodies or more, the forces are non-central and the resulting elastic tensor may be acceptable.

Anisotropy and XKS splitting from geodynamic models of double subduction: testing the limits of interpretation

SUMMARY The analysis of the splitting signature of XKS phases is crucial for constraining seismic anisotropy patterns, especially in complex subduction settings such as outward-dipping double subduction. A natural example of this is found in the Central Mediterranean, where the Apennine and the Dinaride slabs subduct in opposite directions, with the Adriatic plate separating them. To assess the capability of XKS-splitting analysis in revealing anisotropic seismic properties, such as fast polarization directions and shear wave anisotropy (in per cent), we use three-dimensional numerical geodynamic models combined with texture evolution simulations. In these models, two identical outward-dipping oceanic plates are separated by a continental plate. Using the full elastic tensors – directly derived from the texture evolution simulations – we compute anisotropic seismic properties and synthetic teleseismic waveforms. From these waveforms synthetic observables are determined, including apparent splitting parameters (fast polarization directions and delay times) and splitting intensities. Based on these observables, we (1) derive models for a single anisotropic layer (one-layer model), (2) identify regions with significant depth-dependent anisotropic seismic properties, and (3) perform inversions at selected locations in terms of two anisotropic layers (two-layer model). We consider two geodynamic models: one with a strong (M1) and one with a weak (M2) continental plate. Model M1 exhibits significant retreat of the subducting plates with no horizontal stretching of the continental plate, whereas Model M2 shows less retreat, substantial horizontal stretching, and detachment of the subducting plates. These different subduction styles result in distinct flow and deformation patterns in the upper mantle, which are reflected in the anisotropic seismic properties. In Model M1, the fast polarization directions below the continental plate are predominantly trench-parallel, whereas in Model M2, they are mostly trench-normal. In most regions of both models, the one-layer models are sufficient to resolve the anisotropic seismic properties, as these properties are nearly constant with depth. However, for both models, we identify some isolated regions – primarily near the tips of the subducting plates and beneath the continental plate – where fast polarization directions exhibit significant variations with depth. Inverting the apparent splitting parameters in these regions yields multiple two-layer models at each location that excellently fit the observables. However, their anisotropic seismic properties can vary significantly, and not all these two-layer models adequately approximate the true depth variations. This ambiguity can be partially reduced by selecting two-layer models in which the summed shear wave anisotropy closely matches that of one of the one-layer models, as these models better capture the true variations.

First-principle calculations to investigate mechanical and acoustical properties of predicted stable halide Perovskite ABX3

This work examines the predicted stable halide perovskites' elastic, acoustical, and thermal characteristics. The work uses the Full Potential-Linearized Augmented Plane Wave (FP-LAPW) technique through PBE-GGA to compute compounds in the WIEN2K algorithm. The ELATE program for the evaluation of elastic tensors to plot 2D and 3D graphs was also used. The bulk modulus, Young's modulus, shear modulus, anisotropy factors, Cauchy pressure, Pugh's ratio, Poisson's ratio, Kleinman's parameter, Lame's coefficient, Vicker's hardness, sound velocities, Gruneisen parameter and even melting and Debye temperature were computed. The mechanical and elastic properties are reported for the first time for most of the compounds, demonstrating that the investigated HPs—aside from TlBeF3, BaAgBr3, and CsTcl3—are mechanically stable and exhibit weaker resistance against shear distortion than they do to unidirectional compression. The results of Poisson's, Pugh's, and Frantsevich's ratios data prove that all materials are ductile except SrLiF3. The estimated Poisson's ratio data indicates the metallic bonding nature of HPs, whereas only SrLiF3 exhibits covalent behavior with ν = 0.23. Debye temperature for SrLiF3, ZnLiF3, ZnScF3, CsRhCl3, CsRuCl3, and CsBeCl3 is greater than 200 K which signifies their hardness, thermal conductivity, and high sound velocities. The large melting temperature values, make them suitable for high-temperature industrial applications. The anharmonicity effect is highest for CaCuBr3 (3.265) and lowest for SrLiF3 (1.402). The current approach calculates elastic and mechanical properties, providing a practical understanding of various physical processes and enabling technology developers to utilize compounds in diverse applications.

The sensitivity of lowermost mantle anisotropy to past mantle convection

It is widely believed that seismic anisotropy in the lowermost mantle is caused by the flow-induced alignment of anisotropic crystals such as post-perovskite. What is unclear, however, is whether the anisotropy observations in the lowermost mantle hold information about past mantle flow, or if they only inform us about the present-day flow field. To investigate this, we compare the general and seismic anisotropy calculated using Earth-like mantle convection models where one has a time-varying flow, and another where the present-day flow is constant throughout time. To do this, we track a post-perovskite polycrystal through the flow fields and calculate texture development using the sampled strain rate and the visco-plastic self-consistent approach. We assume dominant slip on (001) and test the effect of the relative importance of this glide plane over others by using three different plasticity models with different efficiencies at developing texture. We compare the radial anisotropy parameters and the anisotropic components of the elastic tensors produced by the flow field test cases at the same location. We find, under all ease-of-texturing cases, the radial anisotropy is very similar (difference <2%) in the majority of locations and in some regions, the difference can be very large (>10%). The same is true when comparing the elastic tensors directly. Varying the ease-of-texture development in the crystal aggregate suggests that easier-to-texture material may hold a stronger signal from past flow than harder-to-texture material. Our results imply that broad-scale observations of seismic anisotropy such as those from seismic tomography, 1-D estimates and normal mode observations, will be mainly sensitive to present-day flow. Shear-wave splitting measurements, however, could hold information about past mantle flow. In general, mantle memory expressed in anisotropy may be dependent on path length in the post-perovskite stability field. Our work implies that, as knowledge of the exact causative mechanism of lowermost mantle anisotropy develops, we may be able to constrain both present-day and past mantle convection.

Effect of the underlayer on the elastic parameters of the CoFeB/MgO heterostructures

We investigated the thermally induced surface acoustic waves in CoFeB/MgO heterostructures with different underlayer materials. Our results show a direct correlation between the density and elastic parameters of the underlayer materials and the surface phonon dispersion. Using finite element method-based simulations, we calculate the effective elastic parameters (such as elastic tensor, Young’s modulus, and Poisson’s ratio) for multilayers with different underlayer materials. The simulation results, either considering the elastic parameters of individual layers or considering the effective elastic parameters of whole stacks, exhibit good agreement with the experimental data. This study will help us deepen our understanding of phonon properties and their interactions with other quasiparticles or magnetic textures with the help of these estimated elastic properties.

Oligo-cyclic loading-induced evolution of stress distribution and apparent amorphous modulus in lamellar stacks of high-density polyethylene

Assessing the resistance of high-density polyethylene (HDPE) against earthquake-like loads involves understanding the changes in structure and properties induced by oligo-cyclic loading at various length scales. To study the evolution of stress distribution and intrinsic properties within lamellar stacks from pristine to oligo-cyclic loading pre-conditioned materials, simultaneous in-situ SAXS/WAXS measurements were performed. During the elastic deformation of each pristine and preconditioned sample, crystal strain was tracked using the in-situ WAXS technique. Based on the established elastic tensor of the crystalline structure in polyethylene, we calculated the microscopic stress values within crystalline lamellae. In the pristine sample, lamellar stacks exhibit closely series-like coupling in the equatorial region and parallel-like coupling in the polar region of the spherulite. In the pre-conditioned sample, stress is primarily concentrated in the intra-fibrillar region, where the crystalline and amorphous phases are series-coupled, and strong strain concentration occurs in the inter-fibrillar region. By combining the local strain in the amorphous layer within the lamellar stacks in the equatorial region of the spherulite and the intra-fibrillar region with series-coupled lamellar stacks, measured by in-situ SAXS tests, the apparent amorphous modulus at the lamellar stack scale can be determined. This modulus changes from 71 to 106 MPa in the equatorial region of pristine spherulites to a notable 2000–7000 MPa in the intra-fibrillar region under the influence of oligo-cyclic pre-loading. Importantly, this apparent modulus is affected by both crystallization conditions and molecular structure, with molecular parameters exerting the primary influence.

Invariants of 3D Anisotropic Elasticity Tensor: Measure of Symmetry Defects

Invariants of 3D Anisotropic Elasticity Tensor: Measure of Symmetry Defects

First-Principles Calculations on Electronic, Optical, and Phonon Properties of γ-Bi2MoO6.

The wide band gap γ-Bi2MoO6 (BMO) has tremendous potential in emergent solar harvesting applications. Here we present a combined experimental-first-principles density functional theory (DFT) approach to probe physical properties relevant to the light sensitivity of BMO like dynamic and structural stability, Raman and infrared absorption modes, value and nature of band gap (i.e., direct or indirect), dielectric constant, and optical absorption, etc. We solvothermally synthesized wide band gap Pca21 phase pure BMO (≳3 eV) for two different pH values of 7 and 9. The structural parameters were correlated with the stability of BMO derived from elastic tensor simulations. The desired dynamical stability at T = 0 K was established from the phonon vibrational band structure using a finite difference-based supercell approach. The DFT-based Raman modes and phonon density of states (DOS) reliably reproduced the experimental Raman and infrared absorption. The electronic DOS calculated from Heyd-Scuseria-Ernzerhof HSE06 with van der Waals (vdW) and relativistic spin-orbit coupling (SOC) corrections produced a good agreement with the band gap obtained from diffuse reflectance spectroscopy (DRS). The optical absorption obtained from the complex dielectric constant for the HSE06+SOC+vdW potential closely resembled the DRS-derived absorption of BMO. The BMO shows ∼43% photocatalytic efficiency in degrading methylene blue dye under 75 min optical illumination. This combined DFT-experimental approach may provide a better understanding of the properties of BMO relevant to solar harvesting applications.

Intermitted 3D Diffraction Tomography Combined with In situ Laue Diffraction to Characterize Dislocation Structures and Stress Fields in Microbending Cantilevers

The mechanisms of plastic deformation are investigated using different characterization tools as scanning electron microscopy (SEM), transmission electron microscopy, or synchrotron‐based X‐ray techniques like Laue microdiffraction (μLaue). However, structural information can be limited to the specimen surface (SEM), to extremely thin samples (TEM), or depth averaging (μLaue). Until today, a nondestructive in situ investigation of a dislocation population, and de facto, the determination of the local stress tensor in bulk samples, remain challenging. To decompose the depth‐integrated μLaue signals, the so‐called “differential aperture X‐ray microscopy” (DAXM), allowing the 3D determination of the local structural crystal properties, is used. Using this approach, the local crystallographic phase, orientation, and the elastic strain tensor are obtained with 1 μm3 voxel size. In order to accomplish the experiment, a protocol and a new combined in situ mechanical testing rig with a DAXM microscope is created. The experiment is conducted on a severely bent focused ion beam copper single‐crystal microcantilever (10 × 10 × 25 μm3). The local deviatoric strain tensor and the local lattice curvature in the deformed sample are analyzed in 3D. The advantages and resolution limits of the technique are discussed in detail.

A method to represent the strain hardening effect in the hyper-elastic model within a fully Eulerian framework

We propose a method to treat the strain hardening effect in the hyperelastic model. Coupled with the level set interface capturing method and a real ghost fluid method, it is formulated in a fully Eulerian framework. The HLLD approximate Riemann solver is used to obtain the interface state. We use the elastic inverse deformation gradient tensor to track the material deformation state. Once the solid starts to yield, along the normal direction in stress space, an iterative algorithm is used to relax the elastic inverse deformation gradient tensor to target value where the equivalent stress meets the requirement of consistency condition in plasticity. This approach is intuitive and can avoid the use of the physical delay time, which is difficult to calibrate from experiments. We use the Wilkins flying plate impact, cylindrical shell collapse, Taylor impact and the conical-nosed projectile penetration test cases to validate the method. The results show good agreement with the theoretical analysis and experiments. This approach can also apply without further extension when thermal softening effect or damage models are considered.