Independent Elastic-stiffness Coefficients Research Articles

Ab-initio Investigation of Elastic Properties of Monoclinic ZnAs2 Crystal

Elastic properties of monoclinic ZnAs2 crystal are studied under the PBEsol scheme using the CRYSTAL Program. Independent elastic stiffness coefficients have been computed. Various elastic properties, such as shear modulus, bulk modulus, Young’s modulus and Poisson’s ratio have been analyzed. The directional dependence of the computed Young’s modulus and linear compressibility is studied using ELATE software. Our investigation reveals the finite elastic anisotropy of the monoclinic ZnAs2 crystal.

Interpreting the effects of shale rock properties on seismic anisotropy by statistical and machine learning methods

The elastic anisotropy of unconventional shale reservoirs plays a crucial role in their economic hydrocarbon production. The strong anisotropy of organic-rich shale rocks, characterized by clay platelets, elongated pores, and organic material, presents challenges for traditional reservoir characterization methods, which are designed for isotropic or weakly anisotropic conventional sandstone reservoirs. To effectively describe the geomechanical behaviors of organic-rich shales and estimate changes in stress caused by hydraulic fracturing, it is necessary to develop practical methods for predicting anisotropy parameters from conventional rock property logs. This study proposes a statistical approach for predicting independent elastic stiffness coefficients for vertically transverse isotropic (VTI) shale from three shale rock properties: clay volume fraction, kerogen volume fraction, and total porosity. The results of this study show that the stiffness coefficients, C11 and C33, are related to P-wave anisotropy and porosity has a greater impact on P-wave anisotropy than clay and kerogen volume fractions. In contrast, changes in clay volume have a more sensitive effect on C44 and C66, providing S-wave anisotropy information that can help detect lithology changes. Furthermore, kerogen variation has a greater influence on both C12 and C13 than the effects of porosity and clay variations. From these relationships, we develop exponential nonlinear regression (ENLR) models for predicting independent elastic stiffness coefficients from three shale rock properties. These models are validated with the Midland Basin log data, and we improve the accuracy of the C13 estimation model using the deep feedforward neural network (DFNN) method. The results show that the statistical approach can help understand the effect of shale rock components on seismic anisotropy variations and better predict the variations in lithology, fluid, and organic contents in shales from anisotropic seismic data. Furthermore, the methodology can provide robust estimates of the stress changes from hydraulic fracturing, supporting the efficient and economically viable production of shale hydrocarbons.

Complete determination of elastic stiffness coefficients and local symmetry breaking in the paraelectric barium titanate

The temperature dependence of all independent elastic stiffness coefficients of cubic BaTiO3 was determined over a wide temperature range from 400 °C to the Curie point (Tc ∼ 131 °C) by using Brillouin light scattering spectroscopy. Among the two elastic eigen-modes corresponding to the two measured transverse acoustic waves, (C11-C12)/2 showed a large softening in the paraelectric phase, while the change in C44 was very small. This result caused the elastic anisotropy A, defined as 2C44/(C11-C12), to increase from ∼2.1 to ∼2.7 upon cooling toward the phase transition temperature. The large elastic anisotropy was attributed to the tetragonal instability and the associated anisotropic polarization fluctuations of precursor polar clusters formed by the correlated Ti motions. The elastic anisotropy was fitted by using the Curie-Weiss law resulting in a significant deviation in a certain temperature range between Tc and ∼Tc + 50 °C, similar to the dielectric permittivity.

Complete elastic constants of α-BaB2O4: Schaefer-Bergmann acousto-optic diffraction and resonant ultrasound spectroscopy.

Both Schaefer-Bergmann diffraction and resonant ultrasound spectroscopy were used to measure the six independent elastic-stiffness coefficients of the trigonal, non-piezoelectric crystal α-BaB2O4. The two measurement sets resulted in a root-mean-square variance of 1.2%. This paper provides a detailed analysis of the two different measurement techniques and discusses the similarities and differences.

Ab initio study of the elastic properties of sodium chloride at high pressure

The equation of state and elastic properties for B1- and B2-NaCl up to 160 GPa have been studied by using the density functional simulation within the generalized gradient approximation (GGA). The calculated lattice constants of NaCl agree well with experimental values in a precision of 0.1% over the pressure range studied. It is found that the cell volume decreases 5.5% at the phase transition point. All three independent elastic stiffness coefficients, c11, c12 and c44 for B1‐ and B2-NaCl are evaluated by a calculated stress tensor which was generated by forcing small strain to the optimized unit cell. The calculated zero-pressure elastic moduli, wave velocities, and their initial pressure dependences of B1-NaCl are in excellent agreement with experiments. Systematic investigation on the elasticity of NaCl has been done through four parameters, the Zener anisotropy ratio (AZ), the acoustic anisotropy factor (Aa), the Cauchy deviation (δ), and the normalized elastic constants (cij′). With the pressure, the Zener anisotropy ratio AZ decreases in the B1-phase, but increases in the B2-phase and reaches 1 at about 174 GPa, it suggests that NaCl would become elastic isotropic at this pressure range. The acoustic anisotropy factor Aa shows the similar pressure behavior as AZ. The Cauchy deviation (δ)) increases with pressures, it demonstrates that in the interatomic interaction, the many-body contribution becomes more important at higher pressures. A discussion on the normalized elastic constants is also presented.

Elastic constants and internal friction of martensitic steel, ferritic-pearlitic steel, and α-iron

The elastic constants and internal friction of induction hardened and unhardened SAE 1050 plain-carbon steel at ambient temperatures were determined by resonant ultrasonic spectroscopy. The hardened specimen contained only martensite and the unhardened specimen was ferrite-pearlite. Using an inverse Ritz algorithm with assumed orthorhombic symmetry, all nine independent elastic-stiffness coefficients were determined, and, from the resonance peak widths, all nine components of the internal-friction tensor were determined. Similar measurements and analysis on monocrystalline α-iron were performed. The steel has slight elastic anisotropy, and the isotropically approximated elastic moduli were lower in the martensite than in ferrite-pearlite: shear modulus by 3.6%, bulk modulus by 1.2%, Young modulus by 3.2%, and Poisson ratio by 1.5%. Isotropically approximated elastic moduli of α-iron were 0.6–1.3% higher than ferrite-pearlite. All components of the internal-friction in martensite were higher than those of ferrite-pearlite, but lower than those of α-iron.

Determination of elastic, anelastic, and piezoelectric coefficients of piezoelectric materials from a single specimen by acoustic resonance spectroscopy

We describe an advanced methodology to determine all the independent elastic-stiffness coefficients C ijkl , the associated internal frictions Q ijkl −1, and piezoelectric coefficients e ijk of piezoelectric materials from a single monocrystal specimen using resonant-ultrasound spectroscopy with laser-Doppler interferometry. The mechanical-resonance frequencies of a piezoelectric solid depend on all of the elastic and piezoelectric coefficients, and their accurate measurement allows one to determine the elastic and piezoelectric coefficients simultaneously. Resonance-peak-width measurements yield the internal-friction tensor. Successful determination requires correct vibration-mode identification for the observed resonance frequencies. This is achieved unambiguously by measuring deformation distributions on the vibrating-specimen surface with laser-Doppler interferometry and comparing them with calculated displacement distributions. The methodology is applied to lithium niobate (LiNbO 3) and langasite (La 3Ga 5SiO 14) crystals.

Elastic, anelastic, and piezoelectric coefficients of langasite: resonance ultrasound spectroscopy with laser-Doppler interferometry

This paper presents advanced techniques to determine all independent elastic-stiffness coefficients C/sub ij/, the associated internal friction Q/sub ij//sup -1/, and piezoelectric coefficients e/sub ij/ of monocrystal langasite (La/sub 3/Ga/sub 5/SiO/sub 14/) using a single rectangular parallelepiped specimen. Langasite's crystal structure belongs to the trigonal system with point group 32, and thus possesses six independent C/sub ij/, two e/sub ij/, and two dielectric coefficients /spl epsiv//sub ij/. All of the elastic and piezoelectric coefficients affect the mechanical resonance frequencies of the solid specimen, and measuring them very accurately permits one to determine the C/sub ij/ and e/sub ij/ with known density, dimensions, and e/sub ij/. We developed a piezoelectric tripod to support the specimen upward and measured the free-vibration resonance frequencies with minimum load from its own weight. This weak and stable acoustic coupling ensures accuracy of the frequency measurement better than 10/sup -5/, enough to determine the coefficients reliably. Our C/sub ij/ fall in the range of results measured with previous (conventional) methods. Our e/sub 11/ is smaller than the reported values by 1.2-13%, and e/sub 14/ is larger by 44-97%. For the internal friction measurement, we used a solenoid coil to vibrate the specimen without any contact. The longitudinal-wave internal friction considerably exceeds the shear-wave internal friction, which can be explained by phonon-phonon interactions.

Elastic-stiffness coefficients of a silicon carbide fibre at elevated temperatures: Acoustic spectroscopy and micromechanics modelling

We have determined all five independent elastic-stiffness coefficients Cij of a silicon carbide fibre with transverse isotropy from room temperature up to 873 K. Firstly, we measured the Cij of a titanium-alloy-matrix composite reinforced unidirectionally with the fibres and the matrix alloy alone. Electromagnetic acoustic resonance detected the free-vibration resonance frequencies of the specimens to determine their Cij . Secondly, we applied a micromechanics calculation to deduce the fibre Cij from the measured composite and matrix Cij . The resulting fibre Cij shows strong anisotropy in the temperature derivatives of the Cij ; the temperature derivatives for the fibre-axis-direction Cij are much smaller than the others.

Acoustic spectroscopy of lithium niobate: Elastic and piezoelectric coefficients

We report simultaneous measurement of the complete set of elastic and piezoelectric coefficients of lithium niobate (LiNbO3), which has trigonal crystal symmetry (3m point group) and thus six independent elastic-stiffness coefficients Cij, four piezoelectric coefficients eij, and two dielectric coefficients κij. We used a single specimen: an oriented rectangular parallelepiped about 5 mm in size. Our measurement method, acoustic spectroscopy, focuses on the crystal’s macroscopic resonance frequencies and is sensitive to any property that affects those frequencies. We overcame the principal obstacle to precise measurements—mode misidentification—by using laser-Doppler interferometry to detect the displacement distribution on a vibrating surface. This approach yields unambiguous mode identification. We used 56 resonances ranging in frequency from 0.3 to 1.2 MHz and determined the Cij and eij with known κij. The ten unknowns always converged to the same values even with unreasonable initial guesses. The Cij uncertainty averages 0.09% for the diagonal Cij. The eij uncertainty averages 5%. All our coefficients fall within the (surprisingly wide) error limits of previous (conventional) measurements.

Anisotropic elastic-stiffness coefficients of an amorphous Ni–P film

This study presents a complete set of effective elastic-stiffness coefficients of a Ni80P20 amorphous-alloy thin film deposited on an aluminum-alloy substrate by electroless plating. The film thickness was 12 μm. The electromagnetic-acoustic-resonance method detected resonance frequencies of the triple-layered specimens (film/substrate/film), which enabled us to determine all five independent elastic-stiffness coefficients of the film using known substrate elastic properties. The resulting coefficients were those of a transverse isotropic material. There was strong anisotropy between the in-plane and normal directions; the in-plane Young’s modulus is larger than the normal Young’s modulus by 34%, for example. The anisotropic coefficients can be interpreted by considering a micromechanics model for local incomplete cohesion (thin ellipsoidal voids) aligned parallel to the film surface.

Measurement of elastic-stiffness tensor of an anisotropic thin film by electromagnetic acoustic resonance

This paper presents a contactless methodology for determining all independent elastic-stiffness coefficients C ij of a transverse isotropic thin film: C 11, C 12, C 13, C 33, and C 44. The electromagnetic-acoustic-resonance technique measures the acoustic resonance frequencies of a film-coated specimen with a high accuracy, better than 10 −5, which enables determining the film C ij with the known substrate C ij . The measurement takes two steps. First, through-thickness resonance frequencies of longitudinal and shear modes are measured to determine C 33 and C 44, and the film thickness. Then, remaining three coefficients are deduced from measurements of the free-vibration resonance frequencies of the layer parallelepiped specimen. Simulations and experiments with monocrystal copper and titanium confirm the reliability of the resultant film C ij within 5%, when the film thickness is more than 0.5% of the substrate.

Elastic stiffnesses of an Nb–Ti/Cu-composite superconductive wire

Elastic-stiffness coefficients were determined on a 1.4-mm-diameter wire consisting of superconducting Nb–Ti fibers in a copper matrix, with a polyvinyl-resin coating. The matrix contained 324 Nb–Ti fibers. An electromagnetic-acoustic-resonance method was used to obtain five independent elastic-stiffness coefficients assuming transverse-isotropic symmetry. From these we calculated Young moduli, bulk modulus, and principal Poisson ratios. As a check, we used a mechanical-impulse-excitation method to directly measure the Young modulus in the fiber direction. The three-phase composite wire showed a 10% anisotropy in the Young modulus.

Ultrasonic determination of the anisotropy of Young's modulus of fixed tendon and fixed myocardium.

The linear elastic properties of a soft tissue exhibiting a unidirectional arrangement of reinforcing fibers may be described in terms of the five independent elastic stiffness coefficients C11, C13, C33, C44, and C66. In previous studies, ultrasonic measurements of these coefficients for formalin fixed specimens of bovine Achilles tendon and normal human myocardium were reported. In the present study these results are used to analyze the anisotropy of Young's modulus of these tissues. For formalin fixed tendon a value of 1.37 GPa is obtained for Young's modulus along the fiber axis of the tissue, and a value of 0.0706 GPa is obtained perpendicular to the fibers. For formalin fixed myocardium, values of 0.101 and 0.0311 GPa parallel and perpendicular to the fibers, respectively, are obtained. Based on the results for the angular dependence of Young's modulus from unidirectional specimens of myocardium, a model is introduced to estimate these features for the more complicated fiber architecture of the left ventricular wall.

A generalized method for characterizing elastic anisotropy in solid living tissues

The microstructure of cortical bone may exhibit either transverse isotropic or orthotropic symmetry, thus requiring either five or nine independent elastic stiffness coefficients (or compliances), respectively, to describe its elastic anisotropy. Our previous analysis to describe this anisotropy in terms of two scalar quantities for the transverse isotropic case is extended here to include orthotropic symmetry. The new results for orthotropic symmetry are compared with previous calculations using the transverse isotropic analysis on the same sets of anisotropic elastic constants for bone, determined either by mechanical or by ultrasonic experiments. In addition, the orthotropic calculation has been applied to full sets of orthotropic elastic stiffness coefficients of a large variety of wood species. Although having some resemblance to plexiform bone in microstructural organization, there is a dramatic difference in both the shear and the compressive elastic anisotropy between the two materials: wood is at least one order of magnitude more anisotropic than bone.