Fourth-order Elastic Constants Research Articles

Elastic properties of TiC and TiN crystals in high temperature range

The elastic properties of crystals at high temperature are of significant interest as they affect the mechanical stability and performance of materials at elevated temperatures. In present study a theory for evaluating the temperature variation of second third and fourth order elastic constants for face centred cubic crystal structure solids is used on the basis of Coulomb and Born-Mayer potential using nearest-neighbour distance and hardness parameter. The theory is first applied to get the second order elastic constants (SOECs) and third order elastic contants (TOECs) for TiC and TiN crystals at different temperatures up to 500 K. In addition, the elastic constants thus obtained are used to evaluate the first order pressure derivatives (FOPDs) SOECs and TOECs. The elastic properties of these crystals at high temperature are of significant interest and are related to other important material properties such as thermal expansion, thermal conductivity, phase transformations, creep behaviour, oxidation resistance, and high-temperature strength.The results obtained show that the second order elastic constants and third order elastic constants of TiC are highest, so mechanical properties of TiC will be better than TiN, which makes it stiffer and more resistant to deformation under an applied load. These properties make TiC a popular material for cutting tools, wear-resistant coatings, and other high-performance applications.

Structural parameters of twist-bend nematics and splay-bend nematics in Dozov's theory.

This paper presents the results of numerical calculations revealing how the structural parameters (i.e., the pitch p_{TB}, the spatial period p_{SB}, and the tilt angle θ_{TB} or θ_{SB}) of twist-bend nematics (N_{TB}) and splay-bend nematics (N_{SB}) depend on the values of elastic constants in Dozov's theory [I. Dozov, Europhys. Lett. 56, 247 (2001)10.1209/epl/i2001-00513-x]. Alternative formulas for p_{TB}, θ_{TB}, p_{SB}, and θ_{SB} have been derived and it has been proved that they give more accurate results than the expressions proposed by Dozov. Although the determination of the fourth-order elastic constants C_{1}, C_{2}, and C_{3} is not feasible in a simple way, the order of magnitude of the sum C_{1}+C_{2} has been easily estimated and is equal to 10^{-31}Jm. Moreover, the numerical calculations have shown that twist-bend nematics can exist even when K_{11} is smaller than 2K_{22} and thus Dozov's criterion K_{11}>2K_{22} for the stability of the N_{TB} phase is not strictly satisfied.

Systematic investigations on frequency mixing response of ultrasonic shear horizontal and Rayleigh Lamb waves

The study of self- and cross-interactions of shear horizontal (SH) waves has garnered significant attention in the field of nonlinear ultrasonic practice. This interest stems from the potential high sensitivity of SH waves to microdamage and their low out-of-plane attenuation characteristics. However, due to the dispersive nature and intricate mechanisms governing guided wave propagations and interactions, the nonlinear responses of SH waves, particularly the third-order harmonic generation and wave mixing with Rayleigh Lamb (RL) waves, remain largely unexplored. This study presents a comprehensive investigation into the ultrasonic harmonic wave generation of SH guided waves. The research approach combines theoretical considerations and finite element modeling to investigate and discuss guided wave mixing involving SH and RL waves. Six cases of phase-matched wave mode pairs are developed based on deduced synchronism and symmetry conditions. A nonlinear material constitutive model, encompassing both the third- and fourth-order elastic constants, is incorporated into the simulation. The results of the numerical analysis reveal the existence of newly generated higher and combined harmonic waves in an intuitive manner. Specifically, the interactions between ultrasonic SH and RL guided waves are observed to generate second- and third-order combined harmonic waves at the mixing frequencies under internal resonant conditions. The nonlinear waves have a cumulative effect during the mixing period, with their amplitude remaining constant after passing through the mixing zone. This study provides the first observation of mode conversion and third-order combined harmonic generation of SH waves. The insights gained from the numerical perspective offer understandings of the frequency mixing response of SH waves.

A first-principles method to calculate fourth-order elastic constants of solid materials

A first-principles method is presented to calculate elastic constants up to the fourth order of crystals with the cubic and hexagonal symmetries. The method relies on the numerical differentiation of the second Piola-Kirchhoff stress tensor and a density functional theory approach to calculate the Cauchy stress tensor for a list of deformed configurations of a reference state. The number of strained configurations required to calculate the independent elastic constants of the second, third, and fourth order is 24 and 37 for crystals with the cubic and hexagonal symmetries, respectively. Here, we present conceptual aspects of our method, we provide technical details of its implementation and use, we assess its accuracy, and we discuss several applications. In particular, this method is applied to five crystalline materials with the cubic symmetry (diamond, silicon, aluminum, silver, and gold) and two metals with the hexagonal close packing structure (beryllium and magnesium). Our results are compared to available experimental data and previous computational studies. Calculated linear and nonlinear elastic constants are also used in the context of nonlinear elasticity theory to predict values of volume and bulk modulus over an interval of pressures. These predictions are compared to results obtained from density functional theory calculations to assess the reliability of our method.

Nonlinear Finite Element Calculations of Layered SAW Resonators.

In this work the nonlinear behavior of layered SAW resonators is studied with the help of Finite Element (FE) computations. The full calculations depend strongly on the availability of accurate tensor data. While there are accurate material data for linear computations, the complete sets of higher-order material constants, needed for nonlinear simulations, are still not available for relevant materials. To overcome this problem, scaling factors were used for each available nonlinear tensor. The approach here considers piezoelectricity, dielectricity, eletrostriction and elasticity constants up to fourth order. These factors act as a phenomenological estimate for incomplete tensor data. Since no set of fourth order material constants for LiTaO3 is available, an isotropic approximation for the fourth order elastic constants was applied. As a result, it was found that the fourth order elastic tensor is dominated by one fourth order Lamé constant. With the help of the FE model, derived in two different, but equivalent ways, we investigate the nonlinear behavior of a SAW resonator with a layered material stack. The focus was set to third order nonlinearity. Accordingly, the modeling approach is validated using measurements of third order effects in test resonators. In addition, the acoustic field distribution is analyzed.

Динамика решетки и нелинейный отклик ОЦК-титана на деформации при высокой температуре в методе ab initio молекулярной динамики

The lattice dynamics and the response to deformations of the bcc titanium lattice at high temperatures are studied using the method of ab initio molecular dynamics. The results of calculation of phonon spectra using the harmonic Hamiltonian and the Hamiltonian containing third-order force constants are compared. It is shown that the lattice dynamics of bcc titanium cannot be correctly described without taking into account the anharmonic terms. Stress-strain curves are calculated for a given deformation scheme η1= η11= η, η6=2η12= η. For this deformation scheme, analytical expressions are given for the dependence of the nonzero components of the stress tensor on the magnitude of the deformation, taking into account the elastic constants up to the fourth order inclusive. All second-order elastic constants, three third-order elastic constants, and four fourth-order elastic constants of bcc titanium are calculated.

Cubic nonlinearity parameter measurement and material degradation detection using nonlinear ultrasonic three-wave mixing

Early detection of accumulated damage or material degradation in structures is important to ensure their structural safety. Nonlinear ultrasonic techniques are widely used to measure the quadratic nonlinearity parameter that represents the third-order elastic constants of materials for material degradation detection. In addition, there are ongoing efforts to exploit both the third- and fourth-order elastic constants that describe the cubic nonlinearity parameter to detect material degradation. This study develops a nonlinear ultrasonic three-wave mixing technique to generate and measure third-order combined harmonics (TOCH) in plate-like structures and to measure cubic nonlinearity parameter. The proposed three-wave mixing technique generates three primary Lamb waves in the target structure and measures the TOCH produced by nonlinear cross-interaction of the primary Lamb waves. A theoretical model is developed to describe the generation of TOCH in a nonlinear elastic and homogeneous plate, and the effectiveness of the theoretical model is validated experimentally. Measurements of the TOCH are conducted for intact and degraded aluminum specimens with different degradation levels. Because inherent material nonlinearity and material degradation alter the third- and fourth-order elastic constants of a structure, the three-wave mixing technique for measuring the TOCH can be used to identify the inherent material nonlinearity and material degradation. In particular, the experimental results indicate that the proposed technique is more sensitive to early-stage material degradation than existing nonlinear ultrasonic techniques such as two-wave mixing, and third harmonic generation techniques.

On three-stage temperature dependence of elastic wave velocities for rocks

Abstract For most rocks, the typical temperature behavior of elastic wave velocities generally features a three-stage nonlinear characteristic that could be expressed by a reverse S-shape curve with two inflexion points. The mechanism regulating the slow-to-fast transition of elastic constants remains elusive. The physics of critical points seems related to the multimineral composition of rocks with differentiated thermodynamic properties. Based on laboratory experiments for several rocks with different levels of heterogeneity in compositions, we conduct theoretical and empirical simulations by nonlinear thermoelasticity methods and a S-shape model, respectively. The classical theory of linear thermoelasticity based on the Taylor expansion of strain energy functions has been widely used for crystals, but suffers from a deficiency in describing thermal-associated velocity variations for rocks as a polycrystal mixture. Current nonlinear thermoelasticity theory describes the third-order temperature dependence of velocity variations by incorporating the fourth-order elastic constants. It improves the description of temperature-induced three-stage velocity variations in rocks, but involves with some divergences around two inflexion points, especially at high temperatures. The S-shape model for empirical simulations demonstrates a more accurate depiction of thermal-associated three-stage variations of P-wave velocities. We investigate the physics of the parameters ${a_1}$ and ${b_1}$ in the S-shape model. These fitting parameters are closely related to thermophysical properties by being proportional to the specific heat and thermal conductivity of rocks. We discuss the mechanism that regulates the slow-to-fast transition in the three-stage nonlinear behavior for various rocks.

Anharmonic Properties of Oxides Having Face Centered Cubic Crystal Structure of Fourth Order

A theory has been evolved for obtaining anharmonic properties of oxides having face centered cubic crystal symmetry using long-range Coulomb and short-range Born-Mayer potentials starting from the nearest-neighbour distance and hardness parameter. The fourth order elastic constants are obtained for oxides of Cd, Co, Fe, Nb, Ni, Te, Ti and Zr at elevated temperature. The room temperature data are utilized for obtaining the first order pressure derivatives of second and third order elastic constants, the second order pressure derivatives of second order elastic constants and partial contractions of these oxides. The results are presented and discussed.

First-principles calculation of higher-order elastic constants using exact deformation-gradient tensors

We propose a general and easy-to-use method of the ab initio calculation of the higher-order elastic constants, which is based on the analytical formulas for the deformation-gradient tensors as functions of the Lagrangian strain. The method allows for elimination of the truncation errors in the Taylor expansion series of the deformation gradients and is particularly useful to calculate the fourth-order elastic constants, where large strains have to be applied. It also facilitates the calculation of the Lagrangian stress, which is helpful in determination of the strain-stress relations. To demonstrate the application of our approach, we derive the analytic formulas for the deformation gradients as functions of the Lagrangian strain tensors, which are used in calculations of the third-order elastic constants in trigonal crystals and the fourth-order elastic constants in cubic crystals. Then, we perform the ab initio calculations of the fourth-order elastic constants in face-centered-cubic aluminum. We discuss the results obtained using the strain-energy and strain-stress relations and analyze the errors of the fourth-order elastic constants which would be incurred when approximating the deformation gradients by the Taylor polynomials. We show that the relatively small truncation errors in the Taylor expansion series of the deformation gradients can cause significant deviations of the fourth-order elastic constants. This effect is larger for the strain-energy method than for the strain-stress approach. We find that in both methods, the deviations are particularly significant for the ${C}_{1155}$, ${C}_{1266}$, ${C}_{4455}$, ${C}_{1255}$, and ${C}_{1456}$ elastic constants and are mainly caused by the truncation errors in the nondiagonal elements of the Taylor expansion series of the deformation gradients.

Hyperelastic Ex Vivo Cervical Tissue Mechanical Characterization.

This paper presents the results of the comparison between a proposed Fourth Order Elastic Constants (FOECs) nonlinear model defined in the sense of Landau’s theory, and the two most contrasted hyperelastic models in the literature, Mooney–Rivlin, and Ogden models. A mechanical testing protocol is developed to investigate the large-strain response of ex vivo cervical tissue samples in uniaxial tension in its two principal anatomical locations, the epithelial and connective layers. The final aim of this work is to compare the reconstructed shear modulus of the epithelial and connective layers of cervical tissue. According to the obtained results, the nonlinear parameter A from the proposed FOEC model could be an important biomarker in cervical tissue diagnosis. In addition, the calculated shear modulus depended on the anatomical location of the cervical tissue ( = 1.29 ± 0.15 MPa, and = 3.60 ± 0.63 MPa).

A First-Principles Study of Nonlinear Elastic Behavior and Anisotropic Electronic Properties of Two-Dimensional HfS2.

We utilize first principles calculations to investigate the mechanical properties and strain-dependent electronic band structure of the hexagonal phase of two dimensional (2D) HfS2. We apply three different deformation modes within −10% to 30% range of two uniaxial (D1, D2) and one biaxial (D3) strains along x, y, and x-y directions, respectively. The harmonic regions are identified in each deformation mode. The ultimate stress for D1, D2, and D3 deformations is obtained as 0.037, 0.038 and 0.044 (eV/Ang3), respectively. Additionally, the ultimate strain for D1, D2, and D3 deformation is obtained as 17.2, 17.51, and 21.17 (eV/Ang3), respectively. In the next step, we determine the second-, third-, and fourth-order elastic constants and the electronic properties of both unstrained and strained HfS2 monolayers are investigated. Our findings reveal that the unstrained HfS2 monolayer is a semiconductor with an indirect bandgap of 1.12 eV. We then tune the bandgap of HfS2 with strain engineering. Our findings reveal how to tune and control the electronic properties of HfS2 monolayer with strain engineering, and make it a potential candidate for a wide range of applications including photovoltaics, electronics and optoelectronics.

Higher order elastic tensors of crystal structure under non-linear deformation

The higher-order elastic tensors can be used to characterize the linear and non-linear mechanical properties of crystals at ultra-high pressures. Besides the widely studied second-order elastic constants, the third- and fourth-order elastic constants are sixth and eighth tensors, respectively. The independent tensor components of them are completely determined by the symmetry of the crystal. Using the relations between elastic constants and sound velocity in solid, the independent elastic constants can be measured experimentally. The anisotropy in elasticity of crystal structures is directly determined by the independent elastic constants.

On the nonlinear temperature dependence of elastic constants and wave velocities for solid media with applications to geologic materials.

Temperature-induced variations of elastic moduli in solid media are generally characterized by a strong nonlinear dependence on temperature associated with complex deformations under thermal treatments. Conventional thermoelasticity with third-order elastic constants for the one-order temperature dependence has been extensively studied for crystals, but encountering problems of divergent and limited velocity variations for rocks as a polycrystal mixture, especially at high temperatures. The extension of the theory beyond high-order elastic constants to solid media is addressed in this article to describe the nonlinear temperature dependence of both elastic constants and wave velocities. The total strain is divided into the background component associated with temperature variations and the infinitesimal component induced by propagating waves. A third-order temperature dependence of velocity variations is formulated by taking into account fourth-order elastic constants. Applications to solid rocks (sandstone, granite, and olivine) demonstrate an accurate description of temperature-induced variations, especially for high temperatures. Unlike crystals, the synthetic averaging elastic constants for a solid rock (as a polycrystal mixture) change less than 10% with temperatures. The thermal sensitivity of P-wave velocities is much more than that of S-wave velocities over the vast majority of temperatures examined.

Anharmonic Properties of Oxides Having Face Centered Cubic Crystal Structure

A theory has been evolved for obtaining anharmonic properties of oxides having face centered cubic crystal symmetry using long-range Coulomb and short-range Born-Mayer potentials starting from the nearest-neighbour distance and hardness parameter. The second ,third and fourth order elastic constants are obtained for oxides of Cd, Co, Fe, Nb, Ni, Te, Ti and Zr at elevated temperature. The room temperature data are utilized for obtaining the first order pressure derivatives of second and third order elastic constants, the second order pressure derivatives of second order elastic constants and partial contractions of these oxides. The results are presented and discussed.

Mixed third harmonic shear horizontal wave generation: interaction between primary shear horizontal wave and second harmonic Lamb wave

The third harmonic shear horizontal (SH) waves in a weakly nonlinear plate provide new possibilities for incipient damage detections. The understanding of their generation mechanism, however, is limited to the intuitive process of cubic self-interaction of the primary SH waves. By considering both the third order and fourth order elastic constants, this paper reports the discovery of a new generation mechanism, referred to as mixed generation, which results from the mutual interaction between the primary SH waves and their induced second harmonic Lamb waves. Compared with linearly cumulative third harmonic SH waves induced by the cubic self-interaction of the primary SH waves, the mixed third harmonic SH wave amplitude also increases with the wave propagating distance but in a wavering manner according to the phase velocity matching condition. Upon establishing a complex-domain superposition method which allows a precise extraction of the third harmonic responses, the significance and the propagating characteristics of the mixed third harmonic SH waves are numerically investigated through finite element simulations. Experiments are then conducted with a dedicated subtraction scheme to highlight the material nonlinearity of interest. A gel test is designed and carried out to identify the two types of third harmonic SH wave components from the measured time-domain signals. Both numerical and experimental results confirm the existence and the significance of the mixed third harmonic SH generation mechanism, which may impact on further applications, especially underwater damage inspections.

Nonlinear elastic behavior and anisotropic electronic properties of two-dimensional borophene

Using first-principles density functional theory calculations, the mechanical and electronic properties of the three main (β12, χ3, and striped) phases of single-layer borophene sheets are calculated under in-plane uniaxial/biaxial strain, including the harmonic strain-energy regions of β12, χ3, and striped phases over the strain ranges of −3.5%–3.5%, −4.5%–4.5%, and −2.5%–2.5%, respectively, along the x direction (the direction of the highest bond orientation). We introduce a method by which the nonlinear behavior of these and any other two-dimensional materials can be investigated even above their ultimate strains, beyond which no-uniform plastic deformation occurs. Defining an appropriate deformation, and utilizing both continuum modeling and special equations based on the density functional theory, a method of computing second-, third-, and fourth-order elastic constants of the three different phases of borophene is presented that utilizes rectangular unit cells, which can substitute for any two-dimensional unit cell. Using this new method, 4 independent second-order, 6 third-order, and 9 fourth-order elastic constants are calculated, which is the complete set of elastic constants for two-dimensional structures. The electronic band structure of borophene shows anisotropic electronic behavior. Despite the metallic character of borophene sheets, applying directional strain based on deformation matrices creates a bandgap in some regions of the Brillouin zones, opening up the possibility of mechanical control of electronic properties.

Large acoustoelastic effect for Lamb waves propagating in an incompressible elastic plate.

In this paper, the effect of a large pre-stress on the propagation of small amplitude Lamb waves in an incompressible elastic plate is investigated. Using the theory of incremental elasticity, the dispersion equations, which give the phase velocity of the symmetric and anti-symmetric wave modes as a function of the wavenumber, plate thickness, and pre-stress state, are derived for a general strain energy function. By considering the fourth-order strain energy function of incompressible isotropic elasticity, the correction to the phase velocity due to the pre-stress is obtained implicitly to the second order in the pre-strain/stress, and depends on the second, third, and fourth-order elastic constants. Numerical results are presented to show the dependence of the phase velocity of the Lamb wave modes upon the applied stress. These are compared to the first-order correction, and agree well with the limiting and asymptotic values obtained previously. It is envisaged that the present results may well find important practical applications in various guided wave based ultrasonic techniques utilising gels and rubber-like materials.

Mechanical and strain-tunable electronic properties of the SnS monolayer

By means of first-principles calculations, the mechanical properties and the strain-dependent electronic band structure of the orthorhombic SnS monolayer were investigated. In an attempt to investigate the elasticity of this material, six deformation modes were considered. The stability of this configuration against these external tensions was assessed, and the second-order elastic constants were found to be C11 = 45.2 N/m, C22 = 25.3 N/m, C12 = 18.0 N/m, and C66 = 54.3 N/m. The third- and fourth-order elastic constants, which shed light on the material behavior at strains above the harmonic region, were also determined. The stress-strain relationships imply that the SnS monolayer can withstand tensile strains up to 0.075, 0.29, and 0.12 for uniaxial strain along the a direction, uniaxial strain along the b direction, and biaxial strain, respectively. Moreover, a study of the electronic band structure of the SnS monolayer shows that this indirect band gap semiconductor when exposed to a marginal uniaxial or biaxial strain (about 2%) experiences a transition to a direct band gap semiconductor with a moderate band gap (<2.7 eV). This transition merely occurs when tensile strain is exerted. Our findings suggest that under compressive strain, this semiconducting material maintains its indirect band gap nature, and the band gap predominately declines. Lastly, the band splitting that arises from spin-orbit interactions in most cases vanishes when strain is applied. However, compressive strain along the a direction and tensile strain along the b direction are two exceptions, in which the former strengthens the spin-orbit effects and for the latter, band splitting remains almost unchanged.

Effective properties of flexoelectric fiber-reinforced nanocomposite

The effective flexoelectric coefficients of a multifunctional flexoelectric fiber reinforced composite (FLFRC) have been estimated using an analytical micromechanics based model. The analytical estimation carried out here has been validated with the effective piezoelectric and the fourth-order elastic constants of the composite available in literature. The effective flexoelectric coefficients of the composite show a size-dependent effect, not seen in similar calculations for piezoelectric composites. Also, above a critical fiber volume fraction with a suitable choice of matrix, the effective flexoelectric properties of the FLFRC can be tailored to be significantly superior to those of the monolithic dielectric solid. It is noted that, the FLFRC can be customized to have negative axial flexoelectric coefficients. This study opens new avenues for further research on developing smart nanocomposites characterized with enhanced functional properties.