Analytical and experimental analysis of guided waves in an aluminum plate under bending load

Although guided waves offer great potential for monitoring various structures, interpreting signals from piezoelectric sensors remains a challenging task. One main reason is the significant influence of environmental conditions on the wave propagation. A lot of research has already been done on the influence of temperature effects and recently more attention has been shifted towards loads. While previous publications have mainly focused on uni- or bi-directional loads, this publication expands the developed models to include bending loads. After reviewing the analytical basis of acoustoelasticity, the derived equations are expanded to nonhomogeneous elastic bending loads using the partial wave method. The analysis is completed using recent results developed by C. Hakoda and C. J. Lissenden (2018) [1], that gave more physical insight in the propagation of guided waves in various frequency-bands. The focus of the experimental analysis is around the fundamental S0- and A0-Modes of Lamb waves. To validate the analytical results an aluminum plate is instrumented using piezoelectric transducers and loaded with varying bending loads. The experimental results are in good agreement with the analytical theory and demonstrate the influence of bending prestress on guided wave propagation. Based on these results an innovative measurement method for bending loads is developed, that is robust to small temperature changes.

Monitoring of thermal stress in metal plates by using bonded shear horizontal wave piezoelectric transducers

Thermal stress is one of the major causes of failure of engineering structures and its measurement has attracted more attention in recent years. The ultrasonic wave method is very promising in stress measurement due to its non-destructive nature and easy manipulation. The traditional ultrasonic wave transducers require a coupling medium which would introduce large repeatability errors in travel time measurement and thus in the measured stress. In this work, a methodology based on bonded shear horizontal (SH) guided wave piezoelectric transducers was developed to monitor thermal stress in metal plates. The adhesive bonding between the transducer and the specimen ensures the repeatability in travel time measurements, and the strain gauges are also employed to monitor the wave path length. The dispersive equation of acoustoelastic SH wave propagating in an isotropic medium under the uniaxial stress is derived. Both the uniaxial tension test and thermal modulation test have been performed in aluminum and steel plates. The results show that the acoustoelastic constants of the SH₀ wave are identical to that of the shear bulk wave as predicted by the acoustoelastic theory and the thermal stress measured from -60 °C to 100 °C by the proposed method has a very good repeatability (better than 2 MPa) in both the aluminum and steel plates. Considering the convenience and reliability of the bonded SH₀ wave piezoelectric transducer, the proposed method is very promising for monitoring of thermal stress in engineering structures, such as rails, etc.

A unifying model of weakly nonlinear elastic waves; large on large theory

This article reconsiders traditional topics in nonlinear elastic waves and nonlinear ultrasonics. Herein, higher-order coupling between finite initial deformation and finite amplitude waves are considered. To allow for coupling, a large-on-large deformation model is developed and used to generate the equations of motion relative to the deformed and undeformed material configurations. Thus, the equations of motion provide a single setting to describe topics in nonlinear elastic waves such as acoustoelasticity, second harmonic generation, and coupling relations between these topics. The model is evaluated to recover the traditional linearized acoustoelastic relations and predicted second harmonic amplitudes. Then, the so-called large acoustoelasticity theory is developed for anisotropic materials with specific results given for isotropic materials. Last, the stress influence on second harmonic generation is presented.

Nonclassical nonlinear elasticity of crystalline structures

Hysteretic elastic nonlinearity has been shown to result in a dynamic nonlinear response which deviates from the known classical nonlinear response; hence this phenomenon was termed nonclassical nonlinearity. Metallic structures, which typically exhibit weak nonlinearity, are typically categorized as classical nonlinear materials. This article presents a material model which derives stress amplitude dependent nonlinearity and damping from the mesoscale dislocation pinning and breakaway to show that the lattice defects in crystalline structures can give rise to nonclassical nonlinearity. The dynamic nonlinearity arising from dislocations was evaluated using resonant frequency shift and higher order harmonic scaling. The results show that the model can capture the nonlinear dynamic response across the three stress ranges: linear, classical nonlinear, and nonclassical nonlinear. Additionally, the model also predicts that the amplitude dependent damping can introduce a softening-hardening nonlinear response. The present model can be generalized to accommodate a wide range of lattice defects to further explain nonclassical nonlinearity of crystalline structures.

Influence of residual stress and texture on the resonances of polycrystalline metals

Efficient nondestructive qualification of additively manufactured (AM) metallic parts is vital for the current and future adoption of AM parts throughout several industries. Resonant ultrasound spectroscopy (RUS) is a promising method for the qualification and characterization of AM parts. Although the adoption of RUS in this setting is emerging, the influence of residual stress and texture, which are both very common in AM parts, is not well understood. In this article, a stress- and texture-dependent constitutive relation is used to study the influence on free vibrational behavior in a RUS setting. The results that follow from using the Rayleigh-Ritz method and finite element analysis suggest that residual stress and texture have a significant impact on the resonance frequencies and mode shapes. These results support the potential of using RUS to sense texture and residual stress in AM parts. Additionally, these results suggest that RUS measurements could be misinterpreted when the stress and texture are not accounted for, which could lead to a false positive/negative diagnosis when qualifying AM parts.

Tunable Negative Poisson's Ratio in Van der Waals Superlattice

Negative Poisson's ratio (NPR) materials are functional and mechanical metamaterials that shrink (expand) longitudinally after being compressed (stretched) laterally. By using first-principles calculations, we found that Poisson's ratio can be tuned from near zero to negative by different stacking modes in van der Waals (vdW) graphene/hexagonal boron nitride (G/h-BN) superlattice. We attribute the NPR effect to the interaction of p_z orbitals between the interfacial layers. Furthermore, a parameter calculated by analyzing the electronic band structure, namely, distance-dependent hopping integral, is used to describe the intensity of this interaction. We believe that this mechanism is not only applicable to G/h-BN superlattice but can also explain and predict the NPR effect in other vdW layered superlattices. Therefore, the NPR phenomenon, which was relatively rare in 3D and 2D materials, can be realized in the vdW superlattices by different stacking orders. The combinations of tunable NPRs with the excellent electrical/optical properties of 2D vdW superlattices will pave a novel avenue to a wide range of multifunctional applications.

A reactive molecular dynamics study on the mechanical properties of a recently synthesized amorphous carbon monolayer converted into a nanotube/nanoscroll

Recently, laser-assisted chemical vapor deposition has been used to synthesize a free-standing, continuous, and stable monolayer amorphous carbon (MAC). MAC is a pure carbon structure composed of randomly distributed five, six, seven, and eight atom rings, which is different from that of disordered graphene. More recently, amorphous MAC-based nanotubes (a-CNT) and nanoscrolls (a-CNS) were proposed. In this work, we have investigated (through fully atomistic reactive molecular dynamics simulations) the mechanical properties and sublimation points of pristine and a-CNT and a-CNS. The results showed that a-CNT and a-CNS have distinct elastic properties and fracture patterns compared to those of their pristine analogs. Both a-CNT and a-CNS presented a non-elastic regime before their total rupture, whereas the CNT and CNS underwent a direct conversion to fractured forms after a critical strain threshold. The critical strain values for the fracture of the a-CNT and a-CNS are about 30% and 25%, respectively, and they are lower than those of the corresponding CNT and CNS cases. Although less resilient to tension, the amorphous tubular structures have similar thermal stability in relation to the pristine cases with sublimation points of 5500 K, 6300 K, 5100 K, and 5900 K for a-CNT, CNT, a-CNS, and CNS, respectively. An interesting result is that the nanostructure behavior is substantially different depending on whether it is a nanotube or a nanoscroll, thus indicating that the topology plays an important role in defining its elastic properties.

Beyond linearity: bent crystalline copper nanowires in the small-to-moderate regime

Several models can describe the nonlinear response of 1D objects to bending under a concentrated load. Successive stages consisting of geometrical and, additionally, mechanical non-linearity can be identified in moderately large extensions. We provide an explicit bending moment function with terms accounting for the linearity (Euler-Bernoulli), quasi-linearity, geometrical and finally, mechanical non-linearity as global features of a moderately large elastic deformation. We apply our method, also suitable for other metals, to the experimental data of Cu nanowires (NWs) with an aspect ratio of about 16 under different concentrated loadings. The spatial distribution of strain-hardening/softening along the wire or through the cross-section is also demonstrated. As a constitutive parameter, the strain-dependent stretch modulus represents, undoubtedly, changes in the material properties as the deformation progresses. At the highest load, the Green-Lagrange strain reaches a 12.5% extension with a corresponding ultra-high strength of about 7.45 GPa at the most strained volume still in the elastic regime. The determined stretch modulus indicates a significantly lower elastic response with an approximated Young's modulus (E ≅ 65 GPa) and a third-order elastic constant, C ₁₁₁ ≅ -350 GPa. Surprisingly, these constants suggest a 25-35% of that of the bulk counterparts. Ultimately, the method not only provides a quantitative description of the bent Cu NWs, but also indicates the robustness of the theory of nonlinear elasticity.

Pressure influence on elastic wave attenuation in polycrystalline materials

Traditionally, the acoustoelastic effect refers to the influence of stress in a solid on an elastic wave's phase velocity. Since the phase velocity can be represented by the real part of the complex wave number, a natural question arises regarding the effect of stress on the imaginary part or dissipation of the wave. In this article, the influence of pressure on the elastic wave's attenuation in polycrystalline materials is modeled. The constitutive behavior of an initially stressed solid is coupled into Weaver's scattering-based attenuation model [J. Mech. Phys. Solids 38, 55-86 (1990)]. As a result, the pressure-dependent longitudinal and shear wave attenuation coefficients are unveiled. As the traditional stress-free attenuation coefficients depend on the degree of single-crystal elastic anisotropy, it is shown that the pressure influence on attenuation depends on the anisotropy of the single-crystal's third-order or nonlinear elastic constants. Analysis of the model indicates linkages between pressure derivatives of velocity and attenuation to the material's linear and nonlinear elastic anisotropy, crystal structure, and type of atomic bonding.

Design of a new optical system to generate narrowband guided waves with an application for evaluating the health status of rail material

The Letter presents a new design of Sagnac interferometer-based optical system (SIOS) that emits a line-arrayed pattern generating narrowband high-energy waves in a specimen. The SIOS is further used to excite Rayleigh waves in a pristine rail specimen to evaluate its intrinsic nonlinearity resulting from the lattice anharmonicity and dissolved impurities. Such a nonlinearity appears in the response in the form of a second harmonic that is sensed in this Letter using a scanning laser Doppler vibrometer. In addition to this noncontact measurement, a contact measurement of the nonlinearity of rail steel using wedge transducers is also carried out to compare the performance of the SIOS. Both experimentally evaluated nonlinearities are compared with those obtained using the nonlinear elasticity equations. The close agreement with the theoretically estimated nonlinearity and higher repeatability shows that the SIOS is effective in measuring the intrinsic nonlinearity of the rail steel and, thereby, predicting the health status of the rail specimens before fixing them on the track.

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.

Multipole borehole acoustic field in a transversely isotropic medium induced by stress

A borehole multipole acoustic field in a pre-stressed formation is investigated. The pre-stressed formation is modeled as a transversely isotropic medium induced by uniaxial stress. The formation is assumed to be isotropic in absence of any static stress and then becomes anisotropic due to the applied stress parallel to the borehole axis. The approximate equivalent elastic constants of the stress-induced anisotropic medium are derived from the theory of acoustoelasticity. The nonlinear static stress-strain relation is used for both small and large static deformations. This problem can be solved analytically because of uniformity of deformation induced by static stress applied parallel to the borehole axis. The stress effects on the velocity of guided waves and amplitude of waveforms excited by monopole, dipole, and quadrupole sources are investigated. Numerical results show that the velocities of guided waves increase with uniaxial stress. The uniaxial stress affects both amplitude and arrival time of the acoustic waves in the borehole. The integral amplitude of full waveforms varies almost in a parabolic manner with the increasing stress level and thus shows sensitivity to the uniaxial stress. This result may be helpful for remote stress measurements in boreholes.

An Ab Initio Study of Pressure-Induced Reversal of Elastically Stiff and Soft Directions in YN and ScN and Its Effect in Nanocomposites Containing These Nitrides

Using quantum-mechanical calculations of second- and third-order elastic constants for YN and ScN with the rock-salt (B1) structure, we predict that these materials change the fundamental type of their elastic anisotropy by rather moderate hydrostatic pressures of a few GPa. In particular, YN with its zero-pressure elastic anisotropy characterized by the Zener anisotropy ratio A Z = 2 C 44 / ( C 11 - C 12 ) = 1.046 becomes elastically isotropic at the hydrostatic pressure of 1.2 GPa. The lowest values of the Young's modulus (so-called soft directions) change from 〈100〉 (in the zero-pressure state) to the 〈111〉 directions (for pressures above 1.2 GPa). It means that the crystallographic orientations of stiffest (also called hard) elastic response and those of the softest one are reversed when comparing the zero-pressure state with that for pressures above the critical level. Qualitatively, the same type of reversal is predicted for ScN with the zero-pressure value of the Zener anisotropy factor A Z = 1.117 and the critical pressure of about 6.5 GPa. Our predictions are based on both second-order and third-order elastic constants determined for the zero-pressure state but the anisotropy change is then verified by explicit calculations of the second-order elastic constants for compressed states. Both materials are semiconductors in the whole range of studied pressures. Our phonon calculations further reveal that the change in the type of the elastic anisotropy has only a minor impact on the vibrational properties. Our simulations of biaxially strained states of YN demonstrate that a similar change in the elastic anisotropy can be achieved also under stress conditions appearing, for example, in coherently co-existing nanocomposites such as superlattices. Finally, after selecting ScN and PdN (both in B1 rock-salt structure) as a pair of suitable candidate materials for such a superlattice (due to the similarity of their lattice parameters), our calculations of such a coherent nanocomposite results again in a reversed elastic anisotropy (compared with the zero-pressure state of ScN).

First-Principles Calculation of Third-Order Elastic Constants via Numerical Differentiation of the Second Piola-Kirchhoff Stress Tensor

A general method is presented to calculate from first principles the full set of third-order elastic constants of a material of arbitrary symmetry. The method here illustrated relies on a plane-wave density functional theory scheme to calculate the Cauchy stress and the numerical differentiation of the second Piola-Kirchhoff stress tensor to evaluate the elastic constants. It is shown that finite difference formulas lead to a cancellation of the finite basis set errors, whereas simple solutions are proposed to eliminate numerical errors arising from the use of Fourier interpolation techniques. Applications to diamond, silicon, aluminum, magnesium, graphene, and a graphane conformer give results in excellent agreement with both experiments and previous calculations based on fitting energy density curves, demonstrating both the accuracy and generality of our new methodology to investigate nonlinear elastic behaviors of materials.

Variation of second-order piezoelectric coefficients with respect to a finite strain measure

In this article the consequence of a change of finite strain measure is theoretically considered for nonlinear piezoelectric crystals. Analytical predictions show that second-order piezoelectric coefficients are finite strain measure dependent. Therefore, the use of any finite strain measure in constitutive modelling of piezoelectric materials requires an adequate choice of higher-order piezoelectric coefficients. This allows one to avoid unwanted corrections to the elastic and electric fields in the case of nonlinear modelling of piezoelectric materials, e.g. for piezoelectric heterostructures such as quantum wells or dots. A general transformation formula for second-order piezoelectric coefficients (elastostriction) is derived. As an example, specific transformation formulae for two common crystallographic classes, namely {\bar 4}3m and 6mm, are presented. The piezoelectric coefficients for GaN and GaAs crystals, as representative cases of these crystal classes, are recalculated and their dependence on the strain measure is demonstrated. A further implication of that effect is that a complete set of second-order piezoelectric coefficients should contain additional information about the strain measure applied during calculations or measurements.

Magnetic resonance elastography in nonlinear viscoelastic materials under load

Characterisation of soft tissue mechanical properties is a topic of increasing interest in translational and clinical research. Magnetic resonance elastography (MRE) has been used in this context to assess the mechanical properties of tissues in vivo noninvasively. Typically, these analyses rely on linear viscoelastic wave equations to assess material properties from measured wave dynamics. However, deformations that occur in some tissues (e.g. liver during respiration, heart during the cardiac cycle, or external compression during a breast exam) can yield loading bias, complicating the interpretation of tissue stiffness from MRE measurements. In this paper, it is shown how combined knowledge of a material's rheology and loading state can be used to eliminate loading bias and enable interpretation of intrinsic (unloaded) stiffness properties. Equations are derived utilising perturbation theory and Cauchy's equations of motion to demonstrate the impact of loading state on periodic steady-state wave behaviour in nonlinear viscoelastic materials. These equations demonstrate how loading bias yields apparent material stiffening, softening and anisotropy. MRE sensitivity to deformation is demonstrated in an experimental phantom, showing a loading bias of up to twofold. From an unbiased stiffness of [Formula: see text] Pa in unloaded state, the biased stiffness increases to 9767.5 [Formula: see text]1949.9 Pa under a load of [Formula: see text] 34% uniaxial compression. Integrating knowledge of phantom loading and rheology into a novel MRE reconstruction, it is shown that it is possible to characterise intrinsic material characteristics, eliminating the loading bias from MRE data. The framework introduced and demonstrated in phantoms illustrates a pathway that can be translated and applied to MRE in complex deforming tissues. This would contribute to a better assessment of material properties in soft tissues employing elastography.

Elastic Softness of Hybrid Lead Halide Perovskites

Much recent attention has been devoted towards unraveling the microscopic optoelectronic properties of hybrid organic-inorganic perovskites. Here we investigate by coherent inelastic neutron scattering spectroscopy and Brillouin light scattering, low frequency acoustic phonons in four different hybrid perovskite single crystals: MAPbBr_{3}, FAPbBr_{3}, MAPbI_{3}, and α-FAPbI_{3} (MA: methylammonium, FA: formamidinium). We report a complete set of elastic constants characterized by a very soft shear modulus C_{44}. Further, a tendency towards an incipient ferroelastic transition is observed in FAPbBr_{3}. We observe a systematic lower sound group velocity in the technologically important iodide-based compounds compared to the bromide-based ones. The findings suggest that low thermal conductivity and hot phonon bottleneck phenomena are expected to be enhanced by low elastic stiffness, particularly in the case of the ultrasoft α-FAPbI_{3}.

Higher order longitudinal guided wave modes in axially stressed seven-wire strands

This paper investigates the effect of axial stress on higher order longitudinal guided modes propagating in individual wires of seven-wire strands. Specifically, an acoustoelastic theory for a rod is used to predict the effect of stress on the phase velocity of guided modes in a strand. To this end, the exact acoustoelastic theory for an axially stressed rod is adapted for small deformations. Aside from the exact theory, approximate phase velocity changes (derived from both theory and experiment) are proposed, without the need to solve for dispersion curves. To validate the use of rod theories for strands, a custom-built prestressing bed was designed to apply axial load (up to 50% of yield) to a strand while conducting guided wave measurements. Higher order modes were excited in individual wires, and their phase velocity change under stress is compared to the exact acoustoelastic theory. Furthermore, it is shown that the proposed approximate phase velocity changes derived from theory and experiment only differ by roughly 2% from their exact counterparts. Higher order modes are shown to have stable stress dependence near their peak group velocity, which is beneficial for stress measurement. Additionally, linear stress dependence is observed, which is predicted by rod theories. Due to the unavailability of third order elastic constants for the steel strand, constants for a steel with similar Carbon content (0.6% C Hecla 17) were used as representative values in the theory. Using the Hecla 17 constants, roughly 15% mismatch in the slope of the linear stress dependence was observed when compared to the measurements on a steel strand.

The effect of applied stress on the phase and group velocity of guided waves in anisotropic plates

This paper presents an analytical formulation for the phase and group velocity of acoustoelastic guided waves in anisotropic plates. Uniform in-plane applied stress is considered, with both arbitrary propagation and stress directions. An expression for the energy velocity in a stressed anisotropic plate is derived, from which the group velocity is computed. Since the wavefront and group velocity directions generally differ, the deviation angle between the two is also studied. A method is proposed for verifying the consistency of the formulation, based on the correspondence between a direct and an indirect formulation. Analytical results are presented for a unidirectional fiber-reinforced graphite/epoxy composite plate. The plate is considered homogeneous for large wavelength to fiber diameter ratios. Results for the phase velocity, group velocity, and deviation angle are presented for two uniaxial applied loading cases. These are used to study the effect of stress for various propagation and stress directions. The linearity of the deviation angle with respect to stress is also demonstrated. Exact correspondence between the direct and indirect formulations is observed, which verifies consistency. The importance of accounting for shear strain in the indirect formulation is also demonstrated, which has not been noted in previous guided wave studies.

How to characterize a nonlinear elastic material? A review on nonlinear constitutive parameters in isotropic finite elasticity

The mechanical response of a homogeneous isotropic linearly elastic material can be fully characterized by two physical constants, the Young’s modulus and the Poisson’s ratio, which can be derived by simple tensile experiments. Any other linear elastic parameter can be obtained from these two constants. By contrast, the physical responses of nonlinear elastic materials are generally described by parameters which are scalar functions of the deformation, and their particular choice is not always clear. Here, we review in a unified theoretical framework several nonlinear constitutive parameters, including the stretch modulus, the shear modulus and the Poisson function, that are defined for homogeneous isotropic hyperelastic materials and are measurable under axial or shear experimental tests. These parameters represent changes in the material properties as the deformation progresses, and can be identified with their linear equivalent when the deformations are small. Universal relations between certain of these parameters are further established, and then used to quantify nonlinear elastic responses in several hyperelastic models for rubber, soft tissue and foams. The general parameters identified here can also be viewed as a flexible basis for coupling elastic responses in multi-scale processes, where an open challenge is the transfer of meaningful information between scales.

Poro-acoustoelastic constants based on Padé approximation

Stress-induced velocity variations for porous rocks are generally characterized by a strong nonlinear dependence on stress associated with complex deformations under loading. The classical theory of poro-acoustoelasticity with high-order elastic constants is based on the Taylor expansion of the strain energy function, encountering problems of divergence and limitless elastic wave velocities in describing stress-associated velocity variations, especially for high effective stresses. The extension of the theory beyond the high-order elastic constants based on the Padé approximation to the strain energy function is addressed in this article. The resultant acoustoelastic constants are characteristics of a reasonable theoretical limit in elastic wave velocities with increasing effective stresses, avoiding some of the problems associated with high-order elastic constants such as decreasing moduli with increasing effective pressure at high effective pressure, possibly implying the microstructural dependence of elastic constants. That is, the loading stress increases strain energy and wave velocity, but also induces frame-related attenuation, which in turn reduces stiffness and elastic constants. The Padé nonlinear constants can be reduced for low effective stresses to the conventional acoustoelastic constants based on the Taylor expansion. Theoretical results are compared with ultrasonic measurements for a perfectly elastic crystal, topaz (Al₂SiO₄F₂), and a porous rock, demonstrating that the Padé-approximation-based acoustoelasticity gives a more accurate description of stress-associated velocity variations, especially for higher effective stresses.

Ultrasonic harmonic generation from materials with up to cubic nonlinearity

This letter considers the combined effects of quadratic and cubic nonlinearity on plane wave propagation in generally anisotropic elastic solids. Displacement solutions are derived that represent the fundamental, second-, and third-harmonic waves. In arriving at the solutions, the quadratic and cubic nonlinearity parameters for generally anisotropic materials are defined. The effects of quadratic and cubic nonlinearity are shown to influence the amplitude and phase of the fundamental wave. In addition, the phase of the third-harmonic depends on a simple combination of the quadratic and cubic nonlinearity parameters. Nonlinearity parameters are given explicitly for materials having isotropic and cubic symmetry. Lastly, acoustic nonlinearity surfaces are introduced, which illustrate the nonlinearity parameters as a function of various propagation directions in anisotropic materials.

Scattering of harmonic waves from a nonlinear elastic inclusion

This article considers the scattering of harmonics stemming from the interaction of a primary wave with a heterogeneous and elastically nonlinear inclusion present in an otherwise linearly elastic host medium. The elastodynamic equations of motion are derived for general elastic anisotropy up to a third-order in displacement nonlinearity (cubic nonlinearity). The method of successive approximations is applied in order to decouple the equations of motion into a linear system of equations. The linear equations permit the use of Green's functions to obtain the scattering amplitudes from an arbitrarily shaped inclusion. General forms of the scattering amplitudes are given as functions of scattering-based quadratic and cubic acoustic nonlinearity parameters. Shape factors are offered for some simple geometries in order to arrive at closed-form solutions. An explicit example is given in the case of a spherically shaped inclusion with isotropic elastic moduli. The influence of the second-, third-, and fourth-order elastic stiffnesses, primary and scattered wave mode types, and scattering angles are highlighted. Potential experimental techniques, based on the present scattering model, offer an alternative method of probing the nonlinear elastic properties of materials.

Effect of pressurization on helical guided wave energy velocity in fluid-filled pipes

The effect of pressurization stresses on helical guided waves in a thin-walled fluid-filled pipe is studied by modeling leaky Lamb waves in a stressed plate bordered by fluid. Fluid pressurization produces hoop and longitudinal stresses in a thin-walled pipe, which corresponds to biaxial in-plane stress in a plate waveguide model. The effect of stress on guided wave propagation is accounted for through nonlinear elasticity and finite deformation theory. Emphasis is placed on the stress dependence of the energy velocity of the guided wave modes. For this purpose, an expression for the energy velocity of leaky Lamb waves in a stressed plate is derived. Theoretical results are presented for the mode, frequency, and directional dependent variations in energy velocity with respect to stress. An experimental setup is designed for measuring variations in helical wave energy velocity in a thin-walled water-filled steel pipe at different levels of pressure. Good agreement is achieved between the experimental variations in energy velocity for the helical guided waves and the theoretical leaky Lamb wave solutions.

On the origins of anomalous elastic moduli and failure strains of GaP nanowires

Previous reports suggest that Raman peaks in uniaxially loaded nanowires with diamond cubic and zinc blende crystal structures shift at rates that are significantly different from bulk specimens. We have investigated the first order Raman scattering from individual, free-standing, [111] oriented GaP nanowires ranging from 75 to 180 nm in diameter at uniaxial tensile stresses up to 5 GPa. All of the phonon modes were shifted to frequencies lower than previously reported for bulk GaP, and significant splitting of the degenerate transverse optical mode was observed. A general analysis method using single and double Lorentzian fits of the Raman peaks is presented and used to report more accurate values of the phonon deformation potentials (PDPs) that relate uniaxial strains to Raman peak shifts in GaP. A new set of PDPs determined from the nanowires revealed that the they have elastic moduli and failure strains that are consistent with bulk GaP. The analysis method eliminated the anomalous, inconsistent deformation behavior commonly reported in Raman-based strain measurements of nanowires, and can be extended to other materials systems with degenerate phonons.

Determination of acoustic nonlinearity parameter (β) using nonlinear resonance ultrasound spectroscopy: Theory and experiment

The present article investigates the possibility of using nonlinear resonance ultrasound spectroscopy to determine the acoustic nonlinearity parameter (β) and third order elastic constant by developing an inverse problem. A theoretical framework was developed for nonlinear forced vibration of a cantilever beam using material nonlinearity (stress-strain nonlinearity). The resulting nonlinear equation was solved using method of multiple time scales to obtain the nonlinear frequency shifts. The present works focuses only on classical nonlinearity and, therefore, a diverse group of intact, classic nonlinear materials were chosen. The samples were tested using nonlinear resonance ultrasound spectroscopy, and the developed theory was used to invert the experimental frequency shifts to obtain the nonlinearity parameters. The third order elastic constants and β were calculated using their analytical relationship with the nonlinearity parameter. The experimentally determined C₁₁₁ and β values for all various materials agree well with literature values. In addition to determining β, determination of the sign, or phase of β was also explored theoretically and experimentally.

Dependence of diffuse ultrasonic backscatter on residual stress in 1080 steel

In this article, the effects of residual stress on the ultrasonic scattering in a quenched steel sample are investigated by calculating the change of spatial variance amplitudes of ultrasonic signals after removing residual stress via annealing. The experimental results show that the average spatial variance amplitude decreases by about 11.89% for a scan area on the quenched surface after removing residual stress. This quantity was used to estimate the residual stress based on the developed stress-dependent backscatter model. In addition, the residual stress on the whole scan area was mapped by calculating the change of the spatial variance amplitude for each subarea after annealing, respectively. Diffuse ultrasonic backscatter signals show a high sensitivity to residual stress such that this technique has potential as a non-destructive method for measuring residual stress.

Stress-dependent ultrasonic scattering in polycrystalline materials

Stress-dependent elastic moduli of polycrystalline materials are used in a statistically based model for the scattering of ultrasonic waves from randomly oriented grains that are members of a stressed polycrystal. The stress is assumed to be homogeneous and can be either residual or generated from external loads. The stress-dependent elastic properties are incorporated into the definition of the differential scattering cross-section, which defines how strongly an incident wave is scattered into various directions. Nine stress-dependent differential scattering cross-sections or scattering coefficients are defined to include all possibilities of incident and scattered waves, which can be either longitudinal or (two) transverse wave types. The evaluation of the scattering coefficients considers polycrystalline aluminum that is uniaxially stressed. An analysis of the influence of incident wave propagation direction, scattering direction, frequency, and grain size on the stress-dependency of the scattering coefficients follows. Scattering coefficients for aluminum indicate that ultrasonic scattering is much more sensitive to a uniaxial stress than ultrasonic phase velocities. By developing the stress-dependent scattering properties of polycrystals, the influence of acoustoelasticity on the amplitudes of waves propagating in stressed polycrystalline materials can be better understood. This work supports the ongoing development of a technique for monitoring and measuring stresses in metallic materials.

Stress-dependent second-order grain statistics of polycrystals

In this article, the second-order statistics of the elastic moduli of randomly oriented grains in a polycrystal are derived for the case when an initial stress is present. The initial stress can be either residual stress or stresses generated from external loading. The initial stress is shown to increase or decrease the variability of the grain's elastic moduli from the average elastic moduli of the polycrystal. This variation in the elastic properties of the individual grains causes acoustic scattering phenomenon in polycrystalline materials to become stress-dependent. The influence of the initial stress on scattering is shown to be greater than the influence on acoustic phase velocities, which defines the acoustoelastic effect. This work helps the development of scattering based tools for the nondestructive analysis of material stresses in polycrystals.

Mechanical and thermal properties of γ-Mg2SiO4 under high temperature and high pressure conditions such as in mantle: A first principles study

γ-Mg2SiO4 is an important mineral in mantle, and our knowledge on its mechanical and thermal properties is critical for many areas of geological sciences. In this work, the crystal structure of γ-Mg2SiO4 under high temperature and high pressure conditions is optimized by using the GOMASC method, and the total energy, thermal expansion coefficients, and elastic constants at different temperature and pressure conditions are obtained. On the basis of phonon spectrum, group velocity, phase velocity, Grüneisen parameter, and thermal conductivity are calculated for γ-Mg2SiO4 under high temperature and high pressure conditions. These calculated results can provide an important reference for geological research.

On the acoustoelasticity of polycrystalline materials

A linear relation between the strains and stresses of a crystallite within a polycrystal is used to homogenize the polycrystal's elastic properties. The homogenization parallels the self-consistent method that is used for estimating the polycrystal's linear elastic properties. Acoustoelasticity for a macroscopically isotropic polycrystal is then formulated using a homogenized constitutive equation with initial stress. Simple expressions are given for the phase velocities and polarization directions for a uniaxially stressed polycrystal. The present model is compared with the model of Man and Paroni [J. Elast. 45, 91-116 (1996)]. Strong anisotropy of the crystallite elastic constants causes the present model to differ noticeably from the model of Man and Paroni.

Acoustic nonlinearity parameters for transversely isotropic polycrystalline materials

This article considers polycrystalline materials with macroscopic elastic anisotropy and the effect of the anisotropy on the quadratic nonlinearity parameter used to describe second harmonic generation in solids. The polycrystal is assumed to have transversely isotropic elastic symmetry, which leads to a directional dependence of the nonlinearity parameters. Additionally, the anisotropy leads to second harmonic generation from an input shear wave. Estimates of the longitudinal and shear wave nonlinearity parameters are given as a function of single-crystal elastic constants, macroscopic anisotropy constants, and propagation direction. An inverse model is presented that relates measured nonlinearity parameters to the macroscopic anisotropy constants. The estimates of the nonlinearity parameters can be used to approximate the damage-free or baseline nonlinearity parameter of structural components, which helps the effort toward absolute measures of material damage.

Mechanical properties and stabilities of g-ZnS monolayers

Planar graphene-like ZnS monolayers are mechanically stable under various large strains.

Mechanical properties and stabilities of α-boron monolayers

α-Boron monolayers are mechanically stable under various large strains.

Theoretical foundations of resonant ultrasound spectroscopy at high pressure

The theory of free-vibration acoustic resonance (FVAR) of solids at high pressure is developed within the framework of nonlinear elasticity and the calculus of variations. The FVAR state is formulated as a generalization of the conventional theory that was originally developed by Rayleigh and Ritz in the sense that it includes geometrical and material nonlinearities as well as the potential energy of external pressure. Magnetic point groups and quasi-harmonic approximation are used so as to obtain a natural extension of normal mode phonons in the high-pressure regime. The numerical analysis of eight different cubic-symmetry crystals reveals that FVAR frequencies depend linearly on the pressure, and the slopes vary with the FVAR modes, including the sign. We estimated the mode Grüneisen parameter up to N =2400 and proved that the high-frequency limit γ ∞ is equivalent to the conventional Grüneisen parameter γ . Quantitative agreement of the parameters demonstrates that nearly the entire third-order elastic constants tensor can be determined from high-pressure ultrasound spectroscopy experiments.

Elastic constants of solids and fluids with initial pressure via a unified approach based on equations-of-state

The second and third-order Brugger elastic constants are obtained for liquids and ideal gases having an initial hydrostatic pressure p1. For liquids the second-order elastic constants are C₁₁=A+p₁, C₁₂=A-p₁, and the third-order constants are C₁₁₁=-(B+5A+3p₁), C₁₁₂=-(B+A-p₁), and C₁₂₃=A-B-p₁, where A and B are the Beyer expansion coefficients in the liquid equation of state. For ideal gases the second-order constants are C₁₁=p₁γ+p₁, C₁₂=p₁γ-p₁, and the third-order constants are C₁₁₁=-p₁(γ(2)+4γ+3), C₁₁₂=-p₁(γ(2)-1), and C₁₂₃=-p₁ (γ(2)-2γ+1), where γ is the ratio of specific heats. The inequality of C₁₁ and C₁₂ results in a nonzero shear constant C₄₄=(1/2)(C₁₁-C₁₂)=p₁ for both liquids and gases. For water at standard temperature and pressure the ratio of terms p₁/A contributing to the second-order constants is approximately 4.3×10(-5). For atmospheric gases the ratio of corresponding terms is approximately 0.7. Analytical expressions that include initial stresses are derived for the material 'nonlinearity parameters' associated with harmonic generation and acoustoelasticity for fluids and solids of arbitrary crystal symmetry. The expressions are used to validate the relationships for the elastic constants of fluids.

Bolt axial stress measurement based on a mode-converted ultrasound method using an electromagnetic acoustic transducer

A method is proposed to measure the stress on a tightened bolt using an electromagnetic acoustic transducer (EMAT). A shear wave is generated by the EMAT, and a longitudinal wave is obtained from the reflection of the shear wave due to the mode conversion. The ray paths of the longitudinal and the shear wave are analyzed, and the relationship between the bolt axial stress and the ratio of time of flight between two mode waves is then formulated. Based on the above outcomes, an EMAT is developed to measure the bolt axial stress without loosening the bolt, which is required in the conventional EMAT test method. The experimental results from the measurement of the bolt tension show that the shear and the mode-converted longitudinal waves can be received successfully, and the ratio of the times of flight of the shear and the mode-converted longitudinal waves is linearly proportional to the bolt axial tension. The non-contact characteristic of EMAT eliminates the effect of the couplant and also makes the measurement more convenient than the measurement performed using the piezoelectric transducer. This method provides a promising way to measure the stress on tightened bolts.

Evaluation of third-order elastic constants using laser-generated multi-type ultrasound for isotropic materials

Within the linear elasticity approximation the speed of a small-amplitude sound in conventional linear elasticity is determined only by the second order elastic (SOE) constants and the density of the medium. Subjecting the conveying solid to a static strain of a sufficient magnitude introduces the third-order elastic (TOE) constants in the equation of the sound speed. In this work we applied a homogeneous isotropic deformation caused by a thermal expansion of an aluminum alloy sample. Velocities of three acoustic modes: longitudinal, shear and Rayleigh waves were measured as functions of temperature within a range of 25-100 °C. Two TOE constants C111 and C112 were evaluated in an assumption that the third independent module C144 is far smaller than the former two.

Apparatus for measurement of acoustic wave propagation under uniaxial loading with application to measurement of third-order elastic constants of piezoelectric single crystals

We describe an apparatus for the measurement of acoustic wave propagation under uniaxial loading featuring a special mechanism designed to assure a uniform mechanical load on a cube-shaped sample of piezoelectric material. We demonstrate the utility of the apparatus by determining the effects of stresses on acoustic wave speed, which forms a foundation for the final determination of the third-order elastic constants of langasite and langatate single crystals. The transit time method is used to determine changes in acoustic wave velocity as the loading is varied. In order to minimize error and improve the accuracy of the wave speed measurements, the cross correlation method is used to determine the small changes in the time of flight. Typical experimental results are presented and discussed.

Stress-dependent changes in the diffuse ultrasonic backscatter coefficient in steel: experimental results

In this article, the effects of uniaxial compressive loading on the ultrasonic scattering from polycrystalline grains are shown for 10 MHz ultrasound in annealed, 1018 steel. The results show a decreasing value of the stress-dependent backscatter coefficient for normal incident ultrasound when the compression loading is perpendicular to the scattering direction. The change due to scattering is about 2 orders of magnitude greater than changes observed by others using ultrasonic wavespeed measurements. It is anticipated that this research can serve as the basis for many methods associated with nondestructive determination of stress in structural materials.

Exact second order formalism for the study of electro-acoustic properties in piezoelectric structures under an initial mechanical stress

In this study we develop the exact second order formalism of piezoelectric structures under an external mechanical stress. Indeed, previous models are approximated since they consist in deriving all the equations in the natural coordinate system (corresponding to the pre-stress free case). Hence, our exact formalism proposes to obtain the whole of equations in the current coordinate system (which is the coordinate system after the pre-deformation). Then, this exact formalism is used to derive the modified Christoffel equations and the modified KLM model. Finally, we quantify the correction with the approximate formalism on several transfer functions and electro-mechanical parameters for a non hysteretic material (lithium niobate). In conclusion, we show that for this material, significant corrections are obtained when studying the plane wave velocities and the electrical input impedance (about 4%), whereas other parameters such as coupling coefficient and impulse response are less influenced by the choice of coordinate systems (corrections less than 0.5%).

Modeling of surface acoustic wave strain sensors using coupling-of-modes analysis

SAW devices may be configured as strain sensors, providing passive, wireless strain measurement in demanding conditions. A key consideration is the modeling of the sensors, enabling different device designs to be considered. This paper presents a simulation scheme using coupling-of-modes (COM) analysis which allows both the frequency response of a SAW strain sensor and its bias sensitivity to be evaluated. Example applications are presented to demonstrate the use of the model.