Plane Shear Waves Research Articles

Plane dilatational and shear waves in a chiral porous thermoelastic medium under strain gradient theory

Purpose This study aims to investigate time-harmonic wave propagation in a chiral porous thermoelastic solid under strain gradient theory (SGT), focusing on identifying and characterizing distinct wave modes within the medium. Design/methodology/approach Using Iesan's gradient theory, which incorporates chiral effects and accommodates second sound phenomena, the authors derive mathematical formulations for the velocities and attenuations of eight propagating waves: four dilatational waves and two pairs of coupled shear waves (one left circularly polarized, the other right). Numerical simulations are performed for a specific model, exploring the influence of various parameters on wave propagation. Findings The authors establish that the medium supports four dilatational waves, including a microstretch-associated wave, and four shear waves, distinguished by their chiral-induced characteristics. The results highlight the frequency-dependent dispersive nature of all propagating waves and establish connections with existing theoretical frameworks, demonstrating the broader applicability of our findings. Practical implications The characteristics of wave propagation in chiral media examined here can enhance our understanding of chiral medium behavior. This knowledge is crucial for developing materials with pronounced chiral effects, surpassing those found in natural chiral materials like bone, quartz, sugar and wood. Advances in artificial chiral materials are driven by their superior toughness, durability and other beneficial properties. Consequently, this study has potential applications across various fields, including the design of chiral broadband absorbers and filters, the production of artificial bones and medical devices, aeronautical engineering and beyond. Originality/value This research extends existing theories and deepens the understanding by exploring wave behaviors in chiral media, advancing this emerging field.

Pulse-Echo Ultrasound for Quantitative Measurements of Two Uncorrelated Elastic Parameters

ObjectiveThis paper describes the relationship between elastic tissue properties and strain and presents an initial investigation of pulse-echo ultrasound to measure two uncorrelated elastic parameters in tissue-mimicking phantoms. The two elastic parameters are the shear modulus, related to deformation of shape, and what we in the paper define as the nonlinear compressibility, related to deformation of volume. MethodsWe prepared tissue-mimicking phantoms containing lesions of variable shear modulus and variable nonlinear compressibility. An in-house framework for shear wave imaging was developed using ultrasound radiation force at 4.5 MHz to induce shear waves and plane wave imaging with pulses in a frequency band centered around 12.5 MHz to track the shear waves. For measurements of nonlinear compressibility, co-propagating dual-frequency pulse complexes at 0.7 MHz and 14 MHz were applied. Algorithms were implemented on a Verasonics Vantage ultrasound scanner and a custom-made multi-frequency ultrasound transducer was used. Mechanical indentation measurements were performed to validate ultrasound measurements of the shear modulus. For the nonlinear compressibility, ultrasound measurements were compared to results derived from the literature. ResultsWe found good agreement in elasticity results from ultrasound measurements and mechanical indentation as well as when comparing with results derived from the literature. ConclusionResults of the current investigation were promising. We plan patient studies involving thyroid lesions and liver steatosis to explore whether measurements of elastic parameters related both to shape deformation and volume deformation are useful in clinical practice.

Biomechanical Assessment of Liver Integrity: Prospective Evaluation of Mechanical Versus Acoustic MR Elastography.

Magnetic resonance elastography (MRE) can quantify tissue biomechanics noninvasively, including pathological hepatic states like metabolic dysfunction-associated steatohepatitis. To compare the performance of 2D/3D-MRE using the gravitational (GT) transducer concept with the current commercial acoustic (AC) solution utilizing a 2D-MRE approach. Additionally, quality index markers (QIs) were proposed to identify image pixels with sufficient quality for reliably estimating tissue biomechanics. Prospective. One hundred seventy participants with suspected or confirmed liver disease (median age, 57 years [interquartile range (IQR), 46-65]; 66 females), and 11 healthy volunteers (median age, 31 years [IQR, 27-34]; 5 females). Participants were scanned twice at 1.5 T and 60 Hz vibration frequency: first, using AC-MRE (2D-MRE, spin-echo EPI sequence, 11 seconds breath-hold), and second, using GT-MRE (2D- and 3D-MRE, gradient-echo sequence, 14 seconds breath-hold). Image analysis was performed by four independent radiologists and one biomedical engineer. Additionally, superimposed analytic plane shear waves of known wavelength and attenuation at fixed shear modulus were used to propose pertinent QIs. Spearman's correlation coefficient (r) was applied to assess the correlation between modalities. Interreader reproducibility was evaluated using Bland-Altman bias and reproducibility coefficients. P-values <0.05 were considered statistically significant. Liver stiffness quantified via GT-2D/3D correlated well with AC-2D (r ≥ 0.89 [95% CI: 0.85-0.92]) and histopathological grading (r ≥ 0.84 [95% CI: 0.72-0.91]), demonstrating excellent agreement in Bland-Altman plots and between readers (κ ≥ 0.86 [95% CI: 0.81-0.91]). However, GT-2D showed a bias in overestimating stiffness compared to GT-3D. Proposed QIs enabled the identification of pixels deviating beyond 10% from true stiffness based on a combination of total wave amplitude, temporal sinusoidal nonlinearity, and wave signal-to-noise ratio for GT-3D. GT-MRE represents an alternative to AC-MRE for noninvasive liver tissue characterization. Both GT-2D and 3D approaches correlated strongly with the established commercial approach, offering advanced capabilities in abdominal imaging compared to AC-MRE. 1 TECHNICAL EFFICACY: Stage 2.

Coupling quantitative ultrasound using echo envelop statistics and shear wave propagation to provide new image contrast of mimicked liver lesions

In quantitative ultrasound, biomarkers are sensitive to the scatterers’ positioning and number density, and therefore, shear wave (SW) propagation may modify their time-varying behaviors and provide new image contrast. Herein, we considered echo envelop statistics by using one parameter of the homodyned-K distribution (HKD): the diffuse-to-total signal power ratio 1/(κ+1). A liver lesion-mimicking phantom was simulated with an inclusion and a background of elastic moduli of 40 and 4 kPa, and scatterers’ number densities per resolution cell of 15 and 5, respectively. Ten sets of spatially correlated scatterers were created, and a plane SW was simulated to reposition time and spatially varying scatterers according to the SW motion. Ultrasound images were simulated using MUST with added Gaussian noise to attain 20-dB SNR. Dynamic and static analyses were made over 30 frames with and without SW motion, respectively. Validation was based on the contrast of 1/(κ+1) between the inclusion and the background. HKD image contrast increased with SW motion with values from 0.12 ± 0.10 to 0.48 ± 0.16 (no units) (p &amp;lt; 0.001). Results suggest that the variation in scatterers’ organization under SW propagation may provide new information over its static counterpart to enhance the contrast of liver lesions.

Semi-analytical solution for the 3D response of a tunnel embedded in an elastic half-space subject to seismic waves

This paper presents a semi-analytical solution for the 3D problem of a cylindrical tunnel embedded in an elastic half-space subject to plane harmonic compressional and shear waves. Both the tunnel and soil are modelled as an elastic continuum. Conformal mapping is employed to transform the original physical domain with boundary surfaces of two different types onto an image domain with surfaces of the same type, which makes the problem easier to solve. The total wave field in the half-space consists of incident and reflected (from the half-space surface) plane waves, as well as directly and secondary scattered cylindrical waves, while the total wave field in the tunnel consists of refracted cylindrical waves. The secondary scattered waves, generated when the cylindrical waves directly scattered from the tunnel meet the half-space surface, are represented by cylindrical waves that originate from an image source, which is in line with the spirit of the method of images. The unknown amplitude coefficients of the cylindrical waves are determined from the boundary and continuity conditions of the tunnel–soil system by projecting those onto the set of circumferential modes, which results in a set of algebraic equations. Results show that the present method converges for a small number of circumferential modes. We observe very good agreement between the obtained results and those in literature. In a systematic evaluation, we demonstrate that the method works well for the frequency band of seismic waves, as well as for the complete considered ranges of the tunnel/soil stiffness ratio, the embedded depth of the tunnel, the vertical incident angle and the tunnel thickness. However, the results obtained for a moderate tunnel–soil stiffness contrast under the incident compressional wave are inaccurate when Hankel functions are used to represent the cylindrical waves in the tunnel, which is due to the refracted shear waves in the tunnel transitioning from propagating to evanescent (in the 3D case). These inaccuracies can be perfectly overcome by representing the waves in the tunnel by Bessel functions. We also find that the present method generally works better for the incident compressional wave than for the incident shear wave, as the condition number of the matrix (related to the mentioned algebraic equations) is often larger in the latter case. In view of engineering practice, we conclude that the tunnel is safer when the surrounding soil is stiffer, the tunnel is thicker and the vertical incident angle is larger. Finally, the present method, which is in general fast, elegant and accurate, can be used in preliminary design so as to avoid pronounced resonances, and to assess stress distributions and ground vibrations.

Negative refraction and exceptional point with Parity-Time symmetry in a piezoelectric mechanical metamaterial

In this work, negative refraction and exceptional point (EP) in piezoelectric mechanical metamaterials are considered, which corporates with the Parity–Time (PT) symmetry. The oblique incidence of anti–plane shear wave at the interface between a homogeneous medium and phononic crystal can excite multiple refraction modes. In periodic layers, propagation characteristics are illustrated by the dispersion relation and transfer matrix method. The normal mode decomposition is developed and time average Poynting vector is used to derive the refraction angle at the interface of piezoelectric metamaterials. The beam splitting is shown with different incident angles, as well as the pure negative refraction is achieved. Due to the mechanical and electric coupling, the complex conjugate of Bloch wave number appears in the dispersion relation and degenerates to the exceptional point. In addition, a defect layer is introduced to study the transmission coefficient of two phononic crystals with different arrangements. When the gain and loss are introduced into the metamaterials, the Parity–Time symmetry system is realized and a complete transmission with a unidirectional zero reflection at exceptional point can be achieved.

Mean Radiation Force of Shear Plane Waves on a Sphere in an Elastic Medium

Mean Radiation Force of Shear Plane Waves on a Sphere in an Elastic Medium

A buried piezoelectric spherical particle incident upon by plane shear waves

The dynamic electromechanical coupling in piezocomposites in response to wave propagation and scattering phenomenon is of great importance in engineering sciences, innovative technologies, and nondestructive evaluation. The present work is concerned with the scattering of shear waves by a spherical piezoelectric particle bonded to an infinitely extended isotropic elastic polymer matrix. The particle is assumed to have radial polling direction and spherically isotropic electromechanical properties. An incident shear wave propagates in the positive z-direction, while polarizes along the x-axis. The incident, scattered, and the transmitted electromechanical fields are analytically achieved by the utilization of spherical harmonics followed by the employment of the extended Frobenius series for the interior points of the particle to solve a set of four governing coupled differential equations for the displacement and the electric potentials. The set problem possesses a symmetry with respect to the xz-plane and as it will be shown the mentioned differential equations in conjunction with the particle-matrix interface conditions can be transformed into two systems of linear algebraic equations for the unknown constants which appeared in the Frobenius series. Verification is made in the special case of purely elastic isotropic particle. In the present formulation there is no restriction on the range of the frequency of the incident wave. The effect of the piezoelectric particle properties on the total scattering cross section is investigated. The dynamic stress concentration factors and the dynamic electric displacement concentration factor at the matrix-piezoelectric interface are calculated. Subsequently, the effect of the frequency of the incident wave on the maximum values of the dynamic stress concentration factors and the dynamic electric displacement concentration factor as well as their appurtenant locations are discussed and compared to those of the corresponding static cases.

Novel periodic pile barriers in saturated soil and applications to propagation attenuation of shear plane waves

The frequency attenuation zones (AZs) of periodic pile barriers have attracted extensive attention in recent years due to their superior performance in ambient vibration reduction. However, the research on vibration isolation of pile barriers in saturated soil is very limited. Also, the configuration of the traditional pile barrier is simple and the AZ width is narrow, which limits its application in engineering practice. In this paper, to develop novel periodic pile barriers with the maximum relative attenuation zone width (RWAZ) in saturated soil, the topology optimization on the periodic pile unit is carried out with the genetic algorithm (GA) and FEM. For different site conditions, the effects of the saturated soil porosity, the drained elastic modulus, the drained density and the tortuosity coefficient of soil on the optimization results are analyzed, and the corresponding optimized pile structures are proposed, respectively. Thus, the effect of material distribution on the RWAZ is disclosed, and novel periodic pile structures for ambient vibration isolation in saturated soil are developed. Furthermore, the superior shear plane wave attenuation performances of the optimized pile structure in saturated soil are both demonstrated by the established 2D and 3D numerical models.

Plane nonlinear shear wave propagation in transversely isotropic soft solids.

Nonlinear wave equations are obtained for the two plane shear wave modes in a transversely isotropic soft solid. The material is modeled using a general expansion of the strain energy density up to fourth order in strain. Whereas, in an isotropic soft solid, leading order nonlinearity for plane wave propagation appears at cubic order in strain, elastic anisotropy in a transversely isotropic material introduces nonlinear effects at quadratic order, including interaction between the modes of a wave with two displacement components. Expressions for second harmonic generation in an elliptically polarized wave field illustrate the low efficiency of nonlinear interactions between the two displacement components, which results from the disparity between propagation speeds of the two shear wave modes. Coupled wave equations with up to cubic nonlinearity are presented and then approximated to describe linearly polarized waves by neglecting interaction between modes. Evolution equations are obtained for linearly polarized progressive waves, and explicit expressions are given in terms of elastic moduli and propagation direction for the coefficients of leading order nonlinearity. Expressions are presented for up to third harmonic generation from a time-harmonic source.

Radiation force of stationary elastic compressional and shear plane waves on a cylinder encased in a linear elastic solid

The assembly of acoustically-engineered composite (meta)materials rely on the acousto-elastic radiation force of stationary waves to trap solid particles and form a stable structure. Accurate modeling of the time-averaged force on a particle is needed for experimental design purposes and basic physical understanding of the trapping effect and particle clustering. This work presents a rigorous formalism based on the integration of the time-averaged radial power flow density to derive analytically and calculate numerically the acousto-elastic radiation force induced by compressional (P) and shear (S) plane standing waves incident upon a circular elastic cylinder embedded in a linear lossless solid. Initially, the elastic scattering by the cylinder is determined and then used to derive the acousto-elastic radiation force. Particular emphasis is made on the contributions of elastic mode preservation (P → P, S → S) as well as mode conversion (P → S, S → P) to the radiation force of standing waves. Numerical computations for the dimensionless radiation force efficiency and its components demonstrate the importance of compressional-to-shear and shear-to-compressional mode conversions in scattering that occur in linear elastic media. The analytical formalism and exact solutions of the radiation force can be used to validate numerical methods and the results of the simulations can be utilized a priori in optimizing and designing acousto-elastic radiation force experiments involving standing waves and particle assembly in materials science, elasticity imaging methods, activation of implantable devices, non-destructive evaluation and other related applications.

Plane dilatational and shear waves in a uniformly rotating thermo-viscoelastic material with voids

Effects of uniform rotation on propagating plane waves in a thermo-viscoelastic material with voids are explored. Five distinct modes are found to propagate in the considered rotating medium. Only in some specific directions relative to the axis of rotation, these modes may behave like pure dilatational or pure shear waves. Considering copper as a rotating thermo-viscoelastic material with voids, phase speeds and corresponding attenuations of all the existing pure modes are computed numerically and depicted graphically. Several well-known results are reduced as particular cases from the present formulation. Some facts regarding the directions of the pure modes are pointed out.

Effect of a plane shear wave on cylindrical shell surrounded by elastic medium

Unsteady-state strain of a cylindrical shell on the effect of a stepped shear wave propagating in an external elastic medium is studied in the paper. It is assumed that the shell is infinitely long and the front of the incident wave is parallel to its axis. In the paper under consideration, the expansion in a Fourier series in eigenmodes and finite differences in radial coordinate and time is applied. In calculations, a method of numerical dispersion minimization is used, which gives an almost exact description of frontal discontinuities. An asymptotic solution to the problem is obtained at t → ∞ (t - time) by an analytical method. An integrated approach allows obtaining reliable and fairly complete information about the basic physical laws of unsteady-state strain of a cylindrical shell over the entire time interval of external load impact.

The effect of rotation on the propagation of waves in an incompressible transversely isotropic thermoelastic solid

The present paper is concerned with plane and surface wave solutions of the governing equation for an incompressible, homogeneous, transversely isotropic and rotating thermoelastic medium in the context of the Green–Naghdi theory. The velocity equation for homogeneous plane waves is obtained which indicates the existence of two coupled plane shear waves. The secular equation for Rayleigh waves is derived for thermally insulated as well as isothermal boundaries. The non-dimensional speeds of plane waves are plotted against the angle of incidence, non-dimensional anisotropy parameter and rotation/frequency ratio. The non-dimensional speed of Rayleigh surface waves is also computed against the non-dimensional anisotropy parameter and rotation/frequency ratio.

Elastic Waves in the Half-space of an Incompressible Thermoelastic Material with Transverse Isotropy

This paper investigates the problem of incident and reflected shear waves in an incompressible thermoelastic material having transversely isotropic property using Lord-Shulman theory on generalized thermoelasticity. Two plane shear waves are shown to exist in such medium and the expressions for non-dimensional velocities are given. The amplitude ratios of the reflected waves caused by the incident wave at the stress free surface have been derived by taking suitable boundary conditions. The effect of specific heat and material constants on the amplitude ratios has been analysed numerically and the results are presented graphically.

Mathematical interpretation of seismic wave scattering and refraction on tunnel structures of circular cross-section

Mathematical interpretation of the elastic wave diffraction in circular cylinder coordinates is in the focus of this paper. Firstly, some of the most important properties of Bessel functions, pertinent to the elastic wave scattering problem, have been introduced. Afterwards, basic equations, upon which the method of wave function expansions is established, are given for cylindrical coordinates and for plane-wave representation. In addition, steady-state solutions for the cases of a single cavity and a single tunnel are presented, with respect to the wave scattering and refraction phenomena, considering both incident plane harmonic compressional and shear waves. The last part of the work is dealing with the translational addition theorems having an important role in the problems of diffraction of waves on a pair of circular cylinders.

Surface Effects on the Scattering of SH-Wave Around an Arbitrary Shaped Nano-Cavity

By using the complex variable function theory and the conformal mapping method, the scattering of plane shear wave (SH-wave) around an arbitrary shaped nano-cavity is studied. Surface effects at the nanoscale are explained based on the surface elasticity theory. According to the generalized Yong–Laplace equations, the boundary conditions are given, and the infinite algebraic equations for solving the unknown coefficients of the scattered wave solutions are established. The numerical solutions of the stress field can be obtained by using the orthogonality of trigonometric functions. Lastly, the numerical results of dynamic stress concentration factor around a circular hole, an elliptic hole and a square hole as the special cases are discussed. The numerical results show that the surface effect and wave number have a significant effect on the dynamic stress concentration, and prove that our results from theoretical derivation are correct.

Control of elastic shear waves by periodic geometric transformation: cloaking, high reflectivity and anomalous resonances

A doubly periodic geometric transformation is applied to the problem of out of plane shear wave propagation in a doubly periodic perforated elastic medium. The technique leads to the design of a system of radially anisotropic and inhomogeneous shells surrounding the void inclusions, that can be tuned to give the desired filtering properties.For a regular transformation, the transformed elastic system displays the same dispersion properties than the original homogeneous one, but for overlapping and unfolding transformations new filtering properties can be obtained, which include anomalous resonances at zero and finite frequencies. Low-frequency homogenisation reveals how it is possible to tune the phase and group velocity in the long-wave limit at any value, or to obtain a zero frequency band gap for Neumann boundary conditions. The dispersion properties of the medium are studied both semi-analytically by the multipole expansion and numerically by the finite element methods.Several applications are shown, including the transmission problem throughout a grating of void inclusions and an interface in a waveguide, where the capability of the proposed model is quantitatively demonstrated by computing the transmitted power flow. Finally, we gave a demonstration of defect modes in a waveguide and of tuning the transformation for the research of Dirac points.

Viscoelastic shear modulus measurement of thin materials by interferometry at ultrasonic frequencies

This paper presents a technique for measuring the complex shear modulus of thin slabs of viscoelastic solids based on the measurement of the reflection and transmission of plane shear waves through a sample inserted between two delays lines. Reproducible shear wave transmission through the sample is achieved by inserting bond layers with controlled thickness between the delay lines and the sample and by characterizing beforehand the bond rheology. The frequency dependent complex shear modulus is quantitatively evaluated from the transmission and reflection coefficients using an exact model of interferences within the delay line-bond-sample-bond-delay line sandwich and by selecting the solution among the muliple solutions of the inverse problem from considerations on time of flight and sample thickness. Thanks to its reproducibility and accuracy, this method appears as an original and efficient technique for quantitatively characterizing the high frequency shear modulus of attenuating materials.

Nonlinear travelling strain waves in a gradient-elastic medium

In this paper, we investigate the influence of geometric nonlinearity on the propagation of longitudinal and shear waves in a gradient-elastic medium. It is shown that taking into account surface energy we observe the destruction of travelling plane longitudinal and shear waves.