Pure Shear Waves Research Articles

Shear Mode Ultrasonic Transducers from Flexible Piezoelectric PLLA Fibers for Structural Health Monitoring

AbstractShear mode guided waves are highly demanded for underwater structural health monitoring (SHM) applications due to their simplified non‐dispersive feature and minimal acoustic energy loss in the presence of liquid. Excitation and detection of pure shear wave are challenging using conventional piezoelectric materials used in the current ultrasonic transducers because they have complex piezoelectric responses mixed with multiple longitudinal, transverse, and shear modes. They also suffer from aging issue due to depoling. Here, conformable shear mode ultrasonic transducers are designed and made of flexible piezoelectric poly (L‐lactic acid) (PLLA) fibers on both flat and tubular structures. The electromechanical responses over a macroscopic area of the transducers are evaluated in a wide frequency range up to 500 kHz. The PLLA fiber‐based shear mode ultrasonic transducers exhibit a consistent sensitivity of detecting defects in liquid and air. In addition, the only shear mode in PLLA fibers originates from crystal structure without requiring electrical poling to render piezoelectricity, thus does not depole due to aging. The theoretical analyses including ab initio calculations and experimental results on both flat and tubular structures show the great potential of PLLA material and significant advantage of PLLA fiber‐based shear mode ultrasonic transducers for underwater SHM applications.

Rayleigh Wave-Shear Wave Coupling Mechanism for Large Lateral Deformation in Level Liquefiable Ground

Large lateral deformation in almost level liquefiable ground has been observed during past strong earthquakes, which cannot be explained by the same driving force as that in sloping ground. This study proposes a Rayleigh wave-shear wave coupling mechanism for this phenomenon, which is evaluated on the element and site levels. A simple element level elasto-plastic force–displacement model reveals that the combination of the symmetrical shear force with the asymmetrical shear resistance under Rayleigh wave-shear wave coupling can be a major contributor for the accumulation of lateral soil deformation. Finite element method numerical results for a site level ground model shows that Rayleigh wave-shear wave coupling can induce significantly greater lateral displacement compared to pure shear wave input, and further highlight the importance of soil liquefaction for this phenomenon. The amplitudes and frequencies of the shear wave and the Rayleigh wave and their phase difference are shown to affect the accumulation of lateral deformation. The findings of Rayleigh wave-shear wave coupling induced lateral deformation are consistent with observations in past earthquakes, and should be considered in geotechnical earthquake engineering.

Transient Green's functions of a buried pulse in a constrained transversely isotropic half-space

In this paper, transient surface Green's functions (GFs) excited by a general dynamic pulse buried in a constrained transversely isotropic (TI) half-space are derived in analytical forms. The constrained transverse isotropy is defined by an explicit relation that expresses one of the five elastic constants in terms of the other four independent constants so that no coupled P-SV wave exists. Use is made of the solutions available for general TI materials in Hankel–Laplace transformed domain in order to extract the corresponding response functions for the constrained material. Under this constraint, the coupled fourth-order wave operator, which represents the convolved motion of the quasi-longitudinal and quasi-transverse waves, is decoupled into two separated second-order wave operators describing the motion of the pure dilatational (P) and pure shear (SV) waves. The involved Laplace transform inversion is carried out using the novel Cagniard scheme so that the final displacement GFs are expressed in terms of the elementary line integrals over a finite interval. The GF solution is valid for the entire range of the constrained TI material including material isotropy as its special case, and clearly illustrates the disturbances due to different stress waves including, P, SV, SH, Rayleigh and diffracted SP waves. Moreover, the GFs show that in this constrained TI material, the compressional and horizontally polarized waves, excited by a source located at an arbitrary depth, propagate towards the surface with direction-dependent velocities. The explicit formulas for these velocities of the waves are obtained in terms of the inclination angle of the wave normal with respect to the axis of material symmetry. On the other hand, the vertically polarized wave and the diffracted SP wave propagate with eigen velocity, independent from the direction of the propagation. Two critical epicentral distances, at which mode conversion in ray diagram occurs, are detected for this constrained TI material. One critical distance marks the phenomena of total reflection of SV-waves where the incident SV-wave is totally reflected and the diffracted SP-wave travels along the free surface. The other marks the distance where the SH- and SP-waves become coalescent, and furthermore, before this epicentral distance the SH-wave is faster, and after this distance, the SP-wave is faster. To show the relevance of this constrained TI material in engineering and material modeling applications and its relation to the physics of TI materials, we also demonstrate that, in addition to its mathematical simplification, this reduced anisotropy can well approximate the elastic coefficients of some real TI materials, particularly those encountered in exploration geophysics and seismology.

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.

Slowness vector versus ray direction in polar anisotropic media

SUMMARY The inverse problem of finding the slowness vector from a known ray direction in general anisotropic elastic media is a challenging task, needed in many wave/ray-based methods, in particular, solving two-point ray bending problems. The conventional resolving equation set for general (triclinic) anisotropy consists of two fifth-degree polynomials and a sixth-degree polynomial, resulting in a single physical solution for quasi-compressional (qP) waves and up to 18 physical solutions for quasi-shear waves (qS). For polar anisotropy (transverse isotropy with a tilted symmetry axis), the resolving equations are formulated for the slowness vectors of the coupled qP and qSV waves (quasi-shear waves polarized in the axial symmetry plane), and independently for the decoupled pure shear waves polarized in the normal (to the axis) isotropic plane (SH). The novelty of our approach is the introduction of the geometric constraint that holds for any wave mode in polar anisotropic media: The three vectors—the slowness, ray velocity and medium symmetry axis—are coplanar. Thus, the slowness vector (to be found) can be presented as a linear combination of two unit-length vectors: the polar axis and the ray velocity directions, with two unknown scalar coefficients. The axial energy propagation is considered as a limit case. The problem is formulated as a set of two polynomial equations describing: (i) the collinearity of the slowness-related Hamiltonian gradient and the ray velocity direction (third-order polynomial equation) and (ii) the vanishing Hamiltonian (fourth-order polynomial equation). Such a system has up to twelve real and complex-conjugate solutions, which appear in pairs of the opposite slowness directions. The common additional constraint, that the angle between the slowness and ray directions does not exceed ${90^{\rm{o}}}$, cuts off one half of the solutions. We rearrange the two bivariate polynomial equations and the above-mentioned constraint as a single univariate polynomial equation of degree six for qP and qSV waves, where the unknown parameter is the phase angle between the slowness vector and the medium symmetry axis. The slowness magnitude is then computed from the quadratic Christoffel equation, with a clear separation of compressional and shear roots. The final set of slowness solutions consists of a unique real solution for qP wave and one or three real solutions for qSV (due to possible triplications). The indication for a qSV triplication is a negative discriminant of the sixth-order polynomial equation, and this discriminant is computed and analysed directly in the ray-angle domain. The roots of the governing univariate sixth-order polynomial are computed as eigenvalues of its companion matrix. The slowness of the SH wave is obtained from a separate equation with a unique analytic solution. We first present the resolving equation using the stiffness components, and then show its equivalent forms with the well-known parametrizations: Thomsen, Alkhalifah and ‘weak-anisotropy’. For the Thomsen and Alkhalifah forms, we also consider the (essentially simplified) acoustic approximation for qP waves governed by the quartic polynomials. The proposed method is coordinate-free and can be applied directly in the global Cartesian frame. Numerical examples demonstrate the advantages of the method.

FRI-based sparse sampling for ultrasonic array signals based on pulse-positive sequence delay synthesis

Ultrasonic transducer arrays have been widely applied to nondestructive testing due to their high testing efficiency and abundant echo information regarding defects. However, the considerable sampling data generated in array testing greatly restricts the performance levels of the subsequent processes, such as data storage, transmission, and processing. An finite rate of innovation- (FRI) based sparse sampling method has been recently developed, which greatly reduces the volume of sampling data, but only for single-transducer signals. If a transducer array is designed to use this sampling framework, a new problem of an overly complex sampling circuit system will appear. Therefore, this paper develops a novel array signal sparse sampling framework based on the pulse-positive sequence delay synthesis technique, which combines the ultrasonic array analog signals into a one-channel analog signal with a low degree of information freedom for sparse sampling. Although the degree of information freedom of the synthetic signal is greater than that of a single transducer signal, it still satisfies the conditions of signal sparse sampling and the volume of sampling data is still much smaller than that obtained using the traditional Nyquist sampling technique. An example of the experiment using a pure shear wave in the oblique incidence mode on a steel sample is provided, which indicates that eight-channel transducer array signals can be synthesized into one-channel signal, and then, the time delay and amplitude of the defect echo signals can be accurately estimated for each array element from the synthetic-signal sparse sampling data.

Reduced Linear Constrained Elastic and Viscoelastic Homogeneous Cosserat Media as Acoustic Metamaterials

We consider the reduced constrained linear Cosserat continuum, a particular type of a Cosserat medium, for three different material behaviors or symmetries: the isotropic elastic case, a special type of elastic transversely isotropic case, and the isotropic viscoelastic case. Such continua, in which stresses do not work on rates of microrotation gradients, behave as acoustic metamaterials for the (pure) shear waves and also for one branch of the mixed wave in the considered anisotropic material case. In elastic media, those waves do not propagate for frequencies exceeding a certain threshold, whence these media exhibit a single negative acoustic metamaterial behavior in this range. In the isotropic viscoelastic case, dissipation destroys the bandgap and favors wave propagation. This curious effect is, probably, due to the fact that the bandgap is associated not with the dissipation, but with the wave localization which can be destroyed by the viscosity. The dispersion curve is now decreasing in some part of the former bandgap, above a certain frequency, whence the medium is a double negative acoustic metamaterial. We prove the existence of a boundary wavenumber in the viscoelastic case and estimate its value. Below the characteristic frequency corresponding to the boundary of the elastic bandgap, the wave attenuation (logarithmic decrement) is a growing function of the viscous dissipation parameter. Above this frequency, the attenuation decreases as the viscosity increases.

Anomalous energy transport in laminates with exceptional points

Recent interest in metamaterials has led to a renewed study of wave mechanics in the different branches of physics. Elastodynamics involves a special intricacy, owing to the coupling between the volumetric and shear parts of the elastic waves. Through a study of in-plane waves traversing periodic laminates, we here show that this coupling can result with unusual energy transport. We find that the corresponding frequency spectrum contains modes which simultaneously attenuate and propagate, and demonstrate that these modes coalesce to purely propagating modes at exceptional points in the spectrum. These non-Hermitian degeneracies with propagating modes, which to date were realized by balancing energy gain and loss in systems with parity-time symmetry, are reported here in a purely elastic setting. We show that the laminate exhibits metamaterial features near these points, such as negative refraction, and beam steering and splitting. While negative refraction in laminates has been demonstrated before by considering pure shear waves impinging on an interface with multiple layers, here we realize it for coupled waves impinging on a simple single-layer interface. This feature, together with the appearance of exceptional points, are absent from the model problem of anti-plane shear waves which have no volumetric part, and hence from the mathematically identical electromagnetic waves in materials with positive refractive index. Thereby, our work paves the way for future applications such as asymmetric mode switches using a tangible elastic apparatus.

Boundary effects in discrete element method modeling of undrained cyclic triaxial and simple shear element tests

Cyclic triaxial and simple shear tests are both commonly used to study the behavior of granular assemblies subjected to dynamic loading. These two stress paths have also been simulated with the discrete element method (DEM) using varying boundary conditions. In this work, stiff boundaries and periodic boundaries were implemented in DEM simulations of constant volume cyclic triaxial and simple shear tests. The model response was analyzed at the specimen scale by comparing stress paths, decrease in mean effective stress, and number of cycles to failure across simulations. Particle scale analyses, including quantification of the contact force network, entropy of the local void ratio distribution, and spatial variability in the distribution of void ratio, contact number, and normal contact force are used to provide insight into model response. Simulation results imply that periodic boundaries are more appropriate for consolidating specimens to a homogenous state and minimizing boundary effects during cyclic loading. Significant uneven “squeezing” where particles are pinched between the horizontal and lateral plates in the corners of the specimen with stiff boundaries was observed in the cyclic simple shear test. After liquefaction initiation, increased entropy indicated a more disordered particle assembly in all cases. In addition, specimens in cyclic simple shear tests and cyclic triaxial tests behave significantly differently at the microscale. With the same level of deviatoric shear strain amplitude, the simple shear specimens liquefied in fewer cycles than the triaxial specimens. Cyclic triaxial tests are not able to produce pure shear waves, while cyclic simple shear tests do, micromechanically indicating the better suitability of cyclic simple shear tests to simulate shear wave induced liquefaction. The findings from this work will have implications for the future modeling of cyclic element tests using DEM.

Performance Study of a Torsional Wave Sensor and Cervical Tissue Characterization.

A novel torsional wave sensor designed to characterize mechanical properties of soft tissues is presented in this work. Elastography is a widely used technique since the 1990s to map tissue stiffness. Moreover, quantitative elastography uses the velocity of shear waves to achieve the shear stiffness. This technique exhibits significant limitations caused by the difficulty of the separation between longitudinal and shear waves and the pressure applied while measuring. To overcome these drawbacks, the proposed torsional wave sensor can isolate a pure shear wave, avoiding the possibility of multiple wave interference. It comprises a rotational actuator disk and a piezoceramic receiver ring circumferentially aligned. Both allow the transmission of shear waves that interact with the tissue before being received. Experimental tests are performed using tissue mimicking phantoms and cervical tissues. One contribution is a sensor sensitivity study that has been conducted to evaluate the robustness of the new proposed torsional wave elastography (TWE) technique. The variables object of the study are both the applied pressure and the angle of incidence sensor–phantom. The other contribution consists of a cervical tissue characterization. To this end, three rheological models have fit the experimental data and a static independent testing method has been performed. The proposed methodology permits the reconstruction of the mechanical constants from the propagated shear wave, providing a proof of principle and warranting further studies to confirm the validity of the results.

Waves in non-conducting continuum with frozen-in magnetization

The report reviews the new approach to the ferrohydrodynamics and magnetoelasticity, based on the concept of frozen-in magnetization. Up till now, there were well-known two frozen-in vector fields. The first one is the vorticity field was introduced by Helmgoltz. The second frozen-in field is a magnetic field in a perfectly conducting fluid was introduced by Alfven. The acoustic approximation of the ferrohydrodynamic equations with frozen-in magnetization allowed us to describe the experimental results on the anisotropy of the ultrasonic velocity in magnetized magnetic nanofluids and predict new waves: Alfven-type hydrodynamic wave and slow magnetosonic one. Alfven-type wave is accompanied by oscillations of the magnetization. It is shown that, in a non-conducting solid with a frozen-in magnetization, the propagation of three types of waves are possible: the longitudinal wave, pure shear wave, and new wave which have mixed form of Alfven-type and shear waves. The theoretical results are found to agree well with the experimental data on the dependence of the velocity of longitudinal and transverse waves in polycrystalline nickel on the magnetizing field strength. The set of dynamical equations that we have derived can be used to tackle many problems in ferrohydrodynamics and magnetoelasticity.

Laboratory measurement of elastic waves in Basalt rock

Abstract This paper reports the laboratory measurement of compression and shear waves in Basalt rock. Compression or primary waves were produced using a standard ultrasonic pulse velocity tester. However, it is well accepted that the production of pure shear waves in rock is difficult because of which suitable assumptions are made in the testing. Since bender elements have traditionally been used to produce pure shear at very small strains in soft materials, their use has been extended in this study to Basalt rock. The fabrication and setup of bender element tests is first discussed. The transducers were accommodated in pre-drilled slots at the two flat ends of the samples, the effect of which was carefully investigated using experimental and numerical studies. For 10–12 mm long piezoceramic plates, the shear wave velocity was found to reduce by about 5% because of the slots. The shear wave velocity was estimated using the first arrival method after removing the near field and the crosstalk effects from the output signal. The shear wave velocity ranged between 1.15 and 3.31 km/s and was nearly equal to one-half of the primary wave velocity. The results also show that the ratio of the shear wave velocity and the compression wave velocity was independent of the density and porosity of the Basalt rock. Because most of the rock was unweathered and compact and, its cavities filled with secondary minerals such as zeolites, calcites and silicates, it is unlikely that the high overburden pressures would affect the observed wave velocities.

Computational Feasibility Study of a Torsional Wave Transducer for Tissue Stiffness Monitoring

A torsional piezoelectric ultrasonic transducer design is proposed to measure shear moduli in soft tissue with direct access availability, using shear wave elastography technique. The measurement of shear moduli of tissues is a challenging problem, mainly derived from a) the difficulty of isolating a pure shear wave, given the interference of multiple waves of different types (P, S, even guided) emitted by the transducers and reflected in geometric boundaries, and b) the highly attenuating nature of soft tissular materials. An immediate application, overcoming these drawbacks, is the measurement of changes in cervix stiffness to estimate the gestational age at delivery. The design has been optimized using a finite element model (FEM) and a semi-analytical estimator of the probability of detection (POD) to determine a suitable geometry, materials and generated waves. The technique is based on the time of flight measurement between emitter and receiver, to infer shear wave velocity. Current research is centered in prototype testing and validation. The geometric optimization of the transducer was able to annihilate the compressional wave emission, generating a quite pure shear torsional wave. Currently, mechanical and electromagnetic coupling between emitter and receiver signals are being the research focus. Conclusions: the design overcomes the main described problems. The almost pure shear torsional wave along with the short time of flight avoids the possibility of multiple wave interference. This short propagation distance reduce the effect of attenuation, and allow the emission of very low energies assuring a good biological security for human use. Keywords—Cervix ripening, preterm birth, shear modulus, shear wave elastography, soft tissue, torsional wave.

Electromagnetically driven torsional resonators for viscosity and mass density sensing applications

In this contribution a conceptual study for torsional oscillators, which are electromagnetically driven and read out, is presented. The aim is to experimentally investigate the basic feasibility of a torsional resonator with application to viscosity and mass density sensing in liquids. Such a device is particularly interesting as cylindrical, torsional resonators for fluid sensing applications are hardly reported but unlike many other devices, yield pure shear wave excitation in the liquid. The design of first conceptual demonstrators for measurements in air as well as in liquids and their benefits and disadvantages is discussed in detail. A closed form as well as a reduced order model and measurement results obtained with first demonstrators are presented.

Longitudinal shear wave and transverse dilatational wave in solids.

Dilatation wave involves compression and extension and is known as the curl-free solution of the elastodynamic equation. Shear wave on the contrary does not involve any change in volume and is the divergence-free solution. This letter seeks to examine the elastodynamic Green's function through this definition. By separating the Green's function in divergence-free and curl-free terms, it appears first that, strictly speaking, the longitudinal wave is not a pure dilatation wave and the transverse wave is neither a pure shear wave. Second, not only a longitudinal shear wave but also a transverse dilatational wave exists. These waves are shown to be a part of the solution known as coupling terms. Their special motion is carefully described and illustrated.

Decoupling of plane waves in porous piezoelectric materials

In this paper, the analytical and numerical studies of two-dimensional wave propagation in porous piezoelectric materials (PPMs) are carried out. The decoupling of waves, in such materials for various crystal classes, is studied analytically for different coordinate planes. It is found that, for wave propagation in a plane, the system is decoupled in some crystal classes, whereas it remains coupled in other crystal classes. It is established that the decoupled pure-shear wave, propagating in a PPM, can be stiffened or unstiffened with piezoelectric effects even if the PPM belongs to the same symmetry group but has a different crystal class. The skewing angles and mutual angles between the polarization directions of different waves are also computed numerically.

Concept Studies of Torsional Resonators for Viscosity and Mass Density Sensing Applications

Abstract In this contribution a concept for a torsional oscillators which is electromagnetically driven and read-out is presented. The aim is to experimentally investigate a principle feasibility for a torsional resonator with application for viscosity and mass density sensing. Such a device is particularly interesting as cylindrical torsional resonators for fluid sensing applications are hardly reported but unlike other devices, would yield pure shear wave excitation in the liquid. The design of first conceptual prototypes and measurement results obtained with these devices are presented.

Macro-fiber composite–based structural health monitoring system for axial cracks in cylindrical structures

In cylindrical structures such as pipelines and pressure vessels, cracks are most likely to occur along the longitudinal (axial) direction, and they can be fatal to the serviceability of the structures. However, the conventional wave propagation–based crack detection techniques are not sensitive to this type of cracks. This article focuses on the identification of axial cracks in cylindrical structures using torsional wave generated by piezoelectric macro-fiber composite. The first-order torsional wave, which is a nondispersive pure shear wave propagating at a fixed wave speed, is utilized in this work because, intuitively, it is more sensitive to axial cracks than the longitudinal waves. The crack position is determined by the time of flight of the wave pack, while the crack propagation is monitored by measuring the variation in the crack-induced disturbances. Numerical simulations have been conducted to investigate the feasibility of the proposed method, and experimental tests on aluminum pipes have been carried out for verification. Macro-fiber composite transducers oriented at 45° against the axis of the specimen were used to generate and receive the torsional waves. The experimental results demonstrated that the crack position can be identified, and its growth can be well monitored with the proposed approach.

Annealing effect on the generation of dual mode acoustic waves in inclined ZnO films

ZnO films with different inclined angles on steel substrates were sputter-deposited by changing the substrate tilt angle during deposition and then used to fabricate ZnO film ultrasonic transducers. The ultrasonic performance of those devices was characterized using a standard pulse-echo method. A dual mode wave with both longitudinal and shear wave components was detected from the ZnO device at 0° inclined angle. At a columnar inclined angle of 31°, longitudinal wave excitation was suppressed with a nearly pure shear wave detected. Post annealing of the ZnO film improved the crystallinity and decreased the film stress. The dispersion of the received echoes was observed when the grain sizes of ZnO films were increased after annealing. The frequency components of the waveforms were analyzed and identified using a short time Fourier transform. Post-annealing of the ZnO films changed the primary frequency and enhanced the propagation of the relative high-frequency acoustic wave.

On the use of the Norwegian Geotechnical Institute's prototype seabed-coupled shear wave vibrator for shallow soil characterization - I. Acquisition and processing of multimodal surface waves

SUMMARY Pure shear wave data are only very rarely acquired for offshore site investigations and exploration. Here, we present details of a novel, seabed-coupled, shear wave vibrator and field data recorded by a densely populated, multicomponent ocean-bottom cable, to improve shallow soil characterization. The prototype shear wave vibrator uses vibroseis technology adopted for marine environments through its instalment on top of a suction anchor, assuring seabed coupling in combination with self-weight penetration. The prototype is depth rated to 1500 m water depth, and can be rotated while installed in the seabed. The philosophy is to acquire fully complementary seismic data to conventional P- and P-to-S-converted waves, in particular for 2-D profiling, VSP (vertical seismic profiling) or monitoring purposes, thereby exploiting advantages of shear waves over compressional waves for determining, for example, anisotropy, small-strain shear modulus and excess pore pressures/effective stress. The source was primarily designed for reservoir depths. However, significant energy is emitted as surface waves, which provide detailedgeotechnicalinformationthroughmappingofshearwavevelocitiesinpotentiallyhigh resolution of the upper soil units. To fully utilize pure shear wave content, a proper analysis of surface waves is paramount, due to the proximity of surface wave propagation speed with shear wave velocities. The experiment was carried out in the northern North Sea in 364 m water depth. Cable dragging was necessary to obtain close receiver spacing (2.5 m effective spacing), with total line length of 600 m. Frequency–waveform transforms reveal both Scholte and Love waves. Up to six surface wave modes are identified, that is, fundamental mode and several higher surface wave modes. The occurrence of these two different dispersive surface wave types with well-resolved higher modes allows for a unique analysis and inversion scheme for highresolution mapping of physical properties in the shallow subsurface as well as anisotropy, which is discussed in an accompanying paper. The data presented in this paper are thus a unique (long and densely populated receiver array allows for multimodal Love and Scholte surface waves from the marine environment) but challenging (marine operations) marine data set.