Transverse Shear Waves Research Articles

Rotational waves in layered solids with many sliding layers

Wave propagation in layered materials with free sliding layers is analysed in 2D using anisotropic Cosserat continuum formulation, which accounts for layer bending. It is found that the 2D Cosserat continuum supports propagation of waves of three types: usual longitudinal and transverse (shear) waves and a new one – the rotational wave. The velocities of the longitudinal and transverse waves strongly depend upon the direction of their propagation with respect to the layering, but exhibit only marginal dispersion. Opposite to this the rotational wave is strongly dispersive: its velocity increases proportionally to the wavelength, but marginally depends upon the propagation direction. A prominent feature of the rotational wave is its extremely high velocity that can be considerably greater than the velocity of longitudinal wave propagating in the material of the layers. This high velocity can be used to distinguish the rotational waves. Furthermore, this property allows detecting areas of sliding, which is important in both materials engineering for non-destructive control and in geoscience for monitoring the state of the Earth's crust.

Measurement of anisotropic mechanical properties in porcine brain white matter ex vivo using magnetic resonance elastography.

The mechanical properties of brain tissue, particularly those of white matter (WM), need to be characterized accurately for use in finite element (FE) models of brain biomechanics and traumatic brain injury (TBI). Magnetic resonance elastography (MRE) is a powerful tool for non-invasive estimation of the mechanical properties of soft tissues. While several studies involving direct mechanical tests of brain tissue have shown mechanical anisotropy, most MRE studies of brain tissue assume an isotropic model. In this study, an incompressible transversely isotropic (TI) material model parameterized by minimum shear modulus (μ2), shear anisotropy parameter (ϕ), and tensile anisotropy parameter (ζ) is applied to analyze MRE measurements of ex vivo porcine white matter (WM) brain tissue. To characterize shear anisotropy, "slow" (pure transverse) shear waves were propagated at 100, 200 and 300Hz through sections of ex vivo brain tissue including both WM and gray matter (GM). Shear waves were found to propagate with elliptical fronts, consistent with TI material behavior. Shear wave fields were also analyzed within regions of interest (ROI) to find local shear wavelengths parallel and perpendicular to fiber orientation. FE simulations of a TI material with a range of plausible shear modulus (μ2) and shear anisotropy parameters (ϕ) were run and the results were analyzed in the same fashion as the experimental case. Parameters of the FE simulations which most closely matched each experiment were taken to represent the mechanical properties of that particular sample. Using this approach, WM in the ex vivo porcine brain was found to be mildly anisotropic in shear with estimates of minimum shear modulus (actuation frequencies listed in parenthesis): μ2= 1.04 ± 0.12 kPa (at 100Hz), μ2= 1.94 ± 0.29 kPa (at 200Hz), and μ2= 2.88 ± 0.34 kPa (at 300Hz) and corresponding shear anisotropy factors of ϕ= 0.27 ± 0.09 (at 100Hz), ϕ= 0.29 ± 0.14 (at 200Hz) and ϕ= 0.34 ± 0.13 (at 300Hz). Future MRE studies will focus on tensile anisotropy, which will require both slow and fast shear waves for accurate estimation.

Longitudinal shear wave imaging for elasticity mapping using optical coherence elastography.

Shear wave measurements for the determination of tissue elastic properties have been used in clinical diagnosis and soft tissue assessment. A shear wave propagates as a transverse wave where vibration is perpendicular to the wave propagation direction. Previous transverse shear wave measurements could detect the shear modulus in the lateral region of the force; however, they could not provide the elastic information in the axial region of the force. In this study, we report the imaging and quantification of longitudinal shear wave propagation using optical coherence tomography to measure the elastic properties along the force direction. The experimental validation and finite element simulations show that the longitudinal shear wave propagates along the vibration direction as a plane wave in the near field of a planar source. The wave velocity measurement can quantify the shear moduli in a homogeneous phantom and a side-by-side phantom. Combining the transverse shear wave and longitudinal shear wave measurements, this system has great potential to detect the directionally dependent elastic properties in tissues without a change in the force direction.

The geometric effects of a woodpecker’s hyoid apparatus for stress wave mitigation

In this study a woodpecker’s hyoid apparatus was characterized to determine its impact mitigation mechanism using finite element (FE) analysis. The woodpecker’s hyoid apparatus, comprising bone and muscle, has a unique geometry compared to those of other birds. The hyoid starts at the beak tip, surrounds the woodpecker’s skull, and ends at the upper beak/front head intersection while being surrounded by muscle along the whole length. A FE model of the hyoid apparatus was created based on the geometry, microstructure, and mechanical properties garnered from our experimental measurements. We compared the impact mitigation capabilities of the hyoid apparatus with an idealized straight cylinder and a tapered cylinder. The results showed that the hyoid geometry mitigated a greater amount of pressure and impulse compared to the straight or tapered cylinders. The initially applied longitudinal wave lost its strength from attenuation and conversion to transverse shear waves. This is due to the spiral curvature and tapered geometry, which induced lateral displacement in the hyoid bone. The lateral displacement of the bony hyoid induced strains on the adjacent muscle, where the energy dissipated due to the muscle’s viscoelasticity. Quantitatively, as the stress wave traveled from the anterior to the posterior end of the hyoid apparatus, its pressure decreased 75% and the associated impulse decreased 84%. The analysis of the woodpecker’s hyoid apparatus provides a novel perspective on impact mitigation mediated by a spiral-shaped structure and viscoelastic biocomposite.

Formation and evolution of vortices in a collisional strongly coupled dusty plasma

Formation and evolution of vortices are studied in a collisional strongly coupled dusty plasma in the framework of a Generalized Hydrodynamic model (GH). Here we mainly present the nonlinear dynamical response of this strongly coupled system in presence of dust-neutral collisional drag. It is shown that the interplay between the nonlinear elastic stress and the dust-neutral collisional drag results in the generation of non-propagating monopole vortex for some duration before it starts to propagate like transverse shear wave. It is also found that the interaction between two unshielded monopole vortices having both same (co-rotating) and opposite (counter rotating) rotations result in the formation of two propagating dipole vortices of equal and unequal strength respectively. These results will provide some new understanding on the transport properties in such a strongly coupled system. The numerical simulation is carried out using a de-aliased doubly periodic pseudo-spectral code with Runge–Kutta–Gill time integrator.

Experimental characterization and constitutive modeling of the mechanical behavior of molybdenum under electromagnetically applied compression-shear ramp loading

Magnetically applied pressure-shear (MAPS) is a new experimental technique that provides a platform for direct measurement of material strength at extreme pressures. The technique employs an imposed quasi-static magnetic field and a pulsed power generator that produces an intense current on a planar driver panel, which in turn generates high amplitude magnetically induced longitudinal compression and transverse shear waves into a planar sample mounted on the drive panel. In order to apply sufficiently high shear traction to the test sample, a high strength material must be used for the drive panel. Molybdenum is a potential driver material for the MAPS experiment because of its high yield strength and sufficient electrical conductivity. To properly interpret the results and gain useful information from the experiments, it is critical to have a good understanding and a predictive capability of the mechanical response of the driver. In this work, the inelastic behavior of molybdenum under uniaxial compression and biaxial compression-shear ramp loading conditions is experimentally characterized. It is observed that an imposed uniaxial magnetic field ramped to approximately 10 T through a period of approximately 2500 μs and held near the peak for about 250 μs before being tested appears to anneal the molybdenum panel. In order to provide a physical basis for model development, a general theoretical framework that incorporates electromagnetic loading and the coupling between the imposed field and the inelasticity of molybdenum was developed. Based on this framework, a multi-axial continuum model for molybdenum under electromagnetic loading is presented. The model reasonably captures all of the material characteristics displayed by the experimental data obtained from various experimental configurations. In addition, data generated from shear loading provide invaluable information not only for validating but also for guiding the development of the material model for multiaxial loadings.

Sub- and super-luminar propagation of structures satisfying Poynting-like theorem for incompressible generalized hydrodynamic fluid model depicting strongly coupled dusty plasma medium

The strongly coupled dusty plasma has often been modelled by the Generalized Hydrodynamic (GHD) model used for representing visco-elastic fluid systems. The incompressible limit of the model which supports transverse shear wave mode is studied in detail. In particular, dipole structures are observed to emit transverse shear waves in both the limits of sub- and super-luminar propagation, where the structures move slower and faster than the phase velocity of the shear waves, respectively. In the sub-luminar limit the dipole gets engulfed within the shear waves emitted by itself, which then backreacts on it and ultimately the identity of the structure is lost. However, in the super-luminar limit the emission appears like a wake from the tail region of the dipole. The dipole, however, keeps propagating forward with little damping but minimal distortion in its form. A Poynting-like conservation law with radiative, convective, and dissipative terms being responsible for the evolution of W, which is similar to “enstrophy” like quantity in normal hydrodynamic fluid systems, has also been constructed for the incompressible GHD equations. The conservation law is shown to be satisfied in all the cases of evolution and collision amidst the nonlinear structures to a great accuracy. It is shown that monopole structures which do not move at all but merely radiate shear waves, the radiative term, and dissipative losses solely contribute to the evolution of W. The dipolar structures, on the other hand, propagate in the medium and hence convection also plays an important role in the evolution of W.

Axisymmetric diffraction of a cylindrical transverse wave by a viscoelastic spherical inclusion

In this paper, the scattering and diffraction of a cylindrical transverse shear wave in a viscoelastic isotropic medium by a spherical heterogeneity is analytically solved. The waves are generated by the harmonic longitudinal oscillations of the cylinder walls. The spherical inclusion is located at the radial center of the cylinder and differs from the cylindrical material only in its complex shear modulus. Small amplitude motion is assumed, such that linear system theory is valid. By employing multi-pole expansions, the incident and scattered wave fields are each defined in both cylindrical and spherical coordinates allowing for the satisfaction of the boundary conditions at the surfaces of these multiply connected bodies. The solution involves an infinite sum of improper integrals, which are evaluated numerically. The wave field is determined for a hydrogel (alginate) bead suspended in a different hydrogel (agarose) that fills a glass test tube. Numerical examples showing the effect on displacement fields of varying the stiffness of the inclusion are presented. This solution is further validated with a finite element simulation showing excellent agreement with the analytic results.

Stability of an elliptical vortex in a strongly coupled dusty plasma

The stability of a long scale equilibrium vortex structure to short scale perturbations is studied in a strongly coupled dusty plasma in the framework of a generalized hydrodynamic model. It is shown that the free energy associated with the velocity shear of the vortex can drive secondary instabilities consisting of transverse shear waves when the resonance condition between the vortex rotation frequency and the secondary wave frequency is met. Such a process can transfer energy from the long scale vortex to the short scale secondary wave and thereby provide a saturation mechanism for long scale vortices in plasmas in a manner analogous to that in neutral fluids.

Two-dimensional nonlinear shear waves in materials having hexagonal lattice structure

An asymptotic procedure is developed to obtain governing two-dimensional nonlinear equations as a result of the continuum limits of the original discrete hexagonal lattice model. Possible continualization is analyzed on the basis of linearized discrete equations. New weakly nonlinear continuum equations for plane and weakly transverse disturbed shear waves in a hexagonal lattice are obtained using different continuum limits. It is shown that nonlinear plane shear waves are described different from the model of isotropic continuum.

Turbulence in strongly coupled dusty plasmas using generalized hydrodynamic description

The properties of decaying turbulence have been studied with the help of a Generalized Hydrodynamic (GHD) fluid model in the context of strongly coupled dusty plasma medium in two dimensions. The GHD model treats the strongly coupled dusty plasma system as a visco-elastic medium. The incompressible limit of the GHD model is considered here. The studies carried out here are, however, applicable to a wider class of visco-elastic systems, and are not merely confined to the dusty plasma medium. Our simulations studies show that an initial spectrum that is confined in a limited domain of wave numbers becomes broad, even when the Reynold's number is much less than the critical value required for the onset of turbulence in Newtonian fluids. This is a signature of elastic turbulence, where Weissenberg's number also plays an important role on the onset of turbulence. This feature has been observed in several experiments. It is also shown that the existence of memory relaxation time parameter and the transverse shear wave inhibit the normal process (for 2-D systems) of inverse spectral cascade in this case. A detailed simulation study has been carried out for the understanding of this inhibition.

Exact propagating nonlinear singular disturbances in strongly coupled dusty plasmas

The dynamical response of the strongly coupled dusty plasma medium has recently been described by utilizing the Generalized Hydrodynamic (GHD) model equations. The GHD equations capture the visco-elastic properties of the medium and have been successful in predicting a host of phenomena (e.g., existence of novel transverse shear waves in the fluid medium, modification of longitudinal wave dispersion by elastic effects, etc.) which have found experimental confirmation. In this paper, the nonlinear longitudinal response of the medium governed by GHD equations in strong coupling limit is discussed analytically. The structure of the equations rules out the balance between dispersion and nonlinearity, thereby, forbidding soliton formation. However, a host of new varieties of nonlinear solutions are found to exist, which have singular spatial profiles and yet have conservative properties. For instance, existence of novel conservative shock structures with zero strength is demonstrated, waves whose breaking produces no dissipation in the medium are observed, propagating solutions which produce cusp like singularities can exist and so on. It is suggested that simulations and experiments should look for these novel nonlinear structures in the large amplitude strong coupling limit of longitudinal disturbances in dusty plasmas.

Kelvin-Helmholtz instability in dusty plasma medium: Fluid and particle approach

The Kelvin-Helmholtz (KH) instability is studied in a two dimensional strongly coupled dusty plasma medium using a fluid approach as well as through a molecular dynamic (MD) simulation. For the fluid description the generalized hydrodynamic (GH) model which treats the strongly coupled dusty plasma as a visco-elastic fluid is adopted. For the MD studies the ensemble of particles are assumed to interact through a Yukawa potential. Both the approaches predict a stabilization of the KH growth rate with an increase in the strong coupling parameter. The present study also delineates the temporal evolution and the interaction of transverse shear waves with the collective dynamics of the dusty plasma medium within the framework of both these approaches.

Spatial localization of linear elastic waves in composite materials with defects

AbstractWe study the phenomenon of a spatial localization of elastic waves in periodic composite materials with local defects. The wave spectrum in heterogeneous composite solids includes pass and stop frequency bands. If the frequency of the signal falls within a stop band, the group velocity vanishes and the wave attenuates exponentially. In such a case, a local perturbation of the microstructure may lead to the localization of the wave energy in the vicinity of the defect. Longitudinal tension‐compression waves in a layered composite and transverse antiplane shear waves in a unidirectional fibrous composite are considered. Local perturbations of the density and of the volume fractions of the components are taken into account. The analysis is based on the transfer‐matrix method and on the plane‐wave expansions method. As the result, the frequencies of the wave localization and the corresponding attenuation factors are determined.

Mode-converted diffuse ultrasonic backscatter

Diffuse ultrasonic backscatter describes the scattering of elastic waves from interfaces within heterogeneous materials. Previously, theoretical models have been developed for the diffuse backscatter of longitudinal-to-longitudinal (L-L) wave scattering within polycrystalline materials. Following a similar formalism, a mode-conversion scattering model is presented here to quantify the component of an incident longitudinal wave that scatters and is converted to a transverse (shear) wave within a polycrystalline sample. The model is then used to fit experimental measurements associated with a pitch-catch transducer configuration performed using a sample of 1040 steel. From these measurements, an average material correlation length is determined. This value is found to be in agreement with results from L-L scattering measurements and is on the order of the grain size as determined from optical micrographs. Mode-converted ultrasonic backscatter is influenced much less by the front-wall reflection than an L-L measurement and it provides additional microstructural information that is not accessible in any other manner.

Stress and strain-inertia gradient elasticity in free vibration analysis of single walled carbon nanotubes with first order shear deformation shell theory

A gradient-enriched shell formulation is introduced in the present study based on the first order shear deformation shell model and the stress gradient and strain-inertia gradient elasticity theories are used for dynamic analysis of single walled carbon nanotubes. It provides extensions of the first order shear deformation shell formulation with additional higher-order spatial derivatives of strains and stresses. The higher-order terms are introduced in the formulation by using the Laplacian of the corresponding lower-order terms. The proposed shell formulation includes two length scale size parameters related to the strain gradients and inertia gradients. The effects of the transverse shear, aspect ratio, circumferential and half-axial wave numbers and length scale parameters on different vibration modes of the single-walled carbon nanotubes are elucidated. The results are also compared with those obtained from a classical shell theory with Sanders–Koiter strain-displacement relationships.

Nonlinear shear wave in a non Newtonian visco-elastic medium

An analysis of nonlinear transverse shear wave has been carried out on non-Newtonian viscoelastic liquid using generalized hydrodynamic model. The nonlinear viscoelastic behavior is introduced through velocity shear dependence of viscosity coefficient by well known Carreau-Bird model. The dynamical feature of this shear wave leads to the celebrated Fermi-Pasta-Ulam problem. Numerical solution has been obtained which shows that initial periodic solutions reoccur after passing through several patterns of periodic waves. A possible explanation for this periodic solution is given by constructing modified Korteweg de Vries equation. This model has application from laboratory to astrophysical plasmas as well as in biological systems.

Shear waves in an inhomogeneous strongly coupled dusty plasma

The properties of electrostatic transverse shear waves propagating in a strongly coupled dusty plasma with an equilibrium density gradient are examined using the generalized hydrodynamic (GH) equation. In the usual kinetic limit, the resulting equation has similarity to zero energy Schrodinger’s equation. This has helped in obtaining some exact eigenmode solutions in both Cartesian and cylindrical geometries for certain nontrivial density profiles. The corresponding velocity profiles and the discrete eigenfrequencies are obtained for several interesting situations and their physics discussed.

The analysis of the impact response of a thin plate via fractional derivative standard linear solid model

The impact of a rigid body upon an infinite isotropic plate is investigated for the case when the viscoelastic features of the plate represent themselves only in the place of contact and are governed by the standard linear solid model with fractional derivatives. Thus, the problem concerns the shock interaction of the dropping mass and the target, wherein instead of the Hertz contact law the generalized fractional derivative standard linear solid law is employed as a law of interaction. The part of the plate beyond the contact domain is assumed to be elastic, and its behaviour is described by the equations of motion which take rotary inertia and shear deformations into account. It is assumed that transient waves generate in the plate at the moment of impact, the influence of which on the contact domain is considered using the theory of discontinuities. To determine the desired values behind the transverse shear wave front, one-term ray expansions are used, as well as the equations of motion of the falling mass and the contact region. As a result, we are led to a set of two linear differential equations, the solution of which is found analytically by the Laplace transform and by the Euler substitution method. This allows the contact force to be determined as a function of time.

Shear wave vortex solution in a strongly coupled dusty plasma

The properties of electrostatic transverse shear waves in a strongly coupled dusty plasma are examined using the nonlinear version of the generalized hydrodynamic equation. In the kinetic limit, it is shown that strongly coupled plasmas support localized dipolar vortexlike solutions with amplitude modulated periodically.