Surface Of Half-space Research Articles

Reflection of plane waves from the stress-free boundary of a nonlocal elastic solid half-space containing double porosity

Reflection phenomena of a set of coupled longitudinal waves/shear wave striking obliquely against the mechanically stress-free boundary surface of a uniform nonlocal elastic solid half-space with double porosity structure have been investigated. Employing the appropriate nonlocal stress-free boundary conditions at the surface, the formulae for reflection coefficients and their corresponding energy ratios have been derived for the dissipative and non-dissipative models. Their variations have been computed against the angle of incidence for a specific model. It is found that all the reflection coefficients depend upon the presence of the nonlocality, porosities of both types, frequency and angle of incidence. Their impact on various coefficients is also studied numerically and depicted graphically.

Longitudinal motion of focused shear wave beams in soft elastic media.

Shear waves are employed in medical ultrasound imaging because they reveal variations in viscoelastic properties of soft tissue. Frequencies below 1 kHz are required due to the substantially higher attenuation and lower propagation speeds than for compressional waves. Shear waves exhibiting particle motion in the direction of propagation, referred to as longitudinal shear waves, can be generated with longitudinal motion of a circular disk on the surface of a soft elastic medium. This approach permits imaging of the longitudinal shear wave with a conventional ultrasound transducer that is coaxial with the source of the shear wave. Presented here is a mathematical model describing the complete wave field generated by displacement at the surface of an isotropic elastic half-space. Numerical simulations are shown for longitudinal, transverse, torsional, and radial source polarizations, with emphasis on focused longitudinal shear waves. Predictions are consistent with measurements of light beams revealing that the longitudinal electric field component produces a smaller focal spot than the transverse field component [Dorn, Quabis, and Leuchs, Phys. Rev. Lett. 91, 233901 (2003)]. Simulations are compared with preliminary measurements of a focused longitudinal shear wave beam generated in a soft tissue phantom by longitudinal motion of a spherically concave piston.

Numerical Analysis of Seismic Pounding between Adjacent Buildings Accounting for SSI

The structural pounding caused by an earthquake may damage structures and lead to their collapse. This study is focused on the pounding between two adjacent asymmetric structures with different dynamic properties resting on the surface of an elastic half-space. An exploration of the relationship between the effects of the seismic analysis with the impact response to the torsional pounding between adjacent buildings under different SSI effects has been presented. In this paper, the authors have proposed a procedure for analyzing the response for adjacent buildings subjected to the pounding effects, considering systems with multiple degrees of freedom and modal equations of motion with four types of soil. All the calculations have been performed based on the fourth-order Runge–Kutta method. The novelty of the present study is related to the fact that the rigorous and approximate methods are used to examine the effects of pounding and SSI simultaneously. As a result, these two methods have been thoroughly investigated for both effects and the results have been compared. The results show that the approximate method produces results that are slightly different from those obtained by the rigorous direct integration method in the case of small SSI effects due to an increase in the pounding force. The efficiency of the method is also validated using numerical examples.

On the effective medium model of the interaction of Rayleigh waves with mass–spring oscillators on the surface

The interaction of Rayleigh surface waves with a random arrangement of oscillators on the surface of an elastic half-space is analyzed with the help of the surface elastodynamic Green’s function formalism and the effective medium model valid in the long-wavelength limit is derived. It is shown that in order to apply the central assumption of the model, namely, that the local surface displacement at the contact point of an oscillator is equal to the average effective medium displacement, the spring constant of the oscillators must be corrected to include the stiffness of the contact calculated in the quasistatic approximation. The previous results pertaining to the interaction of Rayleigh and Lamb waves with microspheres are shown to be correct despite the apparent inconsistencies in the assumptions.

High efficiency generation of S-wave via a transmissive binary coding metasurface based on machine learning approach

The well-known Lamb’s problem deals with the generation of surface and cylindrical body waves in an elastic half-space due to vertical point excitations on its surface. Owing to the potentials of S-waves in various engineering applications, the physical response of Lamb’s problem has inspired the authors to innovate a new idea for the generation of such S-waves by applying vertical excitations on an elastic half-space. An analytical model based on Lamb’s problem is established to study the elastic half-space wave motions under a vertical point load. Due to mathematical complexities in attaining the corresponding displacement fields, a machine learning approach is proposed to analyze the entire wavefield of the elastic half-space. Through a trained neural network, it is discovered that exertion of multiple vertical excitations with opposite magnitudes can induce beam splitting of transmitted S-waves. By optimizing the parameters, low R-waves and P-waves but high S-waves are observed in the wavefields predicted by the trained neural network. Analogous to excitations with opposite magnitudes, “0” and “1” coding elements that have out-of-phase response are introduced as a binary coding metasurface to investigate the feasibility of a binary coding metasurface in realizing beam splitting of transmitted S-waves via an input of P-waves. The beam splitting of transmitted S-waves is observed in both aluminum and elastomer thin plates. The outcome is significant and it proves that wave mode conversion of P-waves to S-waves is actualizable in both hard and soft materials through a transmissive binary coding metasurface, thus providing an avenue for various shear wave applications.

Incident fast and slow wave at the surface of porous micropolar thermoelastic half-space

The propagation of plane elastic waves in the porous micropolar thermoelastic medium have been investigated on considering the interaction of wryness tensor and porous distribution. The basic equations of motion are treated to get four coupled fast (i.e. longitudinal) waves and two coupled slow (i.e. shear) waves with their corresponding wavenumbers. The plane waves has been considered to be incident at the impedance boundary surface of porous micropolar thermoelastic half-space considering the cross term between wryness and porous distribution. Using dispersion relation, the nature of dispersive waves after striking the impedance boundary and stress free boundary surface were investigated and formulated to compute the reflection coefficients (i.e. amplitude ratio and energy ratios) of different reflected waves. For particular model, the wavenumbers of the propagated waves and the reflection coefficients of incident coupled fast and slow waves are calculated numerically.

The effect of contact conditions on the performance of flexural seismic metasurfaces

Plane-strain motion of a flexural seismic metasurface in the form of a regular array of thin Kirchhoff plates attached to the surface of an elastic half-space is analysed. Two types of contact conditions, including simply supported plates and plates moving along horizontal rails are studied. Dispersion of time harmonic waves is investigated both asymptotically and numerically. A major effect of the contact conditions on metasurface behaviour is discovered. In particular, it is shown that frequency band gaps are not the feature of the array composed of simply supported plates. It is also demonstrated that the scaling laws, expressed through geometric and material problem parameters, drastically differ from each other for two considered setups.

Effects of surface stress on the indentation response of an elastic half-space

Nanoindentation technique, which is based on Hertzian contact theory and Sneddon's solutions for the contact between a rigid indenter and an elastic half-space, has been widely investigated due to its practical importance in localized mechanical test of submicron structures. However, both the Hertzian contact theory and Sneddon's solutions do not take into account the contribution of surface stress to contact deformation. In this work, we study the effect of surface stress without the out-of-plane term on the indentation deformation of an elastic half-space by rigid, axisymmetric indenters, including flat-ended cylindrical, conical and spherical indenters. In contrast to classical theories, which are based on physically admissible condition of finite normal stress at contact edge, an alternative condition of geometrical continuity at contact edge is used to determine the contact radius. The numerical results reveal the combinational effects of the surface stress and Poisson's ratio on the load-displacement relationship for the indentation of the elastic half-space. The surface stress causes significant change in shear stress and modest variation in normal stress in the direction normal to the surface of the elastic half-space. The numerical method used in this work offers a feasible approach to study the effect of surface stress on the contact deformation of elastic substrates.

Reflection of plane waves at the stress-free/rigid surface of a micro-mechanically modeled Piezo-electro-magnetic fiber-reinforced half-space

The development of fiber-reinforced composites (FRCs) is ever-increasing for better vibrational energy absorption within a material, reducing transmission of noise/vibration to neighboring structures, etc. The integrity of these structural components is tackled using Non-destructive testing (NDT)/Non-destructive evaluation (NDE) methods based on wave reflection through transducers. Owing to such factors, the present work has two primary objectives-firstly, to develop the micro-mechanics model of an orthotropic Piezo-Electro-Magnetic Fiber-Reinforced Composite (PEMFRC), which consists of BaTiO as fiber constituent and CoFeO as matrix constituent. Secondly, to analyze plane wave (qP/qSV) reflection at stress-free/rigid surface of a solid PEMFRC half-space. For an incident plane wave, four waves, namely quasi-longitudinal (qP), quasi-transverse (qSV), electroacoustic (EA), and magnetoacoustic (MA) waves, get reflected. The propagation directions of all reflected waves for incident qP/qSV wave are graphically demonstrated. Electrically Open Magnetically Open (EOMO), Electrically Short Magnetically Short (ESMS), Electrically Open Magnetically Short(EOMS), and Electrically Short Magnetically Open (ESMO) boundaries, for both Stress-free and Rigid surfaces, are considered. Using secular equations along with suitable boundary conditions, the amplitude ratios are derived using Cramer's rule. Furthermore, using the energy flux relation and expressions of amplitude ratios, the energy ratios of all reflected waves, interaction energy, and net energy are derived, which exhibit the influence of the existing parameters. Some special cases of the problem are discussed, and the Law of Conservation of Energy is validated.

A multiple scattering formulation for finite-size flexural metasurfaces

We provide an analytical formulation to model the propagation of elastic waves in a homogeneous half-space supporting an array of thin plates. The technique provides the displacement field obtained from the interaction between an incident wave generated by a harmonic source and the scattered fields induced by the flexural motion of the plates. The scattered field generated by each plate is calculated using an ad-hoc set of Green’s functions. The interaction between the incident field and the scattered fields is modelled through a multiple scattering formulation. Owing to the introduction of the multiple scattering formalism, the proposed technique can handle a generic set of plates arbitrarily arranged on the half-space surface. The method is validated via comparison with finite element simulations considering Rayleigh waves interacting with a single and a collection of thin plates. Our framework can be used to investigate the interaction of vertically polarized surface waves and flexural resonators in different engineering contexts, from the design of novel surface acoustic wave devices to the interpretation of urban vibration problems.

Modelling of third body effect on rolling contact fatigue damage

The contact problem for a rolling elastic sphere and an elastic half-space covered with a viscoelastic layer (third body) is considered. The layer properties in normal and tangential directions are described by the Maxwell model. Based on the macroscopic approach of the contact fatigue damage accumulation the rate of damage accumulation in an elastic half-space is assumed depending on the amplitude values of the principal shear stresses or the equivalent von Mises stress. The influence of the relative longitudinal slippage, the sliding friction coefficient and the properties of the intermediate layer on the internal stresses and at the depth of contact fatigue cracks initiation is studied. The results indicate that the maximum damage can occur both on the surface of the elastic half-space and at a certain depth. The increase of the friction coefficient or relative longitudinal slippage leads to reaching the maximum value of the principal shear stress amplitude on the surface of the half-space, which results in contact fatigue surface fracture (wear). The results also demonstrate that the presence of a viscoelastic layer leads to a decrease of the principal shear stress amplitude within the elastic half-space.

Impact of variable thermal conductivity and diffusivity on two temperature fractional thermodiffusive elastic half space with dual phase lags

The motive of current research is to investigate the impact of variable conductivity and diffusivity on the elasto-thermo-diffusive interactions in a 2D half-space under the two temperature fractional dual phase lag model. At the bounding surface of half-space, thermal and concentration loadings are applied. Variable diffusivity and conductivity are assumed to be linearly dependent on concentration and thermodynamic temperature, respectively. Kirchhoff's transformations are taken into consideration to convert the non-linear heat conduction and diffusion equations into a linear form. Then, reduced governing equations are solved by employing the transform technique. The solutions of different physical quantities are presented in a compact form. Numerical simulation is carried out for copper material. A numerical inversion process is adopted to invert the solutions in the original domain. Consideration of two temperature, dual phase lag fractional thermoelasticity to study the transient responses of half space with specific type of loadings augment the novelty of the present work. The results for conductive and thermodynamic temperature, normal stress and mass concentration are plotted to illustrate the impact of variable diffusivity and conductivity, fractional and two temperature parameters. These parameters are observed to have varying degrees of influence on the physical fields.

ASSESSMENT OF GROUND SURFACE FLUCTUATIONS DURING SHIELD TUNNELING OF SUBWAY TUNNELS

T The problems of evaluation of ground surface vibrations created by underground vibration sources are quite acute, especially in the conditions of urban development. During the construction of subway tunnels by the shield method there is a dynamic impact during the work of working mechanisms. The oscillations during the excavation of the soil by cutting mechanisms of the shield complex, during the ground bailing and pumping of mortar behind the lining can reach the day surface and cause significant deformations. When designing long underground tunnels built next to existing buildings, it is necessary to estimate the parameters of dynamic impact on the buildings caused by the operation of shielding mechanisms. The paper deals with the estimation of dynamic effects of various devices and mechanisms used in underground works. At this stage, the problems of oscillations of the surface of an elastic half-space from various point sources of waves located at different depths, taking into account the waves reflected from the free surface, are solved. The distribution of stresses and displacements during wave propagation in elastic media is a complex picture. In confined media the picture of wave propagation becomes even more complicated due to reflection of waves from boundaries. The reciprocity theorem is used to estimate the levels of ground surface oscillations created by dynamic effects of various devices and mechanisms during shield tunneling. The dynamic effects of working mechanisms are modeled by: the force directed along the axis of the shield, a pair of forces, two pairs of forces without momentum, a combination of a pair of forces and a longitudinal force, a combination of two pairs without momentum and longitudinal force.

Propagation of fundamental and third harmonics along a nonlinear seismic metasurface

To protect civil structures from seismic damage, metasurfaces have been designed to control the propagation of Rayleigh wave along the free surface of an elastic half-space. The introduction of nonlinearity may uncover novel phenomena and provide new avenues in the design of seismic metasurfaces. In this work, the propagation of Rayleigh wave along a nonlinear metasurface consisting of an array of Duffing oscillators attached on the free surface of an elastic half-space, is investigated. In comparison with pervious works on seismic metasurfaces, the generation of third harmonic is taken into account and both the dispersions of fundamental and third harmonics propagating along the nonlinear seismic metasurface are derived via the harmonic balance approach. Then, numerical studies are conducted to reveal the effects of both the hardening and softening nonlinearities on the dispersion and third harmonic generation. Additionally, finite element simulations are carried out to validate our analytical solutions. Our results show that the dispersion of fundamental wave is sensitive to the system's nonlinearity. There exists a band gap for the corresponding linear metasurface induced by the resonance of attached oscillators. Introducing a hardening nonlinearity may close the linear band gap, and introducing a softening nonlinearity can help to obtain a low-frequency band gap. The present work is expected to broaden the design and application of seismic metasurfaces.

Smoke rings re-visited: an analytic solution for the transient electric field produced by a vertical magnetic dipole on or in a homogeneous conductive half-space

For a vertical magnetic dipole (VMD) on or under the surface of a conductive half-space, explicit analytic formulae, not Hankel transform expressions, have been derived for the transient electric field induced below the ground, and the corresponding Hertz potential. These formulae permit rapid calculation of surface and underground dB/dt and B-field components, as well as of the induced current distribution. The solution allows the current produced by the surface reflection to be isolated, revealing how the interaction with the ground surface is responsible for the downward diffusion of the smoke ring. The mathematical form of the solution has also permitted confirmation of a prediction that the current distribution from a buried dipole in the limit of late time is a scaled version of the current distribution produced by a VMD on the ground surface.

Numerical analysis of the dispersion of acoustic Rayleigh waves in a Functionally Graded Piezoelectric half-Space

Considerable attention has been given to the study of the propagation of surface waves in order to improve the efficiency and lifetime of the surface acoustic wave devices such as transducers. In this paper, an investigation of the Rayleigh waves in functionally graded piezoelectric material is presented. The Rayleigh surface wave propagation is assumed to take place in a transversely isotropic graded piezoelectric half-space with material properties varying continuously along the thickness direction. The obtained results have shown that dispersive Rayleigh waves can propagate on the surface of the FGPM half-space with characteristics that depend on the graded variation of the material parameters. Based on the dispersion relations, the phase velocity for both the electrically open and shorted cases at the free surface is deduced. The displacement magnitudes and the corresponding decay variations are plotted and discussed.

Displacements of the surface of a piezoelectric half-space with functionally-graded coating and circle electrode on the surface

A mathematical model describing the deformation of a material caused by the reverse piezoelectric effect is proposed. The model is based on an axisymmetric problem of electroelasticity for a half-space with a functionally-graded coating. An electric potential difference is applied across a circle area on the surface (an electrode whose thickness and elastic properties are neglected) and an infinity boundary of the half-space. Arbitrary independent variation of all electroelastic properties in depth of the coating is considered. The coating is assumed to be perfectly bonded to the substrate. The problem is solved using the bilateral asymptotic method in an approximated analytical form effective for any value of the relative thickness of the coating. Approximated analytical expressions containing finite quadratures are obtained for the distribution of radial and normal displacements and electric potential on the surface under the electrode and outside of it. Numerical calculations are provided for the distribution of radial and normal displacements and electric potential for two typical examples of functionally-graded coatings in a wide range of relative thickness values of the coating. Convergence of the results to those for a homogeneous non-coated half-space is obtained for small and large values of the relative coating thickness. Special attention is paid to the comparison of the results with those for a non-coated half-space. The redistribution of the electromechanical characteristics caused by the presence of the coating is most noticeably observed near the boundary of the electrode especially for thin and intermediate thickness coatings (in comparison with the size of the electrode).

MATHEMATICAL MODELING OF ELASTIC WAVE PROPAGATION IN A BODY WITH A DEFECT

The solution of the dynamic problem of calculation the wave field of displacements on the surface of an elastic half-space caused by the opening of an internal crack under the action of torsional forces is presented. Based on the solutions of the boundary integral equations, the nature of the change in the amplitude-frequency characteristics of elastic oscillations on the surface of a rigid body depending on the size of the defect is shown.

Green’s function analysis of shear wave propagation in heterogeneous poroelastic sandwiched layer influenced by an impulsive source

The present analysis confers the propagation characteristics of horizontally polarized​ shear (SH) wave through a heterogeneous transversely isotropic fluid-saturated poroelastic sandwiched layer of finite width embedded between two heterogeneous isotropic elastic half-spaces due to the impact of an impulsive line source. The dissipation of energy caused by the relative fluid flow is neglected in transversely isotropic fluid-saturated poroelastic sandwiched layer. The heterogeneities in layer and half-spaces are considered with the respective spatial quadratic and exponential type variations in the elastic constants and densities. Green’s functions are obtained for this sandwiched layered structure by considering a charge-density/line-force (Dirac-delta function) at lower bonding surface of layer and half-space. Dispersion relation for the propagation of SH wave in aforesaid structure has been obtained with well-known properties of Green’s functions, Fourier’s transformation and appropriate boundary conditions. The obtained dispersion relation is reduced for a structure with a heterogeneous transversely isotropic fluid-saturated poroelastic layer of finite width overlying a heterogeneous isotropic elastic half-space in the particular case of problem. Moreover, the obtained dispersion relations are validated and matched with pre-established standard and classical results. The effect of various affecting parameters viz. anisotropy of layer, porosity of layer and heterogeneities of layer and half-spaces on phase velocity of propagating wave in said structure are shown graphically through numerical computations. Further, a comparative study of distinct cases of heterogeneous/homogeneous structures is also performed which serves as the major highlight of the present study.

Influence of varying fiber volume fractions on plane waves reflecting from the stress-free/rigid surface of a piezoelectric fiber-reinforced composite half-space

Wave reflection phenomenon at the stress-free/rigid surface of a piezoelectric fiber-reinforced composite (PFRC) half-space is studied. The PFRC is micro-mechanically modeled using Strength of Materials (SM) and Rule of Mixtures (RM). Some of its electro-mechanical data are graphically shown and its advantages are discussed. An incident quasi-longitudinal (qP) wave generates reflected quasi-longitudinal (qP), quasi-transverse (qSV), and electroacoustic (EA) waves, whose propagation angles are graphically demonstrated. Closed-form expressions of energy ratios of reflected waves and interaction energy are obtained, which satisfy the Law of Conservation of Energy, thus validating this problem. Influences of incident angle, varying fiber volume fractions, and stress-free/rigid surfaces on amplitude/energy ratios are discussed.