Surface Of Half-space Research Articles

On Accounting for the Microrotation of Particles of an Elastic Medium According to Cosserat’s Simplified Theory During Electroelastic Wave Propagation in a 6mm Class Piezoelectric Crystal

The article reveals how the micro polarity of a material influences the propagation of a one-dimensional electroelastic wave along different sections of a 6mm class piezoelectric crystal. It is shown that micro polarity leads to dispersion of a one-dimensional electroelastic wave and prohibits the propagation of high-frequency elastic shear waves. In the case of a normally incident electromagnetic wave on the surface of a piezoelectric half-space along the direction of the polarization axis of the piezoelectric crystal, three branches of electroelastic waves localized near the surface appear in the half-space. The frequency range of the existence of these waves is found from the dispersion equation. The behavior of the wave characteristics and frequency zones is numerically studied as a function of the micro polarity of the piezoelectric crystal.

Effect of fractional parameter on thermoelastic half-space subjected to a moving heat source

In this study, we consider a problem for a half space subjected to a moving heat source. The problem is in the context of the fractional order thermoelasticity theory. The half-space surface is taken to be traction free. The medium is assumed to be initially quiescent. The Laplace and Exponential Fourier transform techniques are used. The solution within the transform domain is obtained by employing a direct approach. The inverse of the double transforms is obtained by employing a numerical technique. Numerical results are given and represented graphically. Some comparisons are shown in figures to estimate the effect of the fractional order parameter on all studied fields.

An asymptotic hyperbolic-elliptic model for flexural-seismic metasurfaces.

We consider a periodic array of resonators, formed from Euler-Bernoulli beams, attached to the surface of an elastic half-space. Earlier studies of such systems have concentrated on compressional resonators. In this paper, we consider the effect of the flexural motion of the resonators, adapting a recently established asymptotic methodology that leads to an explicit scalar hyperbolic equation governing the propagation of Rayleigh-like waves. Compared with classical approaches, the asymptotic model yields a significantly simpler dispersion relation, with closed-form solutions, shown to be accurate for surface wave-speeds close to that of the Rayleigh wave. Special attention is devoted to the effect of various junction conditions joining the beams to the elastic half-space which arise from considering flexural motion and are not present for the case of purely compressional resonators. Such effects are shown to provide significant and interesting features and, in particular, the choice of junction conditions dramatically changes the distribution and sizes of stop bands. Given that flexural vibrations in thin beams are excited more readily than compressional modes and the ability to model elastic surface waves using the scalar wave equation (i.e. waves on a membrane), the paper provides new pathways towards novel experimental set-ups for elastic metasurfaces.

Free-field response of a transversely isotropic saturated half-space subjected to incident plane qP1- and qSV-waves

Many natural soils and rocks are not only saturated but are also anisotropic. Due to the fundamental importance of having an understanding of the free-field response of an anisotropic half-space subjected to incident seismic waves, the surface displacements, surface strain, rocking, and pore pressure due to reflection of plane waves in a transversely isotropic (TI) saturated half-space are discussed. The governing equations in u-w formulation are first derived and the plane harmonic wave solution of these equations is then proposed. The free-field response is finally determined by specifying the stress free and surface drainage conditions at the boundary. The half-space surface can be either completely permeable or impermeable, and the excitation includes incident plane qP1- and qSV-waves. The proposed formulations are verified by comparing our results with those for the isotropic half-space. A parametric study is performed to illustrate the influences of the material anisotropy, surface drainage condition and permeability on the surface motions. Numerical results show that the surface motions of the TI half-space are significantly different from those of the isotropic case. The surface motions of the TI half-space are much more dependent on the TI parameters when the incident waves are qSV-waves than when they are qP1-waves because the dynamic responses change sharply as the incident angles approach the critical angles. In contrast with the isotropic case, with an identical angle of 45°, the total reflection angle of the TI half-space varies with the TI parameters. In general, the surface motions of the permeable surface are smaller than those of the impermeable surface, particularly in response to incident qSV-waves.

Two-dimensional FM-IBEM solution to the broadband scattering of elastic waves in a fluid-saturated poroelastic half-space

The fast multi-pole indirect boundary element method (FM-IBEM) is proposed to efficiently solve the high-frequency and large-scale two-dimensional (2-D) elastic wave scattering in a fluid-saturated poroelastic heterogeneous medium. The diffracted wave fields are constructed by applying virtual distributed loads and fluid sources on the boundaries according to the single-layer potential theory, and the multi-pole moment and local expansion coefficients of 2-D Green's function are deduced based on Graf addition theorem. Numerical results illustrated that the proposed method can essentially reduce the calculation CPU time and the memory requirement. The 2-D scattering of elastic waves by a canyon and a group of cracks in a fluid-saturated poroelastic half-space is further investigated. It shows that the scattering characteristics are closely related to the incident frequency, porosity of the medium and special topography. There is a significant ground motion amplification effect around the canyon, while a group of cracks produce an isolation effect on the half-space surface motion.

Finding, by means of a scattered sound, the geometric parameters of a finite elastic cylinder located near the half-space border

The solution of the inverse problem of sound diffraction on a finite circular elastic cylinder is considered. The cylinder T is located near the surface Π of acoustic half-space. The following cylinder parameters are finding according to the scattered sound field: radius, height, distance from the half-space surface, and axis orientation. The solution of the diffraction problem is conducted by expanding the area of the problem to the full space. With that an additional obstacle T′, which is a copy of T, is introduced. The additional obstacle is a mirror reflection of the original one with respect to the surface Π. In addition, a second incident wave, which corresponds to the type of the half-space boundary, is introduced. The solving is based on the linear elastic theory and the model of the small vibrations propagation in ideal fluid. In the external part of the fluid, the solution is sought analytically in the form of expansion in spherical harmonics and Bessel functions. In the ball area, which includes two balls cylinders and an adjacent layer of the fluid, the finite element method (FEM) is used. The cylinder required parameters are detected on the basis of the minimization of the norm of the vector difference () and (pjm). Where () is the vector of measured pressure values in the scattered sound field, and (pjm) is the pressure vector obtained from the theoretical solution of the diffraction problem.

Scattering of transverse surface waves by a piezoelectric fiber in a piezoelectric half-space with exponentially varying electromechanical properties

In the present work, an analytical solution is presented for the scattering of transverse surface waves by a homogeneous piezoelectric fiber contained in a functionally graded piezoelectric half-space with exponential variation. The boundary value problem of interest is solved by constructing an appropriate set of multipole functions which satisfy: (a) the electromechanical field equations in the half-space, (b) the boundary conditions along its free surface, and (c) the far-filed radiation conditions. It is shown that the simple poles of these functions are related to the roots of the pertinent dispersion relation. For the case of electrically short condition along the free surface of the inhomogeneous half-space, the analytical expressions for the scattered electromechanical fields are derived. In the given numerical examples, the effects of such parameters as the frequency, the distance of the fiber to the substrate’s free surface, and the coefficient in the exponent, indicating the variation of the electromechanical properties of the substrate on the scattered fields are addressed in detail. It is seen that these physical parameters have considerable effect on the dynamic response of the medium.

The response of a partially loaded composite half-space weakened by local defects

The elastic field in a fiber-reinforced half-space which consists of distinct constituents and localized internal defects is predicted by employing a multiscale analysis. A normal distributed uniform loading is applied over a rectangular region of the half-space surface, while the other parts are kept traction-free. The effect of internal defects and the half-space traction-free boundary conditions are accounted to by incorporating a damage tensor in the constitutive relations which describe the half-space phases. In the micro level of the micro-to-macro analysis, the effective behavior of the composite half-space is determined by employing a micromechanics analysis. This is followed by a macromechanical analysis in which the triple discrete Fourier transform is applied in the domain of which the formulated problem is solved. The damaged composite half-space response is obtained by inverting the Fourier transform, in conjunction with an iterative procedure. The predicted elastic field in the half-space which is based on the present approach is validated by comparisons with exact and finite element solutions. The offered method is applied to generate the elastic field in a fiber-reinforced half-space with a broken fiber, lost fiber, debonded fiber, missing fiber and a matrix void. Applications of numerous defects of these and other types are discussed.

Deceleration of a Rigid Cylinder Sliding along a Viscoelastic Foundation

The deceleration of an infinite rigid cylinder sliding along a surface of a viscoelastic halfspace is studied (the axis of the cylinder at plane-parallel motion is horizontal). It is assumed that there are no tangential stresses in the contact area. The distribution of contact pressures, the size of contact area, and the dependence of resistance force on the velocity of the cylinder axis are determined from the solution of a quasi-static problem in the theory of viscoelasticity. On the basis of the developed mathematical model, the features of deceleration process are qualitatively and numerically studied along with their dependence on the viscoelastic parameter of the foundation.

Rayleigh surface wave propagation in an orthotropic rotating magneto-thermoelastic medium subjected to gravity and initial stress

The prime objective of the present article is to analyze the effects of rotation and initial stress on the propagation of Rayleigh surface waves in a homogeneous, orthotropic magneto-thermoelastic half space subjected to gravity field. The frequency equations in closed form are derived and the amplitude ratios of surface displacements, temperature change during the Rayleigh wave propagation on the surface of half space have been computed analytically. The highlights of this study are the effects of different parameters (rotation, magnetic field, initial stress, and gravity) on the velocity of Rayleigh waves. Variation in phase velocity of Rayleigh waves against a wave number is shown graphically. Some particular cases have been deduced. Also, the classical Rayleigh wave equation is obtained as a special case of the present study. Numerical example has been carried out and represented by the means of graphs. Impacts of various involved parameters appearing in the solutions are carefully analyzed. In fact, in the absence of various parameters, these equations are in agreement with the results for isotropic medium.

Plastic Deformation of a Rough Surface by the Rigid Rolling Cylinder

Simulation of discrete plastic deformation of a rough surface in an ideal plastic half-space under a rolling rigid cylinder taking into account Prandtl’s contact friction is considered. The roughness of the cylinder is assumed to be insignificant as compared to the roughness of the half-space surface. The regular periodic shape of the rough surface formed after machining with intense cutting conditions is considered. The resistance to the cylinder rolling significantly decreases as compared with the case of the plastic deformation of a rough surface by a sliding tool. The resulting model can be used to estimate the technological parameters for the rolling of a rough surface by wide rollers.

Satisfying Boundary Conditions in Homogeneous, Linear-Elastic and Isotropic Half-Spaces Subjected to Loads Perpendicular to the Surface: Distributed Loads on Adjacent Contact Areas

Abstract This work originates from an experimental program on strain distribution near the loaded surface of an airfield concrete pavement,which provided us with results that contrast with the rheological predictions of Boussinesq for a homogeneous, linear-elastic and isotropic half-space. We already reviewed and extended the original work carried out by Boussinesq in previous papers, to provide a closed form second order solution that enabled us to establish a good match between analytical and experimental findings for point-loads. In this paper, we have explained why Boussinesq’s closed form solution for a homogeneous linear-elastic and isotropic half-space subjected to a point-load is not exact, as believed until now, but approximated. Then, we have shown that our second order solution is the actual solution of Boussinesq’s problem. We have also presented the numerical analysis of second order for rectangular and elliptical contact areas, both loaded by uniform and parabolic laws of external pressure. Moreover, we have evaluated the interaction effect provided on the surface of a concrete half-space by the twin wheels of an aircraft landing gear. Extension of the solution to layered systems is also possible, for improving the knowledge of stress propagation into airfield pavements and promoting more effective design standards.

Reflection of Plane Waves from Surface of a Generalized Thermo-Viscoelastic Porous Solid Half-Space with Impedance Boundary Conditions

Abstract A phenomenon of reflection of plane waves from a thermally insulated surface of a solid half-space is studied in context of Lord-Shulman theory of generalized thermo-viscoelasticity with voids. The governing equations of generalized thermo-viscoelastic medium with voids are specialized in x–z plane. The plane wave solution of these equations shows the existence of three coupled longitudinal waves and a shear vertical wave in a generalized thermo-viscoelastic medium with voids. For incident plane wave (longitudinal or shear), three coupled longitudinal waves and a shear vertical wave reflect back in the medium. The mechanical boundary conditions at free surface of solid half-space are considered as impedance boundary conditions, in which the shear force tractions are assumed to vary linearly with the tangential displacement components multiplied by the frequency. The impedance corresponds to the constant of proportionality. The appropriate potentials of incident and reflected waves in the half-space will satisfy the required impedance boundary conditions. A non-homogeneous system of four equations in the amplitude ratios of reflected waves is obtained. These amplitude ratios are functions of material parameters, impedance parameter, angle of incidence, thermal relaxation and speeds of plane waves. Using relevant material parameters for medium, the amplitude ratios are computed numerically and plotted against certain ranges of impedance parameter and the angle of incidence.

Modeling of wave reflection in gas hydrate-bearing sediments

The content and distribution pattern of the gas hydrate are the key factors affecting the free surface reflection of the gas hydrate-bearing sediment. In this study, a theoretical model of wave reflection in gas hydrate-bearing sediment is discussed for the incidence of P1-wave and S1-wave to analyze the influences of the gas hydrate. The gas hydrate-bearing sediment is modeled by a Biot-type three-phase theory and assumed to be composed of sediment grains with connected pore occupied by the mixture of fluid and hydrate. The incoming wave is split into three reflected compressional waves and two reflected shear waves at the free surface of a gas hydrate-bearing sediment half-space. The analytical solutions for various reflected waves at open-pore and sealed-pore boundaries are obtained in terms of displacement potentials. Then, two-phase type models based on Biot theory are proposed by assuming the gas hydrate and sediment frame are weakly coupled or essentially consolidated to verify the rationality of the theoretical model. A third two-phase type model is given by assuming there is no gas hydrate in the pore. At last, the influences of the content and distribution pattern of gas hydrate, the frequency of the incident wave, and the permeability condition on the horizontal displacements and displacement ratios are analyzed respectively through numerical calculation. It is revealed that the solutions based on the Biot-type three-phase theory are very consistent with the solutions based on Biot theory if the gas hydrate and sediment frame are weakly coupled or essentially consolidated. Furthermore, the model proposed in this study can be used to analyze the influence of the distribution pattern of the hydrate.

Thermal shock fracture of an elastic half-space with a subsurface penny-shaped crack via fractional thermoelasticity

This paper analyzes a thermal shock problem of an elastic half-space with a penny-shaped crack near the surface based on a fractional thermoelasticity theory. The embedded crack is assumed to be insulated. The Hankel transform and Laplace transform are employed to solve an initial-boundary value problem associated with a fractional partial differential equation. Explicit expressions of temperature and thermal stresses induced by the penny-shaped crack are obtained by solving a system of singular integral equations. Numerical results of the thermoelastic fields in the time domain are given by applying the numerical inverse Laplace transform. The temperature jump between the upper and lower crack surfaces and the thermal stress intensity factors at the crack front are illustrated graphically for various relaxation times and fractional orders, as well as the distance between the crack plane and the half-space surface. A comparison of the temperature, thermal stresses, and their intensity factors is made when adopting the fractional heat conduction model and the classical Fourier heat conduction model. Numerical results show that the temperature overshooting phenomenon may occur for the fractional heat conduction model, whereas it does not occur for the classical Fourier heat conduction model.

Surface NMR Responses of Typical 3-D Water-Bearing Structures Evaluated by a Vector Finite-Element Method

Surface nuclear magnetic resonance (SNMR) technique has been widely applied to noninvasive groundwater exploration and aquifer quantitative characterization. Furthermore, it has the potential for studying and detecting hydrocarbon contaminants and for water resource management. To increase the tomography resolution of groundwater and expand the application range of SNMR, we first implemented its 3-D forward modeling using a vector finite-element (FE) method based on the total electric field. The key of SNMR forward modeling is the calculation of excitation magnetic field, and the total-field algorithm is an alternative scheme to numerically simulate the electromagnetic field. In this paper, we viewed the circular loop source as the combination of a certain number of horizontal electric dipoles. The unstructured tetrahedral mesh and local refinement technology were combined to precisely delineate the distribution of circular loop source, which effectively reduces the adverse effects of field source singularity. Then, the vector FE solver based on the total electric field was used to calculate the magnetic field distribution. The accuracy of the calculated results was validated by the analytical solutions for a circular loop laid on the surface of a homogeneous half-space. After calculating the excitation field, SNMR responses of three typical synthetic 3-D groundwater models were obtained with the basic signal response formula. Finally, we studied the effects of some important factors which have significant influences on SNMR signal responses and have practical importance in groundwater issues.

Non-classical laws with conserved quantities from the arbitrary functions included in general solution to elasticity and application

• Noether symmetries are discussed by standard techniques not for the Lagrangian density of an elastic body. • The deep-level symmetries of an elastic field from arbitrary functions included in general solution. • Conserved quantities with arbitrary integer order spatial derivative or integral. • An invariant finite physical parameter can be obtained for arbitrary integer order singularity field. The general solution to static and/or dynamic linear elasticity is a transformation between the displacements and new arbitrary functions, whose conservativeness depends on some independent partial differential equations (PDEs) satisfied by the new arbitrary functions. Zhang's general solutions are mathematically appropriate since the displacements are expressed in terms of two new arbitrary functions, and the sum of the highest order derivative added together from the independent PDEs satisfied by the two new arbitrary functions is the same as that of Navier–Cauchy equations. Therefore, the following points should be emphasized: (i) the independent PDEs come from the Laplace and D'Alembert operators acting on the two new arbitrary functions in static and dynamic general solutions, respectively, and it is found that the two new arbitrary functions are related to the rotations, first strain invariant and distortion; (ii) especially, conservation laws constructed from the equations satisfied by the spatial integrals of functions hold true, although some arbitrary functions of the spatial integrals have been canceled. Based on these facts, since Noether's identity not only can be applied to a Lagrangian but also can be used to construct a functional for widespread PDEs, the functionals relating to the rotations, first strain invariant and distortion are constructed with arbitrary integer order spatial derivative or integral, and the conservation laws follow. This kind of non-classical conservation laws does not come from the Lagrangian density of an elastic body and belongs to the deep-level natures of symmetries of elastic field derived by standard techniques. Availability is shown by two examples, from which the field intensity of a vertical load applied to the surface of an elastic half-space and the path-independent integrals in a coordinate system moving with Galilean transformation are presented for comparison.

On the complexity of seismic waves trapped in irregular topographies

Documented observations from strong seismic events have repeatedly shown that the ground surface topography significantly affects the characteristics of seismic waves (amplitude, frequency and duration) travelling from the deeper layers of the crust compared to what the ground motion would have been on the surface of a flat homogeneous linear elastic half-space. Although numerous theoretical studies have qualitatively corroborated these observations, they systematically underestimate the absolute level of topographic amplification up to an order of magnitude or more in some cases. In this paper, we try to bridge the quantitative gap between previous theoretical studies and observations by systematically studying the role of geometry, stratigraphy, and ground motion characteristics through a series of elaborate numerical analyses. We show a collection of examples that highlight the effects of topography on seismic ground shaking, and we point out what these results suggest in the context of the current state of earthquake engineering practice. Examples range from semi-analytical solutions of wave propagation in infinite wedges to three-dimensional numerical simulations of topography effects using digital elevation map-generated models and layered geologic features. We conclude by demonstrating that topography effects vary strongly with the stratigraphy and material properties of the underlying geologic materials, and thus it cannot be accurately predicted by studying the effects of ground surface geometry alone.

Cyclic Deformation Caused by Repeatedly Contact Loading

An elastic-plastic contact stress analysis is presented to study cyclic plastic deformation of rolling elements under repeatedly contact loadings. The rolling contact is simulated by a Hertzian line contact loading translating over the surface of an elastic-plastic half-space, and the Chaboche nonlinear hardening rule is used to model the cyclic plastic behavior of contact components. A finite element procedure based on the return mapping algorithm is implemented to analyze the evolution plastic strains and residual stresses versus contact cycles. For the contact loading below the shakedown limit, p0/k=4, the plastic deformation occur only at first few contact cycles and become pure elastic in the subsequent cycles due to the existing residual stresses and material hardening. For the contact loading exceeding the shakedown limit, p0/k=7, the plastic strains increase progressively with each pass of contact cycles and result in plastic ratchetting. The normal residual stresses, however, quickly reach a steady state after few contact cycles.

A spectral approach for discrete dislocation dynamics simulations of nanoindentation

We present a spectral approach to perform nanoindentation simulations using three-dimensional nodal discrete dislocation dynamics. The method relies on a two step approach. First, the contact problem between an indenter of arbitrary shape and an isotropic elastic half-space is solved using a spectral iterative algorithm, and the contact pressure is fully determined on the half-space surface. The contact pressure is then used as a boundary condition of the spectral solver to determine the resulting stress field produced in the simulation volume. In both stages, the mechanical fields are decomposed into Fourier modes and are efficiently computed using fast Fourier transforms. To further improve the computational efficiency, the method is coupled with a subcycling integrator and a special approach is devised to approximate the displacement field associated with surface steps. As a benchmark, the method is used to compute the response of an elastic half-space using different types of indenter. An example of a dislocation dynamics nanoindentation simulation with complex initial microstructure is presented.