Orthotropic Half-space Research Articles

Stress-strain state of a two-layer orthotropic body under plane deformation

We consider the problem of determining the stresses and strains of a two-layer body consisting of an orthotropic layer of constant thickness connected to an orthotropic half-space. The surface of the layer is subjected to known external loads, such that the deformation of the body is plane. At infinity, the stresses are zero. The stress-strain state of the body is determined using the method of integral Fourier transforms. The features of solutions of the system of differential equations of the problem are investigated. The solutions of a particular problems are obtained and analyzed.

Bone crack inspired pair of Griffith crack opened by forces at crack faces

The mathematical theory of elasticity helps in the study of physical quantities in the problem of structures. The structures face the problem of crack’s presence, which makes the problem difficult but not impossible to deal. Integral equations are useful in a variety of problems. Integral equations are used to solve problems like fracture mechanics or fracture design. The physical interest in the fracture design criterion is due to stress and crack opening displacement components. We have an accurate form of stress and displacement components for a pair of longitudinal crack propagations in the bone fracture of the human body at the interface of an isotropic and orthotropic half-space that are bounded together in the proposed study. The expression was calculated using the Fourier transform approach near the crack tips, but these components were evaluated using Fredholm integral equations and subsequently reduced to coupled Fredholm integral equations. We employ the Lowengrub and Sneddon problem in this research and reduce it to triple integral equations. The Srivastava and Lowengrub method reduces the solution of these equations to a coupled Fredholm integral equation. The problem is further reduced to a decoupled Fredholm integral equation of the second kind. Triple integral equations are solved, and the problem is reduced to a coupled Fredholm integral equation. The Fredholm integral equation is solved and reduced to a decoupled Fredholm integral equation of the second kind. Stress and crack opening displacement components drive physical interest in fracture design criteria. Finally, the stress and displacement components may be simply calculated in their exact form.

Asymptotic formula of Rayleigh wave speed in an orthotropic half-space coated by a thin piezoelectric film

ABSTRACT Piezoelectric thin films have been widely used in electro-acoustic devices including ones measuring Rayleigh wave speed. The main objective of this work is to find an explicit formula of Rayleigh speed taking into account the effect of the thin piezoelectric film. First the problem of Rayleigh surface waves propagating along an orthotropic substrate coated by a thin piezoelectric film is considered using Stroh’s formalism approach. Then, the exact secular equation of Rayleigh wave is obtained in implicit form. In case when the thickness of the thin layer is much smaller than the considering wavelength of Rayleigh waves, an asymptotic formula of Rayleigh speed is derived from the asymptotic form of the exact secular equation. The obtained asymptotic formula is simple enough and could be used in practice when Rayleigh speed is measured by using piezoelectric device. Numerical illustrations are carried out to show the validity of the obtained approximate formulas and its potential application in inverse problem.

Method of Direct Cutting-Out in Modeling Orthotropic Solids with Thin Elastic Inclusions Under Longitudinal Shear

By the method of direct cutting-out, the problems of longitudinal shear of an orthotropic half space, a layer, and a wedge with thin elastic orthotropic inclusions are reduced to the basic problem of interaction of thin inhomogeneities in the orthotropic space. We establish the conditions of interaction of loaded elastic anisotropic inclusions with the matrix of the body. We study the influence of elastic moduli both of the inclusion and of the body, as well as of the geometric parameters of the problems on the generalized stress intensity factors. The level lines of stresses are plotted in the vicinity of the inclusion.

Elastodynamic response of an orthotropic layered elastic half-space to time-harmonic loading.

The elastodynamics of an orthotropic half-space coated by a thin orthotropic layer is theoretically investigated in this article. We newly propose explicit expressions of free Rayleigh waves in a layered half-space that are dependent on only one unknown constant representing amplitude. The main contribution is on deriving, in a simple manner, the theoretical predictions of far-field Rayleigh wave motion arising from time-harmonic loads using elastodynamic reciprocity theorems. These are the very first closed-form exact solutions found for the forced motion of Rayleigh waves in a layered half-space of orthotropic materials. To demonstrate the theoretical results, computation of Rayleigh wave motion in a jointed rock, including a layer of quartz-schist and a half-space of soil, is considered. We present the phase and group dispersion curves superimposed with the amplitude spectra that provide useful information on wave modes, frequencies, and displacement amplitudes. The inclusion of the amplitude spectra in the dispersion curves is a significant improvement over other dispersion curves currently available in the literature. The analytical predictions are compared with numerical results found by finite element analysis, and they show excellent agreement for the cases of a uniform distributed load and a varying distributed load both applied over a strip on the layer surface. The calculations obtained in the current study could generally be very useful for applications in seismology and materials characterization of coated structures.

Eigenvalue approach to study Rayleigh waves in nonlocal orthotropic layer lying over nonlocal orthotropic half-space with dual-phase-lag model

In this present article, we deal with the propagation of Rayleigh surface waves in the nonlocal orthotropic thermoelastic layer which is lying over a nonlocal orthotropic thermoelastic half-space in the context of the dual-phase-lag model. This problem is considered in the context of Eringen’s nonlocal elasticity theory. A vector-matrix differential equation is obtained which is solved by an eigenvalue approach. The frequency equation of Rayleigh surface waves is derived and various particular cases are discussed. The different characteristics namely attenuation coefficient, specific loss, penetration depth, and phase velocity are presented graphically with respect to frequency.

A Griffith Crack at the Interface of an Isotropic and Orthotropic Half Space Bonded Together

In the past 53 years, many efforts have been contributed to develop and demonstrate the properties of reinforced composite materials. The ever-increasing use of composite materials through engineering structures needs the proper analysis of the mechanical response of these structures. In the proposed work, we have an exact form of Stress components and Displacement components to a Griffith crack at the interface of an Isotropic and Orthotropic half-space bounded together. The expression was evaluated in the vicinity of crack tips by using Fourier transform method but here these components have been evaluated with the help of Fredholm integral equations and then reduce to the coupled Fredholm integral equations. In this paper, we use the problem of Lowengrub and Sneddon and reduce it to dual integral equations. Solution of these equations through the use of the method of Srivastava and Lowengrub is reduced to coupled Fredholm integral equation. Further reduces the problem to decoupled Fredholm integral equation of 2nd kind. We get the solution of dual integral equations and the problem is reduced to coupled Fredholm integral equation. We find the solution of the Fredholm integral equation and reduce it to decoupled Fredholm integral equation of 2nd kind. The Physical interest in fracture design criterion is due to Stress and crack opening Displacement components. In the end, we can easily calculate the Stress components and Displacement components in the exact form.

Rayleigh waves in compressible orthotropic half-space overlaid by a thin un-coaxial orthotropic layer

The problem of Rayleigh waves in compressible orthotropic elastic half-space overlaid by a thin elastic layer of which principal material axes are coincident have been researched by many scientists. However, the problem with the conditions that the half-space and the layer have only one common principal material axis that perpendicular to the layer while the remains are not identical has not gotten enough attention. This paper presents a traditional approach to obtain an approximate secular equation by approximately replacing the thin layer by effective boundary conditions of third-order. The wave then is considered as a Rayleigh wave propagating in an orthotropic half-space, without coating, subjected to the effective boundary conditions. This explicit approximate secular equation is potentially useful in non-damage assessment studies.

Velocity-profile of torsional surface wave in a contorted watery-porous reservoir over a sand-dune deposit

The current study aims to study the velocity profile of torsional surface waves in a three-layer model consisting of sand-dunes deposit sandwiched between the watery-porous layer and the orthotropic half-space. The top and the middle layers exhibit irregular interfaces employing various geometrical shapes such as parabolic, rectangular, and triangular notch. The main objective is to analyze the effect of these irregular interfaces on torsional wave propagation. Furthermore, the three-layer model involves inhomogeneity through gravity, initial stress, rigidity, and medium density. These inhomogeneity parameters, coupled with irregularity, bears a remarkable impact on the velocity-profile. The generalized dispersion relation has been derived, and one particular case has been deduced to validate the current work. Detailed numerical simulation has been performed to characterize the computations, and graphical illustrations have been exhibited to support the mathematical investigation.

Rayleigh waves in porous nonlocal orthotropic thermoelastic layer lying over porous nonlocal orthotropic thermoelastic half space

ABSTRACT The present paper deals with the propagation of Rayleigh surface waves in a nonlocal thermoelastic orthotropic layer which is lying over a nonlocal thermoelastic orthotropic half space. The problem is considered in the context of Green-Naghdi type III model of hyperbolic thermoelasticity and Eringen’s nonlocal elasticity theory in the presence of voids. The problem is solved using eigenfunction expansion method. Frequency equation of Rayleigh waves is derived and different particular cases are also deduced. The effects of voids and nonlocality on different characteristics of Rayleigh waves are presented graphically.

Rayleigh waves propagation in orthotropic solids with two temperature in context of thermoelasticity

The main objective of this paper is to study Rayleigh wave propagation in homogeneous orthotropic half space with two temperature in reference to Three Phase Lag(TPL) model of thermoelasticity. The surface wave solutions are obtained for governing equations of Rayleigh wave. The relevant boundary condition satisfy the solutions and corresponding frequency equation is derived for the Rayleigh wave in orthotropic half-space. A simulation study is conduced for numerical discussion and the results are demonstrated graphically for phase velocity and attenuation coefficient with respect to frequency in the context of Green-Nagdhi, Lord-Shulman and three phase lag models of thermoelasticity.

Reflection of plane waves from the free surface of a rotating orthotropic magneto-thermoelastic solid half-space with diffusion

In this paper, the governing equations of a rotating orthotropic magneto-thermoelastic half-space with diffusion are formulated in the context of Lord-Shulman's theory of generalized thermoelasticity. The plane wave solutions of these equations indicate the existence of four quasi-planar waves namely quasi-longitudinal displacement (qLD) wave, quasi-thermal (qT) wave, quasi-mass diffusion (qMD) wave, and quasi-transverse displacement (qTD) wave. Reflection of these plane waves from a stress-free thermally insulated/isothermal surface is studied to obtain the reflection coefficients and energy ratios of reflected waves. The reflection coefficients and energy ratios of reflected waves are computed for a particular material and effect of diffusion, rotation angular frequency and magnetic parameter against the angle of incidence is observed graphically on reflection coefficients and energy ratios.

Propagation characteristic of Love-type wave in different types of functionally graded piezoelectric layered structure

A mathematical model has been proposed for the analytical study of the propagation of Love-type wave in an initially stressed functionally graded piezoelectric material (FGPM) layer overlying an orthotropic fluid-saturated porous half-space. The expression of dispersion equation is derived in closed form for both electrically open and electrically short conditions. The impact of various affecting parameters viz. wave number, gradient parameter, tensile initial stress on the dispersion curve of Love-type wave has been revealed by means of graphical illustrations. Some remarkable observation has been made through numerical computation for different piezoelectric ceramics of BaTiO, PZT-4 and PZT-5A, respectively. The outcomes of the present study are momentous for both theoretical and practical implication for the surface acoustic wave (SAW) devices.

Rayleigh wave motions in an orthotropic half-space under time-harmonic loadings: A theoretical study

Rayleigh surface wave motion in a homogeneous orthotropic elastic solid half-space under time-harmonic sources is theoretically investigated. In this article, closed-form expressions of the Rayleigh wave field as a result of a time-harmonic concentrated line load are simply determined by using reciprocity theorems. For the verification purpose, analytical predictions are also obtained through Fourier transform technique with the aid of the residue theorem. The solutions found by the two approaches are shown to be mathematically the same. As an example of calculation, surface waves due to a set of buried loadings in the orthotropic half-plane are then examined in detail. A discussion of these results is presented to explore the influence of the loading types on the displacement amplitudes of the generated wave fields.

Effect of two-temperature on the energy ratio at the boundary surface ofinviscid fluid and piezothermoelastic medium

The phenomenon of reflection and transmission of plane waves at an interface between fluid half space and orthotropic piezothermoelastic solid half-space with two-temperature has been investigated. Energy ratios of various reflected and transmitted waves are computed with the use of amplitude ratios. The law of conservation of energy across the interface has been justified. It is found that the energy ratios are the functions of angle of incidence, frequency of independent wave and depend on the different piezothermoelastic material. A piezothermoelastic material has been considered which is in welded contact with water. Variations of energy ratios corresponding to the reflected waves and transmitted waves are computed and shown graphically for the two different models. A particular reduced case of interest is also discussed.

The H/V ratio of Rayleigh waves in layered incompressible orthotropic half-spaces and the incompressible limit method

The main purpose of this paper is to derive exact formulas for the H/V ratio (the horizontal-to-vertical displacement ratio at the upper surface of the layer) of Rayleigh waves propagating in an orthotropic elastic half-space coated by an orthotropic elastic layer. The half-space and the layer are in welded contact and at least one of them is incompressible. These formulas are derived by employing the incompressible limit method. One of the obtained formulas recovers Love's H/V ratio formula. When the layer is absent, the obtained formulas are simplified to the formulas derived previously for the H/V ratio of Rayleigh waves propagating in traction-free orthotropic elastic half-spaces. It is numerically shown that the incompressibility of deposited layers has strong impact on the H/V ratio behavior. As the H/V ratio is a dimensionless quantity and it is easily experimentally measured, the obtained formulas will be a useful tool for nondestructively testing mechanical properties of deposited layers.

Reflection and transmission between an orthotropic thermoelastic half-space and an initially stressed orthotropic rotating thermoelastic diffusive half-space

The purpose of present investigation is to study the reflection and transmission phenomena at the plane interface between an orthotropic thermoelastic half-space and an initially stressed orthotropic thermoelastic rotating half-space with diffusion. The enuciation is applied to generalized thermoelasticity based on Lord-Shulman theory. The amplitude and energy ratios of the reflected and transmitted waves are derived and the numerical computations have been carried out with the help of MATLAB programing. The effects of initial stress, rotation and diffusion parameters on the amplitude ratios are depicted graphically. It has been verified that at each angle of incidence, there is no dissipation of energy at the interface during reflection and transmission phenomena. Moreover, the effect of anisotropy is also depicted on the velocities of various reflected and transmitted waves. The velocities of these waves are found to depend on the angle of propagation in an orthotropic medium. This dependence of velocities on the direction of propagation is shown graphically.

Love Wave Propagation in a Fiber-reinforced Layer with Corrugated Boundaries Overlying Heterogeneous Half-space

Love-type wave generation in a fiber-reinforced medium placed over an inhomogeneous orthotropic half-space is analysed. The upper and lower boundary surfaces of the fiber reinforced medium are periodically corrugated. Inhomogeneity of half-space is caused by variable density and variable shear modules. Displacement components for layer and half-space are derived by applying separable variable technique. Dispersion relation for Love wave is obtained in closed form. Numerical calculations for the achieved dispersion equation are performed. In the numerical examples, the main attention is focused on the effect of corrugation investigation, reinforced parameters and inhomogeneity on the relations between wave number and phase velocity.

Exact formula for the horizontal-to-vertical displacement ratio of Rayleigh waves in layered orthotropic half-spaces.

This paper is concerned with the propagation of Rayleigh waves in a compressible orthotropic half-space coated by a compressible orthotropic layer. The main purpose of the paper is to establish an explicit exact formula for the H/V ratio (the horizontal-to-vertical displacement ratio) of the Rayleigh wave. This formula is derived by using the transfer matrix for an orthotropic elastic layer and by the effective boundary condition technique. The formula recovers the H/V formula by Malischewsky and Scherbaum [(2004). Wave Motion 40, 57-67] for the compressible isotropic case and the H/V formula by Love [(1911). Some Problems of Geodynamics (Cambridge University Press, Cambridge)] for the incompressible isotropic case. The effect of anisotropy on the H/V ratio's properties is examined and it is numerically shown that the anisotropy strongly affects the peak and zero frequency of the H/V ratio and its prograde domain as well. As the H/V ratio is a dimensionless quantity and it is easily experimentally measured, the obtained formula will be a useful tool for the nondestructive evaluation of mechanical properties of thin films deposited onto half-spaces. Two numerical examples are performed to show how the obtained formula can be used to extract the material properties of deposited layers from measured values of the H/V ratio.

A comparative analysis due to the effect of point source on generation of SH wave

The propagation of SH wave in a heterogeneous initially stressed viscoelastic layer lying over a heterogeneous initially stressed orthotropic half-space due to a point source is analysed mathematically. The dispersion equation of SH wave is obtained for the propagation of SH wave in a specified model. The method of Green’s function and Fourier transformation is incorporated to obtain the dispersion equation. The curves of dispersion equation are sketched for various values of heterogeneous parameters and initial stress on angular frequency, phase velocity and damping velocity in respect of wave number. The dispersion equation is derived for some special cases which reduces to the classical equation of Love-type wave. The present study reveals the effect of heterogeneous parameter and initial stress associated with both viscoelastic and orthotropic media.