Homogeneous Elastic Layer Research Articles

On the Receding Contact Between a Homogeneous Elastic Layer and a Half-plane Substrate Coated with Functionally Graded Materials

Mechanical contact threatens the integrity of engineering structures. In particular, receding contact is one of the primary causes that are responsible for the delamination of multilayered elastic structures. This paper aims to analyze the receding contact between a homogeneous elastic layer and a half-plane reinforced by a functionally graded coating. The multilayered structure is indented by a rigid stamp of convex profile. Governing equations and mixed boundary conditions of the double contact problem are converted into a pair of singular integral equations by Fourier integral transforms. The dual integral equations are numerically solved by Gauss–Chebyshev quadrature for the contact pressure and contact length at both interfaces of contact. Taking a circular punch as a case study, the developed algorithm is first validated against classical models available in the literature. Extensive parametric studies are subsequently performed to illustrate the effects of indentation load, geometry and material properties of individual components. Numerical results suggest the possibility of optimizing multilayered structures by the introduction of functionally graded materials (FGMs) as coatings or transitional layers.

Transient responses of nanosandwich structure based on size-dependent generalized thermoelastic diffusion theory

Nowadays, the nanosandwich structure stands as one of the most promising candidates for nanodevices and nanocomposites (e.g., nanogenerator, nano-energy harvester) due to its superior mechanical properties. The size-dependent response analysis of nanosandwich structure under extreme in-service environment is of great importance for its fabrication and exploitation. This motivates us to study the transient responses of nanosandwich structure subjected to the symmetrical transient thermal and chemical potential shock loads, which are both imposed on its upper and lower bounding surfaces. It is further assumed that thermal contact resistance and diffusional contact impedance at the interface of each two adjacent layers are zero with ideal adhesion. For each isotropic homogeneous elastic layer, the governing equations are formulated in the context of size-dependent generalized thermoelastic diffusion theory and then are solved by using Laplace transformation techniques. The effects of elastic nonlocal parameter, fractional order parameters as well as material characteristic parameters on structural transient responses are also analyzed and discussed. The present study not only provides a comprehensive understanding of the size-dependent thermoelastic diffusion coupling of nanosandwich structure, but also offers basic guidelines for optimal design.

A semianalytical and finite-element solution to the unbonded contact between a frictionless layer and an FGM-coated half-plane

In this work, we examine the receding contact between a homogeneous elastic layer and a half-plane substrate reinforced by a functionally graded coating. The material properties of the coating are allowed to vary exponentially along its thickness. A distributed traction load applied over a finite segment of the layer surface presses the layer and the coated substrate against each other. It is further assumed that the receding contact between the layer and the coated substrate is frictionless. In the absence of body forces, Fourier integral transforms are used to convert the governing equations and boundary conditions of the plane receding contact problem into a singular integral equation with the contact pressure and contact size as unknowns. Gauss–Chebyshev quadrature is subsequently employed to discretize both the singular integral equation and the force equilibrium condition at the contact interface. An iterative algorithm based on the method of steepest descent has been proposed to numerically solve the system of algebraic equations, which is linear for the contact pressure but nonlinear for the contact size. Extensive case studies are performed with respect to the coating inhomogeneity parameter, geometric parameters, material properties, and the extent of the indentation load. As a result of the indentation, the elastic layer remains in contact with the coated substrate over only a finite interval. Exterior to this region, the layer and the coated substrate lose contact. Nonetheless, the receding contact size is always larger than that of the indentation traction. To validate the theoretical solution, we have also developed a finite-element model to solve the same receding contact problem. Numerical results of finite-element modeling and theoretical development are compared in detail for a number of parametric studies and are found to agree very well with each other.

The algorithms for calculating synthetic seismograms from a dipole source using the derivatives of Green’s function

The problem of calculating complete synthetic seismograms from a point dipole with an arbitrary seismic moment tensor in a plane parallel medium composed of homogeneous elastic isotropic layers is considered. It is established that the solutions of the system of ordinary differential equations for the motion–stress vector have a reciprocity property, which allows obtaining a compact formula for the derivative of the motion vector with respect to the source depth. The reciprocity theorem for Green’s functions with respect to the interchange of the source and receiver is obtained for a medium with cylindrical boundary. The differentiation of Green’s functions with respect to the coordinates of the source leads to the same calculation formulas as the algorithm developed in the previous work (Pavlov, 2013). A new algorithm appears when the derivatives with respect to the horizontal coordinates of the source is replaced by the derivatives with respect to the horizontal coordinates of the receiver (with the minus sign). This algorithm is more transparent, compact, and economic than the previous one. It requires calculating the wavenumbers associated with Bessel function’s roots of order 0 and order 1, whereas the previous algorithm additionally requires the second order roots.

Mechanical properties and numerical simulation of Sulcata tortoise carapace

Carapace of Sulcata tortoise acts as a natural shield that protects the body from natural predators. The carapace consists of four layers: keratin scutes, dorsal cortex, cancellous interior and ventral cortex. This study aims to investigate the load-bearing mechanisms of the carapace by characterizing the layers, measuring their mechanical properties and relating them to the macroscopic behavior under compression and bending loads. Test results indicate the influence of layers’ morphology, especially porosity and structural compositions, on the macroscopic properties of the carapace. The bending samples are simulated using a commercial finite element code as laminate composite structures with homogeneous elastic layers. The models are shown to capture the load-displacement response of the carapace in the elasticity regime under quasi-static loadings.

Spreadsheets for understanding seismic energy partitioning equations

ABSTRACTSpreadsheets have come a long way and have become a universal computational tool and a powerful platform for performing scientific and engineering calculations. Their inherent strength lies in integrated storage and manipulation of data, together with a wide range of calculations, charting, and graphics presentation capabilities. These facilities make them ideal tools for prototyping mathematical equations and algorithms that can be adapted for instructional purposes to improve the quality of engineering education. This paper presents a set of spreadsheets that help in understanding the mathematical details behind the Zoeppritz equations and their various approximations that describe the partitioning of seismic wave energy at a planar interface between two homogeneous isotropic and elastic layers of rock. © 2017 Wiley Periodicals, Inc. Comput Appl Eng Educ 25:441–448, 2017; View this article online at wileyonlinelibrary.com/journal/cae; DOI 10.1002/cae.21811

Modelling of a continuously inhomogeneous coating of an elastic sphere by a system of homogeneous elastic layers in the problem of sound scattering

Abstract On the basis of analytical solutions of problems on the diffraction of a plane sound wave on a homogeneous elastic sphere with a discretely layered coating and with a continuously-inhomogeneous coating, the direction diagrams of the scattered field are calculated. It is shown that a radially inhomogeneous coating can be modelled by a system of homogeneous elastic layers. For a linear dependence of the inhomogeneity, the number of homogeneous layers in the discretely-layered coating needed to ensure a prescribed accuracy of matching of the direction diagrams of bodies with a discretely layered coating to a continuously-inhomogeneous coating has been determined.

Surface‐wave inversion for a P‐velocity profile with a constant depth gradient of the squared slowness

ABSTRACTSurface waves are often used to estimate a near‐surface shear‐velocity profile. The inverse problem is solved for the locally one‐dimensional problem of a set of homogeneous horizontal elastic layers. The result is a set of shear velocities, one for each layer. To obtain a P‐wave velocity profile, the P‐guided waves should be included in the inversion scheme. As an alternative to a multi‐layered model, we consider a simple smooth acoustic constant‐density velocity model, which has a negative constant vertical depth gradient of the squared P‐wave slowness and is bounded by a free surface at the top and a homogeneous half‐space at the bottom. The exact solution involves Airy functions and provides an analytical expression for the dispersion equation. If the ratio is sufficiently small, the dispersion curves can be picked from the seismic data and inverted for the continuous P‐wave velocity profile. The potential advantages of our model are its low computational cost and the fact that the result can serve as a smooth starting model for full‐waveform inversion. For the latter, a smooth initial model is often preferred over a rough one. We test the inversion approach on synthetic elastic data computed for a single‐layer P‐wave model and on field data, both with a small ratio. We find that a single‐layer model can recover either the shallow or deeper part of the profile but not both, when compared with the result of a multi‐layer inversion that we use as a reference. An extension of our analytic model to two layers above a homogeneous half‐space, each with a constant vertical gradient of the squared P‐wave slowness and connected in a continuous manner, improves the fit of the picked dispersion curves. The resulting profile resembles a smooth approximation of the multi‐layered one but contains, of course, less detail. As it turns out, our method does not degrade as gracefully as, for instance, diving‐wave tomography, and we can only hope to fit a subset of the dispersion curves. Therefore, the applicability of the method is limited to cases where the ratio is small and the profile is sufficiently simple. A further extension of the two‐layer model to more layers, each with a constant depth gradient of the squared slowness, might improve the fit of the modal structure but at an increased cost.

Axisymmetric analysis of a functionally graded layer resting on elastic substrate

This study considers a functionally graded (FG) elastic layer resting on homogeneous elastic substrate under axisymmetric static loading. The shear modulus of the FG layer is assumed to vary in an exponential form through the thickness. In solution, the FG layer is approximated into a multilayered medium consisting of thin homogeneous sublayers. Stiffness matrices for a typical homogeneous isotropic elastic layer and a half-space are first obtained by solving the axisymmetric elasticity equations with the aid of Hankel\'s transform. Global stiffness matrix is, then, assembled by considering the continuity conditions at the interfaces. Numerical results for the displacements and the stresses are obtained and compared with those of the classical elasticity and the finite element solutions. According to the results of the study, the approach employed here is accurate and efficient for elasto-static problems of FGMs.

Influence of multiple scattering in heterogeneous concrete on results of the surface wave inverse problem

This paper computes the phase velocity dispersion curves of coherent surface waves by taking into account multiple scattering in an elastic matrix containing elastic circular inclusions. The computed dispersion curves are inverted with a global neighborhood algorithm that considers a stratified medium with linear homogeneous and isotropic elastic layers. The impact of multiple scattering on the results is quantified. Since the surface wave phase velocity is not constant with frequency, the inversion provides solutions composed of homogeneous layers. The shear wave velocity profiles of the inverted medium and their impact on concrete porosity estimation is the subject of discussion.

Osculations of spectral lines in a layered medium

When the dense web of spectral curves for the normal modes of wave propagation in a layered medium are examined carefully, it is often observed that pairs of curves approach each other to the point that they appear to intersect, yet a closer examination reveals that they do not. In the technical literature these are referred to as avoided crossings, curve veering, mode-splitting, double cusps, diabolical points, or osculation points. We consider herein their presence–or absence–in a homogeneous elastic layer underlain by an elastic half-space, and include the two extreme cases of a homogeneous stratum where the half-space is infinitely rigid, and a free plate when it is infinitely flexible. First we show rigorously that the spectral curves for generalized Love waves (SH waves) in a layered medium will not ever intersect, even if in some cases they may come close and osculate. We then demonstrate that the spectral lines for generalized Rayleigh waves (SV-P waves) in a homogeneous stratum will only intersect when Poisson’s ratio is either exactly 14, or is in some other fractional (but not continuous) ratio below 13, in which case we provide the intersection points in closed form. We also provide the general conditions that must be satisfied by the SVP spectral lines for double roots to exist in a layered medium, whether they are plates, strata, half-spaces of full spaces. We then posit that intersections of SVP spectral lines (i.e. double roots) can only occur when the layered medium exhibits material symmetry with respect to some horizontal plane. Although at first sight the stratum case might seem to contradict this principle we show that ultimately it does not. Thus, neither a non-symmetric free plate nor a half-space will ever have intersecting lines. By contrast, a symmetric full space does have such intersections, as do also symmetrically layered plates, among which the homogeneous free plate and the homogeneous doubly-clamped plate exhibit an abundance of double roots.

The scattering of a plane sound wave by an elastic cylinder with a discrete-layered covering

An analytical solution of the problem of the scattering of a plane sound wave by a homogeneous elastic cylinder with a discrete-layered covering is obtained. The results of calculations of the radiation patterns of the scattered field for a cylinder with continuously-homogeneous and discrete-layered coverings are presented. It is shown that a radially inhomogeneous coating can be modelled by a system of homogeneous elastic layers.

Acoustic microscopy of functionally graded thermal sprayed coatings using stiffness matrix method and Stroh formalism

Acoustic microscopy of multilayered media as well as functionally graded coatings on substrate necessitates to model acoustic wave propagation in such materials. In particular, we chose to use Stroh formalism and the recursive stiffness matrix method to obtain the reflection coefficient of acoustic waves on these systems because this allows us to address the numerical instability of the conventional transfer matrix method. In addition, remarkable simplification and computational efficiency are obtained. We proposed a modified formulation of the angular spectrum of the transducer based on the theoretical analysis of a line-focus transducer for broadband acoustic microscopy. A thermally sprayed coating on substrate is treated as a functionally graded material along the depth of the coating and is approximately represented by a number of homogeneous elastic layers with exponentially graded elastic properties. The agreement between our experimental and numerical analyses on such thermal sprayed coatings with different thicknesses confirms the efficiency of the method. We proved the ability of the inversion procedure to independently determine both thickness and gradient of elastic properties. The perspective of this work is the opportunity to non-destructively measure these features in functionally graded materials.

Seismic characterization of thin beds containing patchy carbon dioxide-brine distributions: A study based on numerical simulations

We studied the seismic attenuation and velocity dispersion effects produced by wave-induced fluid flow in weakly consolidated sandstones containing patchy carbon dioxide [Formula: see text]-brine distributions. The analysis also focuses on the velocity pushdown because of the presence of this gas, as well as on the role of the wave-induced fluid flow (mesoscopic) effects on the amplitude variation with angle (AVA) seismic response of thin layers containing [Formula: see text], such as those found at the Utsira Sand, Sleipner field, offshore Norway. We found that this loss mechanism may play a key role on conventional surface seismic data, suggesting that further data analysis may provide useful information on the characteristics of the fluid distributions in these environments. Numerical experiments let us observe that although mesoscopic effects can be very significant in this kind of media, the seismic response of a given isolated thin layer computed considering such effects is very similar to that of a homogeneous elastic thin layer with a compressional velocity equal to that of the original porous rock averaged in the effective data bandwidth. This suggests that the thin-bed prestack spectral inversion method published by the authors could be used to estimate representative compressional velocities and layer thicknesses in these environments. In effect, results using realistic synthetic prestack seismic data show that isolated [Formula: see text]-bearing thin beds can be characterized in terms of their thicknesses and representative compressional velocities. This information can be qualitatively related to [Formula: see text] saturations and volumes; thus, the prestack spectral inversion method could find application in the monitoring of the evolution of [Formula: see text] plumes at injection sites similar to that at the Sleipner field.

On the theoretical strength limit of the layered elastic and viscoelastic composites in compression

The theoretical strength limit in compression (TSLC) of composite material with two alternating isotropic homogeneous elastic and viscoelastic layers is studied. The investigation is made within the piecewise-homogeneous body model with the use of the Three-Dimensional Linearized Theory of Stability for elastic and viscoelastic materials. It is assumed that there is an initial imperfection in the local curving form of the reinforcing layers along the direction which is inclined to the layers. As the criterion for determination of the TSLC, the case is taken for which the aforementioned imperfections start to increase indefinitely. The numerical results for the TSLC are presented and discussed. According to these numerical results, in particular, the direction (or the surface) along which the fracture in compression takes place, is determined.

On the Evaluation of Settlement Measurements Using Borehole Extensometers

Borehole extensometers are commonly utilized in field monitoring programs to measure settlement. In order to secure the targets of the extensometer, the borehole is typically backfilled with cement-bentonite grout. In the absence of standards for grout mix designs and due to non-universality of the properties of the backfilling grout, the accuracy of settlement furnished by borehole extensometers may be questionable. A preliminary parametric study was conducted to investigate the influence of grout stiffness on settlement measurements using borehole extensometers. The finite element program ABAQUS was used to model a borehole extensometer installed in an isotropic homogeneous elastic soil layer of a finite depth subjected to a uniform surface surcharge. Effect of the grout–soil elastic moduli ratio, soil/grout interface parameters (friction angle and shear strength), and extensometer embedment depth were investigated. Soil/grout interface behavior was assumed to comply with the elastic-perfectly plastic Coulomb friction model. The mechanism of measuring settlement using borehole extensometers was studied. The numerical study revealed that minimum errors in settlement can be obtained if the grout–soil elastic moduli ratio is made equal to about 1.0. Depending on the grout–soil moduli ratio, larger grout/soil interface friction angle and interface shear strength are generally associated with smaller settlement errors. Compared to fully embedded extensometers, partial embedment yields smaller settlement errors for grout–soil moduli ratio significantly above unity. In view of the findings of this preliminary parametric study, it is suggested to adopt the grout–soil moduli ratio in conjunction with grout compressive strength as the controlling criterion rather than the grout compressive strength only (current standard of practice). Additionally, it is evident that cement-bentonite grout mix should be carefully designed as part of the settlement monitoring programs to match the expected deformability (stiffness) of the soil mass under consideration. Future laboratory, field and numerical studies are needed to investigate more complex soil stratifications and to evaluate the behavior of various grout mixes.

Parametric Analysis of Composite Cylinders with an Embedded Exponentially Graded Piezoelectric Layer

An exact solution is obtained for a sandwich cylinder composed of two homogeneous elastic layers and an embedded exponentially graded piezoelectric (EGP) layer under the radial deformation. Based on the displacement method, the governing equations for both the elastic and piezoelectric layers are derived. Frobenius series method is employed to solve the governing equation of EGP layer, which is a second-order nonhomogeneous ordinary differential equation with variable coefficients. With the aid of the obtained solution, the parametric analysis is extensively investigated for a sandwich cylinder subjected to radial mechanical loads at the internal and external surfaces coupled with the applied voltages on the piezoelectric layer. Numerical results show that the Young’s moduli of the elastic layers, the inhomogeneity parameter of the EGP layer, as well as the external applied voltages have significant effects on the mechanical and electric fields in a sandwich cylinder.

Об антиплоской задаче кусочно-однородного упругого слоя, содержащего на линии спая систему щелей и абсолютно жёстких тонких включений.

Об антиплоской задаче кусочно-однородного упругого слоя, содержащего на линии спая систему щелей и абсолютно жёстких тонких включений.

Matrix impedance in the problem of the calculation of synthetic seismograms for a layered-homogeneous isotropic elastic medium

A new method is proposed for calculating synthetic seismograms caused by a force in a plane-parallel medium consisting of homogeneous elastic isotropic layers. The matrix impedance, i.e., the matrix function of depth, by which motion vector must be multiplied in order to obtain the stress vector, is introduced for solving a system of ordinary differential equations with respect to the motion-stress vector, which appears during the separation of variables. An independent nonlinear equation is obtained for the impedance. The propagator for the motion vector is constructed with the aid of the impedance. The closed analytical formulas, which do not contain any exponents with positive indices, are obtained both for the impedance and for the motionvector propagator. The algorithm for the calculation of seismograms, free of limitations on the number and thickness of layers, as well as on the frequency range of interest, is constructed on the basis of these formulas. The algorithm is tested with the aid of an analytical solution.

Pattern instability of functionally graded and layered elastic films under van der Waals forces

This paper investigates surface instability of a functionally graded and layered elastic film interacting with another flat rigid body [or interacting with another functionally graded and layered elastic film (or simply-supported elastic plate)] through surface van der Waals forces under plane strain conditions. The shear modulus in each functionally graded layer is assumed to be exponentially varied in the thickness direction. A homogeneous elastic layer, which is the focus of this research, can be considered as a special case of the functionally graded layer by taking the magnitude of the gradient parameter to be very small. The solution for any functionally graded layer is obtained in terms of the pseudo-Stroh formalism; then the solution for the multilayered system is derived based on the transfer-matrix method. As a result the displacement and traction vectors at the top surface of the layered film (or plate) can be expressed in terms of those at the bottom surface of the layered film (or plate). We can thus obtain simple relationships between the surface normal traction and surface deflection. Expressions for the interaction coefficient as a function of the wave number of the instability mode are therefore obtained. The critical value of the interaction coefficient for surface instability and the associated instability mode can be determined easily by identifying the minimum of the interaction coefficient. An advantage of the present method lies in that it is very convenient to address a film (or a plate) with an arbitrary number of layers. The correctness of the present approach is verified by comparison with known results. The results show that it is possible to find N distinct surface instability modes for an N-layered elastic film interacting with another flat rigid body; and that it is also possible to find that there are at most N1+N2 distinct surface instability modes for an N1-layered elastic film interacting with another N2-layered elastic film. When a multilayered elastic film interacts with a simply-supported multilayered elastic plate, the film-plate system will exhibit the instability mode of the film or that of the plate depending on the stability strength of the plate versus that of the film.