Elastic Layer Of Finite Thickness Research Articles

An adhesive detachment paradox: Does the substrate thickness matter?

A normal adhesive contact between a flat-ended cylindrical punch with elliptical cross-section and the surface of an elastic substrate, which can be regarded as an elastic layer of finite thickness, is considered. The experimental evidence published elsewhere indicates that the initiation of detachment can occur at the ends of the minor axis of the elliptical contact area, which eventually comes into contradiction with the surface energy theory developed so far. This paradox of adhesive detachment is examined in the framework of the JKR-type model formulated in terms of the stress intensity factor (SIF) of the contact normal stress. Based on the asymptotic model for a relatively thick elastic layer, the contact SIFs at the ends of the minor axis are shown to grow with decreasing the relative layer thickness faster than those at the ends of the major axis of the initial contact area. A plausible explanation for the observed inconsistency between experiment and state-of-the-art theory is found in the effect of the nonuniform elastic deformations on the square root singularity of the contact pressure at the punch edge.

Love-type waves in thermoelastic solid with double porosity structure

In the present work, we consider a model consisting of an elastic layer of finite thickness resting over an elastic half-space in perfect contact. The layer and the half-space both are assumed to be composed of different thermoelastic materials possessing different double porosity structures. The dispersion relation for Love-type surface waves has two factors, one of which matches exactly with the dispersion relation of Love waves of classical elasticity, while the second factor arises due to the presence of double porosity and thermal field in the model. Each of these factors (dispersion relation) provides two types of Love waves called ‘first front’ and ‘second front’ of Love waves. For a specific model, these two dispersion relations are computed numerically to obtain the phase speed of corresponding front of propagating Love waves. It is found that the speed of second front is much faster than the speed of first front. The phase speed of fundamental mode of the second front increases with the inclusion of thermal property, while it decreases with the inclusion of either type of voids/porosities in the model. Some earlier known results are also recovered from the present formulation as special cases.

Depth-sensing spherical indentation of an elastic sphere on an elastic substrate

The axisymmetric three-body, double-unilateral contact problem for an elastic sphere is treated by analytical techniques. It is assumed that the elastic sphere is put on the surface of an elastic layer of finite thickness and, afterwards, is indented at the upper pole of the sphere with a rigid spherical punch. Friction is neglected at both contact interfaces. A first-order asymptotic solution for the displacement–force relation, which generalizes the corresponding Hertzian formula, is derived in explicit form in terms of dimensionless asymptotic constants that account for the finite sizes of the elastic sphere and layer. Under the assumption that the sphere/substrate contact diameter is less than the substrate thickness, the presented results can cover a wide range of parameter combinations, which may be used for the purpose of benchmark assessments in the finite element analysis. The constructed asymptotic model is remarkably simple and elegant, and yet can be applied to various contact problems of practical importance (including a very timely problem of indentation of spherical viruses). The underlying theoretical framework is versatile and can be further extended for analysis of multiple contacts involving elastic spheres and elastic layers.

A Screw Dislocation in a Monoclinic Tri-Material

Employing primarily analytic continuation, we derive analytical solutions to the anti-plane problem associated with a screw dislocation located anywhere inside an anisotropic tri-material composed of an intermediate anisotropic elastic layer of finite thickness sandwiched between two semi-infinite anisotropic elastic media. All three phases of the tri-material are monoclinic with the symmetry plane at $x_{3}=0$ in a Cartesian coordinate system. We obtain explicit expressions for each one of the three analytic functions defined in its respective anisotropic elastic phase. This allows for the complete determination of the stresses and displacement in the tri-material. In addition, we present simple yet concise expressions for the image force acting on the screw dislocation. An interfacial Zener-Stroh crack in the tri-material is discussed using the analytical solution developed for an interfacial screw dislocation.

Decohesion of a rigid flat punch from an elastic layer of finite thickness

Interfacial bond strength is a crucial mechanical property of solid adhesive joints that plays an important role in a wide range of applications. A common way of quantifying mechanical strength of an adhered interface is to measure the apparent adhesion strength by finding the maximum pulling force in a tensile bond test and dividing it by the area. Despite the simplicity of this method, there is growing evidence suggesting that the measured quantity is not necessarily an intrinsic attribute of the interface but depends on thickness of the adhesive film and size of the interface. In this work, the critical pull-off force of a flat rigid punch adhered to an elastic film of finite thickness is studied via asymptotic analysis as well as finite element (FE) simulations. The decohesion behavior is found to be governed by two dimensionless parameters η and φ: the former represents the deformation of the interface relative to the interaction distance of interfacial forces, while the latter denotes a correction factor due to finite thickness of the film. As η changes from 0 to infinity, the interface decohesion would take place from uniform detachment (DMT-like) to crack-like propagation (JKR-like). The transition behavior can be well described by an empirical solution, which suggests a new rational procedure to consistently characterize the intrinsic adhesion properties of an adhered interface from tensile bond tests. Finally, the general empirical solution and the adhesion characterization procedure are directly validated by a series of experiments via detaching a stainless-steel flat punch from adhesive Polydimethylsiloxane (PDMS) films. This work not only offers a clear and explicit solution to capture the transition behavior of interface decohesion but also provides a guideline for modulating adhesion performance via geometry rather than chemistry of the adhesives.

On the plate-like and layer-like response of slab foundations to ground-borne vibration

The increasing urban population is leading to the exploitation of building sites, close to surface or underground railways, with considerable levels of ground-borne vibration. An important design consideration regards the levels of perceptible vibration and/or re-radiated noise in the completed buildings. A fundamental question concerns to what extent the mass and stiffness of a building foundation influences these levels. This paper explores this question in relation to a concrete slab foundation.Previous research has explored the influence of the coupling between a thin, flexural plate and an elastic half-space on the free-surface displacements arising from surface Rayleigh waves. Here, a numerical, wave-based approach is used to model the slab foundation as an elastic layer of finite thickness, overlying the half-space. The latter is subjected to incident waves in the form of Rayleigh, P- and SV-waves. It is found that thin-plate theory alone is insufficient for modelling the slab over the full frequency range of interest, and that the assumed soil-slab boundary condition plays a significant role. Design plots are presented in order to summarise the influence of the salient dimensionless parameters, and to help guide the design of a slab foundation to achieve a specific reduction in ground vibration level.

Effect of rotation on Rayleigh waves in a fiber-reinforced solid anisotropic magneto-thermo-viscoelastic media

ABSTRACTThe present paper deals with the of Rayleigh surface waves in fiber-reinforced solid anisotropic magneto-thermo-viscoelastic media, a pre-stressed elastic layer of finite thickness over a homogeneous, pre-stressed elastic half – space subjected to the rotation. The dispersion equation in closed form is derived for a layer over a half-space, when both media are considered as pre-stressed and the effect of rotation shown in earlier investigators, is in general not applicable to the case of pre-stressed media. In order to illustrate the analytical developments, Numerical simulated results are depicted graphically to show the effect of rotation and magnetic field on resulting quantities. Comparison was made with the results obtained in the presence and absence of the rotation and magnetic field. The study shows that there is a variation effect of magneto-thermoelasticity and rotation on the Rayleigh wave velocity. Some special cases are also deduced from the present investigation.

Study on the settlement of raft foundations by different methods

In order to study effects of the size, the stiffness of foundation, and the applied load on the thickness of compressed zone and the settlement of raft foundations, this paper analyze methods of settlement calculation such as the method of summation of partial settlements, the method of elastic layer of finite thickness, and the finite element method with the PLAXIS 3D software. Effects of the size of foundation are considered with many areas of foundation such as 150 m2, 338 m2, 600 m2, 938 m2, 1350 m2, 1838 m2, and 2400 m2; and effects of the applied load of the upper structure are considered with many values of 150 kN/m2, 200 kN/m2, and 250 kN/m2. There are remarkable differences between settlements calculated by the analytical methods and ones simulated by PLAXIS 3D. From the analysis and evaluation of calculated results, this paper proposes a suitable method to determine the thickness of compressed zone, as well as the method of calculation of settlement for raft foundations.

Axisymmetric Contact Problem for a Flattened Cell: Contributions of Substrate Effect and Cell Thickness to the Determination of Viscoelastic Properties by Using AFM Indentation

Nanoindentation technology has proven to be an effective method to investigate the viscoelastic properties of biological cells. The experimental data obtained by nanoindentation are frequently interpreted by Hertz contact model. However, in order to validate Hertz contact model, some studies assume that cells have infinite thickness which does not necessarily represent the real situation. In this study, a rigorous contact model based upon linear elasticity is developed for the interpretation of indentation tests of flattened cells. The cell, normally bonded to the Petri dish, is initially treated as an elastic layer of finite thickness perfectly fixed to a rigid substrate. The theory of linear elasticity is utilized to solve this contact issue and then the solutions are extended to viscoelastic situation which is regarded as a good indicator for mechanical properties of biological cells. To test the present model, AFM-based creep test has been conducted on living human hepatocellular carcinoma cell (SMMC-7721 cell) and its fullerenol-treated counterpart. The results indicate that the present model could not only describe very well the creep behavior of SMMC-7721 cells, but also curb overestimation of the mechanical properties due to substrate effect.

An effective asymptotic method in the axisymmetric frictionless contact problem for an elastic layer of finite thickness

A brief review of asymptotic methods to deal with frictionless unilateral contact problems for an elastic layer of finite thickness is presented. Under the assumption that the contact radius is small with respect to the layer thickness, an effective asymptotic method is suggested for solving the unilateral contact problem with a priori unknown contact radius. A specific feature of the method is that the construction of an asymptotic approximation is reduced to a linear algebraic system with respect to integral characteristics (polymoments) of the contact pressure. As an example, the sixth‐order asymptotic model has been written out. Copyright © 2015 John Wiley & Sons, Ltd.

The Love surface-acoustic wave seen as a guided non-Newtonian quasi-particle

Following along the line of recent works, the paper introduces the notion of quasi-particles that are associated with surface-acoustic waves of the Love type via the canonical conservation laws of Noether’s theory of invariants. The strange point mechanics (velocity-dependent “mass” and energy) is obtained by integrating the conservation laws of “wave momentum” and energy over an element of volume representative of the wave motion. This original but involved point mechanics obtained for the quasi-particle is neither Newtonian nor Lorentzian because of the existence of surface waves restricted to a bounded interval of velocities that is characteristic of Love waves. The reduction of this description to the cases of an isolated elastic layer of finite thickness with pure shear horizontal motion and of the so-called Murdoch case corresponding to a thin-film approximation (half-space with an upper material boundary equipped with mass and surface elasticity) studied independently is exactly proved.

Rotation and Magnetic Field Effect on Surface Waves Propagation in an Elastic Layer Lying over a Generalized Thermoelastic Diffusive Half-Space with Imperfect Boundary

The aim of the present investigation is to study the effects of magnetic field, relaxation times, and rotation on the propagation of surface waves with imperfect boundary. The propagation between an isotropic elastic layer of finite thickness and a homogenous isotropic thermodiffusive elastic half-space with rotation in the context of Green-Lindsay (GL) model is studied. The secular equation for surface waves in compact form is derived after developing the mathematical model. The phase velocity and attenuation coefficient are obtained for stiffness, and then deduced for normal stiffness, tangential stiffness and welded contact. The amplitudes of displacements, temperature, and concentration are computed analytically at the free plane boundary. Some special cases are illustrated and compared with previous results obtained by other authors. The effects of rotation, magnetic field, and relaxation times on the speed, attenuation coefficient, and the amplitudes of displacements, temperature, and concentration are displayed graphically.

Finite-amplitude shear horizontal waves propagating in a pre-stressed layer between two half-spaces

The propagation of finite-amplitude time-harmonic shear horizontal waves, in a pre-stressed compressible elastic layer of finite thickness embedded between two identical compressible elastic half-spaces, is investigated. This is accomplished by combining finite-amplitude linearly polarized inhomogeneous transverse plane wave solutions in the half-spaces and finite-amplitude linearly polarized unattenuated transverse plane wave solutions in the layer. The layer and half-spaces are made of different pre-stressed compressible neo-Hookean materials. The dispersion relation which relates wave speed and wavenumber is obtained in explicit form. The special case where the interfaces between the layer and the half-spaces are principal planes of the left Cauchy–Green deformation tensor is also investigated. Numerical results are presented showing the variation of the shear horizontal wave speed with the pre-stress and the propagation angle.

Interaction of rocking vibration of a buried rigid circular disc and a two-layer transversely isotropic half-space

A linear elastic layer of finite thickness bonded on a half-space each containing a transversely isotropic material is considered. A rigid circular disc attached to the interface of these two domains is considered to be affected by a rocking vibration of constant amplitude. With the aid of a scalar potential function, the equations of motion in each domain are solved using Fourier series and Hankel integral transforms. Because of the involved integral transforms, the mixed boundary value problem is changed to dual integral equations; which are reduced to Fredholm integral equations. Because of the complex integrand function existing in the dynamic case, analytical solution cannot be given in general. However, a closed-form solution is introduced for the static case, which itself degenerates to the solution for an isotropic case existing in the literature. The contact stress in between the rigid disc and the surrounding media, and the related impedance function are analytically determined in the static case. With the help of contour integration, the governing Fredholm integral equations are numerically evaluated in the dynamic case. The dynamic contact pressure and the impedance function are numerically evaluated in a general dynamic case. The shape induced singularity in the contact pressure is investigated in detail. Some numerical evaluations are given for different transversely isotropic materials to show the effect of anisotropy.

Calibrated Analytical Formulas for Foundation Model Parameters

AbstractA set of analytical formulas for the estimation of foundation model parameters is presented by synthesizing mechanical models at three different levels with corresponding variants of a generalized continuum model presented recently. The generalized continuum model was derived using a unified approach on the basis of a subgrade idealized as an elastic layer of finite thickness overlying a rigid base without making prior simplifying assumptions. The spring stiffness in all variants is inversely proportional to the layer thickness, rendering the model sensitive to this parameter with the potential to yield excessively large deflections for thick formations. This problem is alleviated by eliminating the layer thickness through introduction of a calibration factor in the form of a ratio of the layer thickness to the foundation width. It has been demonstrated that the calibration factor for each model type can be established from comparative analytical-numerical studies. Values obtained in this manner a...

Shear-horizontal waves in a rotated Y-cut quartz plate with an isotropic elastic layer of finite thickness

We study shear-horizontal (SH) waves in a rotated Y-cut quartz plate carrying an isotropic elastic layer of finite thickness. The three-dimensional theories of anisotropic elasticity and isotropic elasticity are used for the quartz plate and the elastic layer, respectively. A transcendental frequency equation that determines the dispersion relations of the waves is obtained. The dispersion relations are obtained and plotted by solving the frequency equation using MATLAB. Approximate dispersion relations are also obtained analytically for two special cases. One is for long waves whose wavelength is much larger than the plate thickness. The other is for the case of a very thin elastic layer. The effects of the elastic layer on the dispersion relations are examined. The results obtained are fundamental and useful to acoustic wave sensors for measuring the mechanical and geometric properties of the elastic layer.

Surface Wave Propagation in an Elastic Layer Lying over a Thermodiffusive Elastic Half-Space with Imperfect Boundary

The present investigation is to study the surface wave propagation with an imperfect boundary between a homogenous isotropic thermoelastic diffusive half-space and an isotropic elastic layer of finite thickness. The secular equation for surface waves in compact form is derived after developing the mathematical model. The phase velocity and attenuation coefficient are obtained for stiffness and then deduced for normal stiffness, tangential stiffness, and welded contact. The amplitude of surface displacements, temperature change, concentration, and the specific loss of energy are computed numerically. The dispersion curves for these quantities are illustrated to depict the effect of stiffness and thermal relaxation times. Some special cases of interest are also deduced and compared with known results.

Surface elasticity and residual stress effect on the elastic field of a nanoscale elastic layer

The influence of surface elasticity and surface residual stress on the elastic field of an isotropic nanoscale elastic layer of finite thickness bonded to a rigid material base is considered by employing the Gurtin-Murdoch continuum theory of elastic material surfaces. The fundamental solutions corresponding to buried vertical and horizontal line loads are obtained by using Fourier integral transform techniques. Selected numerical results are presented for the cases of a finite elastic layer and a semiinfinite elastic medium to portray the influence of surface elasticity and residual surface stress on the bulk stress field. It is found that the bulk stress field depends significantly on both surface elastic constants and residual surface stress. The consideration of out-of-plane terms of the surface stress yields significantly different solutions compared to previous studies. The solutions presented in this study can be used to examine a variety of practical problems involving nanoscale/soft material systems and to develop boundary integral equations methods for such systems.

Asymptotic solution of the problem of the action of a stamp on an elastic layer lying on the surface of a compressible fluid of infinite depth

This paper considers a two-dimensional linear unsteady problem of rigid-stamp indentation on an elastic layer of finite thickness lying on the surface of a compressible fluid of infinite depth. The Lame equations holds for the elastic layer, and the wave equation for the fluid velocity potential. Using the Laplace and Fourier transforms, the problem is reduced to determining the contact stresses under the stamp from the solution of an integral equation of the first kind, whose kernel has a logarithmic singularity. An asymptotic solution of the problem is constructed for large times of interaction.

Cell Force Microscopy on Elastic Layers of Finite Thickness

Forces applied by cells to substrates can be measured using soft substrates with embedded displacement markers. Traction forces are retrieved from microscopic images by determining the displacements of these markers and fitting the generating forces. Here we show that using elastic films of 5–10- μm thickness one can improve the spatial resolution of the technique. To this end we derived explicit equations for the mechanical response of an elastic layer of finite thickness to point forces. Moreover, these equations allow highly accurate force measurements on eukaryotic cells on films where finite thickness effects are relevant (below ∼60 μm).