Isotropic Elastic Layer Research Articles

Solution of a three-dimensional problem of the elasticity theory in terms of displacements for an isotropic elastic layer

A new method is proposed for solving a three-dimensional problem of the elasticity theory for a layer in terms of displacements. It is much easier to satisfy the boundary conditions within the framework of this approach than with the known methods. An example brought to numerical results is considered.

Reconstructing the adhesion stiffness distribution in a laminated elastic plate: Exact and approximate inverse scattering solutions

This paper formulates and solves a time harmonic inverse scattering problem to reconstruct the effective stiffness distribution of an adhesive bond in a layered elastic plate. The motivation is based on the assumption that localized adhesion flaws that diminish bond stiffness also tend to diminish bond strength. The formulation is based on the invariant imbedding method, applies to isotropic and anisotropic elastic layers, and is essentially that of identifying embedded acoustic sources in elastic layered structures. This paper presents two solutions for the inverse problem: the Born approximation and the exact solution. The example calculations compare the two solutions and show that when imperfections are too large in either magnitude or extent the accuracy of the Born approximation breaks down. The impact of noise and uncertainties in the background properties in the inversion is also investigated. A regularization strategy is introduced in the exact solution that controls solution sensitivity in regions with low signal to noise ratio.

Transient Piezothermoelasticity of a Two-Layered Composite Hollow Cylinder Constructed of Isotropic Elastic and Piezoelectric Layers Due to Asymmetrical Heating

This article is concerned with the theoretical treatment of transient piezothermoelastic problem involving a two-layered hollow cylinder constructed of isotropic elastic and piezoelectric layers due to asymmetrical heat supply. The transient two-dimensional temperature is analyzed by the method of Laplace transformation. By using the exact solutions for piezoelectric hollow cylinder and isotropic hollow cylinder, the theoretical analysis of transient piezothermoelasticity is developed for a two-layered composite hollow cylinder under the state of plane strain. As an example, numerical calculations are carried out for an isotropic elastic hollow cylinder made of steel, bonded to a piezoelectric layer of cadmium selenide. Some numerical results for the temperature change, the stress and the electric potential distributions in a transient state are shown in figures. Furthermore, the influence of thickness of the piezoelectric layer or the isotropic elastic layer upon the temperature change, stresses and electric potential is investigated.

Rotational microrheology of multilayered finite elastic media

This paper presents derivations of the torque-rotation angle relationship for a rigid spherical bead embedded in a composite medium made of n isotropic and linear elastic layers. Analytical solutions are provided for both compressible and incompressible solids, assuming no-slip conditions between the rigid spherical inclusion and its adjacent medium as well as between elastic layers. Thanks to these general formulas, we investigated the effect of finite size media on the torque-bead rotation response and derived the exact relationship linking apparent and intrinsic elastic moduli of the medium. Thus, this result can be applied to characterize precisely the mechanical heterogeneity and architecture of soft biomaterials, including cells, from real magnetocytometry experiments. This point is exemplified in the case of a bilayer medium which, interestingly, allows distinguishing cellular cortex from deep cytoskeleton. We found that rotational microrheology experiments are well suited to characterize locally the elastic properties of the layer in contact with the probe as soon as the layer thickness is larger than two-bead diameters.

Analysis of time harmonic wave propagation in an elastic layer under heavy fluid loading

This paper concerns assessment of the validity of elementary models of wave propagation in an isotropic elastic layer under heavy fluid loading as well as analysis of coupling effects due to uneven fluid loading. Dispersion curves, which describe propagation of dominantly flexural ‘in-phase’ waves and dominantly longitudinal ‘anti-phase’ waves in an elastic layer loaded by an acoustic medium at both sides, are compared with those obtained in the case of one-sided fluid loading. The latter are also compared with dispersion curves predicted by elementary theories of fluid-loaded plates. It is shown that the compressibility of the fluid dramatically extends the validity ranges of elementary theories and this phenomenon is explained.

Effect of the parameters of a material on the surface displacements caused by waves propagating in a layer

The amplitude—frequency characteristics of the vibrations of the surface of a free elastic layer are determined. These vibrations are caused by propagating Lamb waves under the action of a normal surface load. The changes in the spectra of surface displacements and in the tangential-to-normal displacement-amplitude ratios during changes in the Poisson ratio are studied. Studying the spectrum of surface displacements of an elastic layer is of interest owing to the necessity of discriminating modal components of the wave field [1] and as one of possible methods for determining some of the parameters of a material. The ratio of the amplitudes of the horizontal and vertical displacements as a function of the frequency of a vibration source is sensitive to the Poisson ratio of the material and the conditions of the contact with the base. The amplitude-frequency characteristics of the surface displacements of an elastic isotropic layer are plotted and compared to those obtained in the case of a layer that is in contact with a rigid base.

Numerical solution to the buckling problem for a sandwich plate under uniaxial compression

A thin rectangular sandwich plate with isotropic linear elastic layers is considered. The plate is in a plane-strain state under uniaxial compression. An exact statement of the buckling problem is given. Its approximate solution is found by the finite-difference method. The concept of base scheme is used to formulate discrete problems in explicit and compact form. As an example, the critical parameters of the plate are calculated using a computation optimization procedure. Its efficiency is demonstrated

Deformations in Transform Faults with Rotating Crustal Blocks

A continuum model of deformation of a transform fault is considered. The fault interior is modelled as a part of a plate fragmented into a large number of angular blocks. The blocks are not joined together by any binder, but are rather held together due to external compression such that they can, in principle, rotate independently. Significant numbers of blocks involved in the deformation process permit a continuum description of the deformation, which in view of the possibility of independent rotations, necessitates the application of the Cosserat theory. The crucial point in the described model is the interconnection between the rotations and the normal stresses associated with the angular shape of the blocks: the rotating blocks ‘elbow’ one another. Furthermore, elbowing produces compressive stresses independent of the direction of rotations. Consequently, the constitutive equations become non-linear involving absolute values of the components of the curvature tensor. The paper analyses a simple shear of a fault under constraining compression acting in the direction normal to the fault. An infinite layer subjected to opposite displacements and zero rotations at the edges is considered. It is shown that block rotations can lead to complex deformation patterns. There exists a displacement threshold proportional to the pressure: for imposed displacements below the threshold, the deformation pattern coincides with the conventional one as predicted for a classical elastic isotropic layer with uniform displacement gradient in the absence of block rotations. When the imposed displacement exceeds the threshold value, boundary zones of non-uniform rotations, displacement gradients and dilatation emerge. It is interesting to note that these features, which could be mistaken for indicators of non-elastic or localisation processes, occur in a situation where only elastic processes are acting at the scale of blocks and no friction or other energy dissipation processes take place.

Detecting and classifying adhesive flaws between bonded elastic plate: The inverse problem

A time-harmonic inverse scattering problem to reconstruct the effective stiffness distribution of an adhesive bond in a layered elastic plate was formulated and solved. The motivation is based on the assumption that localized adhesion flaws that diminish bond stiffness also tend to diminish bond strength. The formulation and solution are based on the invariant embedding method, apply to isotropic and anisotropic elastic layers, and are essentially that of identifying embedded acoustic sources in elastic layered structures. Two solutions of the inverse problem are presented: the Born approximation and the exact solution. In the example calculations we compare the two solutions and show that when imperfections are too large the accuracy of the Born approximation breaks down. The impact of noise and uncertainties in the background properties in the inversion is also investigated.

Elementary solution of contact problems for a transversely isotropic elastic layer bonded to a rigid foundation

The contact problem for an arbitrary punch acting on a transversely isotropic elastic layer bonded to a rigid foundation is solved by the generalized images method developed by the author earlier. The problem is reduced to that of an electrostatic problem of infinite row of coaxial charged disks in the shape of the domain of contact. The solution can be obtained by the method of iteration, collocations or any other standard procedure for solving integral equations. Exact inversion can be obtained in the case of a circular domain of contact. The mean value theorem can be used for estimation of the resultant force and tilting moment acting on a punch of arbitrary shape and circular domain of contact. A limiting case of general solution gives the solution for an isotropic layer.

Asymptotic solutions for axisymmetric contact of a thin, transversely isotropic elastic layer

The problem of a rigid sphere in normal contact with a thin, transversely isotropic elastic layer is investigated. The thin elastic layer rests on a rigid foundation and both the limiting cases, wherein the interface is either ideally bonded or unbonded and frictionless, are considered. Approximate analytical solutions for the contact pressure beneath the spherical indenter and for the contact patch radius are presented. By comparing these with those predicted by the finite element method, the accuracy of the solution methods is determined.

Control of the Strain Field Near a Crack in Elastic Layer

In the present paper, we consider the strain analysis near the crack parallel to boundary surfaces of a linear isotropic elastic layer of constant thickness. We pose the problem of minimizing the opening of crack faces when a normal force is applied to the boundary of the layer. The control secondary loading is a normal stress uniformly distributed over the boundary surfaces as to oppose the opening caused by the external applied force. We show that the most dangerous are tangential deformations, and the normal ones are always by many times less dangerous.

Love waves propagation in a piezoelectric layered structure with initial stresses

The propagation behavior of Love waves in a piezoelectric layered structure with inhomogeneous initial stress is studied. Solutions of the mechanical displacement and electrical potential function are obtained for the isotropic elastic layer and transversely isotropic piezoelectric substrate, respectively, by solving the coupled electromechanical field equations. Firstly, effects of the inhomogeneous initial stress on the dispersion relations and phase velocity of Love wave propagation are discussed. Then the influence of the initial stress gradient coefficient on the stress, mechanical displacement and electrical potential distribution in the layer and the substrate is investigated in detail. The results reported in this paper are not only meaningful for the design of surface acoustic wave (SAW) devices with high performance, but also effective for evaluating the residual stress distribution in the layered structures.

Fracture analysis of materials with periodic microstructure by the representative cell method

The periodic structure of some natural and especially man-made materials can be manifested not only on an atomic but also on a larger scale. Investigation of mechanical properties of these materials usually hinges on well-developed homogenization methods. On the other hand, these methods are not suitable for fracture analysis where the knowledge of the local stress-strain fields near a flaw (a crack) is required. The result is obtained by the use of the representative cell method based on the discrete Fourier transform. This method enables one to determine the exact stress distribution in a periodic structure subjected to arbitrary loading. Direct application of the method is impossible since the crack violates the translational symmetry defined by the material microstructure. This obstacle is overcome by application of the fictitious loading to the uncracked body at the line where the crack is to be located. The amplitude of the loading is adjusted in order to fulfill the boundary conditions imposed on the crack faces. The compatibility equation for deriving this amplitude is obtained by the use of the corresponding Green function, which is found in a closed form. Fracture problems for the two types of materials with a periodic microstructure are considered. The first one is a composite material consisting of dissimilar isotropic elastic layers arranged periodically. The second periodic microstructure is a 2D infinite beam lattice modeling a cellular material. The analysis of the failure process in the latter case shows that in contrast to the case of homogeneous material, the crack propagation path is not defined by the condition of zero Mode II stress intensity factor.

Detecting and classifying adhesive flaws between bonded elastic plates

Nondestructively evaluting the quality of an adhesive bond is challenging. The reason is that the only way to measure bond strength is by measuring the force required to break a bond, after which the bond is broken. Localized adhesion flaws that diminish bond strength, however, also tend to diminish bond stiffness. For example, pockets of elevated porosity, microcracking, or other damage in the adhesive layer will simultaneously lower local bond strength and local effective bond stiffness. So guided, we formulate and solve an inverse scattering problem to reconstruct the effective stiffness distribution of an adhesive layer in a layered elastic plate. Our formulation is based on the method of invariant imbedding, and applies to isotropic and anisotropic elastic layers. We present two solutions of the inverse problem: the Born approximation and the exact solution. Both solutions are unstable, as is the nature of inverse scattering problems, and require some regularization in the presence of noise. We present computed examples and discuss the role of regularization.

Elastic solutions for periodically layered strip with perfect bonding or with an interface crack

A non-homogenization approach to the analysis of a strip consisting of a finite number of isotropic elastic layers arranged periodically is presented. The result is obtained by the use of the representative cell method based on the discrete Fourier transform. Two types of problems are addressed. In the problems of the first type the bonding between the layers is perfect. The solution for this case is found in a closed form in terms of the Laplace integrals. The numerical results for different boundary conditions are given and a comparison with plate theory is carried out. In the problems of the second type there is a flaw (crack) at one of the interfaces. The solution in this case hinges on the analytic expression for the Green function corresponding to a single interface dislocation in the uncracked strip. By using this expression the crack problem is reduced to a singular integral equation. The influence of the elastic mismatch and other problem parameters on the fracture characteristics is examined. It is found that, similar to the case of the periodic plane, when the thinner layeres in the strip are stiffer, further increase of their stiffness may lead to the increase or to the decrease of the absolute value of the stress intensity factor depending upon the elastic mismatch between the composite constituents.

Application of the generalized images method to contact problems for a transversely isotropic elastic layer

The idea of image charges, widely used in electrostatics, is applied to solution of a contact problem for an arbitrary punch acting on a transversely isotropic elastic layer resting on a rigid smooth foundation. The governing integral equation obtained is mathematically equivalent to that of an electrostatic problem of infinite row of coaxial charged discs in the shape of the domain of contact. Since the charge sign varies, each truncation gives either the upper or the lower bound to the exact solution. Simple and accurate engineering formulae are obtained for estimation of the resultant force and tilting moment acting on a punch of arbitrary shape and circular domain of contact. The case of isotropy is considered as well.

Mechanical responses of elastic layers with through-the-thickness variation in eigenstrain

A general procedure is developed for the three-dimensional analysis of homogeneous isotropic elastic layers with eigenstrain varying arbitrarily through the thickness. By using the state-space method coupled with asymptotic expansion, the displacement and stress fields are derived and expressed in terms of the displacement components on the mid-plane. The governing equations for these displacement components are formally the same as those in the classic plate theory. Obtaining the displacement of the mid-plane immediately provides a three-dimensional solution to the elastic layer. The solution is exact in Saint Venant's sense that the boundary conditions at the edge of the layer are satisfied in the average manner as in the classic plate theory. As application of the present method, some examples are given and numerical results are shown.

An inverse problem for a layered elastic plate

Under the assumption that an infinite plate consists of a finite number of elastic isotropic layers of various thickness and elastic properties, an inverse problem is considered. It is supposed that a point pulse force is applied at a boundary of the plate and the displacement vector is measured at the boundary. The inverse problem of determining parameters of the elastic plate is studied. It is proved that the given displacement vector determines the elasticity modules and density of layers as well as their thickness uniquely. A procedure of recovering the parameters is given.

Use of Spectra of Normal and Tangential Displacement Components in Lamb Waves for Separating Single-Mode Components

The paper considers one of simple engineering models--an isotropic elastic layer lying on a rigid base where a wave is driven by a vertical periodic load applied to the surface. Using the Hankel transformation and the method of loop integration, formulas have been obtained for the elastic field due to propagating modes generated by an axially symmetrical surface source in a zone where the effect of inhomogeneous waves is insignificant. Calculations of displacement amplitudes and their ratios versus frequency for propagating modes are given.