Thick Elastic Layer Research Articles

Theory of images in plane-layered axisymmetric elastostatics

A fundamental theorem which automatically transforms the solution due to the presence of an arbitrary axisymmetric singularity in an unbounded homogeneous isotropic elastic solid into the corresponding solution for two perfectly bonded isotropic semi-infinite elastic solids is systematically applied in a stepwise fashion to obtain the complete image system when an arbitrary axisymmetric singularity is operative in or near a thick elastic layer which separates two other dissimilar isotropic semi-infinite elastic solids. The solution is closed and well structured. As an illustration, the distant effect in the interface layer produced by an influencing normal point force is luminously revealed to be two-dimensional, consisting of a combination at the origin of a bending hot spot and an infinite line of centres of dilatation. We conclude the paper with a complete theory of images for a free elastic layer under the influence of both axisymmetric and asymmetric singularities. We find that if the influencing displacement field is the gradient of a harmonic function, then the calculation of the induced elastic field reduces to the operation of differentiation only.

Histological and Morphometric Study of the Arterial Route from the Intercostal/Lumbar Artery via the Adamkiewicz Artery to the Anterior Spinal Artery Using Elderly Cadavers with or without Aortic Aneurysms

The Adamkiewicz artery originates from the lower intercostal/lumbar artery, runs along the vertebral roots of the spinal nerve, and communicates with the anterior spinal artery. In graft replacement for an aortic aneurysm, the lower intercostal/lumbar artery should be reconstructed to maintain the arterial blood supply to the lumbar spinal cord via the Adamkiewicz artery. To date, there are few reports on the histological characteristics of the arterial pathway from the aorta to the anterior spinal artery via the Adamkiewicz artery as well as the pathway in the Adamkiewicz artery. In addition, there is a possibility that this arterial pathway was degenerated by the aortic aneurysm. In this study, we attempted to reveal the histological characteristics of the arterial pathway by using elderly cadavers with or without unruptured aortic aneurysm. Specimens were obtained from 190 elderly cadavers (fixed in 10% formalin). These cadavers included 16 cases of aortic aneurysms. We collected fragments of the intercostal/lumbar artery, its dorsal branch, the Adamkiewicz artery, and the anterior spinal artery. After routine histological procedures, several staining procedures were performed for each section. A morphometric study was conducted at the light-microscopic level. The Adamkiewicz artery is composed of a thick intima that contains multiple thick elastic fiber layers and, probably, numerous intimal smooth muscle cells. Among the cadavers, the intimal thickness varied significantly in the intercostal/lumbar artery and the Adamkiewicz artery; it tended to be constant, irrespective of luminar dilatation in the Adamkiewicz artery, in contrast to other arteries. No significant difference was observed between the specimens with or without unruptured aortic aneurysms in all the parameters that were measured. The Adamkiewicz artery appeared to be capable of adapting to acute and chronic changes in systemic status that develop with aging because of its thick intima that contained numerous smooth muscles and elastic fibers. Against various events of systemic circulatory dysfunctions such as hypertension, its unique morphology suggested that this artery may play the critical role of a modulator, or even a barrier, that is intercalated along the arterial route to the lumbar cord.

Asymptotic solution of contact problemsfor a relatively thick elastic layer when there are friction forces in the contact area

The plane contact problem of the theory of elasticity of the interaction between a punch, having a base in the form of a paraboloid,and a layer, taking Coulomb friction in the contact region into account, is considered. It is assumed that either the lower boundary of the layer is fixed or there are no normal displacements and shear stresses on it, and that normal and shear forces are acting on the punch. Here, the punch-layer system is in a condition of limit equilibrium, and the punch does not turn during the deformation of the layer. The case of quasi-statistics, when the punch moves evenly over the layer surface, can be considered similarly in a moving system of coordinates. The problem is investigated by the large-λ method (see [1–3], etc.), which is further developed here, namely, simple recurrence relations are derived for constructing any number of terms of the series expansion of the solution of the corresponding integral equation in negative powers of the dimensionless parameter λ related to the thickness of the layer.

Dynamic analysis of sandwich beams with interlayer slip

AbstractForce induced dynamic bending of sandwich beams composed of three symmetrically arranged layers are studied. Moderately thick elastic layers made of dissimilar transversal isotropic materials are considered. If rigid bond between the layers cannot be achieved, due to relative deformation of the connectors an interlayer slip occurs, that can significantly affect both strength and deformation of the layered structure. Analyzing two special cases of interlaminar connections, i.e., “perfect bond” and “elastic interlayer slip”, a complete analogy between the corresponding initialboundary value problems is derived. Hence, the dynamic response of the beam with elastic interlayer slip can be determined directly from the solution of the perfectly bonded sandwich beam with moderately thick faces.

Comparison between thermal–viscous coupling and frictional sliding

We investigated the similarity between thermal–viscous coupling (TVC) and frictional sliding, proposed by Kameyama and Kaneda [Pure Appl. Geophys. 159 (2002) 2011]. We consider a one-dimensional layer composed of viscous material, which is sandwiched and sheared by two thick elastic layers. The rate of viscous deformation depends on the temperature T c in the viscous layer as well as shear stress τ. The temperature T c changes owing to heating by viscous dissipation and conductive cooling. We carried out velocity-stepping tests for the steady-state deformation both numerically and analytically, and compared the temporal evolution of small perturbations with that of the spring-block model with rate- and state-dependent friction (RSF). We found that, as is the case of frictional slip stability, the manner of temporal evolution is classified into four regimes depending on whether it is stable or not and whether it is monotonous or oscillatory with time. By further interpreting TVC in terms of general RSF theory by Ruina [J. Geophys. Res. 88 (1983) 10359], we obtained the relations between the parameters appearing in the phenomenological RSF law and the nondimensional parameters which characterize the nature of TVC. A further improvement of this approach might be important for estimating the actual values of frictional constitutive parameters at the deeper portion of seismogenic faults of interplate or inland earthquakes where a ductile deformation is expected to be significant.

SANDWICH PANELS WITH INTERLAYER SLIP SUBJECTED TO THERMAL LOADS

Thermally induced bending of sandwich panels composed of three symmetrically arranged layers is studied under the consideration of plane strain. Moderately thick elastic layers made of dissimilar transverse isotropic materials are considered. Analyzing two special cases of interlaminar connections, namely, perfect bond and elastic interlayer slip, a complete analogy between the corresponding initial boundary-value problems is derived. Subsequently, the response of the panel with elastic interlayer slip can be determined directly from the solution of the perfectly bonded sandwich panel with moderately thick faces by substituting effective parameters.

Dynamic Measurement System for Contact Pressure Distribution in Human Joints Utilizing Pressure-Sensitive Conductive Rubber.

We developed a dynamic measurement system for contact pressure distribution in human joints using a thin carbon-particle-impregnated rubber pressure sensor particularly designed for this application together with a simple circuit. The electric resistance in the thin rubber correlates well with the amount of pressure applied. The sensors are flexible and have the ability to adapt to concave surfaces, sufficiently low hysteresis, high test-retest reliability and sufficient endurability. The indentation tests of a thick elastic layer by cylindrical and spherical punches were carried out to make a comparison of the contact pressure distributions measured with those due to a three-dimensional theory of elasticity and to estimate reliability of the sensing system. This system was applied to the patellofemoral joint of human cadaver knees in order to study kinematics of the joint. The results obtained show that the mean pressure on the patella with any flexion angles and the tension in the quadriceps are constant.

Exact solutions for stress analysis of transversely isotropic elastic layers

The general equations for the elastic analysis of transversely isotropic materials are written in a form which allows derivatives in the thickness direction for all orders to be conveniently calculated. The displacements and stresses are expanded in Taylor series of a form suitable for deriving exact solutions for thick elastic layers with stress-free surfaces. This extends the work of Rao and Das for isotropic materials to the transversely isotropic case. A class of exact solutions are used to obtain results for stress concentration factors due to circular holes in layers.

Ridge push and associated plate interior stress in normal and hot spot regions

The stress distribution and plate boundary force produced by ridge push are modelled by elastic/viscoelastic finite element analysis in oceanic lithosphere and in an adjacent continent separated by a passive margin. Models are presented for both a normal slow-spreading ridge and for one underlain by an anomalously low density upper mantle associated with a hot spot which gives rise to 1 km extra elevation of the sea floor. The weak zone at the ridge crest is simulated by the viscoelastic elements extending up to the surface beneath the immediate crest. Using a standard thermal and rheological model of the oceanic lithosphere, a normal ridge gives rise to horizontal deviatoric compression of about 40 MPa in the 25–30 km elastic layer beneath older ocean floor. A much smaller modelled compression of 9 MPa occurs in the 20 km thick elastic layer of the adjacent upper continental crust, as a result of the superimposed tension produced by the thick low-density continental crust (in reality this compression may be up to 30 MPa higher if the sub-continental mantle is denser than that beneath old ocean floor, as the geoid anomalies possibly suggest). A fourfold increase in spreading rate has an insignificant effect on the stresses. Much larger compressions develop adjacent to the modelled hot spot ridge in normal oceanic and especially continental lithosphere, these reaching 100 MPa beneath the old ocean floor and 90 MPa in the adjacent continent. The stresses determined by the modelling are comparable in magnitude but generally slightly lower than values estimated using the density moment function. The ridge push force referenced to 90 Ma old oceanic lithosphere is estimated to be 2.5 × 10 12 N/m for the modelled normal ridge and 6.2 × 10 12 for the modelled hot spot ridge. The modelling shows that the ridge push is produced by the high pressure and shear drag exerted by the asthenosphere on the lithosphere. The high pressure effect predominates at normal ridges where the plates are forced apart by gravitational wedging. The shear drag effect contributes substantially at hot spot ridges. The modelling helps to explain the prominent compressional stress in the continents bordering the North Atlantic, in terms of the hot spot activity beneath the mid-Atlantic ridge.

One-way wave equations for seismoacoustic propagation in elastic waveguides

One-way or parabolic wave equations for time-harmonic propagation in two-dimensional elastic waveguides are considered. It is shown that the direct application of a rational linear approximation with real coefficients to the elastic wave propagation case results in exponential growth in the numerical solutions. Elementary analysis demonstrates that this kind of approximation does not treat properly the modes with complex wavenumber which can exist in elastic waveguides. A new bilinear square-root approximation with complex coefficients is introduced that accommodates all mode types and leads to stable numerical solutions. In the case of thick elastic layers (such as sea-bottom sediments), this new approximation gives accurate total field prediction. When thin elastic layers (such as ice on the sea surface) are present, however, the method introduces excessive damping to modes with wavenumbers significantly different from a reference wavenumber.

The problem of a rigid punch sliding on an elastic or a viscoelastic layer

The problem of a rigid punch moving over a general anisotropic viscoelastic medium is considered both for the case where the medium is a finite layer and where it is an infinite half-space. Formal equations are derived, using Fourier Transform techniques, for the most general case where no assumptions are made concerning the shape of the punch. However, attention is then focussed on the case where the punch is uniformly infinite in one direction. Both the general inertial problem and the non-inertial approximation are considered for the half-plane problem. Previously known results are shown to emerge from the general formalism with relative ease. The general viscoelastic problem is shown to be formally identical to the isotropic problem treated in [1] though algebraically far more complex. In the non-inertial approximation, however, the extra complexity is not great, and in particular, if all the viscoelastic functions are assumed to have similar time behaviour, the resulting equations are essentially no more complex than the isotropic equations solved in [2]. For this case, the relationship between the hysteretic friction coefficients along the two axes of anisotropy on an orthotropic medium is quantified. Displacementtraction relationships are given for an isotropic layer, including inertial effects and some numerical results are given for the thick elastic layer problem.

Quasi-static crustal deformations due to a surface load: Rheological structure of the earth's crust and upper mantle.

In this paper, we treat quasi-static displacement fields due to an axially symmetric load applied on the surface of a composite medium which consists of elastic layers overlying a stratified viscoelastic half-space. Integral representations of the surface displacements are obtained from those in the associated elastic problem by applying the correspondence principle of linear viscoelasticity. General features of the viscoelastic displacements are examined from various aspects through the computation of numerical examples for a three-layered model composed of an elastic surface layer, an intervening low viscosity layer and a viscoelastic substratum with relatively high viscosity.Uplift phenomena in two tectonically different regions, Fennoscandia and Lake Bonneville (the Basin and Range province), are analyzed to investigate regional difference in rheological structure of the earth's crust and upper mantle. The uplift data in Fennoscandia are well interpreted by a three-layered model which consists of a 100km thick elastic surface layer (lithosphere) and a 100-200km thick low viscosity layer (asthenosphere) overlying a relatively high viscous (1.4×1022 poise) substratum. The viscosity of the asthenosphere is estimated as 1.4×1020-1.4×1021 poise. In the case of Lake Bonneville, the uplift data are adequately explained by taking the thickness of the lithosphere as 30-40km and the viscosity of the asthenosphere as less than 1.0-2.0×1021 poise. The lower bound of the asthenosphere is obscure from the present analysis. These results show that there exists notable differences in the thickness of the lithosphere between two tectonically different regions, but not in the viscosity of the asthenosphere.

Fluid loading of plates—Acoustic consequences and coincidences

In view of recent controversy over the “leaky pole” contribution in the pressure field of vibrating fluid‐loaded thin plates, this paper considers the variation with frequency of the modal behavior of a thick elastic layer immersed in fluid. The “thin plate” equation is recovered as a low‐frequency, low‐wavenumber approximation, but is demonstrated that this equation is significantly modified as the frequency increases to the extent that appreciable energy is stored in the fluid nearfield. The consequence is that the flexural mode which these equations describe is slower than the corresponding mode in vacuo by an increasing amount; “coincidence frequency” in the sense of phase speed equal to that of acoustic waves in the fluid is never reached—the flexural mode is subsonic at all frequencies. Ultimately it transforms continuously into the free plate wave described by Crighton [J. Sound Vib. (1979)]; his conclusions on the absence of other physically identifiable “free coupled modes” are confirmed. The role of group velocity in controlling energy transfer is specifically invoked in deriving these conclusions.

Analysis of a circular raft in adhesive contact with a thick elastic layer : Hooper, JA Ove Arup and Partners, London, GB 13F, 4T, 33R. Geotechnique, V24, N4, Dec. 1974, P561–580

Analysis of a circular raft in adhesive contact with a thick elastic layer : Hooper, JA Ove Arup and Partners, London, GB 13F, 4T, 33R. Geotechnique, V24, N4, Dec. 1974, P561–580

Analysis of a circular raft in adhesive contact with a thick elastic layer

The problem considered is that of a circular raft of finite flexibility in adhesive contact with a thick isotropic elastic layer underlain by a rigid base. The applied load is assumed to be either uniform or to vary parabolically from a maximum at the raft centre to zero at the edge. Using the displacement finite element method, solutions are obtained for a homogeneous layer, and also for one particular form of linearly heterogeneous layer. Eight-noded isoparametric elements are used, and the accuracy of the numerical solution is shown to be adequate, provided that certain simple procedures are followed. For rafts on clay, the computations indicate that interfacial adhesion can significantly reduce the consolidation component of differential raft displacement, although, in practice, interfacial slip is likely to diminish this effect. It is shown that differential raft displacements are rather sensitive to the distribution of applied load when the soil is strongly heterogeneous; in particular, the general effect of soil heterogeneity is to increase differential raft displacements for the case of parabolic loading. It is also suggested that appreciable increases in differential raft displacements may result from tension cracking of the concrete. Le problème à étudier est celui présenté par un radier circulaire de flexibilité limitée auquel est affigée une couché épaisse isotrope et élastique reposant sur une assise rigide. Mettons que la charge ainsi appliquée soit ou bien uniforme ou à variation parabolique du maximum au centre du radier jusqu'à zéro à son bord. En utilisant la méthode qui provient de l'élément de déplacement limité on obtient les solutions pour une couche homogène ainsi que pour une forme, qui se distingue entre autres, de couche linéaire hétérogène. A l'aide d'éléments isoparamériques à huit nodules et pourvu que certains proédés de nature simple soient suivis, la solution numérique s'avère plutôt exacte. En ce qui concerne les radiers posés sur l'argile, les calculs indiquent que l'adhérence dièdre peuve réduire très largement le composant à consolidation de déplacement différentiel du radier même si, dans la pratique, le glissement diédre soit susceptible de ralentir telle réduction. D'ailleurs on démontre que les déplacements différentiels du radier sont assez sensibles à la façon dans laquelle une charge appliquée soit distribuée lorsque le sol soit fortement hétérogène. Notamment la portée généale du sol hétérogène agit plutôt dans le sens d'augmenter les d'placements différentiels du radier pour les charges de nature parabolique. D'ailleurs on suggère qu'une tendance se manifeste selon laquelle des fissures de tension dans le béton même peuvent augmenter de façon prononcée les déplacements différentiels du radier.

An improved initial value method for a contact problem with a nonplane punch

Solution of an axisymmetric three dimensional contact problem of a nonplane punch pressing into a finitely thick elastic layer is presented. The method consists in deriving a Cauchy system for two auxiliary functions. This initial value problem is readily solved by modern computers. Complete displacement and stress fields as well as the penetration of the center of the punch, and the force required to push it are determined. This is a direct method, and no successive approximations or series expansions are used. The results are obtained for all radii of contact of the punch with the elastic layer.

Further considerations on the axisymmetric Boussinesq problem

Numerical results are obtained for the analytical solutions developed by the present authors in an earlier paper on the axisymmetric Boussinesq problem for a thick elastic layer under a punch of arbitrary profile. The load and deformation characteristics are plotted for cylindrical, conical, paraboloidal and spherical punch shapes.

The axisymmetric boussinesq problem for a thick elastic layer under a punch of arbitrary profile

A solution of the axisymmetric Boussinesq problem is obtained, for an elastic layer lying over an elastic foundation, from which are deduced simple formulae for the depth of penetration of the tip of a punch of arbitrary profile and the total load which must be applied to the punch to achieve this penetration. Simple expressions are also deduced for the shape of the deformed surface outside the punch and the distribution of pressure under the punch. Particular results are obtained for cylindrical, conical, spherical, paraboloidal and ellipsoidal punch shapes.

On a mixed boundary value problem in elasticity

Stresses and displacements are found in the plane of a small circular stress-free crack which lies at the interface between a thick elastic incompressible layer and a rigid foundation. The layer is assumed to be rigidly bonded to the foundation, except in the interior of the circular crack, and the top face of the layer is deformed by a known pressure loading. Solution of the elastic field equations by Hankel transforms leads to four coupled integral equations which, by means of a general technique due to Noble, are reduced to two coupled Fredholm equations. These are solved by a perturbation technique for the case where the layer's thickness is much greater than the crack radius.