Compression analysis of rectangular elastic layers bonded between rigid plates

An elastic layer bonded between two rigid plates has higher compression stiffness than the elastic layer without bonding. While the finite element method can be applied to calculate the stiffness, the compression stiffness of bonded rectangular layers derived through a theoretical approach in this paper provides a convenient way for parametric study. Based on two kinematics assumptions, the governing equation for the mean pressure is derived from the equilibrium equations. Using the approximate shear boundary condition, the mean pressure is solved and the compression stiffness of the bonded rectangular layer is then established in an explicit single-series form. Through the solved pressure, the horizontal displacements are derived from the corresponding equilibrium equations, from which the shear stress on the bonding surface can be found. It is found that the effect of the rectangular aspect on the compression stiffness is significant only when Poisson’s ratio is near 0.5. For the smaller Poisson’s ratio, the compression stiffness of the rectangular layer can be approximated by the formula for the infinite-strip layer of the same shape factor.     

Flexure analysis of circular elastic layers bonded between rigid plates

An elastic layer of circular cross-section which is bonded between rigid plates and subjected to pure bending moment is analyzed through a theoretical approach. Based on two kinematic assumptions, the governing equations for the two horizontal displacement functions are established from the equilibrium equations. The horizontal displacements are then solved by satisfying the stress boundary conditions in the elastic layer. Through these solved displacements, the vertical stress in the elastic layer, the shear stress on the bonding surfaces, and the tilting stiffness of the bonded layer are derived in closed-forms and are also compared with the results of finite element analysis.     

Tilting stiffness of elastic layers bonded between rigid plates

A theoretical approach to determine the tilting stiffness of an elastic layer bonded between rigid plates is presented and then applied to derive the formulae of tilting stiffness for layers of infinite-strip, circular and square shapes. Based on two kinematics assumptions, the governing equations for the mean pressure are established from the equilibrium equations and the bulk modulus equation. Satisfying the stress boundary conditions, the pressure functions are solved and the formulae for tilting stiffness are derived. The tilting stiffnesses calculated from these formulae are extremely close to the results obtained from the finite element method for an extensive range of shape factor and Poissons ratio.     

A rigid triangular inclusion partially bonded in an elastic matrix

The two-dimensional problem of a rigid rounded-off angle triangular inclusion partially bonded in an infinite elastic plate is studied. The unbonded part of the inclusion boundary forms an interfacial crack. Based on the complex variable method for curvilinear boundaries, the problem is reduced to a non-homogeneous Hilbert problem and the stress and displacement fields in the plate are obtained in closed form. Special attention is paid in the investigation of the stress field in the vicinity of the crack tip. It is found that the stresses present an oscillatory singularity and the general equations for the local stresses are derived. The singular stress field is coupled with the maximum circumferential stress and the minimum strain energy density criteria to study the fracture characteristics of the composite plate. Results are given for the complex stress intensity factors, the local stresses, the crack extension angles and the critical applied loads for unstable crack growth from its more vulnerable tip or two types of interfacial cracks along the inclusion boundary.     

Torsion of two bonded layers by a rigid disk

The problem considered in this paper describes the torsion of a homogeneous isotropic elastic layer (0zd ₁) of finite thickness d ₁, perfectly bonded to another elastic layer (-d ₂z0) of finite thickness d ₂. The problem is reduced to the solution of a Fredholm integral equation of the second kind. The solutions are given for some particular cases.

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Sommario In questo lavoro si considera il problema della torsione di uno strato elastico omogeneo ed isotropo (0zd ₁) di spessore finito d ₁, perfettamente incollato ad un altro strato elastico (-d ₂z0) di spessore finito d ₂. II problema é ricondotto alla soluzione di una equazione integrale di Freedholm del secondo ordine. Le soluzioni sono ottenute per alcuni casi particolari.

     

Compression of solid and annular circular discs bonded to rigid surfaces

Although it is noted in the literature that the presence of a central hole in an elastic layer bonded to rigid surfaces can cause significant drop in its compression modulus, not much attention is given for investigating thoroughly and in detail the influence of the hole on the layer behavior. This paper presents analytical solutions to the problem of the uniform compression of bonded hollow circular elastic layers, which includes solid circular layers as a special case as the radius of hollow section vanishes. The closed-form expressions derived in this study are advanced in the sense that three of the commonly used assumptions in the analysis of bonded elastic layers are eliminated: (i) the incompressibility assumption, (ii) the “pressure” assumption and (iii) the assumption that plane sections remain plane after deformation. Through the use of the analytical solutions derived in the study, the compressive behavior of bonded circular discs is studied. Particular emphasis is given to the investigation of the effects of the existence of a central hole on the compression modulus, stress distributions and maximum stresses/strains in view of three key parameters: radius ratio of the hole, aspect ratio of the disc and Poisson’s ratio of the disc material.     

A NEW VARIATIONAL INEQUALITY FORMULATION FOR SEEPAGE PROBLEMS WITH FREE SURFACES

IntroductionTherearetwoclassesofsolutionsforseepageproblemswithfreesurfaces ,i.e .,theadaptivemeshmethodsandthefixedmeshmethods.Theadaptivemeshmethodsinvolvetoolargeamountofcomputationforinhomogeneoussoilsandoftenleadtodivergentcalculations,andhence,arenowbeingsupercededbythefixedmeshmethods.Thefixedmeshmethodsfallintotwocategories,theintuitivemethodsandthevariationalinequalitymethods.Theintuitivemethods[1- 3]establishusuallytheiterativeproceduresbaseduponthefactthatthereisnodischargebetweenth…     

Crack in functionally graded piezoelectric strip bonded to elastic surface layers under electromechanical loading

Solved is the problem of a crack in a functionally graded piezoelectric material (FGPM) bonded to two elastic surface layers. It is assumed that the elastic stiffness, piezoelectric constant, and dielectric permittivity of the FGPM vary continuously along the thickness of the strip. The outside layers are under antiplane mechanical loading and in-plane electric loading. The solution involves solving singular integral equations by application of the Gauss–Jacobi integration formula. Numerical calculations are carried out to obtain the energy density factors. Their variations with the geometric, loading and material parameters are shown graphically.     

Frictional contact problem of a rigid stamp and an elastic layer bonded to a homogeneous substrate

In this study, the frictional contact problem for a layer bonded to a homogeneous substrate is considered according to the theory of elasticity. The layer is indented by a rigid cylindrical stamp which is subjected to concentrated normal and tangential forces. The friction between the layer and the stamp is taken into account. The problem is reduced to a singular integral equation of the second kind in which the contact pressure function and the contact area are the unknown by using integral transform technique and the boundary conditions of the problem. The singular integral equation is solved numerically using both the Jacobi polynomials and the Gauss?CJacobi integration formula, considering equilibrium and consistency conditions. Numerical results for the contact pressures, the contact areas, the normal stresses, and the shear stresses are given, for both the frictional and the frictionless contacts.     

A two dimensional crack problem for an elastic layer bonded to an infinite elastic medium

In this paper we consider the problem of determining the distribution of stress in the neighbourhood of a crack in an infinitely long strip bonded to semi-infinite elastic planes on either side. By the use of Fourier transforms we reduce the problem to solving a single Fredholm integral equation of the second kind. Analytical expressions up to the order of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabes7aKnaaCaaaleqabaGaeyOeI0IaaGym% aiaaicdaaaaaaa!41AF!\[\delta ^{ - 10} \], where 2 is the thickness of the strip for 1 are derived for the shape of the deformed crack and for the crack energy. Some numerical results have been displayed graphically.

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Zusammenfassung In dieser Arbeit betrachten wir das Problem der Spannungsverteilung in der Nachbarschaft eines Sprunges auf ethem unendlich langen Band welches an beiden Seiten an halbseitig-unendliche elastische Platten aufgeheftet ist. Mit Hilfe von Fourier-Transformationen reduzieren wir das Problem zu einer einzelnen Fredholm Integralgleichung der zweiten Art. Für die Sprung-Energie und die Gestalt des deformierten Sprunges leiten wir analytische Ausdrücke bis zur Ordnung % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabes7aKnaaCaaaleqabaGaeyOeI0IaaGym% aiaaicdaaaaaaa!41AF!\[\delta ^{ - 10} \] her, wobei 2 für 1 die Dicke des Bandes ist. Einige numerische Resultate haben wir graphisch veranschaulicht.

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This work was supported by National Research Council of Canada through NRC-Grant No. A4177. This work was completed while the author was visiting the University of Glasgow.     

Elastic layers bonded to flexible reinforcements

Elastic layers bonded to reinforcing sheets are widely used in many engineering applications. While in most of the earlier applications, these layers are reinforced using steel plates, recent studies propose to replace “rigid” steel reinforcement with “flexible” fiber reinforcement to reduce both the cost and weight of the units/systems. In this study, a new formulation is presented for the analysis of elastic layers bonded to flexible reinforcements under (i) uniform compression, (ii) pure bending and (iii) pure warping. This new formulation has some distinct advantages over the others in literature. Since the displacement boundary conditions are included in the formulation, there is no need to start the formulation with some assumptions (other than those imposed by the order of the theory) on stress and/or displacement distributions in the layer or with some limitations on geometrical and material properties. Thus, the solutions derived from this formulation are valid not only for “thin” layers of strictly or nearly incompressible materials but also for “thick” layers and/or compressible materials. After presenting the formulation in its most general form, with regard to the order of the theory and shape of the layer, its applications are demonstrated by solving the governing equations for bonded layers of infinite-strip shape using zeroth and/or first order theory. For each deformation mode, closed-form expressions are obtained for displacement/stress distributions and effective layer modulus. The effects of three key parameters: (i) shape factor of the layer, (ii) Poisson’s ratio of the layer material and (iii) extensibility of the reinforcing sheets, on the layer behavior are also studied.     

Frictionless contact of two parallel congruent rigid cylindrical surfaces coated with thin elastic transversely isotropic incompressible layers: an analytic solution

The paper deals with a frictionless contact problem of two parallel rigid cylindrical surfaces, one encased in the other, coated with thin elastic transversely isotropic and incompressible layers. The coatings of the two circular cylinders may differ. A simplifying approximation for the displacement in the coating enables the problem to be formulated using stress and strain averaged through the coating thickness (for the method, see [Matthewson, M.J., 1981. Axi-symmetric contact on thin compliant coatings. J. Mech. Phys. Solids 29, 89–113]). Analytical results are obtained for the contact width and contact stress distribution. Given this contact stress distribution, an asymptotic analytical solution for the displacement in the coating is then obtained. The results are applied to the human ankle joint and generalized for articular cartilage with depth-dependent properties.     

A highly redundant coordinate formulation for constrained rigid bodies

The paper presents a formulation for the dynamic analysis of rigid multibodies. An introductory part carries out the kinematic analysis and the definition of the highly redundant differential framework along with the choice of unknowns and equations. From the differential formulation the variational principles, either in Lagrangian or Hamiltonian form, are developed. The Hamiltonian formulation is then used to develop the numerical approximation by applying the finite element method in time. The application of the method in its multifield form is discussed and a solution algorithm is proposed. Some examples are finally presented in order to verify the effectiveness of the formulation.

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Sommario Il lavoro presenta una formulazione per lo studio della dinamica dei sistemi multicorpo rigidi. Nella parte introduttiva viene svolta l'analisi cinematica e si definisce il quadro differenziale con la scelta delle incognite e delle equazioni. Dalla formulazione differenziale vengono poi sviluppati dei principi variazionali nella forma Lagrangiana ed Hamiltoniana. La formulazione Hamiltoniana é quindi utilizzata per sviluppare l'approssimazione numerica col metodo degli elementi finiti di tempo. Viene discussa l'applicazione del metodo nella forma multi-campo e viene proposto un algoritmo di soluzione. Da ultimo, vengono discussi alcuni esempi per verificare la correttezza della formulazione.

     

Stress analysis of an elastic cracked layer bonded to a viscoelastic substrate

The effect of a viscoelastic substrate on an elastic cracked layer under an in-plane concentrated load is solved and discussed in this study. Based on a correspondence principle, the viscoelastic solution is directly obtained from the corresponding elastic one. The elastic solution in an anisotropic trimaterial is solved as a rapidly convergent series in terms of complex potentials via the successive iterations of the alternating technique in order to satisfy the continuity condition along the interfaces between dissimilar media. This trimaterial solution is then applied to a problem of a thin layer bonded to a half-plane substrate. Using the standard solid model to formulate the viscoelastic constitutive equation, the real-time stress intensity factors can be directly obtained by performing the numerical calculations. The results obtained in this paper are useful in studying the problem with bone defects where a crack is assumed to exist in an elastic body made of the cortical bone that is bonded to a viscoelastic substrate made of the cancellous bone.     

A facile method to realize perfectly matched layers for elastic waves

In perfectly matched layer (PML) technique, an artificial layer is introduced in the simulation of wave propagation as a boundary condition which absorbs all incident waves without any reflection. Such a layer is generally thought to be unrealizable due to its complicated material formulation. In this paper, on the basis of transformation elastodynamics and complex coordinate transformation, a novel method is proposed to design PMLs for elastic waves. By applying the conformal transformation technique, the proposed PML is formulated in terms of conventional constitutive parameters and then can be easily realized by functionally graded viscoelastic materials. We perform numerical simulations to validate the material realization and performance of this PML.     

Exact solutions for stress analysis of transversely isotropic elastic layers

Summary 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.

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Exakte Lösungen der Spannungsberechnung von transversal-isotropen elastischen Schichten

Übersicht Die Grundgleichungen der Elastizität von transversal-isotropen Stoffen werden auf eine für die Berechnung beliebig hoher Ableitungen in eine Dickenrichtung zweckmäßige Form gebracht. Die Verschiebungen und Spannungen werden in für exakte Lösungen dicker Schichten mit spannungsfreien Laibungen geeignete Taylor-Reihen entwickelt. Dies ist eine Weiterführung der Arbeiten von Rao und Das für den isotropen Fall auf den transversal-isotropen Fall. Eine Klasse exakter Lösungen wird benutzt, um die Spannungskonzentrationsfaktoren gelochter Schichten zu bestimmen.

     

Interfacial stress analysis of a thin plate bonded to a rigid substrate and subjected to inclined loading

The so-called peel test, in which a thin plate bonded to a substrate is subjected to an inclined pulling force, has been widely used to characterise the bond behaviour of adhesives. This paper presents an analytical solution for the interfacial normal and shear stresses in such a peel test to provide an improved understanding of its underlying mechanism. An approximate closed-form solution is also presented. The effect of the peel angle (i.e. the angle between the applied force and the substrate) on the interfacial stresses is discussed. Apart from being a widely used test for quantifying adhesive characteristics, the process of debonding in a peel test resembles that of intermediate flexural-shear or shear crack induced debonding in flexurally strengthened RC members, where a relative vertical displacement exists between the two sides of the crack, leading to an angle between the external plate and the concrete substrate. Therefore, the results of this study also offer some insight into the latter failure mode which is very important in the flexural strengthening design of RC members.