Plane Elasticity Equations Research Articles

Deformation of rectangular orthotropic laminate resting on thin elastic foundation with nonlinear change of displacements over thickness under edge loading

Deformation of rectangular orthotropic laminate resting on a thin elastic orthotropic foundation with fully fixed bottom plane is studied in this paper. The laminate is subjected to an in-plane uniaxial load uniformly distributed over one of its edges. Models of joint deformation of the laminate and foundation are created based on the equations of plane elasticity for an orthotropic body. The applicability of such equations is justified by the uniformity of edge loading. A laminate model allows for the transverse shear strain, whereas a model created for the foundation considers a nonlinear character of variation of the transverse and axial displacements. A governing system of differential equationsmodelling the joined deformation of the laminate and foundation is derived using static and kinematic contact conditions. The corresponding boundary value problem is solved using the sweep method. Based on this solution, effects of laminate and foundation thicknesses, and elasticity characteristics on the stress–strain state of the jointed structure are investigated. The model created and the method of its analysis enable the strong oscillating behaviour of stresses and strains to be studied. The results obtained can be utilized for verification of solutions of similar problems found by other numerical methods, including finite-element analyses.

Stresses in multi-layer coatings in Hertzian contact with a moving circular punch

This article presents analytical solution procedures for moving Hertzian contact problems involving multi-layer and functionally graded coatings. A frictional circular punch is assumed to slide down the surface of the coating with a specified velocity. Formulation is based on wave equations of plane elasticity, and reduced to a singular integral equation by Galilean and Fourier transformations. Developed procedures are verified through comparisons to the findings available in the literature. Numerical results illustrate influences of factors such as dimensionless punch speed, coefficient of kinetic friction, property variation profile, and contact length-to-thickness ratio upon the contact stresses, and the required contact force.

Symplectic approach for plane elasticity problems of two-dimensional octagonal quasicrystals

The symplectic approach (SA) is used in this paper to analyze the plane elasticity problems of two-dimensional (2D) octagonal quasicrystals (QCs). The equilibrium equations for point group 8mm octagonal QCs are first transferred into Hamiltonian dual equations. Then the symplectic eigenvalue problem of the corresponding Hamiltonian operator matrix is derived by applying the method of separation of variables. Based on the eigenvalue analysis and expansion of symplectic eigenvectors, the exact analytic solutions for point group 8mm octagonal QCs with selected boundary conditions are established. Numerical results for the phonon and phason displacements are presented and validated by the finite integral transform method (FITM), which are useful for validation of other numerical methods. In addition, inspired by the SA, the equilibrium equations of plane elasticity of octagonal QCs with Laue class 15 are simplified, which is an open problem. The approach presented here is rational and systematic with clear step-by-step derivation procedure, thus it has potential for plane elasticity problems of other QCs. The results obtained in this paper can serve as benchmarks for future researches.

Modern interpretation of Saint-Venant’s principle and semi-inverse method

Relevance. The progressive development of views on the Saint-Venant formulated principles and methods underlying the deformable body mechanics, the growth of the mathematical analysis branch, which is used for calculation and accumulation of rules of thumb obtained by the mathematical results interpretation, lead to the fact that the existing principles are being replaced with new, more general ones, their number is decreasing, and this field is brought into an increasingly closer relationship with other branches of science and technology. Most differential equations of mechanics have solutions where there are gaps, quick transitions, inhomogeneities or other irregularities arising out of an approximate description. On the other hand, it is necessary to construct equation solutions with preservation of the order of the differential equation in conjunction with satisfying all the boundary conditions. Thus, the following aims of the work were determined: 1) to complete the familiar Saint-Venants principle for the case of displacements specified on a small area; 2) to generalize the semi-inverse Saint-Venants method by finding the complement to the classical local rapidly decaying solutions; 3) to construct on the basis of the semi-inverse method a modernized method, which completes the solutions obtained by the classical semi-inverse method by rapidly varying decaying solutions, and to rationalize asymptotic convergence of the solutions and clarify the classical theory for a better understanding of the classic theory itself. To achieve these goals, we used such methods , as: 1) strict mathematical separation of decaying and non-decaying components of the solution out of the plane elasticity equations by the methods of complex variable theory function; 2) construction of the asymptotic solution without any hypotheses and satisfaction of all boundary conditions; 3) evaluation of convergence. Results. A generalized formulation of the Saint-Venants principle is proposed for the displacements specified on a small area of a body. A method of constructing asymptotic analytical solutions of the elasticity theory equations is found, which allows to satisfy all boundary conditions.

Exact elasticity solutions to rotating pressurized axisymmetric vessels made of functionally graded materials (FGM)

AbstractIn this study the plane elasticity equations of pressurized and rotating structures, generalized analytical closed‐form solutions are presented for hollow cylinders, spheres and thin disks. Three well known material pairs are graded in radial direction by using the power rule for various boundary and load conditions. Stress and displacement solutions are obtained. The effect of a constant rotational velocity is also investigated separately in absence of the internal and external pressures. Crucial results are also presented in tabular form.

Plane elasticity problems by barycentric rational interpolation collocation method and a regular domain method

SummaryBarycentric rational interpolation collocation method (BRICM) for solving plane elasticity problems with high accuracy is presented. The plane elasticity problems on a circular or rectangular domain can be solved directly by BRICM. Embedded the irregular domain into a regular (circular or rectangular) domain, the governing equations of plane elasticity on regular domain are discretized by the differentiation matrices based on barycentric rational interpolation to form a system of algebraic equations. Discrete boundary conditions are obtained using barycentric rational interpolation. The irregular boundary conditions are imposed by the additional method to form an over‐constraint linear system of algebraic equations. Numerical experiments are presented to illustrate the efficiency and high computing precision of proposed method.

Plane receding contact problem for a functionally graded layer supported by two quarter-planes

In this study, the plane receding contact problem for a functionally graded (FG) layer resting on two quarter-planes is considered by using the theory of linear elasticity. The layer is indented by a rigid cylindrical punch that applies a concentrated force in the normal direction. While the Poisson’s ratio is kept constant, the shear modulus is assumed to vary exponentially through-the-thickness of the layer. It is assumed that the contact at the layer-punch interface and the layer-substrate interface is frictionless, and only the normal tractions can be transmitted along the contact regions. Applying the Fourier integral transform, the plane elasticity equations are converted to a system of two singular integral equations, in which the contact stresses and the contact widths are unknowns. The singular integral equations are solved numerically by Gauss–Jacobi integration formula. Effects of the material inhomogeneity, the distance between quarter-planes and the punch radius on the contact stresses, the contact widths, and the stress intensity factors at the sharp edges are shown. Although the theoretical analysis is formulated with respect to elastic quarter planes, the numerical studies are carried out only for rigid ones.

Contact problem of a graded layer supported by two rigid punches

This paper considers the frictionless contact plane problem of an infinitely graded layer supported by two rigid cylindrical punches and subjected to a concentrated normal force by means of a rigid cylindrical punch. The layer is made of non-homogeneous material with an isotropic stress–strain law with exponentially varying properties. Rather than assuming vanishing displacements or stresses at the bottom of the geometry, this study examines the effect of supports on the graded layer. Using standard Fourier transform and the related boundary conditions, the plane elasticity equations are converted analytically to a system of singular integral equations in which the unknowns are the contact stresses and areas. Gauss–Chebyshev integration formulas are then employed to discretize and solve numerically the derived integral equations. Numerical results for the contact stresses and areas are provided for various dimensionless quantities including material inhomogeneity, distance between the punches, external load and upper and lower punch radii.

A receding frictional contact problem between a graded layer and a homogeneous substrate pressed by a rigid punch

This paper considers the plane problem of a receding frictional nonlinear contact between an elastic graded layer and a homogeneous half-space when they are pressed against each other by a rigid stamp. A non-homogenous isotropic stress-strain law was used to model the graded layer. The contact region is assumed to be under sliding contact conditions with Coulombs law relating the tangential traction to the normal component. Applying Fourier integral transform and the appropriate boundary conditions, the plane elasticity equations are converted analytically into a system of singular integral equations in which the unknowns are the pressures and receding contact lengths in the two contact zones. Ensuring mechanical equilibrium is an indispensable requirement warranted by the physics of the problem and therefore the global force and moment equilibrium conditions for the stamp and the layer are supplemented to solve the problem. The Gauss–Chebyshev quadrature-collocation method is used to convert the singular integral equations into a set of nonlinear equations which are solved with a newly developed iterative algorithm to yield the lengths of the receding contact zones and the associated contact pressures. The main focus of this paper is to investigate the effect of the non-homogeneity parameter of the graded layer, the friction coefficient in the contact zones and the radius of the stamp profile on the contact pressures and lengths of the receding contact zones.

Stabilization of low-order mixed finite elements for the plane elasticity equations

In this paper, we consider the mixed finite element method of the plane elasticity equations based on the Hellinger–Reissner variational principle. Low order mixed finite element spaces are used to approximate the stress and the displacement. Based on the local polynomial projection stabilization method, we present a stabilization scheme for the pairs to overcome the lack of the inf–supcondition. The stability is proved and the error estimate is derived. At last, two numerical examples are implemented to test the stability and effectiveness of the proposed method. These pairs are very convenient in the numerical implementation for the mixed form of the plane elasticity equations.

On the analytical and finite element solution of plane contact problem of a rigid cylindrical punch sliding over a functionally graded orthotropic medium

This study investigates the plane frictional contact problem of a cylindrical punch on a functionally graded orthotropic medium (FGOM). Both analytical and computational methods are developed to obtain the contact stresses. The elastic modulus is assumed to vary exponentially and the principal axes of orthotropy are assumed to be aligned with the global coordinates. In the analytical formulation, plane elasticity equations are converted to Cauchy-type singular integral equation of the second type by using Fourier Transform technique. The resulting integral equations are then solved numerically to obtain the contact stresses throughout the medium for plane stress condition. In the computational model, the elastic modulus of each finite element is specified at its centroid by using the predefined functional variation. The results obtained from the finite element analysis is verified by comparing the results obtained using the analytical formulation. The results of this study can guide tribology engineers in determining the contact stresses that have a great effect on the wear resistance of mating components.

Hybrid stress quadrilateral finite element approximation for stochastic plane elasticity equations

SummaryThis paper considers stochastic hybrid stress quadrilateral finite element analysis of plane elasticity equations with stochastic Young's modulus and stochastic loads. Firstly, we apply Karhunen–Loève expansion to stochastic Young's modulus and stochastic loads so as to turn the original problem into a system containing a finite number of deterministic parameters. Then we deal with the stochastic field and the space field by k−version/p−version finite element methods and a hybrid stress quadrilateral finite element method, respectively. We derive a priori error estimates, which are uniform with respect to the Lamè constant λ∈(0,+∞). Finally, we provide some numerical results. Copyright © 2016 John Wiley & Sons, Ltd.

Sliding contact analysis of functionally graded coating/substrate system

We consider in this paper the fully coupled contact problem between a rigid cylinder and a functionally graded coating bonded to a homogeneous substrate system under plane strain and generalized plane stress sliding conditions. Employing Fourier transform, the plane elasticity equations are converted analytically into Singular Integral Equations, which are then solved numerically to yield the contact stresses as well as the subsurface stresses throughout the medium for plane strain condition. The surface stress distributions as well as the contour plots for normalized von Mises stresses are provided. The effect of the different parameters such as the non-homogeneity coefficient, coating/substrate material properties and the friction coefficient on the surface tractions and von Mises stress distributions are investigated. Also discussed, are the advantages and disadvantages of using functionally graded coatings as compared to conventional homogenous coatings.

Thermal Stress Analysis of an Embedded Crack in a Graded Orthotropic Coating-Substrate Structure

An analysis of a coupled plane thermoelastic problem for a graded orthotropic coating-substrate structure is performed under thermomechanical loading conditions. The crack direction is parallel to the free surface. Applying the superposition principle and Fourier integral transform, the heat conduction and plane elasticity equations lends themselves to the derivation of two sets of Cauchy-type singular integral equations. The thermal stress intensity factors are defined and evaluated. In the numerical results, the effects of the orthotropy parameters, thermoelastic non-homogeneity parameters, and dimensionless thermal resistance on the temperature distribution and the thermal stress intensity factors (TSIFs) are studied. The obtained results can be used to design graded orthotropic coating-substrate structures under thermomechanical loading.

A frictional receding contact plane problem between a functionally graded layer and a homogeneous substrate

This paper investigates the plane problem of a frictional receding contact formed between an elastic functionally graded layer and a homogeneous half space, when they are pressed against each other. The graded layer is assumed to be an isotropic nonhomogeneous medium with an exponentially varying shear modulus and a constant Poisson’s ratio. A segment of the top surface of the graded layer is subject to both normal and tangential traction while rest of the surface is devoid of traction. The entire contact zone thus formed between the layer and the homogeneous medium can transmit both normal and tangential traction. It is assumed that the contact region is under sliding contact conditions with the Coulomb’s law used to relate the tangential traction to the normal component. Employing Fourier integral transforms and applying the necessary boundary conditions, the plane elasticity equations are reduced to a singular integral equation in which the unknowns are the contact pressure and the receding contact lengths. Ensuring mechanical equilibrium is an indispensable requirement warranted by the physics of the problem and therefore the global force and moment equilibrium conditions for the layer are supplemented to solve the problem. The Gauss–Chebyshev quadrature-collocation method is adopted to convert the singular integral equation to a set of overdetermined algebraic equations. This system is solved using a least squares method coupled with a novel iterative procedure to ensure that the force and moment equilibrium conditions are satisfied simultaneously. The main objective of this paper is to study the effect of friction coefficient and nonhomogeneity factor on the contact pressure distribution and the size of the contact region.

Nonlocal Analytical Formulas for Plane Crack Elements and Semi-Analytical Element Method for Mode I Crack

This paper presents the series solutions in Hamiltonian form for plane sectorial domain of the nonlocal linear elasticity originally proposed by Eringen [. Based on the Hamiltonian governing equations of plane elasticity for sectorial domain, the variable separation and eigenfunction expansion techniques were employed to obtain these solutions. These solutions can be used to develop a nonlocal analytical finite element for the model I crack in nonlocal fracture mechanics.

Nonlocal Analytical Formulas for Plane Crack Elements and Semi-Analytical Element Method for Mode I Crack

This paper presents the series solutions in Hamiltonian form for plane sectorial domain of the nonlocal linear elasticity originally proposed by Eringen [. Based on the Hamiltonian governing equations of plane elasticity for sectorial domain, the variable separation and eigenfunction expansion techniques were employed to obtain these solutions. These solutions can be used to develop a nonlocal analytical finite element for the model I crack in nonlocal fracture mechanics.

An embedded crack in a functionally graded orthotropic coating bonded to a homogeneous substrate under a frictional Hertzian contact

In this paper, we consider the elasto-static problem of an embedded crack in a graded orthotropic coating bonded to a homogeneous substrate subject to statically applied normal and tangential surface loading. The crack direction is parallel to the free surface. The coating is graded in the thickness direction and is orthogonal to the crack direction. This coating is modelled as a non-homogeneous medium with an orthotropic stress–strain law. The equivalent crack surface stresses are first obtained and substituted in the plane elasticity equations. Using integral transforms, the governing equations are converted into singular integral equations which are solved numerically to yield the displacement field as well as the crack-tip stress intensity factors. This study presents a complete theoretical formulation for the problem in the static case. A numerical predictive capability for solving the singular integral equations and computing the crack-tip stress intensity factors is proposed. Since the loading is compressive, a previously developed crack-closure algorithm is applied to avoid interpenetration of the crack faces. The main objective of the paper is to investigate the effects of the material orthotropy and non-homogeneity of the graded coating on the crack-tip stress intensity factors, with and without using the crack-closure algorithm, for the purpose of gaining better understanding on the behavior and design of graded coatings.

An Axisymmetric Problem of a Partially Insulated Crack Embedded in a Graded Layer Bonded to a Homogeneous Half-Space Under Thermal Loading

The elastic-static problem of a partially insulated axisymmetric crack embedded in a graded coating bonded to a homogeneous substrate subjected to thermal loading is considered. The coating's gradient is parallel to the axisymmetric direction and is orthogonal to the crack plane. The graded coating is modeled as a nonhomogeneous medium with an isotropic constitutive law. Using Hankel transform, the heat conduction and the plane elasticity equations are converted into singular integral equations, which are solved numerically to yield the temperature distribution and the crack tip stress intensity factors. The Finite Element Method was additionally used to model the crack problem. The main objective of this paper is to study the influence of the material nonhomogeneity, partial insulation of the crack faces and the crack position on the stress intensity factors for the purpose of gaining better understanding on the behavior of graded coatings.

On the plane contact problem of a functionally graded elastic layer loaded by a frictional sliding flat punch

This article is concerned with the contact mechanics of a functionally graded layer loaded by a frictional sliding flat punch. The coefficient of friction is assumed to be constant and the lower side of the graded layer is firmly attached to a rigid foundation. The graded, nonhomogeneous property of the medium is represented in terms of an exponential variation of the shear modulus, while Poisson’s ratio is taken to be constant. Based on the use of plane elasticity equations and the Fourier integral transform technique, the formulation of the current contact mechanics problem lends itself to a Cauchy-type singular integral equation of the second kind for the unknown contact pressure, which is solved numerically. As a result, the effects of several parameters, i.e., the material nonhomogeneity, the friction coefficient, the punch width, and Poisson’s ratio, on the distributions of the contact pressure and the in-plane surface stress component are presented.