Plane Theory Of Elasticity Research Articles

Free vibration analysis of functionally graded beams using an exact plane elasticity approach

Exact natural frequencies of functionally graded beams are determined using plane elasticity theory. The analysis yields infinitely many frequencies. For verification purposes, a comparison with the existing beam theory results is performed and a close agreement is observed for slender members. The elasticity solutions are general in the sense that they are valid for slender members as well as short and thick structural elements. Both flexural and axial free vibration mode shapes are presented for top and bottom surfaces and the effect of the beam thickness is discussed. The exact results presented herein can be used as benchmarks for future research of free vibration behavior of short and thick functionally graded material beams.

Thermo elastic analysis of a functionally graded cylinder under internal pressure using first order shear deformation theory

Thermo elastic analysis of a functionally graded cylinder under mechanical and thermal loads is investigated analytically in the present paper. The first order shear deformation theory (FSDT) is employed for analysis of the problem. FSDT recognizes two components for definition of the deformation. Displacement of the mid-plane and rotation about that are two components of deformation. The modulus of elasticity is assumed to vary continuously along the thickness with a power function. Stress strain relation is developed in the cylindrical coordinate system with regarding the thermal strain. Necessary equations for solving the problem have been obtained by taking variation of the energy equation with respect to four assumed functions using Euler equations. The effect of the temperature rising on the distribution of the displacement and stress for different values of non-homogenous index (n) are investigated in the present paper. Evaluation of the axial displacement of cylinder can be presented as an important result of this paper. Furthermore, the present result is compared with results that were obtained using the plane elasticity theory. Key words: Thermo elastic, shell, temperature, cylinder, energy.

A numerical approach for a crack problem by Gauss–Chebyshev quadrature

In this paper, a problem of a crack in an orthotropic strip is studied under plane strain conditions. It is assumed that normal displacements and shear stresses do not act on neither of the boundaries of the strip. Cauchy-type singular integral equation for the crack problem is derived by using the theory of plane elasticity and the Fourier transformation technique. A quadrature collocation approach is adopted for the numerical solutions of the singular integral equation. The effect of relative thickness and mechanical properties of strip on Mode I stress intensity factors (SIFs) are examined under different loading conditions. Some sample results are given for SIFs; also, material orthotropy and geometrical effects are discussed in detail.

High-cycle fatigue threshold behaviors in notched plates

This article investigates the threshold behavior of fatigue crack emanating from an elliptical hole. The theoretical framework accounts for the presence of plastic zone ahead of the crack tip and the crack-closure development. To predict the threshold behavior, both the crack deriving force and the resisting force are considered. The crack deriving force (i.e. applied stress intensity factor) is calculated based on the plane elasticity theory and linear-elastic fracture mechanics. Based on the proposed computational framework, the fatigue limit, the length of non-propagating crack and the notch-size effect are simulated in a systematic manner. By conducting a series of comparisons between the theoretical predictions and the available experimental data, the predictive capability of the proposed method is examined and discussed.

Contact problem for regular hexagon weakened with full-strength hole

A problem of the plane elasticity theory is addressed for a doubly connected body with an external boundary of the regular hexagon shape and with a 6-fold symmetric hole at the center. It is assumed that all the six sides of the hexagon are subjected to uniform normal displacements via smooth rigid stamps, while the uniformly distributed normal stress is applied to the internal hole boundary. Using the methods of complex analysis, the analytical image of Kolosov-Muskhelishvili’s complex potentials and the shape of the hole contour are determined from the condition that the circumferential normal stress is constant along the hole contour. Numerical results are given and shown in relevant graphs.

COMPLETE CLOSED-FORM SOLUTION FOR PRESSURIZED HETEROGENEOUS THICK SPHERICAL SHELLS

On the basis of plane elasticity theory (PET), the governing equations for axisymmetric thick spherical shells made of nonhomogeneous functionally graded materials (FGMs) subjected to internal and external pressure in general case are derived. It is assumed that the modulus of elasticity varies nonlinearly in the radial direction, and the Poisson’s ratio is constant. The analytical solution of the equations for real, double and complex roots are obtained. The radial stress, meridional stress and radial displacement distributions depending on an inhomogeneity constant are compared with those of the homogeneous case as well as with the solution using finite element method (FEM) and presented in the form of graphs. The obtained result shows that the property of FGMs has a significant influence to the stress distribution along the radial direction. Results are useful for engineers to design a sphere made of FGMs. DOI: http://dx.doi.org/10.5755/j01.mech.18.5.2702

Elastic analysis of heterogeneous thick-walled spherical pressure vessels with parabolic varying properties

On the basis of plane elasticity theory (PET), the displacement and stress components in a thick-walled spherical pressure vessels made of heterogeneous materials subjected to internal and external pressure is developed. The mechanical properties except the Poisson’s ratio are assumed to obey the parabolic variations throughout the thickness. Effect of material inhomogeneity on the elastic deformations and stresses is investigated. The analytical solutions and the solutions carried out through the FEM have a good agreement. The values used in this study are arbitrary chosen to demonstrate the effect of inhomogeneity on displacements, and stresses distributions.

ELECTRO ELASTIC ANALYSIS OF A PRESSURIZED THICK-WALLED FUNCTIONALLY GRADED PIEZOELECTRIC CYLINDER USING THE FIRST ORDER SHEAR DEFORMATION THEORY AND ENERGY METHOD

Shear deformation theory is employed for electro elastic analysis of a functionally graded piezoelectric cylinder as a physical sensor for estimation and controlling the internal pressure. Except Poisson ratio, all mechanical and electrical properties are graded along the radial coordinate system. The present paper develops the previous study using the shear deformation theory as a capable and advantageous theory instead of plane elasticity theory. Two radial and axial deformations are simulated using the first order shear deformation theory. Electric potential is supposed as a quadratic function along the thickness direction and as an unknown function along the longitudinal direction. Comparison between the present results at the regions that are adequate far from two ends of the cylinder with the results of plane elasticity theory justifies accuracy and capability of the present method. Some useful graphical and numerical results are presented in this study. The discussed problem in this paper has many advantageous properties and application in measurement and controlling of the mechanical and electrical components because of using the functionally graded materials instead of an isotropic material. DOI: http://dx.doi.org/10.5755/j01.mech.18.3.1875

On one contact problem of plane elasticity for a doubly connected domain: application to a hexagon

The paper addresses a problem of plane elasticity theory for a doubly connected body whose external boundary is a regular hexagon boundary, and the internal boundary is the required full-strength hole including the origin of coordinates. Hexagon’s two vertices are laid at the axis Oy, and the middle points of its two opposite sides are laid at the axis Ox. This full-strength hole is cycle symmetric. It is assumed that to every link of the broken line of the outer boundary of the given body are applied absolutely smooth rigid stamps with rectilinear bases, which are under action of the force P that applies to their middle points. There is no friction between the surface of given elastic body and stamps. The unknown full-strength contour is free from outer actions. Using the methods of complex analysis, the analytical image of Kolosov–Muskhelishvili’s complex potentials (characterizing an elastic equilibrium of the body) and unknown parts of its boundary are determined under the condition that the tangential normal moment arising at it takes a constant value. Such holes are called full-strength holes. Numerical analysis are also performed and the corresponding graphs are constructed.

COMPREHENSIVE THERMOELASTIC ANALYSIS OF A FUNCTIONALLY GRADED CYLINDER WITH DIFFERENT BOUNDARY CONDITIONS UNDER INTERNAL PRESSURE USING FIRST ORDER SHEAR DEFORMATION THEORY

This paper deals with the thermo elastic analysis of a FG cylinder with different boundary conditions under internal pressure using the first order shear deformation theory. This proposed solution has the ability to solve the cylinder structure with arbitrary boundary conditions. The previous studies have been proposed for the problem with simple boundary conditions (simply supported cylinder). This paper investigates the axial distribution of the displacement and stress for different values of non-homogenous index. This investigation indicates that a support has important influence on the distribution of mechanical components rather than a cylinder with ignoring the effect of support. Simultaneously effect of the temperature and pressure are studied on the deformation and stress of cylinder. Calculation of the stress concentration factor due to end effects is another result of the present paper. The presented results are validated by comparison with previous results that have been evaluated using the plane elasticity theory. DOI: http://dx.doi.org/10.5755/j01.mech.18.1.1273

DETERMINATION OF DISPLACEMENTS AND STRESSES IN PRESSURIZED THICK CYLINDRICAL SHELLS WITH VARIABLE THICKNESS USING PERTURBATION TECHNIQUE

This article presents a closed form analytical solution for clamped-clamped thick cylindrical shells with variable thickness subjected to constant internal pressure. Regarding the problem which could not be solved through plane elasticity theory (PET), the solution based on the first-order shear deformation theory (FSDT) is suggested. Based FSDT on, and the virtual work principle, the differential equations governing axisymmetric thick cylindrical shells with variable thickness have been derived. The governing equations are a system of ordinary differential equations with variable coefficients. Using the matched asymptotic method (MAM) of the perturbation technique, these equations could be converted into a system of algebraic equations with variable coefficients and two systems of differential equations with constant coefficients.DOI: http://dx.doi.org/10.5755/j01.mech.18.1.1274

Elastic Analysis of Heterogeneous Thick Cylinders Subjected to Internal or External Pressure Using Shear Deformation Theory

An analytical formulation based on the first-order shear deformation theory (FSDT) is presented for axisymmetric thick-walled heterogeneous cylinders under internal and external uniform pressure. It is assumed that the material is isotropic heterogeneous with constant Poisson's ratio and radially varying elastic modulus. First, general governing equations of the heterogeneous thick cylinders are derived by virtual work principle, and using FSDT. Then the obtained equations are solved under the generalized plane strain assumptions. The results are compared with the findings of both plane elasticity theory (PET) and finite element method (FEM).

Calculation of settlements for a strip foundation

A solution is presented for calculation of the total settlement of a strip foundation with allowance for additional stresses, which develop in the region under consideration due to displacement of the loaded section of the boundary, and can be determined by solving the second basic boundary problem of the plane theory of elasticity for a half plane. Expressions for the strain components of the second basic boundary problem are presented in closed form, and formulas are derived for calculation of settlement due to a uniformly distributed load. Settlements of the bed, which is composed of sandy and clayey soils, are calculated.

A surface crack in a graded coating subjected to sliding frictional contact

This article presents an analytical method developed to conduct fracture analysis of functionally graded coatings that are under the effect of contact stresses. The geometric model considered comprises a functionally graded coating, which possesses a surface crack and is perfectly bonded to a homogeneous half-plane. A two-step solution procedure is put forward in which the contact and crack problems are treated separately. First, the graded coating is assumed to be in sliding frictional contact with a rigid flat punch and the contact stresses are computed. These stresses are then used as inputs for the crack problem and the mode I and II stress intensity factors are evaluated. Governing partial differential equations for the contact and crack problems are derived by using the theory of plane elasticity and solved by utilizing Fourier transformation techniques. The derivations lead to a single singular integral equation in the case of the contact problem, and two uncoupled singular integral equations in the case of the crack problem. An expansion–collocation approach is adopted for the numerical solutions of the integral equations. Comparisons of the results generated for the contact and crack problems to those available in the literature demonstrate the high levels of accuracy attained by the proposed solution techniques. Further numerical results are presented so as to illustrate the influences of material nonhomogeneity and coefficient of friction on the contact stresses and the mode I and II stress intensity factors.

To the theory of anisotropic plane elasticity

The Lame system of general anisotropic plane elasticity is considered. A representation of a general solution or the system through a so-called Douglis analytic functions is given. The cases of orthotropic and isotropic media are also considered.

On a biharmonic problem in a circular ring

Several problems of plane elasticity theory which can be reduced to different biharmonic boundary value problems are described, and the concept of biharmonic Green functions is discussed. The explicit formula for the particular biharmonic Green function in a circular ring R = { z ∈ C: 0< r <| z |<1} is presented and applied to the solution of the Dirichlet boundary value problem for the bi-Poisson equation (∂ z ∂ z¯ ) 2 w = f in R , w = γ 0 , ;∂ z ∂ z¯ w = γ 1 on ∂ R , for given f ∈ L p ( R ;C), p >2, γ 0 , γ 1 ∈ C (∂ R ;C).

A novel approach to stress analysis of pressurized FGM cylinders, disks and spheres

Axisymmetric displacements and stresses in functionally-graded hollow cylinders, disks and spheres subjected to uniform internal pressure are determined using plane elasticity theory and Complementary Functions method. The material is assumed to be functionally graded in the radial direction. Variations in the material properties such as Young’s modulus and Poisson’s ratio may be arbitrary functions of the radial coordinate. This assumption yields a two-point boundary value problem with a governing differential equation of variable coefficients. General analytical solutions of such equations are not available. Infusion of Complementary Functions method into the analysis of stresses in pressure vessels is a novel approach. Complementary Functions method reduces the boundary value problem to an initial-value problem which can be solved accurately by one of many efficient methods such as Runge–Kutta method. Various material models from the literature are used and corresponding radial displacement and stresses are computed. Verification of the proposed method is done using benchmark solutions available in the literature for some special cases and virtually exact results are obtained using the fifth-order Runge–Kutta method (RK5).

L p -boundedness of a boundary integral operator on a contour with a peak

Results on the solvability of boundary integral equations on a plane contour with a peak obtained in collaboration with V.G. Maz’ya are developed. Earlier, it was proved that, on a contour Γ with an outward peak, the operator of the boundary equation of the Dirichlet boundary value problem maps the space ℒp, β + 1 (Γ) continuously onto \( \mathcal{N}_{p,\beta } (\Gamma ) \). The norm of a function in ℒp, β (Γ) is defined as Open image in new window , provided that the peak is at the origin. In this case, the norms on the spaces \( \mathcal{N}_{p,\beta }^ \mp (\Gamma ) \) are defined by Open image in new window , where q± are the intersection points of Γ with the circle {z: |z| = |q|} and δ > 0 is a fixed small number. On a contour with an inward peak, the operator of the boundary equation of the Dirichlet problem continuously maps ℒp, β + 1 (Γ) onto ℳp, β(Γ), where ℳp, β(Γ) is the direct sum of \( \mathcal{N}_{p,\beta }^ + (\Gamma ) \) (Γ) and the space Open image in new window (Γ) of functions on Γ of the form p(z) = Σk = 0mt(k)Rezk with the parameter m = [μ − β − p−1]. The operator I − 2W of the boundary integral equation of plane elasticity theory, where W is the elastic double-layer potential, is considered. The main result is that the operator I − 2W continuously maps the space ℒp, β + 1 × ℒp, β + 1(Γ) to the space \( \mathcal{N}_{p,\beta }^ - \times \mathcal{N}_{p,\beta }^ - (\Gamma ) \).

Elasticity solution of multi-span beams with variable thickness under static loads

This paper studies the stress and displacement distributions of continuously varying thickness multi-span beams simply supported at two ends and under static loads. The intermediate supports of the beam may be elastic and/or rigid in one or two directions. On the basis of the two-dimensional plane elasticity theory, the general solution of stress function, which exactly satisfies the governing differential equations and the simply supported boundary conditions, is deduced. In the present analysis, the reaction forces of the intermediate supports are regarded as the unknown external forces acting on the lower surface of the beam under consideration. The unknown coefficients in the solutions are determined by using the Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the beam and using the linear relations between reaction forces and displacements of the beam at intermediate supports. The solution obtained is exact and excellent convergence has been confirmed. Comparing the numerical results obtained from the proposed method to those obtained from the Euler beam theory, the Timoshenko beam theory and those obtained from the commercial finite element software ANSYS, high accuracy of the present method is demonstrated.

Complex variable method for plane elasticity of icosahedral quasicrystals and elliptic notch problem

The complex variable method for the plane elasticity theory of icosahedral quasi-crystals is developed. Based on the general solution obtained previously, complex representations of stress and displacement components of phonon and phason fields in the quasicrystals are given. With the help of conformal transformation, an analytic solution for the elliptic notch problem of the material is presented. The solution of the Griffith crack problem can be observed as a special case of the results. The stress intensity factor and energy release rate of the crack are also obtained.