Solution Of Plane Problem Research Articles

Applied element method in the solution of plane problems in the theory of creep

Applied element method in the solution of plane problems in the theory of creep

Bending of a Composite Beam with a Longitudinal Section

Within the context of the nonclassical model of composite beams the analytical solution has been obtained for a beam having a longitudinal section collinear to its median line. Bending load on the beam facilitates opening of the section. The elementary formulas for stress and displacement within all parts of the beam have been obtained. The accuracy of results is confirmed by their comparison with the corresponding data obtained in the solution of plane problem of the theory of elasticity using the boundary element method.

General Solutions of Plane Problem in One-Dimensional Hexagonal Quasicrystals

A theory of general solutions of plane problem is developed for the coupled equations in plane elasticity of one-dimensional (1D) hexagonal quasicrystals (QCs), and three general solutions are presented by an operator method. These solutions are expressed in terms of a displacement function, which satisfies a sixth-order partial differential equation. By utilizing a theorem, a decomposition and superposition procedure is taken to replace the sixth-order function with three second-order displacement functions, and the general solution is simplified in terms of these functions. In consideration of different cases of three characteristic roots, the general solution possesses three cases, but all are in simple forms that are convenient to be used.

Comments on “General solutions of plane problem for power function curved cracks”

Comments on “General solutions of plane problem for power function curved cracks”

On the displacement potential solution of plane problems of structural mechanics with mixed boundary conditions

The present paper describes the advancement of displacement potential approach in relation to solution of plane problems of structural mechanics with mixed mode of boundary conditions. Both the conditions of the plane stress and the plane strain are considered for analyzing the displacement and stress fields of the structural problem. Using the finite difference technique based on the present displacement potential approach for the case of the plane stress and the plane strain conditions, firstly an elastic cantilever beam subjected to a pure shear at its tip is solved and these two solutions (plane stress and plane strain) are compared with Timoshenko and Goodier cantilever beam bending solutions (Theory of elasticity, 2nd edn. McGraw-Hill, New York, 1951); secondly the above-mentioned displacement potential approach for the case of the plane stress and the plane strain conditions are applied to solve a one-end fixed square plate subjected to a combined loading at its tip. Effects of plane stress and plane strain on the elastic field of the plate are discussed in a comparative fashion. Limitations of Timoshenko and Goodier cantilever beam bending solutions (Theory of elasticity, 2nd edn. McGraw-Hill, New York, 1951) over the displacement potential approach for the case of the plane stress and the plane strain conditions are not only discussed but also the superiority of the present displacement potential approach for the case of the plane stress and the plane strain conditions are reflected in the present research work.

The Refined Theory of Thermoelastic Plane Problems

Based on thermoelastic theory, various two-dimensional equations and solutions for plane problems have been deduced systematically and directly from thick plate theory by using Biot's solution and Lur'e method without ad hoc assumptions. These equations and solutions can be used to construct the refined theory for the plane problems. In the case of homogeneous boundary conditions, the exact governing differential equations and solutions for the plate are derived, which consist of four governing differential equations. It is important note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of non-homogeneous boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order terms. The correctness of the stress assumptions in the classic plane stress problems is revised.

A SINGULAR PROBLEM IN INCOMPRESSIBLE NONLINEAR ELASTOSTATICS

This paper represents a contribution to the numerical treatment of problems in incompressible elasticity theory for large deformations. We are especially concerned about the solution of plane problems with corners. A review of the literature on these problems indicates that the behavior of the solution in the vicinity of a corner is given little attention. We investigate the solution of the compressed bonded block problem corresponding to the compression of an incompressible elastic block of rectangular cross-section and infinite transverse length between two opposing bonded rigid surfaces, with the two remaining lateral faces traction-free. We are especially interested in the behavior at a corner where a bonded end is adjacent to a free lateral side. We employ a finite element method based on a reduced and selective integration technique with penalization to construct a numerical solution for this problem. Our computational method converges everywhere except in a small neighborhood of the corner. We appeal to an elementary a priori inequality concerning the angle of shear to show that the numerical calculations in this neighborhood are inaccurate and need a more refined study. Based on the inequality, we offer a conjecture concerning the local shape of the deformed free lateral surface at the corner.

Solution of Plane Problems of the Three-Dimensional Stability of a Ribbon-Reinforced Composite

The plane problem of three-dimensional elastic stability is solved for a ribbon-reinforced composite under lateral compression if its initial state is nonuniform. The net approach is used to numerically solve the problem. The influence of the ratio of the elastic moduli of the matrix and the filler and the ribbon shape factor on the critical load of the material is studied

Finite-element solution of plane problems of the linear elasticity theory of composites

Finite-element solution of plane problems of the linear elasticity theory of composites

Boundary collocation method for multiple defect interactions in an anisotropic finite region

The modified mapping collocation method is extended for the solution of plane problems of anisotropic elasticity in the presence of multiple defects in the form of holes, cracks, and inclusions under general loading conditions. The approach is applied to examine the stress and strain fields in an anisotropic finite region including an elliptical and a circular hole, an elliptical flexible inclusion, and a line crack. It can be readily incorporated into micro-mechanics models, capturing the relative importance of the matrix, the fiber/matrix interface, and reinforcement geometry and arrangement while estimating the effective elastic properties of composite materials. The accuracy and robustness of this method is established through comparison with results obtained from finite element analysis.

Approximate solution of plane problems for ideal elastoplastic inhomogeneous bodies

Approximate solution of plane problems for ideal elastoplastic inhomogeneous bodies

General solution of plane problem of piezoelectric media expressed by ?harmonic functions?

First, based on the basic equations of two-dimensional piezoelectroelasticity, a displacement function is introduced and the general solution is then derived. Utilizing the generalized Almansi's theorem, the general solution is so simplified that all physical quantities can be expressed by three “harmonic functions”. Second, solutions of problems of a wedge loaded by point forces and point charge at the apex are also obtained in the paper. These solutions can be degenerated to those of problems of point forces and point charge acting on the line boundary of a piezoelectric half-plane.

Fundamental solutions for plane problem of piezoelectric materials

Based on the basic equations of two-dimensional, transversely isotropic, piezoelectric elasticity, a group of general solutions for body force problem is obtained. And by utilizing this group of general solutions and employing the body potential theory and the integral method, the closed-form solutions of displacements and electric potential for an infinite piezoelectric plane loaded by point forces and point charge are acquired. Therefore, the fundamental solutions, which are very important and useful in the boundary element method (BEM), are presented.

Bending of a pretwisted bar by terminal transverse load

The problem of bending of a pretwisted bar, cantilevered and loaded by transverse forces at the end, is dealt within the framework of the linear three-dimensional theory of elasticity. The stress components are assumed representable as a suitable series in the pretwist parameter k, whose coefficients can be determined by the solution of plane problems. This is possible if a suitable dependence on the axial coordinate is assigned. The boundary value problems which characterize the coefficients up to the second power are explicitly determined, and the solutions are calculated for an elliptical section. In this way we determine the approximate solution up to terms of order k 2, to account for a moderate rate of pretwist. From the results of the mathematical analysis, some indications can be deduced regarding the effect of pretwist and a comparison can be made with the classical case of a straight bar.

Application of the edge function method to rock mechanics problems

The edge function method is considered as an alternative to conventional numerical schemes for the solution of plane problems in rock mechanics. The essence of the approach is the approximation of the solution by a linear combination of solutions of the field equations. The unknowns in the linear combination are obtained from a system of equations which follows from the approximation of the boundary conditions by a boundary Galerkin energy method. No mesh generation is required over the domain or boundary of the problem. Previous edge function work in anisotropic elasticity is enhanced by the incorporation of a special solution for the effect of gravity. Examples are presented to illustrate the applicability of the method in determining stresses in various rock mechanics problems. A high level of accuracy is achieved with a relatively small number of degrees of freedom. Convergence is rapid because of the inclusion of special analytic solutions to model stress concentrations. The inclusion of the gravity force does, however, lead to a small increase in the number of degrees of freedom needed to achieve acceptable results. The optimum use of the edge function method, at present, may be as a special element within more general finite element or discrete element codes.

An exact element method for plane problem

In this paper, based on the step reduction method[1]and exact analytic method[2]a new method—exact element method for constructing finite element, is presented. Since the new method doesn't need the variational principle, it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficient. By this method, a quadrilateral noncompatible element with 8 degrees of freedom is derived for the solution of plane problem. Since Jacobi's transformation is not applied, the present element may degenerate into a triangle element. It is convenient to use the element in engineering. In this paper, the convergence is proved. Numerical examples are given at the end of this paper, which indicate satisfactory results of stress and displacements can be obtained and have higher numerical precision in nodes.

On the method of reciprocal theorem to find solutions of the plane problems of elasticity

In this paper the method of reciprocal theorem is extended to find solutions of plane problems of elasticity of the rectangular plates with various edge conditions.

Singularity solutions of plane problems in anisotropic solids

Singularity solutions of plane problems in anisotropic solids

The asymptotic analyeical solution of the strain-hardening elasto-plastic plate containing A circular hole under simple tension

In this paper the general asymptotic analytical solution of plane problem of elasto-plasticity with strain-hardening[2] is used in solving the problem of an infinitely large plate containing a circular hole under simple tension, and the analytical expressions of stress components of the first two approximations are given. These results are compared with the numerical and the experimental results given by other authors[4, 5], and a good agreement is obtained. At the end of this paper the authors inspect the correctness of Neuber's formula[9] for this problem.

Solution of plane problems of the three-dimensional theory of elastic stability of plates for inhomogeneous subcritical states

Solution of plane problems of the three-dimensional theory of elastic stability of plates for inhomogeneous subcritical states