Solution Of Elastic Problem Research Articles

Ill-Posed Problems of Geomechanics

The classical solution of elasticity problem on deformation of a plane weakened by a mathematical cut under wedging by constant forces is analyzed. The ill-posedness of the classical failure mechanics statements for problems with angular points is demonstrated. The approximate solution is constructed for deformation of continuum in the vicinity of a cut under small strains.

A Numerical Solution for a Frictional Contact Problem between an Orthotropic Strip and Punch System

In this study, a numerical solution of elasticity problem is examined. This problem is a plane contact problem. The frictional contact problem for an elastic strip under a rigid punch system is considered. The frictional contact problem is related to infinite length elastic strip in contact with N punches under the influence of horizontal and vertical forces. The lower boundary of the strip is hinged. The solution of contact problems is often reduced to the solution of an integral equation. This integral equation system can be derived from contact problem by using the basic equations of elasticity theory and the given boundary conditions. The singular integral equation system is solved with the help of Gauss Jacobi Quadrature Collocation Method. The frictional contact problem for a homogenous and orthotropic elastic layer are investigated numerically the pressure distribution under the punch system due to the geometrical and mechanical properties of elastic layer are examined and the results are shown in the graphics and tabular form.

On the complementary energy in elasticity and its history: the Italian school of nineteenth century

In the formulation of mechanical theories, physicists usually neglect complementary energy, while rational mechanicians and engineers widely use it, especially in continuum physics and structural mechanics. Indeed, in many cases the solutions of elastic problems are found in a simpler way by resorting to complementary, rather than potential, energy. Moseley and Cotterill in England, Menabrea and Castigliano in Italy were among the first to introduce complementary energy in their papers, though implicitly; a more explicit formulation is in Crotti’s papers; and Engesser extended it to non-linear elasticity. In this work we run through the history of complementary energy and search for its possible mechanical meaning.

An Iterative Procedure for the Analysis of Nonlinear Elastic Systems Subjected to Uniaxial Loading and Large Displacements

The paper deals with an iterative procedure for the solution of elastic problems, for which the large displacement theory must be considered. It is shown that the algorithm, based upon a non-traditional iterative scheme, tends to converge to the correct solution and some numerical tests are considered, by assuming a linear and piecewise-linear material model.

Modeling of three-dimensional elastic strain fields by point-source method

The work objective is to investigate the possibility and efficiency of three-dimensional numerical models of the elastic stress fields in the deformed solids. The field point-source method (PSM) designated as the method of fundamental solutions (MFS) in the foreign literature is used to develop these models. The PSM system generation for modeling fields of different physical nature is described. The concept of a point source of the elastic displacement field in the deformed solid is introduced. The research result is a developed PSM equations system that can be applied for solving three-dimensional problems in the elasticity theory, including the classical first and second boundary value problems in the elasticity theory (when either the voltage or bias is set on the boundary), as well as a mixed boundary problem (when on one part of the boundary, the displacement is set, and on the other - the voltage). The PSM properties are studied when solving standard problems, and the Dirichlet problem for a circular domain. The dependences of the numerical solution error on the problem parameters, in particular, on such as the charge number simulating the desired field, the remoteness of the charges from the solution domain boundaries, are obtained. The test problem of calculating the deformation field in the ball domain is solved. Upon the results obtained, the following conclusion is made. In the numerical solution of three-dimensional elasticity problems, a decreasing exponential dependence of the PSM error on the square root of the simulating charge number is observed. This property provides a numerical solution with a very low relative error that implies the PSM application perspectiveness in the numerical solution of the elasticity problems, including three-dimensional problems.

Estimates of deviations from exact solutions for elasticity problems with nonlinear boundary conditions

We present a method of deriving fully guaranteed bounds of the difference be- tween the exact and approximate solutions of variational inequalities gener ated by problems in the theory of elasticity with nonlinear boundary conditions (e.g., unilatera l and friction type boundary conditions). These estimates are obtained with the help of the duality tech- nique in the calculus of variations. They do not contain mesh-dependent constants and are valid for any function from the corresponding energy space compared with the exact so- lution. We prove that the majorants of deviations are continuous and vanish if and only if approximate solutions coincide with the exact one. Several numerical tests demonstrate the quality of the estimates.

Symmetry’s Applications to Bihamonic Equation and the Generalized Solution of Elasticity Problem

The generalized elasticity solutions are obtained in this paper by symmetries from Lie transformations. The symmetries of the bihamonic equation are obtained by mathematica package SYM. The several invariant solutions are found by solving the corresponding ordinary differential equations from Lie transformations. The superposition of invariant solutions, corresponding to an eigenfunction expansion, could yield the generalized elasticity solutions in rectangular coordinates and polar coordinates, where the eigenvalues arises from the invariance of bihamonic equation.

Разложение решения задач теории упругости для полосы в ряд по модам

Разложение решения задач теории упругости для полосы в ряд по модам

On the BIEM solution for a half‐space by Neumann series

AbstractThis paper presents an approach which allows the solution of elastic problems concerning a half‐space (half‐plane) with cavities by the boundary integral equation methods using Neumann's series. To evaluate the series terms at singular points, the regular representations of singular integrals for the external problems were proven and the regular recurrent relationships for the series terms, which can be calculated by any known quadrature rule, are obtained. The numerical proposed procedure was tested by comparison with known theoretical solution and the method convergence was studied for various depths of a buried cavity. Copyright © 2006 John Wiley & Sons, Ltd.

Pseudo Spectral Methods Applied to Problems in Elasticity

Pseudo spectral methods offer an attractive alternative to finite element procedures for the solution of problems in elasticity. Especially for simple domains, questions involving both two and thre...

The Method of R-Functions in the Solution of Elastic Problems on the Basis of Reissner's Mixed Variational Principle

A method is presented for solving boundary-value elastic problems on the basis of the variational–structural method of R-functions and Reissner's mixed variational principle. A mathematical formulation is given to problems on the deformation of elastic bodies under mixed boundary conditions and bodies interacting with smooth rigid dies. Solutions satisfying all the boundary conditions are proposed. For undetermined components of these solutions, the resolving equations are derived and their properties are studied. A posteriori estimation of numerical solutions is made. As examples, solutions are found to a problem on the stress–strain state of a short cylinder and to a contact problem on a cylinder interacting with a smooth die. A numerical method of solving such problems is analyzed for convergence, and the accuracy of the solutions is estimated.

Random Walk on Spheres methods for iterative solution of elasticity problems

Random Walk on Spheres methods for iterative solution of elasticity problems

An Iterative Method for Solving Elasticity Problems for Composite Laminates

An iterative method for approximate analytical solution of elasticity problems in composite laminates is presented. The stress analysis is performed for laminates in the three-dimensional strain state independent of the longitudinal direction. Predictions of the method are compared with results from existing analytical and numerical models. Simple and accurate approximations for stresses are obtained. [S0021-8936(00)02001-8]

A parallel algebraic multigrid solver for problems in solid mechanics discretisized by finite elements

This paper presents an algebraic multigrid solver which can also be applied as a preconditioner for the conjugate gradient method. The solver has been implemented in a parallel version of the finite element program FEAP, see Zienkiewicz O. C. and Taylor R. L. The Finite Element Method, volume 1. McGraw–Hill, London, 4th edition, 1989 [1]. The aim of the paper is to show the performance of these solvers on two different MIMD computers and to present a concept for porting a finite element code to a parallel machine of MIMD class. We discuss the parallel mesh generation and the parallel solution of problems in elasticity.

Spatial decay estimates and upper bounds in elasticity for domains with unbounded cross-sections

In this paper we investigate spatial decay estimates and upper bounds for the solutions of elastic problems when the cross-sections of the three dimensional solid are semi-infinite strips. We obtain spatial decay estimates for the solutions of a static problem in the theories of homogeneous and isotropic linear elasticity and linearised elasticity. Energy bounds and some spatial decay estimates are obtained for the solutions of a dynamical problem in the case of anisotropic linear elasticity. For both kinds of problems we use the energy methods.

Elastic Properties of Polymer Systems Reinforced with Short Organic Fibers

This paper addresses the experimental and theoretical investigations of mechanical properties of polymer resins reinforced with natural fibers. These materials have potentially a wide area of applications and relatively low cost because they are made of inexpensive resins and fibers obtained from plants available in the Mexican desert. We present the results of experimental investigations of elastic properties and strength of composites based on three types of polymer resins reinforced with natural fibers of Palma Zamandoca. Theoretical analysis of effective elastic and strength properties of short-fiber matrix composites is given based on the advanced self-consistent scheme permitting to account for interactions between fibers. The method utilizes the idea of introducing the effective strain field which affects every inclusions in the composite. In distinction to the traditional forms of the effective (i.e., self-consistent) field method we assume that the effective field is not constant and the same for all fibers but rather depends on the orientation of each of them. This allows to take into account the characteristics of the spatial distributions of fibers in the volume of composite. The scheme utilizes the asymptotic solution of elastic problem for rigid cylindrical bar. The results of theoretical analysis are compared with extensive experimental data on composites with natural fibers manufactured and tested by the authors as well as with the data for composites with inorganic fibers (e.g., glass and carbon fibers) available in literature.

The DtN finite element method for elastic domains with cracks and re-entrant corners

A combined analytic-finite element method is proposed for the efficient solution of problems in elasticity involving geometrical singularities. The Dirichlet-to-Neumann (DtN) map is used in the procedure, to enable the replacement of the original singular problem by an equivalent regular problem, which is then solved by a finite element scheme. Problems involving the Saint Venant torsion of long rods with cross-sections containing cracks or re-entrant corners are considered in detail. Mode I and II fracture problems are also discussed. The method yields the stress field in the entire domain, as well as the stress intensity factor. Numerical examples are presented for the torsion problem, which demonstrate the performance of the method.

Procedure for the construction of discrete models of static problems in the theory of elasticity

UDC 539.3 The finite-difference approach is widely used in the solution of elasticity problems. It is implemented on a computer in two stages: the formulation of discrete problems and the solution of those problems. Here we discuss aspects of the construction of discrete models (differencing problems and systems of linear algebraic equations) in application to a set of problems of the linear theory of elasticity of piecewise-homogeneous, anisotropic media in an arbitrary coordinate system subject to various boundary conditions. We propose a procedure by which to obtain an explicit expression for the dynamic problems in the case of an arbitrary continuous problem from the given set. The procedure entails the construction of a differencing scheme (system of equations), called the basic scheme (basic system of equations), on a mesh stencil. For a specific elasticity problem (i.e., for a given region, coordinate system, boundary conditions, and elastic constants of the components of the medium) the expression for the differencing scheme at an arbitrary grid node is written as the sum of the expressions for the basic scheme at certain nodes of the mesh. The corresponding system of linear algebraic equations represents the sum of the basic system of equations over all meshes of the grid with the subsequent correction of certain rows of the system of equations. Basic schemes are constructed for an arbitrary coordinate system. Discrete models are constructed for a specific problem. We investigate three-dimensional = 3) and two-dimensional (p = 2) problems. Unless otherwise stated, summation from 1 to p is implied by repeated subscripts on one side of an equation.

The Thermal expansion of a two-phase composite with body-centred cubic symmetry

A theory is developed using symmetry adapted displacement vectors in Kelvin's method for the solution of problems in elasticity where there is spherical symmetry. For a particle-filled composite with b.c.c. symmetry the filler is assumed to be spherical and the matrix to fill the Wigner-Seitz (W-S) unit cell. Suitable displacement vectors are chosen from the Kubic harmonics of Von der Lage and Bethe and the first two odd-ordered harmonics used to derive an explicit expression for the composite expansion coefficient. The predictions of the present theory are compared with those from the spherical shell theory and found to be in better agreement with published data for silica particles in epoxy resin. Calculated stresses show a pronounced variation with angle and position in the W-S cell.

Solution of elasticity problems by a self‐adaptive mesh refinement technique for boundary element computation

AbstractA self‐adaptive mesh refinement technique is developed for boundary element solutions of the two‐dimensional Navier's equation. The method is based on error reduction and applied on the element and global level to estimate the residual error associated with each mesh. Results for several problems employing constant two‐dimensional isotropic elastostatic boundary elements are presented.