Plane Elasticity Problems Research Articles

Caustics, pseudocaustics and the related illuminated and dark regions with the computational method of quantifier elimination

The method of caustics is a powerful experimental method in elasticity and particularly in fracture mechanics for crack problems. The related method of pseudocaustics is also of interest. Here we apply the computational method of quantifier elimination implemented in the computer algebra system Mathematica in order to determine (i) the non-parametric equation and two properties of the caustic at a crack tip and especially (ii) the illuminated and the dark regions related to caustics and pseudocaustics in plane elasticity and plate problems. The present computations concern: (i) The derivation of the non-parametric equation of the classical caustic about a crack tip through the elimination of the parameter involved (here the polar angle) as well as two geometrical properties of this caustic. (ii) The derivation of the inequalities defining the illuminated region on the screen in the problem of an elastic half-plane loaded normally by a concentrated load with the boundary of this illuminated region related to some extent to the caustic formed. (iii) Similarly for the problem of a clamped circular plate under a uniform loading with respect to the caustic and the pseudocaustic formed. (iv) Analogously for the problem of an equilateral triangular plate loaded by uniformly distributed moments along its whole boundary, which defines the related pseudocaustic. (v) The determination of quantities of interest in mechanics from the obtained caustics or pseudocaustics. The kind of computations in the applications (ii) to (iv), i.e. the derivation of inequalities defining the illuminated region on the screen, seems to be completely new independently of the use here of the method of quantifier elimination. Additional applications are also possible, but some of them require the expansion of the present somewhat limited power of the quantifier elimination algorithms in Mathematica. This is expected to take place in the future.

New solution of the plane problem for an equilibrium crack

We consider the classical plane problem of elasticity about a crack in an isotropic elastic unbounded plane resulting in a singular solution for the stresses near the crack edge. Relations of generalized elasticity with a small parameter characterizing the medium microstructure are derived, and the higher order of these relations permits eliminating the singularity of the classical solution. An experimental method for determining the medium parameter is proposed, and the corresponding experimental results are given.

Integral equations of plane static boundary value problems of the elasticity theory for an inhomogeneous anisotropic medium

The static boundary value problems of plane elasticity for an inhomogeneous anisotropic medium in a simply connected domain are reduced to the Riemann–Hilbert problem for a quasianalytic vector. Singular integral equations over the domain are obtained, and their solvability is proved for a sufficiently wide anisotropy class. In the case of a homogeneous anisotropic body, the solutions of the first and second boundary value problems are obtained in closed form.

A new strain-based finite element for plane elasticity problems

Purpose In this paper, a new quadrilateral strain-based element is developed. The element has five nodes, four at the corners as well as an internal node. Design/methodology/approach Through the introduction of the internal node, the numerical performance of the element proved to be superior to existing elements in the literature, even though a static condensation is required. Findings From several numerical examples, it is shown that convergence can be achieved with the use of only a small number of finite elements. The proposed element can be used to solve general plane linear elasticity problems resulting in excellent results. Originality/value The results obtained are comparable with those given by the robust element Q8.

Numerical solution of Eshelby's elastic inclusion problem in plane elasticity by using boundary integral equation

This paper provides a numerical solution for Eshelby's elastic inclusions in an plane elasticity based on the complex variable boundary integral equation (CVBIE) method. An inclusion with arbitrary shape is embedded in the infinite matrix. In the inclusion, the constant eigenstains are assumed. No remote loading is applied to the matrix. The displacements from the assumed eigenstrains are evaluated exactly, which in turn are the generalized loading in the problem. The proposed problem is reduced to solve interior and exterior boundary value problems simultaneously. For the elliptical inclusion case, the computed stresses along the interface of the inclusion side are nearly uniform. One more numerical example is devoted to a square-type inclusion with round corner.

Adaptive isogeometric analysis based on a combined r-h strategy

ABSTRACTIn the present work, an r-h adaptive isogeometric analysis is proposed for plane elasticity problems. For performing the r-adaption, the control net is considered to be a network of springs with the individual spring stiffness values being proportional to the error estimated at the control points. While preserving the boundary control points, relocation of only the interior control points is made by adopting a successive relaxation approach to achieve the equilibrium of spring system. To suit the noninterpolatory nature of the isogeometric approximation, a new point-wise error estimate for the h-refinement is proposed. To evaluate the point-wise error, hierarchical B-spline functions in Sobolev spaces are considered. The proposed adaptive h-refinement strategy is based on using De-Casteljau’s algorithm for obtaining the new control points. The subsequent control meshes are thus obtained by using a recursive subdivision of reference control mesh. Such a strategy ensures that the control points lie in the physical domain in subsequent refinements, thus making the physical mesh to exactly interpolate the control mesh and thereby allowing the exact imposition of essential boundary conditions in the classical isogeometric analysis (IGA). The combined r-h adaptive refinement strategy results in better convergence characteristics with reduced errors than r- or h-refinement. Several numerical examples are presented to illustrate the efficiency of the proposed approach.

On one problem of the plane theory of elasticity for a circular domain with a rectangular hole

The paper considers a plane problem of elasticity for a circle with a rectangular hole. To find a solution, the use is made both of the method of conformal mappings and of boundary value problems of analytic functions. In particular, relying on the well-known Kolosov–Muskhelishvili’s formulas, the problem formulated with respect to unknown complex potentials is reduced to the two Riemann–Hilbert problems for a circular ring, and the solutions of the latter problems allow us to construct potentials effectively (analytically). The estimates of the obtained results in the neighborhood of angular points are given. Analogous results (as a particular case) are obtained for a circular domain with a rectilinear cut.

Stochastic spline fictitious boundary element method for modal analysis of plane elastic problems with random fields

Mathematical formulation and computational implementation of the stochastic spline fictitious boundary element method (SFBEM) are presented for modal analysis of plane elastic problems with structural parameters modeled as random fields. Two sets of governing differential equations with respect to the means and deviations of displacement modes are derived by including the first order terms of deviations. These equations are in similar forms to those of deterministic plane elastostatic problems, and can be solved using deterministic elastostatic fundamental solutions, resulting in the means and covariances of the eigenvalues and mode shapes. For the effective treatment of the domain integrals involved in the deviation solution, the random fields considered are represented by Karhunen–Loeve (KL) expansion in conjunction with the Galerkin projection. Numerical examples indicate that the results of the present method are in good agreement with those from the Monte Carlo simulation (MCS) with small variations, and the present approach is more efficient than the perturbation stochastic finite element method (FEM) with the same KL expansion technique.

Nonlinear stability and dynamics of composite skew plates under nonuniform loadings using differential quadrature method

The buckling, postbuckling and postbuckled vibration behaviour of composite skew plates subjected to nonuniform inplane loadings are presented here. The skew plate is modelled using first order shear deformation theory accounting for von-Kármán geometric nonlinearity and initial geometric imperfections. The different types of nonuniform loads that have been considered in this study are concentrated load, partial load and parabolic load. The explicit analytical expressions for prebuckling stress distributions within composite skew plate subjected to three different types of nonuniform inplane loadings are developed by solving plane elasticity problem using Airy's stress function approach. It is observed that the inplane normal stress distributions within the skew plate due to above nonuniform loadings do not become uniform even at mid-section. The generalized differential quadrature (GDQ) method is used to solve the nonlinear governing partial differential equations. It is observed that the postbuckled load carrying capacity of skew plate under concentrated loading is the lowest compared to other nonuniform and uniform loadings.

Uniform Convergence Analysis of a Higher Order Hybrid Stress Quadrilateral Finite Element Method for Linear Elasticity Problems

Abstract.This paper derives a higher order hybrid stress finite element method on quadrilateral meshes for linear plane elasticity problems. The method employs continuous piecewise bi-quadratic functions in local coordinates to approximate the displacement vector and a piecewise-independent 15-parameter mode to approximate the stress tensor. Error estimation shows that the method is free from Poisson-locking and has second-order accuracy in the energy norm. Numerical experiments confirm the theoretical results.

The Newton-Kantorovich method in analytical solution of plane elasticity problems under superimposed finite strains

The Newton-Kantorovich method in analytical solution of plane elasticity problems under superimposed finite strains

A Numerical Method Based on Daubechies Wavelet Basis and B-Spline Patches for Elasticity Problems

The Daubechies (DB) wavelets are used for solving 2D plane elasticity problems. In order to improve the accuracy and stability in computation, the DB wavelet scaling functions in0,+∞)comprising boundary scaling functions are chosen as basis functions for approximation. The B-spline patches used in isogeometry analysis method are constructed to describe the problem domain. Through the isoparametric analysis approach, the function approximation and relevant computation based on DB wavelet functions are implemented on B-spline patches. This work makes an attempt to break the limitation that problems only can be discretized on uniform grids in the traditional wavelet numerical method. Numerical examples of 2D elasticity problems illustrate that this kind of analysis method is effective and stable.

Fast multipole singular boundary method for large-scale plane elasticity problems

The singular boundary method (SBM) is a recent meshless boundary collocation method that remedies the perplexing drawback of fictitious boundary in the method of fundamental solutions (MFS). The basic idea is to use the origin intensity factor to eliminate singularity of the fundamental solution at source. The method has so far been applied successfully to the potential and elasticity problems. However, the SBM solution for large-scale problems has been hindered by the operation count of O(N3) with direct solvers or O(N2) with iterative solvers, as well as the memory requirement of O(N2). In this study, the first attempt was made to combine the fast multipole method (FMM) and the SBM to significantly reduce CPU time and memory requirement by one degree of magnitude, namely, O(N). Based on the complex variable representation of fundamental solutions, the FMM-SBM formulations for both displacement and traction were presented. Numerical examples with up to hundreds of thousands of unknowns have successfully been tested on a desktop computer. These results clearly illustrated that the proposed FMM-SBM was very efficient and promising in solving large-scale plane elasticity problems.

The sensitivity of the adaptive algorithm with a posteriori error control to marking criteria

The aim of this work is to compare different marking strategies, their influence on the work of adaptive algorithms with a posteriori error control for plane elasticity problems. The error control was performed using a functional error majorant. The implemented adaptive algorithms were based on the functional error majorant with no symmetry limitation on the free tensor, computed using the zero-order Raviart–Thomas approximations on triangular meshes. The four most commonly used element-marking criteria were used in adaptation. Numerical results for several plane-strain problems have been presented, including the case of different materials and geometry. A comprehensive analysis of the obtained results was given.

Method of fundamental solutions without fictitious boundary for plane time harmonic linear elastic and viscoelastic wave problems

This study makes the first attempt to apply a recently developed modified method of fundamental solutions (MFS) without fictitious boundary, which is named as the singular boundary method (SBM), to the solution of plane linear elastic and viscoelastic wave problems. Like the standard MFS, the SBM applies the fundamental solutions of the governing equations of interest as the basis functions. Unlike the standard MFS, the SBM, however, does not require the fictitious boundary outside physical domain to avoid the singularity of the fundamental solution and instead directly places the source points on the physical boundary coinciding with collocation points via the concept of origin intensity factors. To demonstrate the effectiveness of the SBM for plane elastic and viscoelastic wave problems, several numerical examples are given in comparison with analytical solutions, and numerical results of the MFS and the finite element method.

Stresses and displacements in a circular ring loaded by symmetrically spaced radial forces

This paper presents the study of a plane elasticity problem for a concentric circular ring loaded by a system of equal radial forces equally spaced around its outer ring boundary. The problem is solved by the Kolosov–Muskhelishvili complex variables method. Explicit formulas are derived for calculating components of the stress tensor and of the displacement vector. Numerical values for circumferential stress and radial displacement are given for various ratios of inner and outer ring radii and various numbers of forces.

Numerical solution for the degenerate scale problem in plane elasticity using null field CVBIE

This paper provides a numerical solution for the degenerate scale problem in plane elasticity using the null field complex variable boundary integral equation (CVBIE). After performing the coordinate transformation, the CVBIE can be formulated in the normal scale. After making discretization, a linear algebraic equation is obtained. The influence matrix in the normal scale is invertible. By introducing two basic solutions, the degenerate scale problem is finally solved. Several numerical examples are given.

Dynamic instability of damped composite skew plates under non-uniform in-plane periodic loading

In this paper, the non-linear dynamic instability of damped composite skew plates under non-uniform in-plane periodic loadings is studied using analytical methods. The structural model for composite skew plate is based on first order shear deformation theory considering von-Kármán geometric nonlinearity. The analytical expressions for stress distributions within the composite skew plate subjected to three different types of non-uniform in-plane loadings are developed by solving, plane elasticity problem. Subsequently, using these stress distributions and via Hamilton principle, the equations governing the dynamic instability of composite skew plates are derived in terms of displacement (u–v–w) and rotation (φx,φy) variables. The generalized differential quadrature method is used to reduce the governing partial differential equations into a set of ordinary differential (Mathieu type) equations. The dynamic instability regions are traced by the periodic solutions (with period T and 2T) to Mathieu-type differential equations. Numerical results are presented to demonstrate the influence of skew angles, boundary conditions, non-uniform loadings and damping on the dynamic instability regions. Furthermore, the characteristic features of linear and nonlinear response in stable and unstable regions are studied. This brings out various features of instability problem such as existence of beats, effect of non-linearity and damping on response and its dependence on forcing frequency.

Transverse compression of a two-layered elastic plane with a circular hole on the border between layers

The results are presented on the stress distribution in a transversally compressed two-layered elastic plane with a circular hole on the border between layers, received by using a variational fi- nite-difference procedure for solving plane elastic problems for the holed rectangular domains. The effects of the layers' elastic constants on the level of stresses around the hole are examined.

Stress fields around two pores in an elastic body: exact quadrature domain solutions.

Analytical solutions are given for the stress fields, in both compression and far-field shear, in a two-dimensional elastic body containing two interacting non-circular pores. The two complex potentials governing the solutions are found by using a conformal mapping from a pre-image annulus with those potentials expressed in terms of the Schottky–Klein prime function for the annulus. Solutions for a three-parameter family of elastic bodies with two equal symmetric pores are presented and the compressibility of a special family of pore pairs is studied in detail. The methodology extends to two unequal pores. The importance for boundary value problems of plane elasticity of a special class of planar domains known as quadrature domains is also elucidated. This observation provides the route to generalization of the mathematical approach here to finding analytical solutions for the stress fields in bodies containing any finite number of pores.