Two-dimensional Linear Elastic Problems Research Articles

Applications of multi-region Trefftz-collocation to fracture mechanics

This paper describes the application of a multi-region indirect Trefftz method in the analysis of two-dimensional linear elastic fracture mechanics problems. The technique allows for the use of subregions with the necessary continuity conditions being appropriately enforced at the interfaces. The system of equations is assembled by collecting all the required terms from the collocation of the displacement and the traction equations at a number of boundary points. Numerical tests were carried out which included the stress intensity factor computation for cracked plates (centre and edge cracks) and for a notched plate.

An rh-method for efficient adaptive finite element analysis

An rh-method, which combines r- and h-methods, is proposed for cost-effective adaptive FE analysis in two-dimensional linear elastic problems. Through various numerical test examples, the rh-method is compared with the h-method. From these examples it is concluded that the rh-method has the advantages of both the r- and h-methods, so that the rh-method can minimize the computational time required for the adaptive analysis. The algorithm of the rh-method can be implemented easily in existing h-adaptive FE codes, since the r-method is independently combined with the h-method.

Intelligent finite element mesh generation

A knowledge-based and automatic finite element mesh generator (INTELMESH) for two-dimensional linear elasticity problems is presented. Unlike other approaches, the proposed technique incorporates the information about the object geometry as well as the boundary and loading conditions to generate an a priori finite element mesh which is more refined around the critical regions of the problem domain. INTELMESH uses a blackboard architecture expert system and the new concept of substracting to locate the critical regions in the domain and to assign priority and mesh size to them. This involves the decomposition of the original structure into substructures (or primitives) for which an initial and approximate analysis can be performed by using analytical solutions and heuristics. It then uses the concept of wave propagation to generate graded nodes in the whole domain with proper density distribution. INTELMESH is fully automatic and allows the user to define the problem domain with minimum amount of input such as object geometry and boundary and loading conditions. Once nodes have been generated for the entire domain, they are automatically connected to form well-shaped triangular elements ensuring the Delaunay property. Several examples are presented and discussed. When incorporated into and compared with the traditional approach to the adaptive finite element analysis, it is expected that the proposed approach, which starts the process with near optimal initial meshes, will be more accurate and efficient.

A methodology for crack tip mesh design

A methodology for crack tip mesh design is developed which consists of comparing the mesh geometric parameters against the accuracy of the finite element solution. By successive changes in the mesh parameters a near optimal mesh can be obtained. This was done here two-dimensional linear elastic single mode problems. The direct displacement extrapolation method for stress intensity factor estimation is used.

Crack tip displacement factor-D: an application to two dimensional linear elastic fracture mechanics

A fracture parameter, crack tip displacement factor D is introduced to characterize the two-dimensional linear elastic fracture mechanics problems. Similar to the conventional COD approach, the D factor can be determined directly from the crack surface displacement. The major advantage of using this fracture control parameter is that one can obtain it by evaluating the crack surface displacement from a numerical analysis or an experiment without the need for considering the stress field at the crack tip. Unlike the COD method, the introduced fracture parameter may be used for not only the simple opening mode but also for the sliding and tearing modes as well as for the mixed mode fracture problems. This article presents a detailed derivation of the D factor as well as some sample applications.

A SIMPLE PATH-INDEPENDENT INTEGRAL FOR CALCULATING MIXED-MODE STRESS INTENSITY FACTORS

A path-independent integral is introduced for calculating stress intensity factors. The derivation of the integral is based on the application of the known Bueckner's fundamental field solution for a crack in an infinite body and on the reciprocal theorem. The method was applied to two-dimensional linear elastic mixed-mode crack problems. The key advantage of the present path-independent integral is that the stress intensity factor components for any irregular cracked geometry under any kind of loading can be easily obtained by a contour integral around the crack tip. The method is simple to implement and only the far field displacements and tractions along the contour must be known. The required stress analysis can be made by using any analytical or numerical method, e.g. the finite element method, without special consideration of the modelling of crack tip singularity. The application of this integral is also independent of the crack type, that is, there is no difference between an edge crack and an embedded crack, provided that the crack tip asymptotic behaviour exists.

A New Boundary Integral Equation Formulation for Linear Elastic Solids

A new boundary integral equation formulation is presented for two-dimensional linear elasticity problems for isotropic as well as anisotropic solids. The formulation is based on distributions of line forces and dislocations over a simply connected or multiply connected closed contour in an infinite body. Two types of boundary integral equations are derived. Both types of equations contain boundary tangential displacement gradients and tractions as unknowns. A general expression for the tangential stresses along the boundary in terms of the boundary tangential displacement gradients and tractions is given. The formulation is applied to obtain analytic solutions for half-plane problems. The formulation is also applied numerically to a test problem to demonstrate the accuracy of the formulation.

Adaptive h- p procedures for high accuracy finite element analysis of two-dimensional linear elastic problems

Two adaptive procedures of the h- p version for a high accuracy finite element analysis of two-dimensional elastic problems are studied. These are based on a strategy of first using an h-version to predict a nearly optimal mesh up to a certain accuracy and then following up with a p-version to achieve a higher accuracy. The h-version, using linear triangular elements, is developed by coupling a code, ADMESH, with an error estimation in energy norm. Following the h-version, two alternative procedures of non-uniform p-refinements are then performed. In procedure I, p-refinements are made in one step by selectively adding hierarchical shape functions of order p = 2 and 3 based on the estimated error in energy norm. In procedure II, p-refinements are made in a step-by-step way by which, in the kth step of the p-refinements, hierarchical shape functions of order p = k + 1 are selectively introduced. In the first step of p-refinements of procedure II, hierarchical variables are selected by means of the estimated errors in energy norm, whereas in the later steps, they are selected with a guidance of an error estimate which evaluates the local average error of stresses. The performances of both procedures and the rate of convergence are studied in numerical examples. Numerical tests for the error estimates being used are also made. Obtained results indicate that both procedures can achieve a high accuracy (say, error below 5% measured in energy norm) in an exponential rate of convergence.

Local shape optimization of a two-dimensional elastic body

This study concerns an efficient computational method for local shape optimization in structural design by a combined use of the finite and boundary elements. Two types of discrete elements are employed for the analysis, i.e., boundary element discretization on the specific boundary of the body undergoing successive geometrical change and finite element discretization for the remaining region which is invariant during iteration for optimization. The system equation has smaller, fully-populated submatrices, and therefore the scheme becomes more effective. Numerical applications to the two-dimensional linear elastic problems demonstrate the utility of the proposed approach for local optimal shape synthesis under the specified optimization condition.

L∞-error estimates for linear elasticity problems

Two-dimensional linear elasticity problems are approximately solved by a mixed finite-element method based on Raviart—Thomas—Nedelec spaces; maximum-norm error estimates of optimal rate and almost optimal regularity are derived for the displacement field, and of optimal regularity and almost optimal rate for the stress tensor. The superconvergence of the approximate displacement field to the L 2-projection of the actual one is established.

Analysis of two-dimensional mixed-mode crack problems by finite element alternating method

A finite element alternating method is presented and applied to analyze two-dimensional linear elastic mixed-mode fracture problems with single or multiple cracks. The method involves the iterative superposition of the finite element solution of a bounded uncracked plate and the analytical solution of an infinite two-dimensional plate with a crack subjected to arbitrary normal and shear loadings. The normal and shear residual stresses evaluated at the location of fictitious cracks are fitted by appropriate polynomials through the least-squares method. Based on those coefficients of the determined polynomials, the mixed-mode stress intensity factors can be calculated accurately. The interaction effects among cracks are also considered. This method provides a highly efficient way to deal with two-dimensional fracture problems.

Analysis of mixed mode fracture and crack closure using the boundary integral equation method

A multi-domain boundary integral equation method, employing an isoparametric quadratic representation of geometries and functions, is developed for the analysis of two-dimensional linear elastic fracture mechanics problems. The multi-domain approach allows the two faces of a crack to be modelled in independent sub-regions of the body, avoiding singularity difficulties and making it possible to analyse crack closure problems with contact stresses over part of the cracked faces. Problems solved include slanted cracked plate mixed mode and crack closure examples, also crack closure situations involving fully reversed bending of an edge cracked strip, both with and without a superimposed tensile loading.

Comments on "A Note on the General Solution of the Two-Dimensional Linear Elasticity Problem in Polar Coordinates"

Comments on "A Note on the General Solution of the Two-Dimensional Linear Elasticity Problem in Polar Coordinates"

Comment on "A Note on the General Solution of the Two-Dimensional Linear Elasticity Problem in Polar Coordinates"

Comment on "A Note on the General Solution of the Two-Dimensional Linear Elasticity Problem in Polar Coordinates"

Comments on "A Note on the General Solution of the Two-Dimensional Linear Elasticity Problem in Polar Coordinates"

Comments on "A Note on the General Solution of the Two-Dimensional Linear Elasticity Problem in Polar Coordinates"