Fixed rigid line problem in antiplane elasticity

In this paper the properties of eigenfunction expansion form (EEF) in the fixed rigid line problem in antiplane elasticity are discussed in detail. After using Betti's reciprocal theorem for a body containing a fixed rigid line, several new path-independent integrals are obtained. Finally, many solutions for this problem are proposed in a closed form.     

Determination and representation of the stress coefficient terms by path-independent integrals in anisotropic cracked solids

Because the elastic T-stress and other coefficients of the higher-order terms play an important role in fracture mechanics such as the stability of crack kinking, crack path, and two-parameter characterization of elastic-plastic crack tip fields, determination of all the coefficients in the crack tip field expansion in an anisotropic linear elastic solid is presented in this paper. Utilizing conservation laws of elasticity and Betti's reciprocal theorem, together with selected auxiliary fields, T-stress and third-order stress coefficients near the crack tip are evaluated first from path-independent line integrals. To determine the T-stress terms using the J-integral and Betti's reciprocal work theorem, auxiliary fields under a concentrated force and moment acting at the crack tip are used respectively. Through the use of Stroh formalism in anisotropic elasticity, analytical expressions for all the coefficients including the stress intensity factors are derived in a compact form that has surprisingly simple structure in terms of one of the Barnett-Lothe tensors, L. The solution forms for degenerated materials, monoclinic, orthotropic, and isotropic materials are also presented.     

On path-independent integrals within the linear theory of elasticity

Conservation laws and associated path-independent integrals play a dominant role in field theories ranging from theoretical physics to applied engineering. Especially, material conservation laws are widely used to assess structural components with flaws like defects or cracks. Within the linear theory of elasticity, a complete set of conservation laws are derived by employing the so-called Neutral-Action method. An illustrative application is discussed.     

Periphractic construction of elasticity and plasticity on Gurtin's generalized stress functions and the criterion for yielding

An alternative approach is made to Gurtin's stress functions of non-Beltrami feature to paraphrase its anholonomic periphractic standpoint and to provide a basis to reconstruct the Theory of Yielding. The stress is defined with reference to a three-dimensional version of Einstein's coordinates. The meaning and construction of elasticity is attributed to the aeolotropic periphracticies of material continua. The multidimensional picture for the yielding manifold is reached in terms of average metric involving Killing's field. Account is taken of a microphysical restriction on variational degrees of freedom to assume the form of Klein-Goldon field. The fourth order partial differential equation is so secured for the field equation of yielding. The final forms of the boundary condition equations have to be fixed by taking account of the periphracticy effects, especially of the standpoint of the plasticity tensor $\text{B}{}^{\mathrm{klmn}}$, two components of which are independent for isotropic materials, owing to integration constants. That these equations agree with only a slight modification with the ones that have been proposed many years since is checked with reference to the isotropic uniform material.The modification is concerned also with the microphysical and thermodynamical implications. A spontaneous unification with the postulate of the principle of relativity is included summarizing also the relation between spatially-temporally fluctuating harmonic and biharmonic fields.     

Locating a straight crack in an infinite elastic medium by using complex path-independent integrals

Summary The Cauchy integral theorem and the relevant formula (or, equivalently, complex path-independent integrals) have been used in a long series of papers for the determination of zeros and poles of analytic and meromorphic functions. Here this approach is generalized to become applicable to the problem of location of a straight crack inside an infinite plane isotropic elastic medium. The complex path-independent integrals used here contain the first complex potential (z) of Kolosov-Muskhelishvili, which can be obtained experimentally. The present method can be modified to apply to a variety of problems where discontinuity intervals of analytic (or, rather, sectionally analytic) functions are sought.     

Conservation laws and the material momentum tensor for the elastic dielectric

It is shown that a Langrangian formulation of continuum mechanics can provide not only the equations of motion, but the conservation laws related to the material symmetries in a perfect continuum interacting with an external electric field. These conservation laws in the presence of defects lead to the path-independent integrals widely used in fracture mechanics. They are basically related to the “material force” on a defect in a continuum. The quantity playing the role of the physical stress tensor in this formulation is the material momentum tensor. A material force in the form of a path-independent integral for the elastic dielectric is derived employing Toupin's [1] formulations.     

Advanced Kantorovich method for biharmonic problems

In this paper a method for solving biharmonic problems involving a mixed numerical-analytical approach is described. The algorithm of this method is given and the efficiency of its application for the solution of biharmonic problems is discussed. The recommendations about an application of this method for solving stationary three-dimensional problems in the theory of elasticity are given.     

Non-axial-symmetric thermal stresses in circular discs or cylinders

A solution is presented for the computation of the transient thermoelastic stresses in a hollow cylinder with temperature boundary conditions given as a circumferential variation of surface heat transfer coefficient. The temperature distribution is solved explicitly. The problem is set up using the Airy stress function which leads to the biharmonic equation. This approach requires the satisfaction of three Michell integrals at the inner boundary in order to ensure single-valued displacements and rotation. An iterative method is described in which these integrals are all simultaneously satisfied and thus provide the necessary non-zero boundary conditions for the solution of the biharmonic equation which is rapidly solved by Gaussian elimination. Results are presented for the general case where the temperature is a function of r and θ. The computer program is checked by assuming a constant value of the surface heat transfer coefficients. In this case a closed form solution is obtained.     

A posteriori pointwise upper bound error estimates in the finite element method

An error estimate for the finite element method is presented in this paper. The error is identified as the response to a set of residual forces, and a complementary analysis provides an upper bound estimate of the global energy of the error. The inequality proposed by Babu?ka and Miller¹ is then employed to bound the error in stress and displacement at a point. The formula is derived for two-dimensional elasticity, but the procedure is general; and can be applied to three-dimensional and other problems. Numerical experiments using the procedure are carried out and the results are given for the four-node bilinear compatible element and plane stress.     

A new boundary integral equation method for analysis of cracked linear elastic bodies

Abstract

A novel integral equation method is developed in this paper for the analysis of two‐dimensional general anisotropic elastic bodies with cracks. In contrast to the conventional boundary integral methods based on reciprocal work theorem, the present method is derived from Stroh's formalism for anisotropic elasticity in conjunction with Cauchy's integral formula. The proposed boundary integral equations contain boundary displacement gradients and tractions on the non‐crack boundary and the dislocations on the crack lines. In cases where only the crack faces are subjected to tractions, the integrals on the non‐crack boundary are non‐singular. The boundary integral equations can be solved using Gaussian‐type integration formulas directly without dividing the boundary into discrete elements. Numerical examples of stress intensity factors are given to illustrate the effectiveness and accuracy of the present method.     

On the corner tensor in three-dimensional linear elasticity

The subject of this paper is the corner tensor C that appears in the free term in a boundary integral equation formulation for three-dimensional linear elasticity. A general corner, locally composed of piecewise flat and curved surfaces, is considered in explicit fashion. The solid angle at the corner appears in the expression for C. A new formula for the solid angle at a general corner, in terms of line integrals, is derived in this paper. Finally, examples for cones are presented and discussed.     

The biharmonic problem and progress in the development of analytical methods for the solution of boundary-value problems

A comparative analysis of harmonic and biharmonic boundary-value problems for 2D problems on a rectangle is given. Some common features of two types of problems are emphasized and special attention is given to the basic distinction between them. This distinction was thoroughly studied for the first time by L. N. G. Filon with respect to some plane problems in the theory of elasticity. The analysis permits to introduce an important aspect of the general solution of boundary-value problems. The procedure for solving the biharmonic problem involves both the method of homogenous solutions and the method of superposition. For some cases involving self-equilibrated loadings on one pair of sides of the rectangle, the complete solution, including calculation of the quantitative characteristics of the displacements and stresses, is given. The efficiency of the numerical implementation of the general solutions is shown. The analysis of the quantitative data allows to elucidate some main points of the Saint-Venant principle.     

Path-independent H integrals for three-dimensional fracture mechanics

In fracture mechanics, a number of real applications have intrinsically three-dimensional crack geometries, thereby requiring a means of extracting stress intensity factors under such circumstances. Two approaches to this end are examined here: one, a three-dimensional J-integral; the other, three-dimensional H integrals for each mode. The first integral is well accepted by the fracture mechanics community; the second integrals are newly developed herein. The two are compared on three-dimensional test problems with closed-form solutions that are constructed for this purpose. Analysis is via quarter-point elements on two successively refined grids for each test problem. The results demonstrate that both types of path-independent integral can furnish estimates of stress intensity factors which converge to good levels of accuracy in return for reasonable levels of computational effort. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dual reciprocity boundary element method using compactly supported radial basis functions for 3D linear elasticity with body forces

A new computational model by integrating the boundary element method and the compactly supported radial basis functions (CSRBF) is developed for three-dimensional (3D) linear elasticity with the presence of body forces. The corresponding displacement and stress particular solution kernels across the supported radius in the CSRBF are obtained for inhomogeneous term interpolation. Subsequently, the classical dual reciprocity boundary element method, in which the domain integrals due to the presence of body forces are transferred into equivalent boundary integrals, is formulated by introducing locally supported displacement and stress particular solution kernels for solving the inhomogeneous 3D linear elastic system. Finally, several examples are presented to demonstrate the accuracy and efficiency of the present method.     

A numerical study of the use of path independent integrals in elasto-dynamic crack propagation

The use of the path independent J' integral for dynamic crack propagation, which has the physical meaning of energy release rate is numerically studied by the finite element method. Other path independent integrals are also investigated along with the J' integral. Numerical results show that the combined use of the J' integral and the finite element method is a useful tool to obtain the fracture parameters such as the stress intensity factors and the energy release rates. The use of the several other types of path-independent integrals, despite their lack of a direct interpretation as energy release rates, is also demonstrated. This is so, because the alternate path-independent integrals have been explicitly expressed in terms of the time-dependent K-factors, or the energy release rate, in the present work.     

Representation of stress intensity factors by path integrals of the first stress invariant or anti-plane displacement for general two-dimensional static problems

Determining stress intensity factors (SIFs) is a difficult task either analytically or experimentally. The difficulty arises from the fact that there is no simple and accurate expression for the SIFs under general circumstances. As a result, the determination of the SIFs is usually a complex process. For finding a suitable expression for the SIFs, the first stress invariant and anti-plane displacement are analyzed, and Green's theorem is used. It is found that the stress intensity factors can be represented by path integrals involving only the first stress invariant or anti-plane displacement for general two-dimensional static problems. K _I and K _II are represented by path integrals of the first stress invariant and its partial derivative. K _III is represented by a path integral of the anti-plane displacement as well as its partial derivative. The integrals are path-independent and valid for an arbitrarily shaped elastic medium with stationary cracks of arbitrary shape. They are also valid for a body containing isolated inhomogeneities such as holes and inclusions. If a crack is straight near its tip, and if the straight portion of the crack can be treated as a cut along the radius of a simply connected circular disk, there exists another kind of integrals representation that does not include the partial derivative terms in the representation for K _I. The representation by these integrals provides a new approach to determine the SIFs experimentally, which is simpler and more accurate. This is because the integrals are exact expressions for the SIFs and involve only the first stress invariant or anti-plane displacement.     

Crack tip and associated domain integrals from momentum and energy balance

A unified derivation of crack tip flux integrals and their associated domain representations is laid out in this paper. Using a general balance statement as the starting point, crack tip integrals and complementary integrals which are valid for general material response and arbitrary crack tip motion are obtained. Our derivation emphasizes the viewpoint that crack tip integrals are direct consequences of momentum balance. Invoking appropriate restrictions on material response and crack tip motion leads directly to integrals which are in use in crack analysis. Additional crack tip integrals which are direct consequences of total energy and momentum balance are obtained in a similar manner. Some results on dual (or complementary) integrals are discussed. The study provides a framework for the derivation of crack tip integrals and allows them to be viewed from a common perspective. In fact, it will be easy to recognize that every crack tip integral under discussion can be obtained immediately from the general result by appropriately identifying the terms in the general flux tensor. The evaluation of crack tip contour integrals in numerical studies is a potential source of inaccuracy. With the help of weighting functions these integrals are recast into finite domain integrals. The latter integrals are naturally compatible with the finite element method and can be shown to be ideally suited for numerical studies of cracked bodies and the accurate calculation of pointwise energy release rates along a curvilinear three-dimensional crack front. The value of the domain integral does not depend on domain size and shape — this property provides an independent check on the consistency and quality of the numerical calculation. The success of the J-based fracture mechanics approach has led to much literature on pathindependent integrals. It will be shown that various so-called path-independent integrals (including path and area integrals) are but alternate forms of the general result referred to above and do not provide any additional information which is not already contained in the general result. Recent attempts to apply these ‘newer’ integrals to crack growth problems are discussed.     

Dynamic stress intensity factors due to scattering of harmonic waves by a crack in an infinite medium

An analytical method for calculating dynamic stress intensity factors in the mixed mode (combination of opening and sliding modes) using complex functions theory is presented. The crack is in infinite medium and subjected to the plane harmonic waves. The basis of the method is grounded on solving the two‐dimensional wave equations in the frequency domain and complex plane using mapping technique. In this domain, solution of the resulting partial differential equations is found in the series of the Hankel functions with unknown coefficients. Applying the boundary conditions of the crack, these coefficients are calculated. After solving the wave equations, the stress and displacement fields, also the J‐integrals are obtained. Finally using the J‐integrals, dynamic stress intensity factors are calculated. Numerical results including the values of dynamic stress intensity factors for a crack in an infinite medium subjected to the dilatation and shear harmonic waves are presented.     

Biharmonic analysis of rectilinear plates by the boundary element method

An improved boundary element formulation (BEM) for two-dimensional non-homogeneous biharmonic analysis of rectilinear plates is presented. A boundary element formulation is developed from a coupled set of Poisson-type boundary integral equations derived from the governing non-homogeneous biharmonic equation. Emphasis is given to the development of exact expressions for the piecewise rectilinear boundary integration of the fundamental solution and its derivatives over several types of isoparametric elements. Incorporation of the explicit form of the integrations into the boundary element formulation improves the computational accuracy of the solution by substantially eliminating the error introduced by numerical quadrature, particularly those errors encountered near singularities. In addition, the single iterative nature of the exact calculations reduces the time necessary to compile the boundary system matrices and also provides a more rapid evaluation of internal point values than do formulations using regular numerical quadrature techniques. The evaluation of the domain integrations associated with biharmonic forms of the non-homogeneous terms of the governing equation are transformed to an equivalent set of boundary integrals. Transformations of this type are introduced to avoid the difficulties of domain integration. The resulting set of boundary integrals describing the domain contribution is generally evaluated numerically; however, some exact expressions for several commonly encountered non-homogeneous terms are used. Several numerical solutions of the deflection of rectilinear plates using the boundary element method (BEM) are presented and compared to existing numerical or exact solutions.     

The local boundary integral equation (LBIE) and it's meshless implementation for linear elasticity

 The meshless method based on the local boundary integral equation (LBIE) is a promising method for solving boundary value problems, using an local unsymmetric weak form and shape functions from the moving least squares approximation. In the present paper, the meshless method based on the LBIE for solving problems in linear elasticity is developed and numerically implemented. The present method is a truly meshless method, as it does not need a “finite element mesh”, either for purposes of interpolation of the solution variables, or for the integration of the energy. All integrals in the formulation can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. The essential boundary conditions in the present formulation can be easily imposed even when the non-interpolative moving least squares approximation is used. Several numerical examples are presented to illustrate the implementation and performance of the present method. The numerical examples show that high rates of convergence with mesh refinement for the displacement and energy norms are achievable. No post-processing procedure is required to compute the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.