Complex Path-independent Integrals Research Articles

Computer-generated formulae for the location of straight cracks

Complex path-independent integrals have been widely used for the location of cracks, holes and inclusions in plane isotropic elasticity problems. Here a much simpler and more direct approach, applicable to particular inverse straight crack problems in infinite elastic media, is suggested. This approach is based on the use of experimental data for the derivative of the first complex potential of Kolosov-Muskhelishvili from only a few points of the elastic specimen (not along a whole closed contour) and, next, on the construction of appropriate polynomial equations for the determination of the position of the crack tips, the stress-intensity factors, etc. These equations can be derived by using computer algebra methods (e.g. Gröbner bases and characteristic sets) and the related commercially available software (e.g. Maple V). The present elementary approach is illustrated in two simple cases of straight cracks and the related formulae are displayed. Further possibilities are also suggested in brief.

Locating a crack of arbitrary but known shape by the method of path-independent integrals

The solution of a crack problem of an arbitrary, but known, shape inside an infinite plane isotropic elastic medium can be achieved in general by the method of complex singular integral equations and their numerical solution by using the Gauss- or the Lobatto-Chebyshev methods. In a few special cases, like straight or circular-arc-shaped cracks, this solution is available in closed form. In this paper we will not contribute to the above methods of solution of crack problems, but we will propose a method for the determination of the exact position of such a crack inside a closed contour in the elastic medium by gathering and using information along this contour only (by experimental techniques) and applying the method of complex path-independent integrals for the location of the crack. This paper constitutes a nontrivial generalization of relevant previous results by the author and it is of quite general applicability in fracture mechanics for nondestructive testing. Numerical results for the particular case of a straight crack are displayed for the illustration of the efficiency of the method. The generalization of the present results to the determination of additional geometric and loading parameters of the crack is also suggested very briefly and related numerical results, concerning the length of the crack and the pressure distribution on it, in the aforementioned numerical application are also presented.

Application of complex path-independent integrals to problems of bending of thin elastic plates

Complex path-independent integrals have been already widely applied to problems of plane and antiplane elasticity for the determination of a variety of quantities of interest including stress intensity factors, loading intensities and the positions of geometrical characteristic lengths of singularities in the elastic field (like cracks, holes and inclusions). In this paper, we show that the same results apply also to the case of problems of thin isotropic elastic plates under bending where the complex-variable formulation is also valid. We make reference to the experimental methods which are appropriate for these integrals in an engineering environment and, finally, we apply this approach to the location of a circular hole in the problem of bending of a thin plate. Numerical results are also presented.

The location of discontinuity intervals of sectionally analytic functions: Application to the interface crack problem

Complex path-independent integrals have been extensively used in applied mathematics for the location of zeros and poles of analytic and meromorphic functions, but also in applied mechanics for the computation of stress intensity factors and the location of simple cracks and related discontinuity intervals of sectionally analytic functions, as well as for the computation of stress intensity factors. Here we generalize this approach to the problem of location of discontinuity intervals corresponding to interface cracks in plane elasticity (or to ordinary cracks, but under mixed loading conditions). The general case of this problem is also considered. The computer algebra system DERIVE was found appropriate for use in the formal (symbolic) computations of the present paper.

Application of the conformal mapping and the complex path-independent integrals to the location of elliptical holes and inclusions in plane elasticity problems

The classical method of conformal mapping, which was successfully used for the solution of problems of elliptical holes and inclusions (either rigid or elastic), is combined with the more recent method of complex path-independent integrals for the location of an elliptical hole or inclusion inside an infinite plane isotropic elastic medium by using information about one or more complex potentials of plane isotropic elasticity along a closed contour surrounding the elliptical hole or inclusion. The method constitutes one more generalization of the relevant elementary methods for the location of poles of meromorphic functions in the complex plane and is simply based on the Cauchy residue theorem of complex analysis. The complex path-independent integrals used here are evaluated in the physical plane of the specimen. A numerical application of the method is made, extensive numerical results are presented and future generalizations of the method are reported in brief.

Mixed mode crack analysis using complex path-independent integrals

In this paper, complex path-independent integrals are rederived and their expression in terms of stress and displacements is given. The integrals are evaluated along elliptical paths using Gauss integration rule. The stress and displacement field is obtained by the finite element method where an automatically generated grid is used. The grid around the crack is composed by confocal elipses and hyperbolas, so the data for the path independent integrals are easily obtained. The proposed method presents some very essential advantages against the previous tentatives. The numerical results comfirm the accuracy, the reliability and the computational efficiency of the method.

On the practical application of the method of complex path-independent integrals to problems of fracture mechanics

On the practical application of the method of complex path-independent integrals to problems of fracture mechanics

On the location of straight discontinuity intervals of arbitrary sectionally analytic functions by using complex path-independent integrals

The classical method of locating zeros and poles of analytic and meromorphic functions in the complex plane via the evaluation of complex path-independent integrals on closed contours surrounding the sought zeros and poles is generalized to become applicable (for the first time) to the location of straight discontinuity intervals of completely general sectionally analytic functions possessing such an interval. Such is the case e.g. in crack problems in plane elasticity or in airfoils in plane fluid dynamics. This generalization makes use of appropriate or arbitrary quadrature rules and, in this way, it leads to a completely numerical method. The method is illustrated in the case of a simple sectionally analytic function, where the classical Chebyshev and Legendre polynomials are used and numerical results are presented. Possible generalizations of the present results are also reported in brief and are strongly expected in future.

Application of complex path-independent integrals to the solution of the problem of a straight crack in a finite plane isotropic elastic medium

The classical problem of a straight crack in a finite, plane, isotropic, elastic medium of arbitrary shape is reconsidered by the well-known method of Muskhelishvili for such a crack (but in an infinite medium). Both the crack and the boundary of the medium are assumed loaded in an arbitrary way. It is shown that this problem can be completely solved if the numerical values of the first complex potential Φ(z) of Muskhelishvili are known along a closed contour surrounding the crack, probably along the boundary of the medium. To this end, complex path-independent integrals associated with Φ(z) and Chebyshev polynomials have been used. Numerical results for the stress intensity factors are displayed in an application. Generalizations of the method are also proposed and the second fundamental crack problem, the problem of a crack in an anisotropic medium and the problem of an interface crack between two isotropic media are considered in some detail.

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

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 Φ(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.

Stress-Intensity Factors and Complex Path-Independent Integrals

Path-independent integrals about crack tips may be used to estimate stress-intensity factors at crack tips in plane and antiplane elasticity problems. In this paper a new class of such integrals is established by using complex stress functions and the trivial application of the Cauchy theorem of complex analysis. Both the simple Westergaard complex potentials of plane and antiplane elasticity and the more general Muskhelishvili complex potentials will be used for the construction of appropriate path-independent integrals. Two applications of these integrals to the theoretical determination of stress-intensity factors at crack tips are presented. An optical method for the experimental determination of stress-intensity factors at crack tips, based on the use of appropriate complex path-independent integrals, is also proposed.

On the evaluation of stress intensity factors in interface crack problems by using complex path-independent integrals

On the evaluation of stress intensity factors in interface crack problems by using complex path-independent integrals