Solution Of Crack Problems Research Articles

Boundary integral equation methods for the solution of crack problems

Boundary integral equation methods for the solution of crack problems

Symbolic computations: A powerful method for the solution of crack problems in fracture mechanics

Computer algebra (or symbolic computations or manipulations) is a powerful tool available for more than twenty years to mathematicians, engineers and scientists, through appropriate computer software packages, both with mainframes and minicomputers, as well as with personal microcomputers. Five such computer programs were reviewed in [1] last February. We can add that today the best of these programs, MACSYMA, is also available for microcomputers based on INTEL's 80386 chip. Moreover, other competitive packages, DERIVE, REDUCE and MATHEMATICA, are advertised (as is the case with MACSYMA as well) in ASME's Mechanical Engineering and/or SIAM's SIAM News. A book on the theory of computer algebra, by Akritas, appeared in 1989 [2], and the practical application of computer algebra can be found in Wolfram's book Mathematica [3], which was published in 1988 and is commercially available independent of the associated software (that is, the computer program MATHEMATICA).

Solute distribution in crack tips under mixed mode I/II conditions

Concentration solutions of crack problems under steady state conditions are expressed in terms of the Westergaard stress function. The problem of a central crack in an infinite plate subjected to a biaxial stress field at any angle of inclination with respect to the crack axis is considered in detail. The concentration distribution in the vicinity of the crack tip is obtained and is expressed in terms of the opening-mode and sliding-mode stress intensity factors. Constant terms, usually omitted, are incorporated into the concentration solution. The crack growth criterion based on the maximum concentration of diffusing species in front of the crack tip is reformulated by incorporating the constant terms of the concentration solution. It is shown that the omission of the constant terms may result in a significant error in the prediction of the critical quantities for crack growth.

The normal displacement of a crack surface with arbitrary shape

This research has basically solved the problem of determining the normal displacement field of crack surface with arbitrary shape in an infinite elastic body under uniform pressure or more general loads, exactly or approximately. All the solutions for crack problems are originated from a general solution of a two dimensional integro-differential equation which has been found by taking advantage of the constitutive function and the specific expansions. The results show that for the crack whose boundary contour π(θ) has the forms π( θ) = Θ 1(1, cos 2 θ,…, cos 2 kθ) π(θ) = Θ 2(1,¦cosθ¦,…,¦coskθ¦) π(θ) = Θ 3(1,¦sinθ¦,…,|sinkθ¦) and the forms combined by the three cases, exact or approximate solutions of polymonial forms attached √1-( r/ ρ) 2 or other closed forms for the normal displacement of the crack surface can be obtained easily. As on part of the research work, this paper has established a general theory and has given the complete exact solution of an elliptic crack subject to polymonial loads of even and odd powers.

Three-dimensional finite element analysis of circumferentially-cracked notched and un-notched cylindrical components

A comprehensive three-dimensional computer-aided finite element study is made in an attempt to calculate the stress intensity factor ( K I ) for a variety of V-notched and un-notched circumferentially-cracked geometries made of mild steel subjected to remote uniform bending moment, M. Throughout the study, the solid modelling facility of SDRC† (GEOMOD) was interactively utilised to create the different cracked geometries. Prior to the analysis, these geometries were transferred to the pre-processor (SUPERTAB) for model creation, mesh definition and model checking. The appropriate remote bending moment and imposed restraints were also applied interactively. In this study, the stress-intensity factor calculations for the different configurations were based upon the resulting displacement field just behind the crack tip. The effect of crack depth, notch-tip radius and included angle of the V-notched specimens upon the stress-intensity factor K I were examined. The work also included the effect of circumferentially cracked un-notched specimens upon K I and comparisons with the earlier solutions of Benthem and Koiter ( Methods of Analysis and Solutions of Crack Problems, pp. 131–178, 1973) and Harris ( J. Basic Engng. 89, 49–54, 1967) were made and the results show good agreement. The finite element study was also extended to evaluate the effect of complete and partial contacts of the cracked surfaces in the compressive side of the specimen upon the resulting stress intensity factor, K I . The current results agree with the general trends of the experimental findings observed earlier by Sawaki, Aoyama and Kawaski ( Int. J. Fracture 26, 43–54, 1984).

A hybrid method for the solution of crack problems based on the cauchy integral formula and the Riemann-Hilbert problem

Using the values of a complex potential on a closed contour surrounding a crack (derived either by an experimental or by a numerical method), this potential (as well as the corresponding stress intensity factors) is determined inside the contour by using the classical theory of the Riemann-Hilbert boundary value problem together with the well-known Cauchy integral formula. The case of an array of periodic collinear cracks is considered in detail and the corresponding numerical results (based on the trapezoidal quadrature role) are displayed.

Uncoupled superposition of finite element and integral equation methods for the solution of crack problems

A direct superposition procedure is used in order to solve the problem of a generally cracked body: The regular field is generated by the finite element method whereas the singular field is by the singular integral equation method. This process, combined in a one-step technique, was applied to three typical plane problems of cracked sheets and the results are compared with those from other numerical methods.

An improved boundary element Galerkin method for three-dimensional crack problems

In this paper we analyze the solution of crack problems in three-dimensional linear elasticity by equivalent integral equations of the first kind on the crack surface. Besides existence and uniqueness we give sharp regularity results for the solution of these pseudodifferential equations. Two versions of Eskin's Wiener-Hopf technique are presented: the first one requires the factorization of matrix-valued symbols which is avoided in the second case. Based on these regularity results we show how to improve the boundary element Galerkin method for our integral equations by using special singular trial functions. We apply the approximation property and inverse assumption of these elements together with duality arguments and derive quasi-optimal asymptotic error estimates in a scale of Sobolev spaces.

Exact expression for a two‐dimensional finite‐part integral appearing during the numerical solution of crack problems in three‐dimensional elasticity

AbstractAn exact expression is derived for the finite‐part integral over a triangular domain S, where r denotes the distance of the points of the triangle from one of its vertices and f is a linear function of the Cartesian co‐ordinates. The more general case where r denotes the distance of the points of the triangle from another constant point inside (or outside) the triangle is also studied and numerical results are presented. These integrals appear during the numerical solution of plane crack problems (under constant or variable pressure distribution) in three‐dimensional elasticity. The availability of exact expressions for these integrals will increase the accuracy of the numerical results and, simultaneously, it will permit us not to use cubature formulae over triangular domains for the numerical solution of the previous class of problems.

A modified algorithm for the numerical solution of singular integral equations with index equal to 1

A modification of the direct collocation and quadrature methods for the numerical solution of singular integral equations with Cauchy type kernels and index equal to 1 is proposed. This method is based on the replacement of the original singular integral equation by another one and the solution of the latter by the direct collocation, or quadrature, method. The advantage of the proposed algorithm is just that the condition supplementing the singular integral equation is no more separately considered and one more collocation point is used instead of it. The present algorithm is useful particularly for the solution of crack problems in the theory of plane or antiplane, static or dynamic elasticity. The equivalence of the proposed modification with the direct collocation and quadrature methods for solving the original singular integral equation is also proved and numerically verified.

Further remarks on an exact solution for crack problems

Further remarks on an exact solution for crack problems

The crack problem in a specially orthotropic shell with double curvature

In this paper the crack problem of a shallow shell with two nonzero curvatures is considered. It is assumed that the crack lies in one of the principal planes of curvature and the shell is under Mode I loading condition. The material is assumed to be specially orthotropic. After giving the general formulation of the problem the asymptotic behavior of the stress state around the crack tip is examined. The analysis is based on Reissner's transverse shear theory. Thus, as in the bending of cracked plates, the asymptotic results are shown to be consistent with that obtained from the plane elasticity solution of crack problems. Rather extensive numerical results are obtained which show the effect of material orthotropy on the stress intensity factors in cylindrical and spherical shells and in shells with double curvature. Other results include the stress intensity factors in isotropic toroidal shells with positive or negative curvature ratio, the distribution of the membrane stress resultant outside the crack and the influence of the material orthotropy on the angular distribution of the stresses around the crack tip.

A natural quadrature formula for the numerical evaluation of the Macgregor-Westergaard complex potentials es crack problems

Abstract The method of singular integral equations can be used for the numerical solution of crack problems in plane and antiplane elasticity. Here we consider the problem of the subsequent numerical evaluation of the stress components in the whole cracked medium by using the MacGregor-Westergaard complex potentials. To this end we use a modified quadrature formula for Cauchy type (but not principal value) integrals and their derivatives, where the poles of the integrands are properly taken into consideration. This is achieved by using a natural interpolation-extrapolation formula for singular integral equations and, for this reason, the new term ‘natural quadrature formula’ is proposed. Two simple applications to specific crack problems, based on the Gauss- and Lobatto-Chebyshev quadrature formulas, show the efficiency of the suggested quadrature formula.

The practical evaluation of stress intensity factors at semi-infinite crack tips

The solution of crack problems in plane or antiplane elasticity can be reduced to the solution of a singular integral equation along the cracks. In this paper the Radau-Chebyshev method of numerical integration and solution of singular integral equations is modified, through a variable transformation, so as to become applicable to the numerical solution of singular integral equations along semi-infinite intervals, as happens in the case of semi-infinite cracks, and the direct determination of stress intensity factors at the crack tips. This technique presents considerable advantages over the analogous technique based on the Gauss-Hermite numerical integration rule. Finally, the method is applied to the problems of (i) a periodic array of parallel semi-infinite straight cracks in plane elasticity, (ii) a similar array of curvilinear cracks, (iii) a straight semi-infinite crack normal to a bimaterial interface in antiplane elasticity and (iv) a similar crack in plane elasticity; in all four applications appropriate geometry and loading conditions have been assumed. The convergence of the numerical results obtained for the stress intensity factors is seen to be very good.

The numerical solution of crack problems in plane elasticity in the case of loading discontinuities

The direct quadrature method of numerical solution of Cauchy type singular integral equations encountered in plane elasticity crack problems is applied to the case where the loading distribution along the crack edges presents jump discontinuities. This is made by using a well-known modification of the quadrature method which is free of undesirable errors due to the loading discontinuities. Hence, the method is ideal to treat the aforementioned class of crack problems and, particularly, crack problems where the Dugdale-Barenblatt elastic-perfectly plastic model is adopted. Finally, a numerical application of the method to the problem of a periodic array of cracks with a loading distribution presenting a jump discontinuity is made. The numerical results obtained in this problem compare favorably with the corresponding theoretical results available in this special problem.

Approximate representation of elastic-plastic small scale yielding solution for crack problems

Approximate representation of elastic-plastic small scale yielding solution for crack problems

Elastic-plastic fracture analysis and design using the finite element method

Published stress analysis solutions for crack problems are reviewed under three headings — elastic, elastic-plastic, and finite element. Emphasis is laid on results which are of practical importance for fracture analysis. The review section is concluded with a critical survey of some commonly used fracture design procedures. Results obtained by the author using elastic-plastic finite element analysis are then presented. Correlations are given for J and COD in a range of edge and centre cracked tension geometries. Comparison is made with various estimation procedures. Problems involved in defining J and COD are fully enumerated and their significance and inter-relationship investigated. It is emphasised that J is not an available energy release rate in the presence of irreversible plastic deformation, and that its possible use as a fracture criterion rests solely on its ability to characterise the stress-strain environment at the crack tip. The relationship between J and near tip stress, strain, and displacement is examined. The author concludes that JIC is unlikely to be a geometry invariant fracture criterion, but that it may have considerable use as a scaling parameter for predicting the fracture behaviour of a large specimen from a much smaller but geometrically similar specimen. The possibility of providing a complete description of the crack tip environment by using J together with the geometry dependent non-singular stresses is indicated. Elastic-plastic analyses are presented for a number of structural geometries, including cracks emanating from a hole in a plate, cracks approaching a hole in a plate, and an internal crack in a thick walled cylinder. These results indicate that, provided net section yielding does not occur, J may be estimated from the nominal elastic stress distribution even if this locally exceeds yield. Recommended fracture design procedures are presented in the light of the above findings.

Methods of analysis and solution of crack problems. : Sih, GC Lehigh Univ. Bethlehem, PA. USA Textbook. Vol. 1, in the series Mechanics of Fracture. 22OF, Refs. Noordhoff Internat. Publ. Co. 1973, 517P

Methods of analysis and solution of crack problems. : Sih, GC Lehigh Univ. Bethlehem, PA. USA Textbook. Vol. 1, in the series Mechanics of Fracture. 22OF, Refs. Noordhoff Internat. Publ. Co. 1973, 517P

Mechanics of fracture 1. Methods of analysis and solutions of crack problems. Herausgegeben von G. C. sih, Professor of Mechanics and Director of the Institute of Fracture and Solid Mechanics, Lehigh University, Bethlehem, Pennsylvania

Mechanics of fracture 1. Methods of analysis and solutions of crack problems. Herausgegeben von G. C. sih, Professor of Mechanics and Director of the Institute of Fracture and Solid Mechanics, Lehigh University, Bethlehem, Pennsylvania

On Papkovich-Fadle solutions of crack problems relating to an elastic strip

Papkovich-Fadle eigenfunctions are employed to study a class of crack problems of an elastic strip. A system of series relations is obtained which is reduced to a Fredholm integral equation of the second kind by the use of the generalized orthonormality of the eigenfunctions and the calculus of residues. Four types of crack configuration are considered. Based on the numerical solutions of the integral equations, stress intensity factor and crack energy for two types of edge cracks are reported. Some crack problems of the strip have been attempted by integral transforms which have been widely used in elasticity by Sneddon. It is not clear, however, how they may be used to tackle the problems considered here. The present approach is quite general and straightforward and, may be applied to a wide class of mixed boundary problems.