Three-dimensional Crack Problems Research Articles

DC Potential-Drop Technique for the Evaluation of 3-D Back-Wall Cracks in Steel Structures

A direct-current potential-drop NDE technique has been developed for the evaluation of a 3-D back-wall crack in thick-walled structures. The measuring system contains two pairs of probes – one for current supply and the other for the measurement of potential-drop, where the currentsupply and measuring probes are in close proximity to each other. The optimum arrangement of the four-point-probes system is determined from the finite-element simulation of the three-dimensional electric crack problem. The predicted potential-drop across the crack has been verified experimentally using the newly developed measuring sensor. Measured results of semi-elliptical cracks in stainless steel plates establish the practicality as well as reliability of the present potentialdrop method of testing for the evaluation of 3-D back-wall cracks in thick-walled structures.

A virtual crack closure-integral method (VCCM) for three-dimensional crack problems using linear tetrahedral finite elements

A virtual crack closure-integral method (VCCM) for three-dimensional crack problems using linear tetrahedral finite elements

A Simple Galerkin Boundary Element Method for Three-Dimensional Crack Problems in Functionally Graded Materials

This paper presents a Galerkin boundary element method for solving crack problems governed by potential theory in nonhomogeneous media. In the simple boundary element method, the nonhomogeneous problem is reduced to a homogeneous problem using variable transformation. Cracks in heat conduction problem in functionally graded materials are investigated. The thermal conductivity varies parabolically in one or more coordinates. A three dimensional boundary element implementation using the Galerkin approach is presented. A numerical example demonstrates the eáciency of the method. The result of the test example is in agreement with ßnite element simulation results.

Weight functions for T-stress for semi-elliptical surface cracks in finite-thickness plates

This paper presents the application of the weight function method for the calculation of elastic T-stress for semi-elliptical surface cracks. First, the weight function method for the calculation of T-stress previously developed for two-dimensional crack problems was extended for the T-stress calculation for three-dimensional crack problems. Then, the T-stress weight functions for the deepest point (corresponding to the parametric angle ϕ = 90°) and for any general point (5° ≤ ϕ < 90°) along the crack front of semi-elliptical surface cracks in finite-thickness plates for wide ranges of crack aspect ratios a/c and relative depths a/t were derived. The resulting weight functions were validated using available finite element results for non-linear stress fields and remote tension and bending cases, and very good agreement was achieved. The weight functions are suitable for the calculation of the T-stress under complex loading conditions for any general point (5° ≤ ϕ ≤ 90°) of surface cracks with wide ranges of aspect ratios, 0 ≤ a/c ≤ 1, and relative depths, 0 ≤ a/t ≤ 0.8.

Three-dimensional BEM for piezoelectric fracture analysis

A boundary element approach with quadratic isoparametric elements, quarter-point elements and singular quarter-point elements for three-dimensional crack problems in piezoelectric solids under mechanical and electrical loading conditions, is presented in this paper for the first time. The procedure is based on Deeg's fundamental solution for anisotropic piezoelectric materials, and the classical extended displacement boundary integral equation. Stress and electric displacement intensity factors are directly evaluated as system unknowns, and also as functions of the computed nodal displacements and electric potentials at crack faces. Special attention is paid to the fundamental solution evaluation. Several three-dimensional crack problems in transversely isotropic bodies under mechanical and electrical loading conditions are analysed. Numerical solutions computed for prismatic cracked 3D plate problems with a plane strain behaviour are in very good agreement with their corresponding 2D BE solutions. Results for a penny shape crack in a piezoelectric cylinder are presented for the first time. The proposed approach is shown to be a simple, robust and useful tool for stress and electric displacement intensity factors evaluation in piezoelectric media.

Boundary element analysis of three-dimensional crack problems in two joined transversely isotropic solids

The present paper presents a boundary element analysis of penny-shaped crack problems in two joined transversely isotropic solids. The boundary element analysis is carried out by incorporating the fundamental singular solution for a concentrated point load in a transversely isotropic bi-material solid of infinite space into the conventional displacement boundary integral equations. The conventional multi-region method is used to analyze the crack problems. The traction-singular elements are employed to capture the singularity around the crack front. The values of the stress intensity factors are obtained by using crack opening displacements. The numerical scheme results are verified with the closed-form solutions available in the literature for a penny-shaped crack parallel to the plane of the isotropy of a homogeneous and transversely isotropic solid of infinite extent. The new problem of a penny-shaped crack perpendicular to the interface of a transversely isotropic bi-material solid is then examined in detail. The crack surfaces are subject to the three normal tractions and the uniform shear traction. The associated stress intensity factor values are obtained and analyzed. The present results can be used for the prediction of the stability of composite structures and the hydraulic fracturing in deep rock strata and reservoir engineering.

Analysis of an elliptical crack parallel to graded interfacial zone of bonded bi-materials

This paper analyzes an elliptical crack parallel to the functionally graded interfacial zone between two fully bonded solids. The functionally graded interfacial zone is treated as a nonhomogeneous solid layer with its elastic modulus varying in the thickness direction. A generalized Kelvin solution based boundary element method is employed for the calculation of the stress intensity factors associated with the three-dimensional crack problem. The elliptical crack surface is subject to either uniform normal traction or uniform shear traction. The stress intensity factors are examined by taking into account the effects of the nonhomogeneity parameter and thickness of the functionally graded interfacial zone, as well as the crack distance to the zone. The numerical results are in very good agreement with the existing solutions under degenerated conditions. These stress intensity factor values can be further used to predict the crack growth in the functionally graded materials.

Elliptical crack normal to functionally graded interface of bonded solids

This paper presents an analysis of an elliptical crack that is perpendicular to a functionally graded interfacial zone between two fully bonded solids. The functionally graded interfacial zone is treated as a non-homogeneous solid layer with its elastic modulus varying in the thickness direction. A generalized Kelvin solution based boundary element method is employed for the calculation of the stress intensity factors associated with the three-dimensional crack problem. The elliptical crack surface is subject to either uniform normal traction or uniform shear traction. The stress intensity factors are examined by taking into account the effects of the non-homogeneity parameter and thickness of the functionally graded interfacial zone, as well as the crack distance to the zone. The SIF values are further incorporated into the S-criterion for prediction of crack growth. The paper presents the most possible direction and location of the elliptical crack growth under an inclined tensile (or compressive) load. The paper further presents results of the critical external loads that would cause the elliptical crack to grow at the most possible location and along the most possible direction. The paper also examines the effects of external load direction and material and geometrical parameters on the critical loads.

Potential theory method for 3D crack and contact problems of multi-field coupled media: a survey.

This paper presents an overview of the recent progress of potential theory method in the analysis of mixed boundary value problems mainly stemming from three-dimensional crack or contact problems of multi-field coupled media. This method was used to derive a series of exact three dimensional solutions which should be of great theoretical significance because most of them usually cannot be derived by other methods such as the transform method and the trial-and-error method. Further, many solutions are obtained in terms of elementary functions that enable us to treat more complicated problems easily. It is pointed out here that the method is usually only applicable to media characterizing transverse isotropy, from which, however, the results for the isotropic case can be readily obtained.

Application of Hypersingular Integral Equation Method to a Three-Dimensional Crack in Piezoelectric Materials

Using the Green's functions, the general solutions of a three-dimensional crack problem in piezoelectric materials under mechanical and electrical loads is derived by boundary element method. Then this crack problem is reduced to solve a set of hypersingular integral equations coupled with boundary integral equations. The unknown functions are the discontinuities of the elastic displacements and electrical potential of the crack surface. The singularity of the unknowns at the crack front is analyzed by the main-part analysis method of two-dimensional hypersingular integral equations, and the exact analytical solution of the singular stresses and electrical displacements near the crack front in a transversely isotropic piezoelectric solid is given.

比例法による三次元き裂の応力拡大係数の解析(G.S. 変形・疲労特性・破壊力学,O.S.1:先進材料の力学解析と強度・機能性評価)

比例法による三次元き裂の応力拡大係数の解析(G.S. 変形・疲労特性・破壊力学,O.S.1:先進材料の力学解析と強度・機能性評価)

Dynamic BE analysis of 3-D cracks in transversely isotropic solids

A boundary element approach for the analysis of three-dimensional crack problems in transversely isotropic solids under time harmonic dynamic loading conditions is presented in this paper. For the first time a single region approach based on the tractions integral representation for three-dimensional anisotropic solids under dynamic loads is presented. Hypersingular and strongly singular kernels appearing in the formulation are regularized prior to any discretization. Two different crack geometries under time harmonic impinging waves are studied. Stress intensity factors are computed from the crack opening displacements at quarter-point quadratic elements. To the authors’ knowledge, no previous results for elliptical cracks in transversely isotropic solids under dynamic loading conditions exist in the literature. The results obtained in this paper for situations where previous solutions exist, show a good agreement with those other results. The number of elements required for the analysis is rather small.

Time-domain BEM for three-dimensional fracture mechanics

The present paper deals with time-domain analysis of three-dimensional transient dynamic crack problems. The time-domain formulation of the boundary element method for 3-D elastodynamic problems is used. Quarter-point and singular quarter-point elements represent displacements and tractions, respectively, near the crack front. Special attention is paid to integration and algorithms to preserve stability. Cracks in finite and unbounded regions under single and mixed mode dynamic loading conditions are studied. To the authors’ knowledge, no previous BE approach for 3-D elastodynamic crack problems based on the time-domain displacement representation exists.

Principles of fracture mechanics

Most chapters include an Introduction, Summary, References and Exercises. 1. Introduction to Fracture Mechanics. Historical Overview of Brittle Fracture. Elementary Brittle-Fracture Theories. Crack Extension Behavior. 2. Elements of Solid Mechanics. Concepts of Stress and Strain. Equations of Elasticity in Cartesian Coordinates. Equations of Elasticity in Polar Coordinates. Solution of the Biharmonic Equation. The Problem of the Elliptical Hole. 3. Elasticity of Singular Stress Fields. Overview. The Williams Problems. The Generalized Westergaard Approach. The Central Crack Problem. Single-Ended Crack Problems. The Effect of Finite Boundaries. Determining the Geometric Stress Intensity Factor. The Three-Dimensional Crack Problem. 4. Numerical Methods for K Determination. Boundary Collocation. The Finite Element Method. 5. Experimental Methods for K Determination. Overview. Classical Photoelesatic Methods. The Method of Caustics. Strain Gages. Multi-Parameter Full-Field Methods: Local Collocation. Interference Patterns. Moire Patterns. Photoelasticity. 6. A Stress Field Theory of Fracture. The Critical Stress-State Criterion. Crack-Tip Plasticity. The Effect of Variables on Fracture Toughness. R-Curves. 7. The Energy of Fracture. Griffith's Theory of Brittle Fracture. A Unified Theory of Fracture. Compliance. 8. Fracture Toughness Testing. Fracture Toughness Standards. Nonstandard Fracture-Toughness Tests. 9. Fatigue. Stages of Fatigue Crack Growth. Mathematical Analysis of Stage II Crack Growth. The Effects of Residual Stress on Crack Growth Rates. Life Prediction Computer Programs. Measuring Fatigue Properties: ASTM. 10. Designing against Fracture. Fracture Mechanics in Conventional Design. The Role of NDE in Design. U.S. Air Force Damage-tolerant Design Methodology. Designing by Hindsight: Case Studies. 11. Elastoplastic Fracture. Nonlinear Elastic Behavior. Characterizing Elastoplastic Behavior. Comments on the J-Integral in Elastoplastic Fracture Mechanics. Appendix A: Comprehensive Exercises. General Comments. Appendix B: Complex Variable Method in Elasticity. Complex Numbers. Complex Functions. Appendix C: An Abbreviated Compendium of Westergaard Stress Functions. Appendix D: Fracture Properties of Engineering Materials. Appendix E: NASGRO 3.0 Material Constants for Selected Materials. Index.

Numerical Weight Function Method for Structural Analysis

The weight function method was originally derived for two- and three-dimensional crack problems to calculate stress intensity factors for arbitrary loading conditions. In the present paper the weight function method has been generalized to the response analysis of structures basing upon the Maxwell-Betti's reciprocal theorem, where the weight function method is formulated and applied to two-dimensional elasticity and plate bending problems with illustrative examples. The method is very useful for the analysis of structures subjected to a variety of loading conditions, because once the weight function is calculated the responses such as displacements and stresses may simply be calculated by the integration of the inner product of the universal weight function and a load vector.

A Novel Method For The Solution Of The Three-dimensional Dynamic Crack Problems

A Novel Method For The Solution Of The Three-dimensional Dynamic Crack Problems

Generalized Kelvin solution based boundary element method for crack problems in multilayered solids

This paper presents a new boundary element method (BEM) for linear elastic fracture mechanics in three-dimensional multilayered solids. The BEM is based on a generalized Kelvin solution. The generalized Kelvin solution is the fundamental singular solution for a multilayered elastic solid subject to point concentrated body-forces. For solving three-dimensional elastic crack problems in a finite region, a multi-region method is also employed in the present BEM. For crack problems in an infinite space, a large finite body is used to approximate the infinite body. In addition, eight-node traction-singular boundary elements are used in representing the displacements and tractions in the vicinity of a crack front. The incorporation of the generalized Kelvin solution into the boundary integral formulation has the advantages in elimination of the element discretization at the interfaces of different elastic layers. Three numerical examples are presented to illustrate the proposed method for the calculation of stress intensity factors for cracks in layered solids. The results obtained using the proposed method are well compared with the existing results available in the relevant literature.

Finite element analyses of three-dimensional crack problems in piezoelectric structures

In this paper, electromechanical fracture mechanics and finite element techniques for crack analyses are extended to three-dimensional crack configurations. Penny-shaped cracks and elliptical cracks are analyzed, subjected to combined mechanical tension and electric fields. For the penny-shaped crack, exact solutions originating from different resources are compared with numerical results. Some errors in the literature concerning the analytical solution for the elliptical crack are corrected. Numerical results of the stress-intensity factors and energy release rates for these crack configurations are presented.

Stress intensity factors for three‐dimensional surface cracks using enriched finite elements

AbstractThe analysis of three‐dimensional crack problems using enriched crack tip elements is examined in this paper. It is demonstrated that the enriched finite element approach is a very effective technique for obtaining stress intensity factors for general three‐dimensional crack problems. The influence of compatibility, integration, element shape function order, and mesh refinement on solution convergence is investigated to ascertain the accuracy of the numerical results. It is shown that integration order has the greatest impact on solution accuracy. Sample results are presented for semi‐circular surface cracks and compared with previously obtained solutions available in the literature. Good agreement is obtained between the different numerical solutions, except in the small zone near the free surface where previously published results have often neglected the change in the stress singularity at the free surface. The enriched crack tip element appears to be particularly effective in this region, since boundary conditions can be easily imposed on the stress intensity factors to accurately represent the correct free surface condition. Copyright © 2002 John Wiley & Sons, Ltd.

THE INFLUENCE OF CRACK-FACE FLUID PRESSURE ON THE FATIGUE CRACK PROPAGATION DUE TO ROLLING CONTACT WITH FRICTIONAL HEAT

A three-dimensional surface planar crack problem in a half-space is considered under rolling/sliding contact with frictional heat and hydraulic pressure by the entrapped fluid within the crack. Rolling contact is simulated as a line load with both normal and shear components, moving with constant velocity over the surface of the half-space. The body force method for three-dimensional fracture mechanics is utilized to determine the three modes of stress intensity factors along the crack contour. To account for mixed-mode propagation, the modified Paris power law is used. Numerical results for the stress intensity factors and the simulations of fatigue crack propagation are given for 30-degree inclined planar surface cracks of semicircular shape. The effects of the frictional coefficient, sliding/rolling ratio, and the crack-face fluid pressure on the crack propagation life are considered for a high carbon-chromium bearing steel.