Mode I weight functions for an orthotropic double cantilever beam

Approximate weight functions are proposed and validated numerically for an orthotropic double cantilever beam (DCB) loaded in mode I. They define the stress intensity factor at the crack tip due to a pair of point forces acting on the crack surfaces and have been deduced from the corresponding isotropic result using an orthotropy rescaling technique. The weight functions allow mode I large scale bridging problems in beams and plates to be formulated as integral equations, in terms of stress intensity factors at the crack tip, without the limitations imposed on accuracy by beam theory approximations. The proposed functions are applied to investigate the influence of the orthotropy of the material on the fracture behavior of DCBs in the presence of large scale bridging.     

Calculation of stress intensity factors for functionally graded materials by using the weight functions derived by the virtual crack extension technique

The weight function method provides a powerful approach for calculating the stress intensity factors for a homogeneous cracked body subjected to mechanical loadings. In this paper, the basic equations of weight function method for mode I and mixed mode crack problems in a two-dimensional functionally graded crack system are derived based on the Betti’s reciprocal theorem. The weight functions derived by the virtual crack extension technique are further used to calculate the stress intensity factors of functionally graded materials (FGMs). The practicability and accuracy of this proposed method has been confirmed by the comparison with theoretical or numerical solutions available in the literatures. On account that the repeated extractions of the distributions of normal stress and shear stress in the uncracked component along the prospective crack line under different loadings can be avoided using the method presented in this paper, this method can be potentially used for optimal design for FGMs under multiple-load cases.     

A generalised solution for the crack surface displacement of mode I two-dimensional part-elliptical crack

A generalised approximate crack surface displacement solution for the two-dimensional part-elliptical mode I crack was developed. This solution includes the surface crack, corner crack and embedded crack, which is subjected to the arbitrary crack surface pressure. The crack surface displacement is derived from stress intensity factor solution and corresponding crack surface pressure distribution. Comparisons of the solution with accurate solutions showed that rather high accuracy has been achieved with the developed solution for various surface, embedded and corner crack problems. This solution can be used to derive three-dimensional weight functions as long as the stress intensity factor and the corresponding crack surface pressure are available for arbitrary mode I problems.     

Wide‐range and accurate mixed‐mode weight functions and stress intensity factors for cracked discs

The Wu‐Carlsson displacement‐based weight function method is extended to obtain the mode I and mode II weight functions for the edge‐ and centre‐cracked discs. Compared with Fett's direct adjustment weight functions for the edge‐cracked discs, the present weight functions are more accurate and are applicable for a wider range of crack lengths. Using the present weight functions, extensive and highly accurate mixed‐mode stress intensity factors are obtained for the cracked discs subjected to diametrically compressive forces. Assuming perfect contact and using Coulomb friction law and the present weight functions, the mode II stress intensity factors for the cracked discs with consideration of friction are obtained and widely compared with the corresponding results from finite element analyses.     

Limits of crack growth stability in the double cleavage drilled compression specimen

A theoretical investigation of the double cleavage drilled compression specimen was undertaken to define the limits of stable crack growth for a range of geometries and fracture toughnesses typically used in fracture testing of brittle materials. Two-dimensional large displacement solutions for the mode I stress intensity factor were derived using energy methods. Comparisons with finite element results indicate that these models maintain a high level of accuracy well past the onset of unstable crack growth. Crack growth stability was assessed by differentiating the semi-analytical solutions and assembling the results in the form of design curves.     

Computation of stress intensity factors (KI, KII) and T-stress for cracks reinforced by composite patching

To increase the operational life of defected structures, a repairing method using composite patches has been used to reinforce cracked components. Due to various advantages of composite materials, this method has received much attention from researchers and engineers. Considerable investigations have been performed to highlight the effect of bonded composite patches on the fracture parameters such as stress intensity factors (SIF) and J-integral. However the effect of composite patches on the T-stress, the constant stress term acting parallel to the crack, has not been investigated in the past. In this paper, the finite element method is carried out to analyze the effect of bonded composite patches for repairing cracks in pure mode I and also mixed mode I/II conditions, by computing the stress intensity factors and the T-stress, as functions of the crack length, the crack inclination angle and the type of composite material. In pure mode I condition, the finite element analysis is carried out for three different specimens: centre crack, double edge crack and single edge crack specimens. For mixed mode I/II condition the analysis is conducted on an inclined central crack of various slant angles. For both pure mode I and mixed mode I/II, the numerical results show that composite patching has considerable effect on the T-stress.     

The stress intensity factors for a short crack partially penetrating an inclusion of arbitrary shape

Some approximate solutions for predicting the stress intensity factor of a short crack penetrating an inclusion of arbitrary shape have been developed under mode I and mode II loading conditions. The derivation of the fundamental formula is based on the transformation toughening theory. The transformation strains in the inclusion are induced by the crack-tip field and remotely applied stresses, and approximately evaluated by the Eshelby equivalent inclusion theory. As validated by detailed finite element (FE) analyses, the developed solutions have good accuracy for different inclusion shape and for a wide range of modulus ratio between inclusion and matrix material.     

Experimental and numerical investigations on fracture process zone of rock–concrete interface

A crack propagation criterion for a rock–concrete interface is employed to investigate the evolution of the fracture process zone (FPZ) in rock–concrete composite beams under three‐point bending (TPB). According to the criterion, cracking initiates along the interface when the difference between the mode I stress intensity factor at the crack tip caused by external loading and the one caused by the cohesive stress acting on the fictitious crack surfaces reaches the initial fracture toughness of a rock–concrete interface. From the experimental results of the composite beams with various initial crack lengths but equal depths under TPB, the interface fracture parameters are determined. In addition, the FPZ evolution in a TPB specimen is investigated by using a digital image correlation technique. Thus, the fracture processes of the rock–concrete composite beams can be simulated by introducing the initial fracture criterion to determine the crack propagation. By comparing the load versus crack mouth opening displacement curves and FPZ evolution, the numerical and experimental results show a reasonable agreement, which verifies the numerical method developed in this study for analysing the crack propagation along the rock–concrete interface. Finally, based on the numerical results, the effect of ligament length on the FPZ evolution and the variations of the fracture model during crack propagation are discussed for the rock–concrete interface fracture under TPB. The results indicate that ligament length significantly affects the FPZ evolution at the rock–concrete interface under TPB and the stress intensity factor ratio of modes II to I is influenced by the specimen size during the propagation of the interfacial crack.     

Wide range data for crack tip parameters in two disc-type specimens under mixed mode loading

Disc-type specimens are among favorite test samples for determining mode I and mixed mode fracture toughness in brittle materials like rocks, brittle polymers, ceramics, etc. In this research, the finite element method is used to analyze two disc-type specimens: a semi-circular disc specimen containing an edge crack and subjected to three-point-bend loading (SCB specimen), and a centrally cracked circular disc subjected to diametral compressive loading, often called the Brazilian disc specimen. The crack parameters K_I, K_II and T are calculated for different mode mixities from pure mode I to pure mode II. Although the stress intensity factors K_I and K_II are presented mainly for validation of the analyses, they are also used for determining the crack angle corresponding to pure mode II for each specimen. It is shown that in general the T-stress increases for larger crack angles. While the T-stress in the Brazilian disc specimen is always negative for any combinations of mode I and mode II, the sign of T-stress in the SCB specimen depends on the mode mixity. A very good agreement is shown to exist between the calculated results for T and those very limited data presented in other papers.     

Stress intensity factor computation using the method of fundamental solutions

In this paper, we study the application of the method of fundamental solutions to the computation of stress intensity factors in linear elastic fracture mechanics. The displacements are approximated by linear combinations of the fundamental solutions of the Cauchy–Navier equations of elasticity and the leading terms for the displacement near the crack tip. The applicability of two formulations of the method is demonstrated on two mode I crack problems, where it is shown that accurate approximations for the stress intensity factors can be obtained with relatively few degrees of freedom. Parts of this work were undertaken while the first author was a Visiting Professor in the Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado 80401, U.S.A.     

ROUTINE FE DETERMINATION OF STRESS INTENSITY FACTORS USING A MÜLLER-BRESLAU INFLUENCE FUNCTION TECHNIQUE

Abstract A remarkably simple and accurate one-step application of the finite element (FE) method is suggested as a means for the designer's routine determination of stress intensity factors in linear fracture mechanics for complicated non-symmetric geometries. The vector-valued influence functions (Green functions) introduced here can be seen as a special kind of weight functions. Each of them is numerically found as the displacement field resulting from a certain unit deformation singularity being implanted at the crack tip through a prescribed set of mutual nodal displacements between the crack surfaces. Mode separation is inherent to the procedure. Plane model, mode II and mixed mode I and II numerical examples demonstrate the ease and accuracy of the method. Detailed guidance to the design of the FE mesh at the crack tip is given and is related to accuracy. Any standard FE code can be used. The literature in the field of computational fracture mechanics is surveyed, and some suggestions for further work are made. The present method draws on a classical technique for the calculation of influence lines in structural mechanics. The method is believed to have an added value in that it promotes an overview and understanding of how different load combinations on a given cracked body contribute to a stress intensity factor. Field plots of a calculated influence function are given in one of the examples.     

Analysis of a new specimen for mixed mode fracture tests on brittle materials

Numerical and experimental studies were performed on a new fracture test configuration called the diagonally loaded square plate (DLSP) specimen. The mode I and mode II stress intensity factors were computed for different crack lengths and crack orientation angles using finite element analysis. The numerical results show that the DLSP specimen is able to provide pure mode I, pure mode II and any mixed mode loading conditions in between. Fracture experiments were also conducted on Plexiglas using the DLSP specimen. It is shown that the results obtained from the fracture tests are consistent very well with mixed mode fracture theories.     

Analytical wide-range weight functions for various finite cracked bodies

Closed-form wide-range weight functions have been presented for various finite plane cracked bodies. A unified analytical procedure was used in the derivation. First, accurate crack face displacement expressions for center and edge cracks were determined for the polynomial type reference load case. These displacements were then used to derive analytical weight functions, whose accuracy was critically assessed using the related Green's functions. Stress intensity factors formulae for a number of basic load cases including concentrated forces, polynomial as well as a band of linearly varying stress, have been obtained. These basic solutions combined with superposition method enable stress intensity factors to be rapidly determined for complex loadings, as demonstrated by example engineering crack problems. Discussions were made on the reference load case dependence of the weight functions, and the significance of the number of terms contained in the crack face displacement representation on the solution accuracy at extended crack lengths. The analytical wide-range weight functions have been proved versatile, very cost-saving, easy-to-use, and accurate.     

A generally applicable approximate solution for mixed mode crack-inclusion interaction

Summary Based on the transformation toughening theory an approximate solution is developed for predicting the stress intensity factor for a crack interacting with an inclusion of arbitrary shape and size under I/II mixed mode loading conditions. The transformation strains in the inclusion induced by the crack tip field and the remotely applied stresses are evaluated based on the Eshelby equivalent inclusion theory. As validated by detailed finite element analyses, the solution is applicable with good accuracy for the inclusion of arbitrary shape and large size under mixed mode loadings.     

Effect of cooperative grain boundary sliding and migration on dislocation emitting from a semi-elliptical blunt crack tip in nanocrystalline solids

A theoretical model is established to investigate the interaction between the cooperative grain boundary (GB) sliding and migration and a semi-elliptical blunt crack in deformed nanocrystalline materials. By using the complex variable method, the effect of two disclination dipoles produced by the cooperative GB sliding and migration process on the emission of lattice dislocations from a semi-elliptical blunt crack tip is explored. Closed-form solutions for the stress field and the force acting on the dislocation are obtained in complex form, and the critical stress intensity factors for the first dislocation emission from a blunt crack under mode I and mode II loadings are calculated. Then, the influence of disclination strength, curvature radius of blunt crack tip, crack length, locations and geometry of disclination dipoles, and grain size on the critical stress intensity factors is presented detailedly. It is shown that the cooperative GB sliding and migration and the grain size have significant influence on the dislocation emission from a blunt crack tip.     

A weight function method for mixed modes hole‐edge cracks

A weight function approach is proposed to calculate the stress intensity factor and crack opening displacement for cracks emanating from a circular hole in an infinite sheet subjected to mixed modes load. The weight function for a pure mode II hole‐edge crack is given in this paper. The stress intensity factors for a mixed modes hole‐edge crack are obtained by using the present mode II weight function and existing mode I Green (weight) function for a hole‐edge crack. Without complex derivation, the weight functions for a single hole‐edge crack and a centre crack in infinite sheets are used to study 2 unequal‐length hole‐edge cracks. The stress intensity factor and crack opening displacement obtained from the present weight function method are compared well with available results from literature and finite element analysis. Compared with the alternative methods, the present weight function approach is simple, accurate, efficient, and versatile in calculating the stress intensity factor and crack opening displacement.     

Stress intensity factor solutions for estimation of fatigue lives of laser welds in lap-shear specimens

Fatigue behavior of laser welds in lap-shear specimens of high strength low alloy (HSLA) steel is investigated based on experimental observations and two fatigue life estimation models. Fatigue experiments of laser welded lap-shear specimens are first reviewed. Analytical stress intensity factor solutions for laser welded lap-shear specimens based on the beam bending theory are derived and compared with the analytical solutions for two semi-infinite solids with connection. Finite element analyses of laser welded lap-shear specimens with different weld widths were also conducted to obtain the stress intensity factor solutions. Approximate closed-form stress intensity factor solutions based on the results of the finite element analyses in combination with the analytical solutions based on the beam bending theory and Westergaard stress function for a full range of the normalized weld widths are developed for future engineering applications. Next, finite element analyses for laser welded lap-shear specimens with three weld widths were conducted to obtain the local stress intensity factor solutions for kinked cracks as functions of the kink length. The computational results indicate that the kinked cracks are under dominant mode I loading conditions and the normalized local stress intensity factor solutions can be used in combination with the global stress intensity factor solutions to estimate fatigue lives of laser welds with the weld width as small as the sheet thickness. The global stress intensity factor solutions and the local stress intensity factor solutions for vanishing and finite kinked cracks are then adopted in a fatigue crack growth model to estimate the fatigue lives of the laser welds. Also, a structural stress model based on the beam bending theory is adopted to estimate the fatigue lives of the welds. The fatigue life estimations based on the kinked fatigue crack growth model agree well with the experimental results whereas the fatigue life estimations based on the structural stress model agree with the experimental results under larger load ranges but are higher than the experimental results under smaller load ranges.     

K‐dominance of static crack tip in functionally gradient materials

K‐dominance of static crack tip in functionally gradient materials (FGMs) with a crack oriented along the direction of the elastic gradient is studied through coherent gradient sensing (CGS), digital speckle correlation method (DSCM) and finite element method (FEM). In the direction of crack propagation, the shear modulus has a linear variation with constant mass density and Poisson's ratio. First, the CGS and DSCM governing equations related to the measurements and the elastic solutions at mode I crack in FGMs are obtained in terms of the stress intensity factor, material constants and graded index. Secondly, two kinds of FGMs specimens and one homogenous specimen are prepared to observe the influences of the property variation on the K‐dominance. Then, CGS and DSCM experiments using three‐point‐bending of FGMs and homogenous beams are performed. Thirdly, based on the results of the experiments, the stress intensity factors of three kinds of specimens are calculated by CGS and DSCM. Meanwhile, the stress intensity factors are obtained by FEM. Finally, comparing the results from CGS, DSCM and FEM, the K‐dominance of mode‐I static crack tip in FGMs is discussed in detail.     

Finite element study of a penny-shaped crack along the interface in a bi-material cylinder

A contemporary approach to the analysis of interface cracks in bi-material cylinders using finite elements is presented. From results obtained with a commercial finite element code using regular and singular isoparametric elements, three fracture mechanics techniques are considered to study the interface crack problem and are presented in a fundamental manner. These are the stress intensity factor evaluation by the crack opening displacement method, the strain energy release rate evaluation using the modified crack closure integral method, and the J-integral evaluation using the virtual crack extension technique. Only the finite element results in the vicinity of the crack are then needed. The accuracy of the proposed approach is assessed by solving standard test problems with known solutions. In particular, the mode I problem of a penny-shaped crack in a homogeneous isotropic cylinder under remote tension loading is used as a standard test case. Finally, the mixed-mode (I and II) problem of a penny-shaped crack along the interface in a bi-material cylinder under three loading conditions is studied in detail. Numerical results are presented to quantify the combined effects of geometry and material discontinuities on the strain energy release rate.