Crack propagation from a pre-existing flaw at a notch root – II: Detailed form of the stress intensity factors at the initial crack tip and conclusion.

This paper pursues the study of crack kinking from a pre-existing crack emanating from some notch root. It was shown in Part I that the stress intensity factors at the tip of the small initial crack are given by universal (that is, applicable in all situations, whatever the geometry of the body and the loading) formulae; they depend only on the `stress intensity factor of the notch' (the multiplicative coefficient of the singular stress field near the apex of the notch in the absence of the crack), the length of the crack, the aperture angle of the notch and the angle between its bisecting line and the direction of the crack. Here we identify the universal functions of the two angles just mentioned which appear in these formulae, by considering the model problem of an infinite body endowed with a notch with straight boundaries and a straight crack of unit length. The treatment uses Muskhelishvili's complex potentials formalism combined with some conformal mapping. The solution is expressed in the form of an infinite series involving an integral operator, which is evaluated numerically. Application of Goldstein and Salganik's principle of local symmetry then leads to prediction of the kink angle of the crack extension. It is found that although the direction of the crack is closer to that of the bisecting line of the notch after kinking than before it, the kink angle is not large enough for the crack tip to get closer to this line after kinking, except perhaps in some special situations.     

Crack tip stress intensity factors for a crack emanating from a sharp notch

Accurate calibrations are provided for the crack tip stress intensity factor for a crack of finite length emanating from the symmetric tip of a sharp notch, of arbitrary angle, in terms of the generalised stress intensity quantifying remote loading of the notch. The solution is applied to example problems and shown to be accurate for cases where the crack is much shorter then the notch depth.     

Automatic crack tip detection and stress intensity factors estimation of curved cracks from digital images

With the development of full‐field measurement techniques, it has been possible to analyze crack propagation experimentally with an increasing level of robustness. However, the analysis of curved cracks is made difficult and almost unexplored because the possible analysis domain size decreases with crack curvature, leading to an increasing uncertainty level. This paper proposes a digital image correlation technique, augmented by an elastic regularization, combining finite element kinematics on an adapted mesh and a truncated Williams' expansion. Through the analysis of two examples, the proposed technique is shown to be able to address the experimental problems of crack tip detection and stress intensity factors estimation along a curved crack path. Copyright © 2015 John Wiley & Sons, Ltd.     

Influence of effective stress intensity factor range on mixed mode fatigue crack propagation

ABSTRACT The behaviour of fatigue crack propagation of rectangular spheroidal graphite cast iron plates, each consisting of an inclined semi‐elliptical crack, subjected to axial loading was investigated both experimentally and theoretically. The inclined angle of the crack with respect to the axis of loading varied between 0° and 90°. In the present investigation, the growth of the fatigue crack was monitored using the AC potential drop technique, and a series of modification factors, which allow accurate sizing of such defects, is recommended. The rate of fatigue crack propagation db/dN is postulated to be a function of the effective strain energy density factor range, ΔS_eff. Subsequently, this concept is applied to predict crack growth due to fatigue loads. The mixed mode crack growth criterion is discussed by comparing the experimental results with those obtained using the maximum stress and minimum strain energy density criteria. The threshold condition for nongrowth of the initial crack is established based on the experimental data.     

The application of asymptotic solutions to a semi-elliptical crack at the root of a notch

This paper presents an approximate method based on asymptotic solutions for estimating the stress intensity factor K for semi-elliptic surface cracks at stress concentrations. The proposed equation for estimating K makes use of the near-notch and remote-notch solution to interpolate over the entire range from shallow to deep cracks. The near-notch solution is obtained by means of the stress concentration factor. For cracks located in the remote stress field, K is obtained by considering the crack to be located in a smooth plate with a crack depth equal to the sum of the notch depth and the actual crack depth. The accuracy of the predictions is assessed using numerical calculations and solutions found in the literature.     

On the numerical evaluation of stress intensity factors for an interface crack of a general shape

A numerical algorithm is presented for the problem of a crack along the interface of an elastic inclusion embedded in an elastic plane subjected to uniform stress at infinity. The algorithm is based on a Fredholm integral equation of the second kind and allows for fast and accurate solutions to geometries of great complexity. In an example crack opening displacement and stress intensity factors are computed for a crack in the interface of an inclusion with nineteen protruding arms. Copyright © 1999 John Wiley & Sons, Ltd.     

Strain energy-based evaluations of plastic notch stress intensity factors at pointed V-notches under tension

An analytical study is carried out on the existing link between elastic and plastic notch stress intensity factors at pointed V-notches in plates under tension.The frame is developed on the basis of the elastic and plastic energy concentration factors of the notch defined here as the ratio between the local and the nominal strain energy densities. The link varies under plane stress and plane strain conditions. The local strain energy density is evaluated over a control volume drawn by the energy contour lines ahead of the notch and allows plastic notch stress intensity factors to be predicted on the basis of an ideally linear elastic analysis, both under small and large scale yielding.     

Stress intensity factors for a mixed mode center crack in an anisotropic strip

An approach based on the continuous dislocation technique is formulated and used to obtain the Mode I and II stress intensity factors in a fully anisotropic infinite strip with a central crack. First, the elastic solution of a single dislocation in an anisotropic infinite strip is obtained from that of a dislocation in an anisotropic half plane, by applying an array of dislocations along the boundary of the infinite strip, which is supposed to be traction-free. The dislocation densities of the dislocation array are determined in such a way that the traction forces generated by the dislocation array cancel the residual tractions along the boundary due to the single dislocation in the half plane. The stress field of a single dislocation in the infinite strip is thus a superposition of that of the single dislocation and the dislocation array in the half plane. Subsequently, the elastic solution is applied to calculate the stress intensity factors for a center crack in an anisotropic strip. Crack length and material anisotropy effects are discussed in detail.     

Closed‐form thermal stress intensity factors for an internal circumferential crack in a thick‐walled cylinder

In this paper the method of weight functions is employed to calculate the stress intensity factors for an internal circumferential crack in a thick‐walled cylinder. The pressurized cylinder is also subjected to convection cooling on the inner surface. Finite element method is used to determine an accurate weight function for the crack and a closed‐form thermal stress intensity factor with the aid of the weight function method is extracted. The influence of crack parameter and the heat transfer coefficient on the stress intensity factors are determined. Comparison of the results in the special cases with those cited in the literature and the finite element data shows that the results are in very good agreement.     

Evaluation of fatigue crack propagation by mode I and mixed mode in 1050 aluminium

The mechanism of mixed‐mode fatigue crack propagation was investigated in pure aluminum. Push‐pull fatigue tests were performed using two types of specimens. One was a round bar specimen having a blind hole, one was a plate specimen having a slit. The slit direction cut in the specimen was perpendicular or inclined 45 degrees relative to the centre of the specimen axis. In both cases, cracks propagated by mode I or by the mixed mode combining mode I and shear mode, depending on the testing conditions. In these cases the crack propagation rate was evaluated with a modified effective stress intensity factor range. Crack propagation retardation was observed in some specimens. However, it was found that the crack propagation rate could also be evaluated by the effective stress intensity factor range independent of the crack propagation mode.     

Thermal stress intensity factors of an infinite orthotropic layer with a crack

Stress intensity factors are determined for a crack in an infinite orthotropic layer. The crack is situated parallel to the plane surfaces of the layer. Stresses are solved for two kinds of the boundary conditions with respect to temperature field. In the first problem, the upper surface of the layer is heated to maintain a constant temperature T ₀, while the lower surface is cooled to maintain a constant temperature –T ₀. In the other problem, uniform heat flows perpendicular to the crack. The surfaces of the crack are assumed to be insulated. The boundary conditions are reduced to dual integral equations using the Fourier transform technique. To satisfy the boundary conditions outside the crack, the difference in temperature at the crack surfaces and differences in displacements are expanded in a series of functions that vanish outside the crack. The unknown coefficients in each series are evaluated using the Schmidt method. Stress intensity factors are then calculated numerically for a steel layer that behaves as an isotropic material and for a tyrannohex layer that behaves as an orthotropic material.     

Mixed mode crack tip parameters for different wheel positions relative to a vertical crack at the rail foot

Finite element method is used to analyze a rail with a vertical bottom up crack at its foot, under the axle load and surface traction of a wheel. The possibility of crack formation at the foot of the rail in the neighborhood of a welding connection is discussed. A brief review on the importance of T‐stress in brittle fracture is presented. Seven cases with different locations of the crack relative to rail's sleeper contact region are considered. Numerous positions of the wheel are considered, and in each case, 3 crack parameters K_I, K_II, and T‐stress are calculated. Then, the biaxiality ratio and the mixity parameter for each loading and crack condition are calculated. It is shown that the location of crack and wheel can create mixed mode loading in the cracked rail and that the magnitude of crack tip parameters are strongly dependent on these geometric variables. In particular, the magnitudes of T‐stress and biaxiality ratio are significant in some cases. The effect of friction between the crack faces in the presence of compressive mode I loading on the mode II stress intensity factor is studied. Under mixed mode loading, due to the axle load and surface traction, the most critical condition is the formation of vertical cracks near the sleeper contact region.     

Evaluating stress intensity factors of a surface crack in lubricated rolling contacts

The three-dimensional finite element method and the least-squares method were used to find the stress intensity factors (SIFs) of a surface crack in a lubricated roller. A steel roller on a rigid plane was modeled, in which a semi-elliptical surface crack is inclined at an angle ψ to the vertical axis. A distance c is set between the crack base and the roller edge. The results indicate that the mode-I SIF reaches the maximum value when the angle θ is equal to 0° (on the roller surface), and the mode-II SIF reaches the absolute maximum value when the angle θ is near or equal to 90° (inside the roller), where θ is the angle of the semi-ellipse from 0° to 180°. The influence of mode-III SIFs in this model is minor since they are much smaller than the mode-I and mode-II SIFs. The SIFs increase greatly when the crack location approaches the uncrowned edge. At this time, a crowned profile can be used to significantly reduce the SIFs near the roller edge. This revised version was published online in July 2006 with corrections to the Cover Date.     

The effect of material combinations and relative crack size to the stress intensity factors at the crack tip of a bi-material bonded strip

Several types of singular stress fields may appear at the corner where an interface between two bonded materials intersects a traction-free edge depending on the material combinations. Since the failure of the multi-layer systems often originates at the free-edge corner, the analysis of the edge interface crack is the most fundamental to simulate crack extension. In this study, the stress intensity factors for an edge interfacial crack in a bi-material bonded strip subjected to longitudinal tensile stress are evaluated for various combinations of materials using the finite element method. Then, the stress intensity factors are calculated systematically with varying the relative crack sizes from shallow to very deep cracks. Finally, the variations of stress intensity factors of a bi-material bonded strip are discussed with varying the relative crack size and material combinations. This investigation may contribute to a better understanding of the initiation and propagation of the interfacial cracks.     

Stress intensity factors of a surface crack near an interface end

Due to the singular behavior of the stress field near the interface edge of bonded dissimilar materials, fracture generally initiates near the interface edge, or just from the interface edge point. In this paper, an edge crack near the interface, which can be considered as being induced by the edge singularity and satisfying two conditions, is analyzed theoretically, based on the singular stress field near the interface edge and the superposition principle. It is found that the stress intensity factor can be expressed by the stress intensity coefficient of the edge singular stress field, the crack length, the distance between the interface and the crack, as well as the material combination. Boundary element method analysis is also carried out. It is found that the theoretical result coincides well with the numerical result when the crack length is small. Therefore, the theoretical representation obtained by this study can be used to simply evaluate the stress intensity factor of an edge singularity induced crack for this case. However, when the crack length becomes larger than a certain value, a significant difference appears, especially for the case with large edge singularity.     

Experimental and finite element analyses on stress intensity factors of an elliptical surface crack in a circular shaft under tension and bending

Experimental backtracking technique and finite element analysis have been employed to evaluate the stress intensities along the front of an elliptical surface crack in a cylindrical rod. The finite element solution covers a wide range of crack shapes loaded under end-free and end-constrained axial tension and pure bending. Convenient closed form stress intensity expressions along the whole crack front for each of the loading cases have been given in terms of the crack aspect ratio, crack depth ratio and place ratio.The closed form solutions have been compared against a number of representative solutions collected from the literature. It has been found that different finite element results for the interior points are generally in good mutual agreement, while solutions derived from other methods may sometimes indicate different trends. At the surface interception point agreement is less good because of a complication in the interpretation of stress intensity there.Experimental backtracking results on the end-constrained axial tension case corroborate well with the closed form solution presented. It suggests that the current closed form solution is adequate in describing the stress intensities along the whole crack front of real surface cracks in cylindrical rods.     

Distribution of dislocations at a mode I crack tip and their shielding effect

Distribution of dislocations at a finite mode I crack tip is formulated. Closed form solutions for the dislocation distribution function, the dislocation-free zone (DFZ), the local stress intensity factor and the crack tip stress field are obtained. The dislocation distribution has similar features to a mode III crack model. Under a given applied stress, there may exist different configurations of plastic zone and DFZ. Crack tip shielding by dislocations depends on both applied stresses and the configuration.     

Stress singularities near the tip of a two-dimensional notch formed from several elastic anisotropic materials

In this paper, we studied the stress singularities near the tip of a two-dimension notch, which could be a crack tip, formed from several elastic materials, each of them may be generally anisotropic. By introducing the dual variables in the state space, the basic equations governing the posed problem equations were established. We also proposed a numerical method to solve the governing equations. It was shown that the mathematical formulations advanced are quite simple and the numerical method proposed is easy and highly accurate. Finally, a few examples of problems in this topic were solved and presented in order to demonstrate the accuracy and the potential possibilities of applications of the present method.     

Fatigue crack propagation rates and threshold stress intensity factors for welded joints of HT80 steel at several stress ratios

Fatigue crack propagation rates and threshold stress intensity factors were measured for welded joints and base metal by using 200 mm wide centre-cracked specimens. The fatigue crack propagation properties of welded joints were similar in spite of the different zones in which the cracks propagated (ie, in the heat-affected zone and in the weld metal) and the different welding process used (submerged arc welding and gas metal arc welding). They were, however, inferior to those of the base metal. It was revealed by observation of the crack closure that the fatigue cracks were fully open during the whole range of loading, due to the tensile residual stress distribution in the middle part of the welded joints. This observation also explains the lack of a stress ratio effect on the fatigue crack propagation properties of welded joints, and their inferiority to those of the base metal.