The effect of crack surface interaction on the stress intensity factor in Mode III crack growth in round shafts

Turbine-generator shafts are often subjected to a complex transient torsional loading. Such transient torques may initiate and propagate a circumferential crack in the shafts. Mode III crack growth in turbo-generator shafts often results in a fracture surface morphology resembling a factory roof. The interaction of the mutual fracture surfaces results in a pressure and a frictional stress field between fracture surfaces when the shaft is subjected to torsion. This interaction reduces the effective Mode III stress intensity factor.The effective stress intensity factor in circumferentially cracked round shafts is evaluated for a wide range of applied torsional loading by considering a pressure distribution between mating fracture surfaces. The pressure between fracture surfaces results from climbing of asperities respect to each other. The pressure profile not only depends on the fracture surface roughness (height and width (wavelength) of the peak and valleys), but also depends on the magnitude of the applied Mode III stress intensity factor. The results show that asperity interactions significantly reduce the effective Mode III stress intensity factor. However, the interactions diminish beyond a critical applied Mode III stress intensity factor. The critical stress intensity factor depends on the asperities height and wavelength. The results of these analyses are used to find the effective stress intensity factor in various Mode III fatigue crack growth experiments. The results show that Mode III crack growth rate is related to the effective stress intensity factor in a form of the Paris law.     

Dependence of sub-critical crack growth on effective stress intensities

Sub-critical crack growth in high strength steels exhibit three regions of growth rates. In region I and III the growth is dependent on the stress intensity (computed for a sharp crack tip), while in region II the growth rate appears insensitive to the stress intensity. An investigation of the behavior in region II indicates that the insensitivity may be due to blunting of the crack tip resulting in a lower effective stress intensity. Based on experimental results a ratio for approximation of the effective stress intensity at the crack tip is proposed, and the cause of the anomalous behavior in region II is suggested.     

Plastic yielding at the tip of a blunt notch during static and fatigue loading

Rice's analytical Mode III solution for the relationship between anti-plane stress and anti-plane strain was used to determine the small scale plastic yielding at the tip of a two-dimensional blunt notch. The results were applied to fatigue loading. The plastic zone size and crack opening displacement derived in the present analysis were determined as functions of applied stress, geometric factors (notch radius and length) and material properties (yield stress and the work hardening rate). The minimum stress intensity required for plastic yielding at a blunt notch tip was postulated to be the experimentally observed threshold stress intensity for fatigue crack initiation. The threshold stress intensity so determined depends not only on the notch geometry but also on material properties. There is good agreement with calculated and measured values of the threshold stress intensity for fatigue crack initiation.     

THRESHOLD STRESS INTENSITY BEHAVIOUR OF CRACKED STEEL STRUCTURAL COMPONENTS

Abstract— The effects of stress variables on the fatigue design of steel structural components (SAE 1010 steel) in the threshold region are investigated. The threshold and the closure threshold stress intensity ranges both decreased linearly as the stress ratio is increased. The threshold and opening threshold stress intensities also decreased linearly as the magnitude of compressive peak stress is increased. Crack opening stress measurements using a mechanical extensometer showed that the crack is not fully closed throughout the stress cycle at the threshold level. The crack opening stress is found to be independent of the crack length up to a certain crack length depending on the loading conditions. It is also found that the threshold stress intensity consists of two components: opening or closure stress intensity required to overcome crack closure, and intrinsic stress intensity range required to grow the crack. Linear relationships are obtained for the intrinsic stress intensity range as a function of stress ratio or compressive peak stress.     

Stress intensity factors for large arrays of radial cracks in thick-walled steel cylinders

Mode I stress intensity factors for large arrays of up to 512 radial cracks emanating from the inner surface of a pressurized thick-walled cylinder are evaluated. Furthermore, for cylinders that underwent autofrettage, the negative stress intensity factors due to the compressive residual stresses are also calculated. Both stress intensity factors are evaluated, for numerous crack arrays (2–512), for a wide range of crack lengths and for a fully autofrettaged cylinder, via the finite element (FE) method. The present results accentuate the considerable influence of the number of cracks in the array, as well as that of the autofrettage on the actual stress intensity factor prevailing at the tip of these cracks.     

Determination of stress intensity factor solutions for cracks in finite-width functionally graded materials

A generalized method to determine the stress intensity factor equations for cracks in finite-width specimens of functionally graded materials (FGMs), based on force balance in regions ahead of the crack tip is provided. The method uses the Westergaard's stress distribution ahead of the crack in an infinite plate and is based on the requirement of isostrain deformation of layers of varying moduli ahead of the crack tip. It is shown that the modified Westergaard equation describes the normal stress distribution and the singular stress state ahead of the crack tip in a reasonably accurate manner. Based on this, closed-form analytical equations for the stress intensity factors of cracks in finite-width center cracked specimens were derived. Comparisons of the K values from the analytical equations with that obtained from FEM simulations indicate that the derived stress intensity factor equations for FGMs are reasonably accurate. For the finite-width center-cracked-tension (CCT) specimen, the errors are less than 10% for most of the crack lengths for materials with the outer layer modulus ratios varying from 0.2 to 5. The stress intensity factors were found to be sensitive to the absolute values of moduli of the layers, the modulus ratio of the outer layers as well as the nature of gradation including the increasing and the decreasing functional forms. The stress intensity factor equations are convenient for engineering estimates of stress intensity factors as well as in the experimental determinations of fracture toughness of FGMs.     

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.     

Effect of loadline shift on applied crack tip stress intensity in WOL specimens

A change in applied stress intensity due to shifting of load line from the pin hole to a wedge located on the outside edge of the notch has been investigated by: (1) finite element analysis, (2) measurements of front face crack opening displacement and (3) strain relaxation near the crack tip.

Results show that this wedge loading procedure will result in a significant drop, up to a factor of two, in applied stress intensity. The drop in stress intensity is inversely related to the crack length (expressed by a/W). This drop in stress intensity is due to overall specimen distortion because of load line shift and local deformation of the wedge and notch surfaces. Implications of this drop on Stress Corrosion Cracking results are discussed. For reliable stress corrosion testing, modifications in specimen geometries and loading procedures are suggested.     

Fracture mechanics analysis of geometrically nonlinear shear deformable plates

This paper presents boundary integral equations for fracture mechanics analysis of geometrically nonlinear shear deformable plates. A radial basis function and dual reciprocity method are utilized to evaluate the derivative terms and the domain integrals that appear in the formulations, respectively. Numerical examples of the clamped and simply supported plates containing a center crack subjected to uniform transversal loadings are presented. Displacement extrapolation technique is used to compute the stress intensity factors (SIFs). Stress intensity factors of mode I for plate bending and membrane problems are presented. The normalized stress intensity factors in membrane significantly increase after few increments of the load while the normalized stress intensity factors in bending decrease. Less displacement and rotational constraints in cracked plates under uniform transversal loadings will raise the stress intensity factors. The bending stress intensity factors of a central crack in clamped square plate were found to be the highest values compared to those for clamped non-square plates.     

THREE-DIMENSIONAL CONSTRAINT EFFECTS ON STRESS INTENSITY DISTRIBUTIONS IN PLATE GEOMETRIES WITH THROUGH-THICKNESS CRACKS

Three-dimensional finite element analyses of through-thickness cracks in plate geometries are presented. It is shown that the local constraint variation towards the crack tip causes an increase in stress intensity which is neglected in analyses which assume a constant stress state throughout the geometry. This means that stress intensity factors for these types of crack geometries are a factor (1 — v²)^-0.5 higher than generally reported in stress intensity handbooks. Analyses on crack tunnelling situations show that if plane stress dominates the global behaviour, an almost constant stress intensity across the thickness is reached for a relative tunnelling depth of a 2.5% of the plate thickness. For geometries which are not predominantly in plane stress this value will be somewhat larger. Crack front tunnelling does not influence the mean stress intensity across the thickness. Shear lips are shown to reduce the mean equivalent stress intensity (the mode I stress intensity which has the same energy release rate as the actual combined mode I, II and III loading) by a factor equal to the square root of the ratio between plate thickness and projected length of the crack front. This may explain the reduction in crack growth rate caused by shear lips during fatigue crack growth experiments. The reduction of the mode I stress intensity factor is considerably larger, which may cause a further reduction of fatigue crack growth rate for crack growth mechanisms that depend primarily on the mode I crack tip loading. Analyses on CNT and SENB specimens show that the conclusions reached on infinite plate models also hold for real structures. However for an SENB specimen with an uncracked ligament equal to the plate thickness, the overall constraint is larger than that of a pure plane stress situation, and the effect of stress intensity increases due to the constraint transition is less pronounced.     

A periodic array of cracks in functionally graded materials subjected to transient loading

A periodic array of cracks in an infinite functionally graded material under transient mechanical loading is investigated. In-plane normal (mode I) and shear (mode II) loading conditions are considered. For each individual loading mode, a singular integral equation is derived, in which the crack surface displacements are unknown functions. Numerical results are obtained to illustrate the variation of the stress intensity factors as a function of the crack periodicity for different values of material inhomogeneity, either at the transient state or steady state. The material inhomogeneity can increase or decrease the mode I and mode II stress intensity factors. Compared with the single crack solution, it is also shown that multiple cracking may decrease the mode I stress intensity factors, but enhance the mode II stress intensity factors significantly.     

Numerical methods for the determination of multiple stress singularities and related stress intensity coefficients

This paper proposed numerical methods to determine the multiple stress singularities (including the oscillatory stress singularities) and the related stress intensity coefficients, by the use of common numerical solutions (stresses or displacements) obtained by an ordinary numerical tool such as finite element method (FEM) or boundary element method (BEM). To verify the efficiency of the present methods, two models of bonded dissimilar materials under the plane strain state are analyzed by BEM, and the orders of the stress singularities and the related stress intensity coefficients are examined numerically. The results show that all the orders of the stress singularities at an interface edge can be determined precisely by the present method, and the related stress intensity coefficients can be determined by the extrapolation method with a very good linearity. It is found that the methods presented in this paper are very simple and efficient. Moreover, they can be easily extended to any singular problem.     

Increase of stress intensity near interface edge of elastic-creep Bi-material under a sustained load

The transition from small-scale creep to large-scale creep ahead of a crack tip or an interface edge with strong elastic stress singularity at the loading instant causes stress relaxation and the decrease of stress intensity in general. However, this study shows that the stress near the interface edge of bi-material with no or weak elastic stress singularity increases after the loading instant and brings about the stress concentration during the transition. In addition, the creep strain distribution of this bi-material after the loading instant is different from that occurred in the transition of an interface edge with strong elastic stress singularity or a crack tip (notch root). The criterion for the increase or decrease of stress intensity near the interface edge proved by the finite element method is proposed in this study. The stress intensity near the interface edge increases when the elastic stress singularity is lower than the creep stress singularity (λ^el < λ^cr) and vice versa.     

Computation of membrane and bending stress intensity factors for thin,cracked plates

The crack tip stress fields for plate bending and membrane loading problems are reviewed and the four stress intensity factors that determine these fields are defined. These four stress intensity factors arise from use of Kirchhoff plate theory to account for the bending loads and two dimensional plane stress elasticity to account for the membrane loads. The energy release rate is related to the stress intensity factors and to the stress resultants of plate theory. Virtual crack extension, nodal release and modified crack closure integral methods are discussed for computing components of the energy release rate from finite element analyses of cracked plates. Sample computations of stress intensity factors for single and mixed mode cases are presented for a crack in an infinite plate. Sample computations of stress intensity factors for a double edge notched tension-torsion test specimen are given as well.School of Civil and Environmental Engineering, Cornell University     

An analytical method for mixed-mode propagation of pressurized fractures in remotely compressed rocks

An analytical method for mixed-mode (mode I and mode II) propagation of pressurized fractures in remotely compressed rocks is presented in this paper. Stress intensity factors for such fractured rocks subjected to two-dimensional stress system are formulated approximately. A sequential crack tip propagation algorithm is developed in conjunction with the maximum tensile stress criterion for crack extension. For updating stress intensity factors during crack tip propagation, a dynamic fictitious fracture plane is used. Based on the displacement correlation technique, which is usually used in boundary element/finite element analyses, for computing stress intensity factors in terms of nodal displacements, further simplification in the estimation of crack opening and sliding displacements is suggested. The proposed method is verified comparing results (stress intensity factors, propagation paths and crack opening and sliding displacements) with that obtained from a boundary element based program and available in literatures. Results are found in good agreements for all the verification examples, while the proposed method requires a trivial computing time.     

Influence of anisotropy,porosity and initial stresses on crack propagation due to Love‐type wave in a poroelastic medium

An analytical solution has been attained to establish the closed form expression of stress intensity factor at the tip of a semi‐infinite crack, dynamically propagating in an initially stressed transversely isotropic poroelastic strip due to Love‐type wave for the case of concentrated force of constant intensity as well as for the case of constant load. The study presents the sound effect of various affecting parameters viz. speed of the crack, length of the crack, horizontal compressive/tensile initial stress, vertical compressive/tensile initial stress, porosity parameter and anisotropy parameter on the stress intensity factor. In order to delineate the effects of these aforementioned parameters on the stress intensity factor graphically, numerical simulations have been accomplished. One of the major highlight of the paper is the comparative study carried out for horizontal compressive/tensile initial stress, vertical compressive/tensile initial stress, porosity parameter and anisotropy parameter with the case when the strip is isotropic, non‐porous and free from initial stresses. Wiener–Hopf technique and the Fourier integral transform has been effectuated for the procurement of the closed form expression (exact solution) of stress intensity factor.     

利用模拟衍射线计算X射线应力测定中的误差

利用模拟衍射线计算了X射线应力测定中不同线形衍射线的峰位角和残余应力的误差,讨论了它们与半高宽（HW）、净峰强度（IP）和峰背比（IP／IB）之间的关系。根据分析结果,分别推导出不同线形衍射线的峰位角和残余应力误差的计算公式,并利用实测衍射线考察了该公式的可靠性。     

Theoretical and experimental determination of magnetic stress intensity factors of a crack in a double cantilever beam specimen

This paper presents the results of an experimental and theoretical investigation of the magnetic fracture behaviour of double cantilever beam (DCB) specimens. DCB tests were conducted on ferritic stainless steel SUS430 in the bore of a superconducting magnet at room temperature. A simple experimental technique using strain gauges was used to determine the stress intensity factor. The experiments show the predicted increase in the stress intensity factor with increasing magnetic field. The theoretical analysis is based on a beam‐plate theory for magnetoelastic interactions in a soft ferromagnetic material. Numerical calculations are carried out, and the stress intensity factor is obtained for several values of magnetic field. A comparison of the stress intensity factor is made between theory and experiment, and the agreement is good for the magnetic field considered.     

Strong interactions of morphologically complex cracks

Previous works on crack morphology have focused on such cracks as a kinked crack, a branched crack, and an inclined array of identical branched cracks. In this paper, the strong interactions between two cracks in two-dimensional solids under remote tension are investigated. Three morphological types are considered: kinks, branches and zigzags. The method of analysis follows the singular integral equation approach in which the deviations from the main cracks are modeled by distributions of dislocations. Investigations are made on the dependence of the stress intensity factors on the asymmetry of the crack configuration, the crack separation, and the shape of the cracks. The results show that (i) strong interactions can have significant effects on the mode mixity of the stress intensity factors, (ii) a small asymmetry of the crack configuration can cause significant changes to the stress intensity factors, and (iii) zigzag cracks with rectangular steps reduce the stress intensity factors more efficiently than those with triangular or trapezoidal steps.     

Stress intensity factor analysis of elliptical corner cracks in mechanical joints by weight function method

Mixed-mode stress intensity factors at the surface and deepest points of quarter elliptical corner cracks in mechanical joints such as bolted or riveted joints are analyzed by weight function method. The weight function method is an efficient technique to calculate the stress intensity factors using uncracked stress field. The extended form of the weight function method for 2D mixed-mode problems to 3D mixed-mode is presented and the accuracy due to the number of terms included in the weight function is examined. The effects of the amount of clearance between the hole and the bolt or rivet on the stress intensity factors are investigated, and the critical angle causing the mode I stress intensity factor to be maximized is determined by analyzing the variation of the stress intensity factors along incline angle of crack.