Calculation of stress intensity factors for straight cracks in grooved and ungrooved shafts

Stress intensity factors have been calculated by finite element methods for a straight edged crack in a plane normal to the axis of rotation of a circular cylinder under tension and bending, and for a similar crack at the base of a symmetric groove in a shaft under tension and bending and under compressive stresses on the cylindrical surface. Satisfactory agreement with results of simplified approximations was found in most cases. A simple formula for determining the effect of the crack on the stiffness of the shaft in bending has also been derived.For the crack in an ungrooved cylinder, the stress intensity factor due to tensile or bending loads was highest at the centre, and lower than that due to a semi-elliptical notch in a slab of the same thickness under the same load. For bending loads on cracks of depths up to a tenth of the cylinder diameter, the stress intensity factor is approx. $\text{1}\text{1}\text{2}\text{σ√l}$ (where $\text{l}$ is the maximum crack depth and σ the maximum stress in the absence of the crack). For cracks in shallow grooves, the maximum stress intensity under tensile and bending loads is also at the centre but that for cracks in deep grooves is at the surface. The maximum stress intensity factor due to shrink fit is near the surface for both types of groove.     

Mixed mode fatigue crack growth predictions

This work is aimed at developing a predictive capability for the quantitative assessment of crack growth under fatigue loadings. The crack growth rate relation, $\text{Δa}\text{ΔN}$, may involve all three stress intensity factors k₁-k₃ such that the direction of crack growth may not be known in advance and must be predicted from a preassumed criterion. In principle, both the stress amplitude and the mean stress level should be included in the original expression for $\text{Δa}\text{ΔN}$.The strain energy density factor range, ΔS, is found to be a convenient parameter for predicting fatigue crack growth and can be applied expediently to examine the combined influence of crack geometry, complex loadings and material properties. Assumed is the accumulation of energy, $\text{ΔW}\text{ΔV}$, stored in an element ahead of the crack which triggers subcritical crack growth upon reaching a number of loading cycle, say ΔN. The proposed $\text{δa}\text{ΔN}$ relationship includes both the stress amplitude and mean stress effects.     

Stress intensity factors for half-elliptical surface cracks subjected to complex crack face loadings

The disagreement in the literature on the stress intensity factors for surface cracks is considerable. It is also noted that not enough attention has been given to the behaviour of surface cracks in stress fields more complex than uniform tension and bending, although such solutions are needed for crack problems in, e.g. thermal and residual stress fields.In the present paper, stress intensity factors (Mode I) are presented for nine crack geometries in combination with six load cases. The finite element method using 20-node collapsed quarter point singular elements was employed. By a proper modelling of the problem, the number of degrees of freedom was significantly reduced and high accuracy achieved. For the cases of uniform stress and bending, the results agree very well with those of Refs. [8,10], (1–3% for most cases) except one, ($\text{a}\text{c = 0.2}$, $\text{a}\text{t = 0.75}$) for which the present results are 10–17% lower than those in [8]. (For this case other published results deviate up to ± 50%). The K-factors for the more complex loadings will be useful in analysing surface cracks in complex stress fields.     

An examination of mixed fatigue-tensile surface crack growth in rails

The fracture instability associated with alternating periods of fatigue and tensile growth of surface cracks was investigated in steel rails. Three different steels were tested. The instabilities commenced when the maximum stress intensity factor $\text{K}$ exceeded the fracture toughness $\text{K}{}_{\mathrm{IC}}$ and resulted in crack jump or total rail failure. The conditions for the establishment of fatigue-tensile crack jump and arrest are described. The load level, residual stresses, crack geometry and fracture toughness effects are analysed. The fatigue surface cracks were penetrated in both stress relieved and stress unrelieved rails. The effective stress intensity factors including the contribution of the applied load and residual stresses were calculated. For both the fatigue-tensile and tensile-fatigue transitions the stress intensity factors were almost the same with the value for the tensile-fatigue transition being slightly lower. Both calculated stress intensity factors were close to the fracture toughness $\text{K}{}_{\mathrm{IC}}$.     

Effect of specimen thickness on the crack propagation and crack branching in delayed failure

The crack propagation and crack branching behaviors in delayed failure have been investigated on the specimens with various thickness ($\text{B = 1.5–10}\text{mm}$).The crack propagation velocity reveals a maximum value at a medium specimen thickness ($\text{B = 5}\text{mm}$). This fact can be understood by assuming the compound effect of two factors that the triaxiality of stress at crack tip as a driving force for hydrogen diffusion increases with increase of specimen thickness $\text{B}$, and that the invasion of hydrogen atoms from specimen surface increases with decrease of $\text{B}$.The stress intensity factor at crack branching, $\text{K}{}_{\mathrm{IB}}$, increases with decrease of specimen thickness $\text{B}$, and when B is 1.5 mm, the specimen fractures without showing the crack branching. The latter fact can be explained by connecting the necessary and sufficient conditions for crack branching with the decrease in height of plastic region at the crack tip in thin specimens.     

Transient shear waves in a wedge subjected to rapid tearing

The surface of an elastic wedge is subjected to sudden antiplane surface tractions and displacements sufficient to cause tearing. The subsequent crack instability is investigated. The wedge faces subtend an angle κπ with the line of antisymmetry, along which the crack propagates with a constant velocity v. For the externally applied disturbances that are considered here, and for constant crack tip velocities, the particle velocity and $\text{?t}{}_{\mathrm{\theta z}}$ are functions of $\text{r}\text{t}$ and θ only, which allows Chaplygin's transformation and conformai mapping to be used. The theory of analytic functions is then used. For various values of the crack propagation velocity, the dependence of the elastodynamic stress intensity factor, and energy flux into the crack tip, on the wedge angle 2κπ is investigated.     

On the influence of rubbing fracture surfaces on fatigue crack propagation in mode III

Fatigue crack growth has been studied under fully reversed torsional loading (R = ?1) using AISI 4340 steel, quenched and tempered at 200°, 400° and 650°C. Only at high stress intensity ranges and short crack lengths are all specimens characterized by a microscopically flat Mode III (anti-plane shear) fracture surface. At lower stress intensities and larger crack lengths, fracture surfaces show a local hill-and-valley morphology with Mode I, 45° branch cracks. Since such surfaces are in sliding contact, friction, abrasion and mutual support of parts of the surface can occur readily during Mode III crack advance. Without significant axial loads superimposed on the torsional loading to minimize this interference, Mode III crack growth rates cannot be uniquely characterized by driving force parameters, such as ΔK_III and ΔCTD_III, computed from applied loads and crack length values. However, for short crack lengths (?0.4 mm), where such crack surface interference is minimal in this steel, it is found that the crack growth rate per cycle in Mode III is only a factor of four smaller than equivalent behaviour in Mode I, for the 650°C temper at $\text{ΔK}{}_{\mathrm{III}}\text{= 45 MPa m}\text{1}\text{2}$.     

A simple model of stress intensity range threshold and crack closure stress

The cyclic stress intensity threshold (Δ$\text{K}{}_{\mathrm{TH}}$) below which cracks will not propagate varies with length for short cracks. A model is proposed which relates Δ$\text{K}{}_{\mathrm{TH}}$ to the crack closure stress arising from fracture surface roughness. This is used to predict a variation in Δ$\text{K}{}_{\mathrm{TH}}$ with crack length for surface cracks in Ti 6Al-2Sn-4Zn-6Mo alloy, based upon measured values of crack opening displacement arising from roughness. The predicted variation in Δ$\text{K}{}_{\mathrm{TH}}$ with crack length is found to be similar to that obtained from the empirical model of Δ$\text{K}{}_{\mathrm{TH}}$ proposed by El Haddad et al.[5]. The application of the new model to estimate the value of crack closure stress arising from crack tip plasticity for short surface cracks is also discussed.     

A fracture mechanics analysis of ultrasonic fatigue

The method of ultrasonic fatigue finds increasing interest in materials science. Especially, fatigue crack growth rates near the threshold stress intensity range, $\text{ΔK}{}_0$, can be determined with this method in reasonable times providing no frequency and corrosion effects exist. But for an accurate application of this technique it is necessary to improve the testing systems and also the determination of the dynamic cyclic stress intensity range, $\text{ΔK}$. In this paper, fatigue crack growth experiments at ultrasonic frequencies with different mean stresses and also the calculation of the dynamic stress intensity range with finite elements are treated. On this basis fatigue crack growth curves at room temperature of the alloys Hastelloy X and IN 800 were measured and compared with results obtained at low frequencies. No significant influence of frequency could be found in these materials.     

Stress intensity factor at a bilaterally-bent crack in the bending problem of thin plate

An infinite plate with an asymmetric bilaterally-bent crack is analyzed as a bending problem of a thin plate. Stress distributions and stress intensity factors are obtained for some angles and length of bent crack. These are obtained for the some Poisson's ratio. Influence of the initial crack width on the stress intensity factor are also investigated. Three loading conditions are taken into consideration: uniform out of plane bending at infinity in the $\text{x}$ and $\text{y}$ directions and uniform out of plane twist. The rational mapping function in the form of a sum of fractional expressions and the complex variable method are used for the analysis.     

Impact response of a finite crack in a finite strip under anti-plane shear

The response of a through-the thickness crack with finite dimensions to impact in a finite elastic strip is investigated in this study. The elastic strip is assumed to be subjected to anti-plane shear deformation. Laplace and Fourier transform were used to formulate the mixed boundary value problem. The dynamic stress intensity factor and crack opening displacement are obtained as a function of time and the strip width to crack length ratio, h/a. The results indicate that the intensity of the crack-tip stress field reaches a peak very quickly and then decreases in magnitude oscillating about the static value. In general, the dynamic stress intensity factor is higher for small $\text{h/a}$. Similar behavior has also been found for the crack surface displacement.     

Experimental research on surface crack propagation laws for low alloy steel

Surface crack propagation experiments were performed for low alloy steel. The testing result shows that the data for the crack propagation rate in the surface in the direction of the width may be treated by using Paris and Erdogan formula [1] for the crack propagation rate, and Shah and Kobayashi's formula for the stress intensity factor[2]. For the crack propagation rate in the direction of the depth, the data obtained cannot be treated in this way.It has been found that the data can also be treated by using the experimental formula suggested by Kawahara et al.[3].In addition, a test for investigating through-thickness crack propagation was made. It was found that the propagation rate of the through-thickness crack is much greater than those of the surface crack both in the direction of the width and in the direction of the depth. When $\text{ΔK(= 100 Kg/mm}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$) is the same, the propagation rate of through-thickness crack da/dN is five times as great as that of surface crack in the direction of the width.During the propagation of the crack, the relationship between the crack length b and the crack depth a is $\text{a}\text{b}\text{= A?B}\text{a}\text{h}$, where A = 0.97, B = 1.29.With db/dN determined, the prediction of fatigue life can be calculated by

$\text{N=}\text{∫}\text{a}{}_0\text{a}\text{A}\text{A?B}\text{a}\text{h}{}^2\text{d}\text{N}\text{d}\text{b}\text{d}\text{a}$

.     

Mean stress effects on fatigue crack growth and failure in a rail steel

Over a limited range, the effect of mean stress has been studied on fatigue crack propagation and on the critical fatigue crack size associated with sudden fast fracture in centre-notched plate specimens of a rail steel under pulsating loading. The results have been presented in terms of the stress intensity factor range $\text{ΔK}$ and the ratio $\text{R}$ of the minimum to maximum stress. Increasing $\text{R}$ was found to both accelerate cracking and reduce the critical crack size at instability. The data have been correlated with three crack growth equations currently used in the literature and it was found that the equation of Forman et al. relating crack growth rate to $\text{ΔK}$ and $\text{R}$ gave the best fit. This equation was used to predict life in the finite range of the S-N curve. Fractographic examination revealed that the fracture surfaces were complex and a number of fracture modes contributed to cracking.     

Near-tip behavior of a crack in a plane anisotropic elastic body

The asymptotic form of the stress and displacement components near the tip of a straight crack in a generally rectilinear anisotropic plane elastic body are resolved. As in the isotropic analysis, the solutions for the stresses display a $\text{r}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$ dependence, where r is the distance from the tip, while the angular dependence depends upon the anisotropy in a complicated way. The effect of some special anisotropies upon these solutions is fully explored. Finally, these solutions are used to solve the problem of a finite length straight crack in an anisotropic elastic plane when uniform stresses are applied far from the crack. This solution includes obtaining the stress intensity factors, and the nature and magnitude of the crack face displacements.     

Stiffness derivative finite element technique to determine nodal weight functions with singularity elements

The use of the stiffness derivative technique coupled with “quarter-point” singular crack-tip elements permits very efficient finite element determination of both stress intensity factors and nodal weight functions. Two-dimensional results are presented in this paper to demonstrate that accurate stress intensity factors and nodal weight functions can be obtained from relatively coarse mesh models by coupling the stiffness derivative technique with singular elements.The principle of linear superposition implies that the calculation of stress intensity factors and nodal weight functions with crack-face loading, σ($\text{r}{}_{\mathrm{s}}$), is equivalent to loading the cracked body with remote loads, which produces σ($\text{r}{}_{\mathrm{s}}$) on the prospective crack face in the absence of crack. The verification of this equivalency is made numerically, using the virtual crack extension technique. Load independent nodal weight functions for two-dimensional crack geometry is demonstrated on various remote and crack-face loading conditions. The efficienct calculation of stress intensity factors with the use of the “uncracked” stress field and the crack-face nodal weight functions is also illustrated.In order to facilitate the utilization of the discretized crack-face nodal weight functions, an approach was developed for two-dimensional crack problems. Approximations of the crack-face nodal weight functions as a function of distance, ($\text{r}{}_{\mathrm{s}}$), from crack-tip has been sucessfully demonstrated by the following equation: $\text{h(a, r}{}_{\mathrm{s}}\text{,) =}\text{A(a)}\text{√r}{}_{\mathrm{s}}\text{+ B(a) + C(a)√r}{}_{\mathrm{s}}\text{+ D(a)r}{}_{\mathrm{s}}$.Coefficients $\text{A(a)}$, $\text{B(a)}$, $\text{C(a)}$ and $\text{D(a)}$, which are functions of crack length ($\text{a}$), can be obtained by least-squares fitting procedures. The crack-face nodal weight functions for a new crack geometry can be approximated using cubic spline interpolation of the coefficients $\text{A, B, C}$ and $\text{D}$ of varying crack lengths. This approach, demonstrated on the calculation of stress intensity factors for single edge crack geometry resulted in total loss of accuracy of less than 1%.     

Acoustic emissions of fatigue crack growth

Acoustic emissions of fatigue crack growth have been monitored and quantitatively correlated with growth rate and the applied range of stress intensity for high cycle fatigue of 2024-T851 aluminum alloy. The data suggest a more cogent relationship for acoustic emissions and the applied range of stress intensity rather than between acoustic emissions and the average crack growth rate. Since nearly all crack growth is expected during the maximum load portion of the fatigue cycles, only the emissions from the acoustic events in the vicinity of the peak load were incorporated in correlations with $\text{d}\text{a/}\text{d}\text{n}$ and $\text{ΔK}$. Large amplitude emissions in the proximity of the minimum cyclic load were also detected. Because of their characteristics, these emissions are attributed to crack surface interference and, consequently, were not included in the correlation analyses.     

Fatigue crack propagation from a crack inclined to the cyclic tensile axis

Cyclic stresses with stress ratio R = 0.65 were applied to sheet specimens of aluminium which have an initial crack inclined to the tensile axis at angles of 30°, 45°, 72° or 90°. The threshold condition for the non-propagation of the initial crack was found to be given by a quadratic form of the ranges of the stress intensity factors of modes I and II. The direction of fatigue crack extension from the inclined crack was roughly perpendicular to the tensile axis at stress ranges just above the threshold value for non-propagation. On the other hand, at stress ranges 1.6 times higher than the threshold values the crack grew in the direction of the initial crack. The rate of crack growth in the initial crack direction was found to be expressed by the following function of stress intensity factor ranges of mode I, K₁, and mode II, K₂: $\text{dc}\text{dN}\text{= C(K}{}_{\mathrm{eff}}\text{)sum}$, where $\text{K}{}_{\mathrm{eff}}\text{= [K}{}_1{}^4\text{+ 8K}{}_2{}^4\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}$. This law was derived on the basis of the fatigue crack propagation model proposed by Weertman.     

Stress intensity factors for coalescing and single corner flaws along a hole bore in a plate

The objective of this paper is to describe the effects of crack interaction on stress intensity factors for two symmetric coplanar corner flaws located along a hole bore. This numerical analysis employes the Finite Element-Alternating Method to determine Mode I stress intensity factors for single and coalescing corner flaws. Using single flaw stress intensity factors as a reference, analysis of crack size and shape effects on $\text{K}{}_{\mathrm{I}}$ for coalescing corner flaws indicates the stress intensity factor for crack points along the hole bore increases as the crack tip separation distance decreases. Interaction effects are not experienced by hole bore crack points when the crack tip separation distance is equal to or greater than half of the largest corner flaw dimension.     

The general solutions of the doubly periodic cracks

The purpose of this paper is to use the basic theorem of Jacobi elliptic functions and residues to find the general solutions of the stress intensity factor2 of doubly periodic cracks subjected to concentrated forces $\text{P}$, $\text{T}$ and $\text{Q}$ on the surfaces of each crack. In this paper the general solutions are expressed in simply closed forms, and they can also be applied to solve practical problems of rectangular sheet with central crack.The general solutions in this paper may be used as a fundamental Green's function for finding other solutions of the doubly periodic cracks involving arbitrary surface forces. It is easily proved that all problems of the isolated crack and the singly periodic cracks loaded by arbitrary surface forces, are the special cases of the general solutions in this paper.