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.     

Penny-shaped crack in a sphere embedded in an infinite medium

We consider the problem of determining the stress intensity factor and the crack energy in an Isotropie, homogeneous elastic sphere embedded in an infinite Isotropie, homogeneous elastic medium when there is a diametrical crack in the sphere. We assume that the crack is opened by an internal pressure and the sphere is bonded to the surrounding material. The problem is reduced to the solution of a Fredholm integral equation of the second kind in the auxiliary function φ($\text{t}$). Expressions for the stress intensity factor and the crack energy are obtained in terms of φ($\text{t}$). The integral equation is solved numerically and the numerical values of the stress intensity factor and the crack energy are graphed.     

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.     

A fracture mechanics approach to high temperature fatigue crack growth in Udimet 700

Crack growth behavior under high temperature fatigue in Udimet 700 has been analyzed using both linear and non-linear elastic fracture mechanics concepts. It is shown that crack growth data for various loads in a compact tension specimen correlate well with the stress intensity factor, even at temperatures as high as 850°C. Using these results, a self consistent procedure has been developed for the determination of the $\text{J-}\text{integral}$ parameter under load-controlled fatigue and is shown to be compatible with data based on the stress intensity factor. The spread in the crack growth data is smaller in terms of $\text{J-}\text{integral}$ as compared to stress intensity or crack opening displacement parameters. Also based on a detailed fractographic analysis, it is suggested that the micromechanism of crack growth in Stages I and II is the environmentally assisted cleavage process, whereas in Stage III creep assisted crack growth processes are superimposed on the cleavage mode of crack growth. Effects of stress and temperature on the fatigue crack growth behavior of the Udimet alloy are discussed in detail.     

The analysis of stress-strain field in a finite plate with a blunt crack

The expressions of stress-strain field in a finite plate with crack-tips of different radii of curvature have been derived by the method of conformal mapping-weighted residues. The stress-strain field has been evaluated for the LY12-M plate of dimensions 300mm × 150mm × 2mm under uniaxial tension. Values of σ_v computed by the present method and by Creager's formula are in agreement within 5% in the distance range of $\text{(x ? a)}\text{a}\text{?}\text{1}\text{50}$ from the crack-tip. The radius of curvature of the crack-tip has a significant effect on the stress σ_y at the tip. While it has little effect on the stress σ_x and almost no effect on the stress σ_y in the elastic region, it has a remarkable effect in the plastic region for small-scale yielding and very little effect for large-scale yielding. The effect of the radius of curvature (for $\text{ρ ?}\text{2L}\text{250}$) on the stress intensity factor at the crack-tip is negligible.Uniaxial tensile tests on 300mm × 150mm × 2mm specimens of LY12-M with crack-tips of different radii of curvature have been performed, and strain measurements made by Moire's method and electrical-resistance gauges. Experimental results show that the radius of curvature has no effect on the strain ?_y in the elastic region; measured values of ?_x and ?_y in front of the crack from the boundary of the plastic region to the edge of the specimen agree with the computed values to an accuracy of 5%; the measured values of K_I from specimens with cracks of different radii of curvature (ρ ? 2L/250) are consistent with the computed values.     

Growth of fatigue cracks in metals and polymers

Empirical data on the propagation of tensile fatigue cracks in metals and thermoplastics have been examined. It was found that a cyclic crack propagation relationship, based on the stress intensity factor concept, exists which can be successfully utilised for both types of materials.The proposed equation has a form $\text{/.a}{}_{\mathrm{x}}\text{= MA}{}^{\mathrm{n}}$ where $\text{A}$ is a function of$\text{ΔK}$ and mean $\text{K}$. The analysis of results suggests that this equation incorporating the influence of mean stress intensity factor provides an excellent fit to the investigated data. The possible modified forms of such a relationship in terms of strain energy release rate, the crack tip yielding and the crack opening displacement concepts are also indicated.     

An elastic-plastic finite element analysis of crack tip fields under biaxial loading conditions

A square plate containing a central crack and subjected to biaxial stresses has been studied by a finite element analysis. An elastic analysis shows that the crack opening displacement and stress of separation ahead of the crack tip are not affected by the mode of biaxial loading and therefore the stress intensity factor adequately describes the crack tip states in an elastic continuum.

An elastic-plastic analysis involving more than localized yielding at the crack tip provides different solutions of crack tip stress fields and crack face displacements for the different modes of biaxial loading.

The equi-biaxial loading mode causes the greatest separation stress but the smallest plastic shear ear and crack displacement. The shear loading system induces the maximum size of shear ear and crack displacement but the smallest value of crack tip separation stress.

     

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%.     

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.     

Estimation of fatigue crack propagation life of steel plates and of structural member model of ship hull

The authors carried out experimental and analytical investigation for the purpose of finding out a method of estimating the fatigue crack propagation life of large flat steel plate and of ship hull structure model quantitatively.We theoretically derived a formula indicating that fatigue crack propagation rate is in proportion to the $\text{m-}\text{th}$ power of the plastic displacement of the tip of a crack based on a B.C.S. dislocation model. The crack propagation rate is proportional to $\text{2m-}\text{th}$ power of stress intensity factor in the case stress is small.We proved experimentally that this relation holds generally, from fatigue crack propagation tests for flat plates with a center notch (mild steel, high tensile strength steel), large flat plate with an edge notch and ship hull corner model (mild steel), and from the $\text{K-}\text{value}$ calculation by the finite element method for these specimens. The fatigue crack propagation life is obtained by integrating the reciprocal number of crack propagation rate from the initial crack length to the final crack length. The life calculated agreed well with the one observed. But for the two stress level test, the life calculated was smaller than the experimental value due to slackened progress of crack. We also stated the general characteristics of the rate curve.     

A mixed-mode crack analysis of rotating disk using finite element method

This paper presents a rigorous elastodynamic hybrid-displacement finite element procedure for a safety analysis of fast rotating disks with mixed-mode cracks. Based on a modified Hamilton's principle, the finite element model is derived such that the proper crack-tip singularities are taken into consideration and the interelement displacement compatibility conditions are still satisfied. Thus, the specimen can be represented by a finite element assemblage in which “singular” elements are used around the crack-tip and high-order isoparametric “regular” elements are taken elsewhere.To determine the mixed-mode stress intensity factors, the modified $\text{J}\text{?}{}_{\mathrm{k}}$ integrals for rotating cracked disks have been established taking into account the effect of centrifugal force. Using the “strain-energy-density factor” concept, the direction of crack growth of a rotating disk with an arbitrary internal crack is predicted. To provide a method of non-destructive testing in evaluating the integrity of structures, natural vibrations of cracked disk are then studied. Lastly, the influence of inertia effects due to rotating speed changes in determining the dynamic stress intensity factors is examined.For verification purposes, the simple case of a rotating disk with radial cracks is first solved. Excellent correlations between the computed results and available referenced solutions are drawn. New solutions for the circular disk with circumferential or arbitrarily-oriented cracks are also presented.     

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.     

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.     

A method for determining the stress intensity threshold level for fatigue crack propagation

The influence of a decreasing rate of stress intensity factor with crack propagation, $\text{d}\text{K/}\text{d}\text{a}$, on a stress intensity threshold level, $\text{δK}{}_{\mathrm{th}}$, below which fatigue crack propagation becomes insignificant is investigated. Specimens, 200 mm wide, 10 mm thick with a 40 mm-long central crack, are fatigued at the decreasing rates, $\text{∥}\text{d(δK)}\text{d}\text{a∥}$, of 2,44, 5 and 10 kg/mm^5/2 with a peak load control system and a pair of crack followers. In this range of $\text{d}\text{(δK)/}\text{d}\text{a}$, the stress intensity threshold levels, $\text{δK}{}_{\mathrm{th}}$, have the same value regardless of $\text{d}\text{K/}\text{d}\text{a}$. Therefore, the present method of decreasing the stress intensity factor at a constant rate is suitable for determining the characteristic $\text{δK}{}_{\mathrm{th}}$ of materials. Furthermore, the influence of stress ratio, $\text{R}$, is investigated at the decreasing rate, $\text{∥}\text{d}\text{(δK)/}\text{d}\text{a]}$, of 10 kg/mm^5/2.     

A study of crack closure in fatigue

Crack closure phenomenon in fatigue was studied by using a Ti-6Al-4V titanium alloy. The occurrence of crack closure was directly measured by an electrical potential method, and indirectly by load-strain measurement. The experimental results showed that the onset of crack closure depends on both the stress ratio, $\text{R}$, and the maximum stress intensity factor, $\text{K}{}_{\max }$. Crack closure was not observed for stress ratio, $\text{R}$, greater than 0.3 in this alloy.A two-dimensional elastic model was used to explain the behavior of the recorded load-strain curves. Closure force was estimated by using this model. Based on the estimated closure force, the crack opening displacement was calculated. This result showed that onset of crack closure detected by electrical-potential measurement and crack-opening-displacement measurement is the same.The implications of crack closure on fatigue crack are considered. The experimental results show that crack closure cannot fully account for the effect of stress ratio, $\text{R}$, on crack growth, and that it cannot be regarded as the sole cause for delay.     

Studies on crack growth rate under high temperature creep,fatigue and creep-fatigue interaction—II: On the role of fracture mechanics parameter aeffσg

For high temperature creep, fatigue and creep-fatigue interaction, several authors have recently attempted to express crack growth rate in terms of stress intensity factor $\text{K}{}_{\mathrm{I}}\text{= α}\text{aσ}{}_{\mathrm{g}}$, where a is the equivalent crack length as the sum of the initial notch length a₀ and the actual crack length $\text{a}{}^1$, that is, $\text{a = a}{}_0\text{+ a}{}^1$. On the other hand, it has been shown by Yokobori and Konosu that under the large scale yielding condition, the local stress distribution near the notch tip is given by the fracture mechanics parameter of $\text{aσ}{}_{\mathrm{g}}\text{?(σ}{}_{\mathrm{g}}$), where a is the cycloidal notch length, σ_g is the gross section stress and $\text{?(σ}{}_{\mathrm{g}}$) is a function of σ_g. Furthermore, when the crack growth from the initial notch is concerned, it is more reasonable to use the effective crack length a_eff taking into account of the effect of the initial notch instead of the equivalent crack length a. Thus we believe mathematical formula for the crack growth rate under high temperature creep, fatigue and creep-fatigue interaction conditions may be expressed at least in principle as function of $\text{a}{}_{\mathrm{eff}}\text{σ}{}_{\mathrm{g}}\text{, σ}{}_{\mathrm{g}}$ and temperature.In the present paper, the geometrical change of notch shape from the instant of load application was continuously observed during the tests without interruption under high temperature creep, fatigue and creep-fatigue interaction conditions. Also, the effective crack length a_eff was calculated by the finite element method for the accurate estimation of local stress distribution near the tip of the crack initiated from the initial notch root. Furthermore, experimental data on crack growth rates previously obtained are analysed in terms of the parameter of $\text{a}{}_{\mathrm{eff}}\text{σ}{}_{\mathrm{g}}$ with gross section stresses and temperatures as parameters, respectively.     

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.     

On crack opening stress level in fatigue by Eddy current technique

Linear elastic fracture mechanics relates fatigue crack growth with the stress intensity factor at the crack tip. Presence of residual deformations at the tip of a fatigue crack reduces the crack tip stress intensification such that effective stress intensity range $\text{ΔK}{}_{\mathrm{e}}\text{= U · ΔK}$. In this paper use of eddy current technique is exhibited to find the values of test value of effective stress range factor $\text{U}{}_{\mathrm{test}}$. A reasonable comparison between computed and experimental results of $\text{U}{}_1$ and $\text{U}{}_{\mathrm{test}}$ on two Al alloys 6061-T6 and 6063-T6 has recommended the Eddy Current Technology for finding out the values of crack opening stress level under given loading conditions.     

A multilayer hybrid-stress finite element analysis of a through-thickness edge crack in A ± 45° laminate

A solution is given for the three-dimensional stress field near a through-thickness edge crack in a thin ± 45° laminate having elastic ply moduli typical of graphite/epoxy. The stress distribution was obtained by a three-dimensional multilayer finite element analysis based on the hybrid stress model, formulated through the minimum complementary energy principle. The results indicate that the in-plane stresses of each individual ply follow the classical $\text{1}\text{√r}$ stress singularity, but that the shape of isostress contours in the crack tip region is strongly distorted from predictions based on two-dimensional anisotropic fracture mechanics theory. The interlaminar shear stresses increase rapidly as the crack tip is approached, but are restricted to a local region around the crack tip and flanks. The interlaminar normal stress is assumed to be negligible in the formulation of the analysis.     

Fatigue crack tip strain loop under single overload in Fe3Si steel

For the problem of the retardation of fatigue crack growth caused by single overload, cyclic strain changes in one cycle of the local crack tip region were investigated by using fine-grid-method. From the application of Crack Tip Strain Loop (C.T.S.Loop), the correlation between effective stress intensity factor range (ΔK_eff^T) defined from the stress range of C.T.S.Loop and crack growth rate ($\text{d}\text{a}\text{d}\text{N}$) was discussed. Moreover, from the strain range of C.T.S.Loop, the correlation between crack-tip strain range (Δ_?^T) and $\text{d}\text{a}\text{d}\text{N}$ was also investigated.