Optimum strain gage location for evaluating stress intensity factors in single and double ended cracked configurations

A general methodology has been proposed for calculation of optimum radial strain gage locations for measurement of the stress intensity factors using strain gage technique. The upper bound (r_max) of strain gage locations for complex single ended and double ended cracked configurations has been determined using the proposed method. Further, dependency of the r_max on the crack length to width ratio and on the state of stress is investigated. Numerical results obtained from the present investigation are observed to be in accordance with the theoretical predictions. Using the proposed approach the correctness of strain gage locations used by the earlier researchers is also verified.     

Three-dimensional stress intensity factors of a central square crack in a transversely isotropic cuboid with arbitrary material orientations

In this paper, we present the dual boundary element method (dual-BEM) or single-domain BEM to analyze the mixed three-dimensional (3D) stress intensity factors (SIFs) in a finite and transversely isotropic solid containing an internal square crack. The planes of both the transverse isotropy and square crack can be oriented arbitrarily with respect to a fixed global coordinate system. A set of four special nine-node quadrilateral elements are utilized to approximate the crack front as well as the outer boundary, and the mixed 3D SIFs are evaluated using the asymptotic relation between the SIFs and the relative crack opening displacements (COD) via the Barnett–Lothe tensor.Numerical examples are presented for a cracked cuboid which is transversely isotropic with any given orientation and is under a uniform vertical traction on its top and bottom surfaces. The square crack is located in the center of the cuboid but is oriented arbitrarily. Our results show that among the selected material and crack orientations, the mode-I SIF reaches the largest possible value when the material inclined angle ψ₁=45° and dig angle β₁=45°, and the crack inclined angle ψ₂=0° and dig angle β₂=0°. It is further observed that when the crack is oriented vertically or nearly vertically, the mode-I SIF becomes negative, indicating that the crack closes due to an overall compressive loading normal to the crack surface. Variation of the SIFs for modes II and III along the crack fronts also shows some interesting features for different combinations of the material and crack orientations.     

Fracture toughness and crack morphology in indentation fracture of brittle materials

The relationship between the indentation fracture toughness, K _c, and the fractal dimension of the crack, D, has been examined on the indentation-fractured specimens of SiC and AIN ceramics, a soda-lime glass and a WC-8%Co hard metal. A theoretical analysis of the crack morphology based on a fractal geometry model was then made to correlate the fractal dimension of the crack, D, with the fracture toughness, K _IC, in brittle materials. The fractal dimension of the indentation crack, D, was found to be in the range 1.024–1.145 in brittle materials in this study. The indentation fracture toughness, K _c, increased with increasing fractal dimension, D, of the crack in these materials. According to the present analysis, the fracture toughness, K _IC, can be expressed as the following function of the fractal dimension of the crack, D, such that $$In K_{IC} = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\{ In[2\Gamma E/(1 - \nu ^2 )] - (D - 1)In r_L \}$$ Where Γ is the work done in creating a unit crack surface, E is Young's modulus, v is Poisson's ratio, and r _L is r _min/r _max, the ratio of the lower limit, r _min, to the upper limit, r _max, of the scale length, r, between which the crack exhibits a fractal nature (r _min ?r?r _max). The experimental data (except for WC-8%Co hard metal) obtained in this study and by other investigators have been fitted to the above equation. The factors which affect the prediction of the value of K _IC from the above equation have been discussed.     

Shakedown of a cracked body consisting of kinematic hardening material

In this paper, the shakedown behaviour of a cracked body is studied. The main idea is to consider the crack as a notch. Then no singular stresses appear at crack tip. Due to the local character of the problem, Melan's shakedown theorem is used. By solving shakedown as an optimization problem, the limited stress intensity factor (SIF) for shakedown K_sh is obtained. It is found that the shakedown limit SIF of a cracked body is proportional to the initial yield stress σ_y of the material times the square root of the effective crack tip radius π, i.e. K_sh ∝ σ_y√ϱ. Comparison of shakedown limit SIFs with fatigue thresholds for certain materials, so far as can be found in literature, shows that these two quantities agree well with each other. This agreement indicates that shakedown of the cracked body is one of the reasons for arrest of the crack under cyclic loads. Shakedown investigation is then a new method for predicting the fatigue threshold of a cracked body. Thus, a transition from shakedown to cyclic fracture mechanics has been achieved.     

Numerical models verification of cracked tubular T, Y and K-joints under combined loads

This paper summarizes the key steps involved in the construction of an accurate and consistent finite element model for general cracked tubular T, Y and K-joints. The joint under consideration contains a surface crack which can be of any length and located at any position along the brace-chord intersection. Welding details along the brace-chord intersection, compatible with the American Welding Society (AWS) specifications [American Welding Society. Structural welding code-steel, ANSI/AWS, 15th ed., Miami, 2000], are included in the geometrical model. In order to develop a systematic and consistent modelling procedure, the whole process is divided into four key steps. They are, namely, (1) construction of a consistent geometrical model of the joint with welding details, (2) determination of cracked surface to define the semi-elliptical surface crack profile, (3) generation of well-graded finite element meshes, and (4) stress intensity factor studies around the crack front. To produce a well-graded finite element mesh, a sub-zone technique is used in the mesh generation whereby the entire structure is divided into several sub-zones with each zone consisting of different types of elements and mesh densities. The stress intensity factors (SIFs) are evaluated using the standard J-integral method. Two full-scale T and K-joint specimens were tested to failure under axial load (AX), in-plane bending (IPB), and out-of-plane bending (OPB). In the tests, the rate of crack propagation was monitored carefully using the alternating current potential drop (ACPD) technique. Using the known material parameters C and m, the experimental SIFs were obtained, and they are found to be in complete agreement with the computed SIFs obtained from the generated models. Hence, the proposed finite element models are both efficient and reliable.     

The fatigue crack growth rate and crack opening displacement in 18G2A-steel under tension

In the paper, the results of crack tip opening displacement (CTOD) and crack opening displacement (COD) in place of crack initiation as well as the fatigue crack growth rate in higher strength steel are presented. The investigation were carried out on flat specimens with central notch under constant amplitude tensile fatigue loading at stress ratio R = 0.2 and different value of the stress σ_max. The test results showed that with growth of crack length l grew values of the CTOD and COD. In the work, it was proposed calculation of the CTOD value on basis various dependence of plastic zone radius on crack tip.     

3-D stress intensity factors for arrays of inner radial lunular or crescentic cracks in a typical spherical pressure vessel

Many spherical pressure vessels are manufactured by methods such as the integrated hydro-bulge forming (IHBF) method, where the sphere is composed of a series of double curved petals welded along their meridional lines. Such vessels are susceptible to multiple radial cracking along the welds. For fatigue life assessment and fracture endurance of such vessels one needs to evaluate the stress intensity factors (SIFs) distribution along the fronts of these cracks. However, to date, only two 3-D solutions for the SIF for one inner semi-elliptical crack in thin or thick spheres are available, as well as 2-D SIFs for one through-the-thickness crack in thin spherical shells. In the present paper, mode I SIF distributions for a wide range of lunular and crescentic cracks are evaluated. The 3-D analysis is performed, via the FE method employing singular elements along the crack front, for a typical spherical pressure vessel with outer to inner radius ratios of η = R_o/R_i = 1.1. SIFs are evaluated for arrays containing n = 1-20 cracks; for a wide range of crack depth to wall thickness ratio, a/t, from 0.025 to 0.95; and for various ellipticities of the crack, i.e., the ratio of crack depth to semi crack length, a/c, from 0.2 to 1.5. The obtained results clearly indicate that the SIFs are considerably affected by the three-dimensionality of the problem, and the following parameters: the number of cracks in the array-n, the relative crack depth a/t, and the crack ellipticity a/c.     

Crack closure under high load-ratio conditions for Inconel-718 near threshold behavior

Fatigue-crack-growth (FCG) rate tests were conducted on compact specimens made of an Inconel-718 alloy to study the behavior over a wide range in load ratios (0.1 ? R ? 0.95) and a constant K_max test condition. Previous research had indicated that high R (>0.7) and constant K_max test conditions near threshold conditions were suspected to be crack-closure-free and that any differences were attributed to K_max effects. During a test at a load ratio of 0.7, strain gages were placed near and ahead of the crack tip to measure crack-opening loads from local load-strain records during crack growth. In addition, a back-face strain (BFS) gage was also used to monitor crack lengths and to measure crack-opening loads from remote load-strain records during the same test. The BFS gage indicated that the crack was fully open (no crack closure), but the local load-strain records indicated significant amounts of crack closure. The crack-opening loads were increasing as the crack approached threshold conditions at R = 0.7. Based on these measurements, crack-closure-free FCG data (ΔK_eff against rate) were calculated. The ΔK_eff-rate data fell at lower ΔK values and higher rates than the constant K_max test results. In addition, constant R tests at extremely high R (0.9 and 0.95) were also performed and compared with the constant K_max test results. The constant R test results at 0.95 agreed well with the ΔK_eff-rate data, while the R = 0.9 data agreed well with constant K_max test data in the low-rate regime. These results imply that the R = 0.7 test had a significant amount of crack closure as the threshold was approached, while the R = 0.9 and K_max test results may have had a small amount of crack closure, and may not be closure free, as originally suspected. Under the high load-ratio conditions (R ? 0.7), it is suspected that the crack surfaces are developing debris-induced crack closure from contacting surfaces, which corresponded to darkening of the fatigue surfaces in the near-threshold regime. Tests at low R also showed darkening of the fatigue surfaces only in the near-threshold regime. These results suggest that the ΔK_eff against rate relation may be nearly a unique function over a wide range of R in the threshold regime.     

The effect of mean stress on delay in fatigue crack growth under load stepdown

The effect of mean stress together with decreasing stress range on fatigue crack propagation behaviour in mild steel is investigated. The delay period between crack arrest and reprogation is found to be a function of the maximum stress intensity factor stepdown ration, K_2max/K_1max. Delay only occurs when this ratio is less than unity. For specimen thicknesses of 1.6 to 6.4 mm, non-propagating cracks, where the affected delay cycles are 500 000 cycles or greater, appear to occur when K_2max/K_1max has a value of approximately 0.7 and the stepdown plastic zone size is about half the initial load plastic zone size, which is approximately equal to the affected crack length.     

Measuring overload effects during fatigue crack growth in bainitic steel by synchrotron X-ray diffraction

In this work we present the results of in situ synchrotron X-ray diffraction measurements of fatigue crack-tip strain fields following a 100% overload (OL) under plane strain conditions. The study is made on a bainitic steel with a high toughness and fine microstructure. This allowed a very high (60 μm) spatial resolution to be achieved so that fine-scale changes occurring around the crack-tip were captured along the crack plane at the mid-thickness of the specimen. We have followed the crack as it grew through the plastic/residually stressed zone associated with the OL crack location. We observed two effects; one when the enhanced plastic zone is ahead of the crack and one after it has been passed. Regarding the former it was found that the compressive stress at the crack-tip initially falls sharply, presumably due to the increased plastic stretch caused by the OL. This is associated with a concomitant fall in peak tensile stress at K_max, the elastic excursion between K_min and K_max remaining essentially unchanged from before OL. Subsequently discontinuous closure as seen previously for plane stress caused by crack face contact at the OL location limits the elastic strain range experienced by the crack tip and thereby retards crack growth.     

Modeling of fatigue crack growth threshold development for a microstructurally small crack

A method for predicting the fatigue crack growth threshold using finite element analysis is investigated. The proposed method consists of monitoring the plastic strain hysteresis energy dissipation in the crack tip plastic zone, with the threshold being defined in terms of a critical value of this dissipated energy. Two-dimensional plane-strain elastic-plastic finite element analyses are conducted to model fatigue crack growth in a middle-crack tension M(T) specimen. A single-crystal constitutive relationship is employed to simulate the anisotropic plastic deformation near the tip of a microstructurally small crack without grain boundary interactions. Variable amplitude loading with a continual load reduction is used to generate the load history associated with fatigue crack growth threshold measurement. Load reductions with both constant load ratio R and constant maximum stress intensity K_max are simulated. In comparison with a fixed K_max load reduction, a fixed R load reduction is predicted to generate a 35% to 110% larger fatigue crack growth threshold value.     

Experiments under pure shear and rolling contact fatigue conditions: Competition between tensile and shear mode crack growth

In the present paper, the mechanism of shear crack growth under both pure torsion and mixed mode loadings, simulating rolling contact fatigue testing conditions, has been investigated for a bearing steel and the role of the superimposed compressive stress in subsurface RCF has been clarified both numerically and experimentally. In particular a previous data set of fatigue tests on micro-notched specimens subjected to torsion and out-of-phase loads with |σ_min|/τ_max ≅ 3.5 (LP1) has been complemented with the new tests onto micro-notched specimens loads with |σ_min|/τ_max ≅ 0.7 (LP2) and a test under pure compression. The same tests have been also simulated numerically with a non-linear FE analysis of crack advance. The numerical analyses have been conducted with the aim of demonstrating that the compressive stress fully suppresses the tendency to tensile mode growth as the crack extends.Eventually, the competition between tensile and shear mode growth during a fatigue cycle has been investigated theoretically in terms of local branch SIFs. In particular, the conditions for the branch crack growth have been examined on the basis of the effective SIFs: the crack tip shielding effects due to the crack surface interference (both the mode I contribution caused by the asperity mismatch and the shear attenuation produced by the frictional stresses) have been quantified by employing a model for crack sliding interaction under pure mode III and mixed mode I + III loadings.     

A numerical investigation of 3-D small-scale yielding fatigue crack growth

The 3-D small-scale yielding (SSY) model provides a computational framework to study fatigue crack growth in thin, metallic components (and test specimens) containing an initially sharp, straight-through crack. This work describes a finite element study of plasticity-induced crack closure in the 3-D SSY model under mode I, constant amplitude cyclic loading with a ratio R=K_min/K_max. A purely kinematic hardening model with constant modulus represents the material constitutive behavior. This paper first addresses key computational issues and proposes modeling guidelines leading to 3-D numerical results for fatigue crack growth in SSY that exhibit convergence with mesh refinement. Specifically, computed crack opening loads show an independence of finite element mesh refinement when (a) the plastic zone at peak load encloses more than 10 eight-noded brick elements (b) the reverse plastic zone encloses at least two elements, and (c) the half-thickness has at least five element layers. The paper also describes stress and deformation fields at the crack front for a growing fatigue crack and provides an understanding of localized 3-D effects on the normalized remote opening load value K_op/K_max. In addition, the computational studies demonstrate that the similarity scaling relationship established for R=0 [Roychowdhury and Dodds, Fatigue Fract. Engng. Mater. Struct. (accepted for publication)] also holds for the non-zero ratio R=0.1--a value commonly adopted in experimental programs. In particular, K_op/K_max, at each location along the crack front remains unchanged when the peak load (K_max), thickness (B) and material flow stress (σ₀) all vary to maintain a fixed value of .     

R-curve behaviour of a structural steel

The ductile tearing behaviour-of BS4360 50D structural steel has been studied using two bending specimen geometries: square (W = B) cross section three point bend specimens with different initial fatigue crack length, ranging from a/W = 0.20 to about 0.75, and rectangular cross section (W = 2B) specimens with initial fatigue crack length a/W = 0.3.With the first set of specimens it was intended to investigate the possible dependence of the tearing behaviour on initial crack length. It was concluded that specimens with shorter values of initial crack length present higher resistance to ductile tearing. Two different methods of calculating the J-resistance curve, one based on the approximate equation J = 2A/(W?a)B and a more elaborate calculation procedure introduced by Garwood were used. The curves determined following the more accurate procedure were used to determine the value of the tearing modulus T, which is related to the occurrence of tearing instabilities in structures. It was found that T decreases slightly with increasing value of initial crack length.The tests using the wider specimen geometry were intended to examine the tearing behaviour over a larger amount of crack growth allowed by the wider geometry used. It was found that the resistance curves, using the COD concept, present a maximum after which the resistance begins to drop. This maximum value of the COD-resistance curve is higher than the conventional δ_max measured at max load. These tests were performed using the compliance technique of evaluating the crack length. Tests performed with 10% unloading presented slightly lower values of resistance than tests performed with complete unloading.It was found that the relationship J = mσ_flowδδ presents increasing values of m with crack growth. This is attributed to an increasing local yield stress, caused by the through thickness deformation.     

A Study of X-Ray analysis of fatigue fracture surface

An X-ray analysis was made of the fatigue fracture surface of SM50A steel. The residual stress and the half-value breadth of the diffraction intensity curve were examined on and under the fracture surface. An analytical study was also made of the residual stress and strain near the fatigue crack surface by using the finite element method. Emphasis was on finding the correlation between the residual stress or the half-value breadth and the applied stress intensity factor of K_max or 2K. It was found that both the residual stress and the halfvalue breadth distributions under the fracture surface were useful for the estimation of the monotonie plastic zone size or K_max.The correlation of the absolute value of the residual stress with K_max or 2K, however, was not clear in the present study. From the analytical study, it was found that the effect of the surface contact due to fatigue crack closure on the residual stress was less important and limited to the thin layer near the crack surface.     

A CTOD equation based on the rigid rotational factor with the consideration of crack tip blunting due to strain hardening for SEN(B)

Crack tip opening displacement (CTOD) from national and international standards was shown to give different values. This paper investigates the feasibility of CTOD determined based on the concept of rigid rotational factor in single‐edge notched bend (SENB) specimens. Based on validated modelling methods, finite element (FE) models were simulated for crack ratios 0.3 ≤ a₀/W ≤ 0.7 and yield‐to‐tensile ratio 0.44 ≤ σ_ys/σ_uts ≤ 0.98. This covers cases of shallow to deeply cracked specimens and a wide range of strain hardening properties. CTOD obtained from the FE models was used as the basis of a newly implemented strain hardening corrected rotational factor, which considers the effects of crack tip blunting due to strain hardening, r_{p sh}. An improved equation considering strain hardening was implemented based on the r_{p sh}. The equation gives accurate estimation of CTOD from the FE models compared with the equation from BS 7448‐1, ASTM E1820, and WES 1180.     

Some problems in experimental fracture mechanics

A standard procedure for the determination of fracture toughness K_IC is discussed. The insufficiency of the existing K_ic determination confidence criteria is stressed and the following criteria are proposed instead: φ_max ? 1.5%; $\text{σ}{}_{\mathrm{fr}}{}^{\mathrm{net}}\text{σ}{}_{\mathrm{0.2\; ?\; 0.8}}$, in conjunction with the old criterion $\text{P}{}_{\max }\text{P}{}_{\mathrm{Q}}\text{? 1.1}$. Determination of K_IC from P_max should be used instead of from P_Q.A method for the determination of a point on the “force-displacement” diagram corresponding to crack growth initiation is set forth. The method is based on specimen compliance tests under repeated load-relief cycles. The crack growth initiation point is used to determine both the critical crack opening and plane strain fracture toughness. The indefinite effect of the growing crack (in the ease of crack opening or Cherepanov-Rice integral calculations) is thereby eliminated. Necessity is emphasized to determine the share of the J-integral which contributes to fracture process. A method for plotting the elastic displacement diagram is proposed which allows on the basis of preliminary estimates to determine fracture toughness of small-sized specimens without using special setups. The area ratio between the plastic and elastic strain diagrams is proposed to be adopted as fracture type criterion. Certain experiments to determine crack resistance of material specimens are described and discussed.     

Progressive edge cracked aluminium plate repaired with adhesively bonded composite patch under full width disbond

In this study, the crack growth behaviour of an aluminium plate cracked at the tip and repaired with a bonded boron/epoxy composite patch in the case of full-width disbond was investigated. This effect is the imperfection which could result during the bonded patch of the repaired structure. Disbonds of various sizes and situated at different positions with respect to the crack tip as well as the effect of adhesive and patch thickness on repair performance were examined. An analysis procedure involving the efficient finite element modelling applied to cracked plate, adhesive and composite patch was used to compute the stress intensity factors. The crack growth rate is dominated by the stress intensity factor near the location and size of the pre-existing disbonds. The cracked plate and disbond propagation result in an increase in the patch deformation. The patch does not have an influence on the crack growth when the ratio 2a/d_R exceeds 0.8.     

Representation of stress intensity factors by path integrals of the first stress invariant or anti-plane displacement for general two-dimensional static problems

Determining stress intensity factors (SIFs) is a difficult task either analytically or experimentally. The difficulty arises from the fact that there is no simple and accurate expression for the SIFs under general circumstances. As a result, the determination of the SIFs is usually a complex process. For finding a suitable expression for the SIFs, the first stress invariant and anti-plane displacement are analyzed, and Green's theorem is used. It is found that the stress intensity factors can be represented by path integrals involving only the first stress invariant or anti-plane displacement for general two-dimensional static problems. K _I and K _II are represented by path integrals of the first stress invariant and its partial derivative. K _III is represented by a path integral of the anti-plane displacement as well as its partial derivative. The integrals are path-independent and valid for an arbitrarily shaped elastic medium with stationary cracks of arbitrary shape. They are also valid for a body containing isolated inhomogeneities such as holes and inclusions. If a crack is straight near its tip, and if the straight portion of the crack can be treated as a cut along the radius of a simply connected circular disk, there exists another kind of integrals representation that does not include the partial derivative terms in the representation for K _I. The representation by these integrals provides a new approach to determine the SIFs experimentally, which is simpler and more accurate. This is because the integrals are exact expressions for the SIFs and involve only the first stress invariant or anti-plane displacement.     

The effect of crack tunneling on crack growth: experiments and CTOA analyses

This paper compares experimental crack-front shapes recorded at various stages of crack growth with area-average crack growth values during fracture tests conducted on 2024-T351 aluminum alloy plate. Crack-front shapes were determined by fracturing the specimen to a predetermined amount of crack growth and fatigue cycling the specimen for about 4000 cycles at a high stress ratio (P_min/P_max) to mark the crack-front location. For each shape, the area-average crack length was determined. The evolution of tunneling was used to create a calibration curve that could be used to adjust surface measured crack-length values, for a more representative comparison with analyses that use a straight crack-front approximation. The analysis compares much more favorably with the average crack growth than with the surface measured values near maximum load. However, the area-average technique tends to over correct crack growth near the crack initiation load. Crack tunneling results show that the area-average technique produces more representative crack-length measurements compared to optical based surface measurements.