Formula For Stress Intensity Factor Research Articles

Investigation of the Fitness for Service (FFS) of Cracks in API 5L X70 Pipeline Steel using Failure Assessment Diagram (FAD)

In this study, the fitness for service of crack propagation in API 5L X-70 stee1 was investigated using a model called Failure Assessment Diagram (FAD) to determine how fit a crack can be under certain operational pressure. It is a known fact that during the production of pipes, there are tendencies for flaws such as inclusions and cracks to occur in the pipes. When these flaws are subjected to stresses, there are tendencies for failures to occur starting from where the cracks or flaws are located. The failure due to the propagation of the cracks leads to oil spillage causing pollution to the environment which had negatively impacted livelihood of the host communities and aquatic lives. This had resulted in Government spending huge amount of money maintaining the pipelines and remediation. The purpose of this paper is to investigate the fitness for service of crack lengths 12 mm, 17 mm, 22 mm and 27 mm using the Failure Assessment Diagram (FAD) mode1. The material used in this paper is API 5L X 70 steel in the form of a Compact Tension (CT) specimen machined according to ASTM E1820 – 13. API 5L X-70 stee1 is a low-carbon stee1 with a carbon content of as low as 0.04 %. It is used in the production of pipelines for conveying crude oil and natural gas from the place of production to the place of refining or export. In the investigation of the fitness for service of the cracks, a charpy V-notch impact test was carried out to determine the energy required to fracture the steel, which was later inputted numerically into a critical stress intensity factor formula in accordance with BS 7910 – 13 standards to obtain the critical stress intensity factor (KIC or KQ). The stress intensity factor (KI) was obtained from formula also according to BS 7910 – 13 standards. The ratio of KI to KQ was used in the FAD analysis. Subsequently, a monotonic tensile test was conducted to obtain the yield stress ( ys) and the reference stress ( ref) was obtained numerically according to BS 7910 – 13. The ratio of ref) to ys) was also used in the FAD analysis. The FAD analysis was used to determine the fitness for services and fracture behaviour of each crack. Scanning electron microscopy (SEM) was used to confirm the fracture behaviour obtained from the FAD. The results obtained show that the energy from the charpy V-notch impact test was 302.9 J and the critical stress intensity factor (KQ) correlated numerically according to BS 7910 – 13 was determine as 246.73 MPa . The yield stresses obtained from the monotonic test for crack lengths of 12 mm, 17 mm, 22 mm and 27 mm were 132.51 MPa, 109.10 MPa, 114.36 MPa and 118.21 MPa, respectively. In the FAD analysis, it was observed that the safe operational stress to ensure fitness for service decreases with an increase in crack length. The fracture behaviour shows a ductile fracture behaviour since the FAD lies within the plastic collapse region. This fracture behaviour was confirmed by the image obtained from the scanning electron microscopy (SEM), which showed a cup and cone image suggesting ductile fracture behaviour. This FAD method will ensure that safe operational stresses are maintained for various crack length to prolong the life span of the pipeline. It is a novel method that can also be used to properly schedule the rate of inspection in pipelines alongside Ultrasonic sound, Liquid penetrant, Magnetic particle and radiographic methods of inspection.

SIF formula based on exact SIF distribution for semi‐elliptical surface cracks subjected to mode I, II, III uniform loading

AbstractThe stress intensity factor (SIF) distribution of a semi‐elliptical surface crack is often used for evaluating the fatigue strength of structures. Virtually exact distributions of SIFs can be provided along the crack front by solving the hypersingular integral equation of the body force method. In this paper, to create a very accurate SIF variation formula, the elliptical crack SIF solutions , , are used and the SIF ratios , , are mainly focused. Paying attention to the corner point singularity, by applying the least squares method to the ratio , in the whole range of parametric angle , and formulas can be proposed. Instead, formula can be proposed by applying the method of least squares to the ratio of in the range but applying directly to in the range . In this way, all of formulas proposed in this paper provide the SIFs with better than 1.0% accuracy.

Theoretical analysis and fracture process of elliptical tunnel with symmetric cracks subjected to water pressure and ground stress

The water diversion tunnel played an important role in the allocation of water resources in the water diversion project. In the process of actual use, some cracks or joints at the edge of the tunnel could produce further cracking under the action of internal water pressure and in-situ stress, which was dangerous to the safety of the tunnel. The analytical solution formulas of stress intensity factors (SIFs) of the elliptical tunnel with two cracks were derived by using the complex function method, and the external factors affecting the stability of the elliptical tunnel were analyzed. The failure process of the specimens was analyzed by using the high-speed camera and digital image correlation (DIC) method to obtain the evolution laws of crack initiation behavior and tunnel fracture modes. The results can conclude that both sides of the elliptical tunnel and the crack tip were stress concentration areas. As the inclination angle α of the elliptical tunnel is varied from 0° to 45°, tensile cracks occur at the top and bottom of the elliptical tunnel caused by tensile stress, and the compressive strength is smallest when the elliptical tunnel with α is 45°. As the inclination angle α is 75°, two penetrating cracks appear at the edge of the elliptical tunnel and crack tips. The initiation pressure increases with the inclination angle α. The SIF of mode II rises with the ellipse major semi axis and the crack length.

Theoretical analysis of two cracks in sandstone under the simultaneous action of pressure and shear stress

AbstractThe cracked rock mass could slide along the weak side under the action of compressive loads and shear loads in landslides; thus, the stability evaluation of the crack in rock mass media is of particular importance. The formulas of stress intensity factors (SIFs) for each crack tip are theoretically derived by implementing complex function theory. The results of the analytical solutions are then compared with the predicted results of the finite element analysis via ABAQUS software, revealing a slight relative error. Photo elastic experiment and static compression tests are conducted to verify the correctness of theoretical results. The obtained results reveal that the internal pressure leads to the increasing SIFs of mode II at crack tips. Further, the larger the difference between the horizontal and vertical compressive stresses, the more likely the occurrence of shear failure in the cracked rock mass. For the two cracks, the most dangerous point is the inner crack tip of the longer crack.

Study on the global stress field of the rock mass under compression and shear considering three kinds of crack parameters

AbstractThe mechanical behavior of the cracked rock mass is controlled by the crack initiation and propagation under compression and shear. The key to predicting the wing crack initiation angle is to correctly describe the distribution of the stress field at the crack tip. First, the previous studies are reviewed, and their inadequacies are discussed in detail. Next, according to the experimental results, three kinds of crack parameters (i.e., geometry, strength, and deformation) are introduced, and a stress transfer model on the crack face under compression and shear considering crack parameters is proposed. Then, the relevant formulae of stress intensity factors (KI and KII) and non‐singular term (T‐stress) at the crack tip are obtained. Finally, the proposed model is verified by other researchers' test results, and the effects of three kinds of crack parameters and the rock property on the distribution of tangential stress at the crack tip are discussed.

Crack growth life calculation approach of surface cracked landing gear strut under eccentric compression load by bulging effect

The landing gear strut is mainly subjected to eccentric compression load during service, and the calculation of its stress intensity factor is often considered as a challenge to calculate the surface crack growth life. Based on the bulging effect, we modify the stress intensity factor of the surface cracked cube, and obtain the stress intensity factor formula of the surface cracked strut. It calculates the stress intensity factor by using geometric characteristics and load characteristics, and calculates the crack growth life without meshing finite element model. In the study of surface fatigue crack growth of steel strut under eccentric compression load, the proposed approach to calculate the crack growth life of surface cracked strut is proved to be feasible. The method is simple and conservative.

Experimental study on fatigue crack propagation in balcony opening corners of a cruise ship

The balcony opening corner in a cruise ship is subjected to highly stress concentration and alternating loading during operation, which is prone to cause fatigue crack initiation and further crack propagation. These initiated cracks are difficult to find in time due to these opening corners being hidden in the decoration materials. It is necessary to assess the residual fatigue lives accurately and provide reasonable repairing plans for these cracked opening corners to ensure safety of the cruise. In this paper, in order to study the fatigue crack propagation (FCP) law of the balcony opening corner, make up the lack of the FCP experiment of the balcony opening corners, correlate the FCP behavior of the full scale model with the reduced scale model, and provide an accurate FCP live prediction method for balcony opening corners, reduced scale models based on the similarity theory and the corresponding fixtures are designed. FCP tests of these reduced scale models with small initial cracks are carried out. a-N curves of the cracks at different notch locations in balcony opening corners under different stress ratios are obtained. Formulas of stress intensity factors (SIFs) of cracks at the midpoint of the three types of corners are given by fitting the data obtained by a serial of FE analysis of the cracked corners. Further, the fatigue lives of the tested specimens are calculated and compared with the test data. It is shown that the prediction results fit well with the model test results. The procedure of model test and experimental data can provide a reference for detailed design and precise residual FCP life assessment of opening corners in ship hulls.

Fracture Characteristics of Sliding Crack in Brittle Rock: Analysis Based on an Improved Equivalent Crack Model

Under low confinement axial compression, the failure of brittle rock is mainly caused by tensile fracture. Many scholars adopt the sliding crack as an idealized model to present the intrinsic mechanism of the tensile failure but due to the complex configuration of sliding crack, its stress intensity factor (SIF) calculation has always been a difficult problem. In this study, an improved model of sliding crack is proposed; in the context of linear elastic fracture mechanics (LEFM) and weight function method, the expression of stress intensity factor (SIF) has been derived. The propagation manners of sliding crack under axial loading and lateral unloading conditions have been further analyzed. The extended finite element method (XFEM) is employed to verify the correctness of the theoretical SIF formulation and its inferences. The formula of SIF shows that a sliding crack is highly sensitive to the change of the lateral stress, which theoretically explains compressive failure characteristics of brittle rock as follows: 1) under the condition of axial compression, increasing the lateral stress has a very strong no-linear impact on the strength of brittle rock; 2) under the condition of lateral unloading, the destruction of rock is more abrupt and ferocious than that of the loading case. In order to confirm that micro-fractures in rocks are notably influenced by confining pressure, as the former theoretical fracture analysis predicted, the tri-axial compression test combined with the acoustic emission monitoring technique has been conducted on basalt samples. According to the acoustic waveform parameter method, it shows that increasing the confining pressure will greatly reduce the proportion of tension-type fractures, which indirectly proves the correctness of the sliding crack hypothesis and fracture analysis.

Stress intensity factors and T-stress for an edge cracked Brazilian disk specimen under diametrically distributed load

Analytical formulae of stress intensity factors (SIFs) and T-stress in an edge cracked Brazilian disc (ECBD) subjected to diametrically uniformly and parabolic distributed pressure were firstly derived with the weight function method (WFM). Meanwhile, these analytical expressions of SIFs and T-stress were verified to be very accurate through the comparative analysis with the existing results, which could be expediently used to calculate the SIFs and T-stress in ECBD specimens with normalized crack length less than 0.50. Furthermore, load distribution form and load distribution angle affecting SIFs and T-stress was also investigated, which indicated that their influences were closely related to the crack length and crack inclination angle. The values of YI and T* decreased with increasing load distribution angle, however, YII was increased for θ = 30°, a/D < 0.25. It was also found that load distribution form and distribution angle had almost no effect on SIFs for a/D > 0.35, but their effects on T-stress should be considered for large crack inclination angles. Moreover, the influence of uniformly distributed pressure on SIFs and T-stress was more significant than that of parabolic distributed pressure for a larger load distribution angle.

Effects of rivet hole arrangement on fatigue performance of thin sheets for fuselage: DIC and numerical calculation

The fatigue performance and fracture mechanics of 2198-T8 Al-Li thin sheet with multi-diameter rivet holes were studied experimentally and numerically, which is mainly used in the new generation of C919 aircrafts. Firstly, the fatigue crack propagation mode under uniaxial fatigue load was investigated based on digital image correlation and virtual extensometer. Then, the Stress Intensity Factor (SIF) formula was modified and the applicable range was determined based on the tests and finite element method simulation results for the specimen with single rivet hole. Finally, the modified SIF formula was applied to predict the fatigue life of specimen with multi-diameter rivet holes. The results show that the crack propagation mode transforms from mode I to mode I+II, and the modified SIF formula could be used to obtain a more conservative fatigue result for single rivet hole when the angle of pre-crack is less than 5°. For the specimen with multi-diameter rivet holes, the fatigue life decreases with the rise of diameter of the larger rivet hole Φ1, and it also decreases with the increase of difference between larger and smaller hole Φ1−Φ2 when larger diameter Φ1 is constant. The fatigue life of HH-type specimen is generally smaller than that of the SS-type under the same rivet hole arrangement. The prediction result is conservative with a maximum relative error less than 7.0%

A novel methodology for determining creep crack initiation and growth properties using FEM with notched small punch specimens

Determination of material properties on creep crack initiation (CCI) and creep crack growth (CCG) using small punch tests is still a virgin field to be exploited. Spanish and Czech researchers have carried out pioneering work and put forward an approach to correlate the notched specimen with the standard CT specimen. The advantage is that, the formulas of C*-integral for CT specimen are available and can be used directly. However, building a big data pool with both SP and CT tests within the creep region is a daunting task and hence it is worthwhile to derive C*-integral formulas for notched specimens themselves. The research is novel in that it proposes a methodology to derive the key parameter C*-integral for notched small punch specimens directly without supporting known CT tests. A 3D finite element model (FEM) for through-wall notched specimen with different crack lengths is build, and a general formulas for stress intensity factor (SIF) K, and limit load PL are obtained to calculate C*-integral for notched specimen. Experiments on notched specimens are carried out for stainless steel Sanicro 25 at temperature 700 °C. By using general formulas derived in this study, the key parameter C*-integral is obtained. The relation between C*-integral and the creep initiation time ti can be drawn as a straight line in a double logarithmic coordinate. The relation between C*-integral and the creep crack rate ȧ appears to follow the same tendency. Material constants for CCI and CCG can be derived according to the power law using least square fitting. The proposed method is not fully developed and further validating experiments will be necessary to complete the research.

Shear fracture (Mode II) toughness measurement of anisotropic rock

Fracture toughness of anisotropic shale is an important parameter in shale-gas exploitation technology. Currently available literature is mainly focused on Mode I fracture toughness (KIC) of anisotropic rock under tensile loading. Although there are the cracked straight through Brazilian disk and crack ring disk used to determine Mode II fracture toughness (KIIC) of anisotropic rock under pure shear stress, their fracture trajectories are deviated from the original crack plane and cannot be regarded as the true Mode II fracture. In this paper, shear-box test was firstly adopted to measure KIIC of anisotropic shale. New physical factors YI0 and YII0 were proposed to describe effects of both the geometry and the material parameters on stress intensity factors (SIFs) of the original crack plane (KI0 and KII0) and to derive the calculation formulae of SIFs on arbitrary (KIθ and KIIθ). Calculated results show that KIIθ reaches its maximum absolute value at θ = 0–10° where KIθ is negative, which promotes occurrence of Mode II fracture. The predicted planes of Mode II fracture agree well with the tested fracture trajectories (amostly along the original crack plane). The KIIC is increased with increase of β (β = 0–90°). KIIC is 3–4 times as large as KIC and can be regarded as the true Mode II fracture toughness of anisotropic shale. The shear-box test is an effective method for measuring KIIC of anisotropic rock.

A study on rock mass crack propagation and coalescence simulation based on improved numerical manifold method (NMM)

As an emerging numerical calculation method, the numerical manifold method (NMM) has incomparable advantage in solving continuous and discontinuous deformation problems in a unified framework. In this study, an improved numerical manifold method was deployed to solve the crack propagation and coalescence problem with cracking friction effect in fractured rock mass. Firstly, based on the basic theory of NMM, the structural surface roughness JRC coefficient was introduced into the friction matrix, and the numerical manifold matrix derivation of the friction matrix is embedded into the original software package, which enhanced the NMM calculation program module. A new method of numerical manifold displacement method for solving crack propagation problem is proposed. The calculation formula of stress intensity factor in crack propagation process are given, and the program module used to simulate crack propagation process was developed. Furthermore, two case study was introduced to verify the reliability and applicability of the calculation package. Through comparison of the experiment result with the numerical solution of a rectangle flawed red sandstone specimen with prefabricated hole and fissure, it could be certified that the improved method can acquired the accuracy to an extent, and also enhanced the effectiveness of the proposed method for fracture propagation and coalescence behavior under complex geometry.

Extraction formulas of stress intensity factors for the biharmonic equations containing crack singularities

We derive formulas extracting stress intensity factors of the biharmonic equations on cracked domains with clamped (or simply supported or free) boundary conditions along the crack faces. Each of these formulas can be written in terms of the integral of the given source function multiplied by the cut-off dual singularity and the integral of the unknown true solution multiplied by the cut-off dual singularity over the unit disk. The unknown true solution in the extraction formulas is calculated by either the Implicitly Enriched Galerkin Method or the Iteration Methods. The former was developed by the authors (Kim, Oh, Palta, Kim) and the latter proposed in this paper is the sum of the solution of a regular biharmonic equation and singular functions with iteratively estimated stress intensity factors as coefficients. We show the Iteration Methods quickly converge and the proposed Enrichment Method yields highly accurate stress intensity factors. We also demonstrate that for a known true solution, the extraction formulas yield exact stress intensity factors.

A practical stress intensity factor formula for CFRP-repaired steel plates with a central crack

In this paper, the effects of carbon fiber reinforced polymer (CFRP) repairing on fatigue life improvement of center cracked steel plates was intensively studied. Generally, stress intensity factors (SIFs) were calculated using numerical approaches and employed to predict fatigue cycles via classical crack propagation models. We proposed a practical SIF formula based on thorough finite element (FE) analysis of 1254 CFRP-repaired specimens. The analyses were carried using Python Script in order to achieve a high FE modeling and calculating efficiency. A correction coefficient Fc, representing the CFRP repairing effect, was introduced in the formula to calculate SIFs at the crack tip. Thereafter, the formula were employed to predict the fatigue lives of CFRP repaired steel plates together with the Elber model. Good consistency was achieved with test results.

Non-Fourier thermal shock fracture of solids with shallow semi-elliptical surface crack

In this study, a thick plate with a shallow semi-elliptical surface crack subjected to a transient thermal load was investigated based on the non-Fourier heat conduction law. Using a Laplace transform and its numerical inversion transform methods, the non-Fourier temperature fields and associated thermal stress formulas in the solid uncracking were obtained. Then, using the weight function and analogy methods, the thermal stress intensity factor formula at the crack tip was determined. The effects of the thermal shock time, crack depth, and crack shape on the thermal stress intensity factor and the crack growth behavior were explored numerically by comparing the Fourier model with a non-Fourier model. The results demonstrate that the non-Fourier model is much safer for studying the thermal shock cracking and crack growth behavior of solids under high-temperature applications.

Investigation on metal diaphragm of pulse separation device in dual-pulse solid rocket motor

The theories of circular plates with large deflection and fracture mechanics were employed to investigate the calculating formula of stress intensity factor on prefab gap. The relationship of the opening pressure with physical dimensions of hard pulse separation device (PSD) was analyzed to achieve the feasible configuration of metal diaphragm, which could play a very important role in dual-pulse solid rocket motor. Moreover, six specific single-term tests of the metal diaphragm were conducted to validate the computing precision of designing formula. In addition, three kinds of scale experiments, e.g. the bearing test, opening test, and associated test were performed to explore the working characters of hard pulse separation device. The results indicate that the error between the experimental results and designing results is 4.4%, and the method can be applied to design the physical dimensions of metal diaphragm. The performances of bearing ability, sealability, opening and melt of PSD have been demonstrated. The results show that this kind of PSD can satisfy the requirements of dual-pulse solid rocket motor very well.

Clarification of formulas for stress intensity factor for DCDC specimens

A correction is provided to the formula for the stress intensity factor for the double cleavage drilled compression (DCDC) specimen derived by Plaisted et al. in Int J Fract 141: 447–457, 2006.

Determination of Double-K Fracture Parameters of Concrete Using Split-Tension Cube: A Revised Procedure

This paper presents a revised procedure for computation of double-K fracture parameters of concrete split-tension cube specimen using weight function of the centrally cracked plate of finite strip with a finite width. This is an improvement over the previous work of the authors in which the determination of double-K fracture parameters of concrete for split-tension cube test using weight function of the centrally cracked plate of infinite strip with a finite width was presented. In a recent research, it was pointed out that there are great differences between a finite strip and an infinite strip regarding their weight function and the solution of infinite strip can be utilized in the split-tension specimens when the notch size is very small. In the present work, improved version of LEFM formulas for stress intensity factor, crack mouth opening displacement and crack opening displacement profile presented in the recent research work are incorporated. The results of the double-K fracture parameters obtained using revised procedure and the previous work of the authors is compared. The double-K fracture parameters of split-tension cube specimen are also compared with those obtained for standard three point bend test specimen. The input data required for determining double-K fracture parameters for both the specimen geometries for laboratory size specimens are obtained using well known version of the Fictitious Crack Model.

The mutual effects between two unequal collinear cracks under compression

For two unequal collinear cracks under compression, which crack would propagate first, the longer one or the shorter one, and how they affected each other was studied. By using complex stress function theory and considering crack surface friction, the analytical formula of stress intensity factors (SIFs) for an infinite plane containing two unequal collinear cracks was obtained and the analytical results have been validated through numerical simulation by employing ABAQUS code, photoelastic experiments, and compressive tests. The results show that the numerical, photoelastic, and compressive test results agree well with the analytical result. Finally, the effects of crack length, crack surface friction, and crack interval distance between two crack tips on SIFs were analyzed, and the results show that the SIF values at the longer crack tips are always higher than those at the shorter crack tips; for each crack, the SIF value at the internal tip is always larger than that at the external tip.