Asymptotics of ‘stress intensity factors’ for solutions to wave equation at a crack tip close to external boundary

Elastic interaction between screw dislocations and collinear edge and internal cracks

This paper describes a new stress-intensity factor (SIF) solution for an external surface crack in a sphere that expands capabilities previously available for this common pressure vessel geometry. The SIF solution employs the weight function (WF) methodology that enables rapid calculations of SIF values. The WF methodology determines SIF values from the nonlinear stress variations computed for the uncracked geometry, e.g., from service stresses and/or residual stresses. The current approach supports two degrees of freedom that denote the two crack tips located normal to the surface and the surface of the sphere. The geometric formulation of this solution enforces an elliptical crack front, maintains normality of the crack front with the free surface, and supports two degrees of freedom for fatigue crack growth from an internal crack tip and a surface crack tip. The new SIF solution accommodates spherical geometries with an exterior diameter greater than or equal to four times the thickness. This WF SIF solution has been combined with stress variations common for spherical pressure vessels: uniform internal pressure on the interior surface, uniform tension on the crack plane, and uniform bending on the crack plane. This paper provides a complete overview of this solution. We present for the first time the geometric formulation of the crack front that enables the new functionality and set the geometric limits of the solution, e.g., the maximum size and shape of the crack front. The paper discusses the bivariant WF formulation used to define the SIF solution and details the finite element analyses employed to calibrate terms in the WF formulation. A summary of preliminary verification efforts demonstrates the credibility of this solution against independent results from finite element analyses. We also compare results of this new solution against independent SIFs computed by finite element analyses, legacy SIF solutions, API 579, and FITNET. These comparisons indicate that the new WF solution compares favorably with results from finite element analyses. This paper summarizes ongoing efforts to improve and extend this solution, including formal verification and development of an internal surface crack model. Finally, we discuss the capabilities of this solution’s implementation in NASGRO® v10.0.

Cracked plate analysis with the dual boundary element method and Williams׳ eigenexpansion

Using the conformai mapping technique, the elastic interaction between a screw dislocation and an internal crack in a thin plate material is investigated. The stress field, the stress intensity factor at the crack tip, and the force on the dislocation are derived. The stress intensity factor at the crack tip is dependent on the positions of the lattice screw dislocations, the dislocations inside the crack, the applied stress and the geometry of the medium. From the image force, the zero‐force points between the crack tip and free surface are obtained. The number of zero‐force points varies from one to five depending on the applied stress and dislocations inside the internal crack. Finally, our results can be reduced to several special cases.

This paper describes a new stress-intensity factor (SIF) solution for an internal surface crack in a sphere that expands capabilities for this common pressure vessel geometry. The SIF solution employs the weight function (WF) methodology that enables rapid calculations of SIF values. The WF methodology determines SIF values from the nonlinear stress variations extracted in the uncracked geometry, e.g., from service stresses and/or residual stresses. The current approach supports two degrees of freedom that denote the two crack tips located at the deepest location and the surface of the sphere. The geometric formulation of this solution enforces an elliptical crack front, maintains normality of the crack front with the free surface, avoids non-physical crack shapes with protruding “ears”, and supports two degrees of freedom for fatigue crack growth from an internal crack tip and a surface crack tip. The new SIF solution enables all spherical geometries with the exterior diameter greater than or equal to four times the thickness. This WF SIF solution has been combined with stress variations common for spherical pressures vessels: internal pressure on the interior surface, uniform tension on the crack plane, and bending on the crack plane. These stress variations facilitate solution usability. This work builds on earlier efforts for an external surface crack (PVP2021-61397[1]) and provides a comprehensive solution capability for spheres. This paper provides a complete overview of this solution. We present for the first time the geometric formulation of the crack front that enables the new functionality and set the geometric limits of the solution, e.g., the maximum size and shape of the crack front. The paper discusses the bivariant WF formulation used to define the SIF solution and details the finite element (FE) analyses employed to calibrate terms in the WF formulation. A summary of verification efforts demonstrates the credibility of this solution against independent results from FE analyses. We also compare results of this new solution against independent SIFs computed by FE analyses, legacy SIF solutions, API 579, and FITNET. These comparisons indicate that the new WF solution compares favorably with results from FE analyses. Finally, we summarize the capabilities of this solution in NASGRO®.

Fracture mechanics-based failure theory has been used for analyzing structural integrity in Fitness-for-Service assessments of structures containing cracks. The stress intensity factors (SIFs) along the crack front are the key information in order to assess the remaining service life of a cracked component. In the case of a multiply cracked component, according to Fitness-for-Service (FFS) standards, these cracks must be first identified as to whether they are on the same cross-sectional plane, to be considered aligned cracks, or whether they are on parallel planes and thus be considered non-aligned parallel cracks. Extensive studies have been carried out on the mutual influence of adjacent parallel cracks. However, the scenario of a semi-elliptical surface crack under the influence of a quarter corner circular crack under remote bending has never been addressed. The present analysis addresses this problem by evaluating the effect of a corner circular crack of length a2 on the SIF of an adjacent nonaligned parallel semi-elliptical surface crack of length 2a1 and depth b1. A parametric study of the effect on the SIF as a function of the horizontal separation (S) and vertical gap (H) distances between the two cracks and the crack length ratio a2/a1 is conducted. Mode I SIFs are evaluated for a wide range of the normalized crack gaps of H/a2 = 0.4∼2, and normalized crack separation distances S/a2 = −0.5∼2. As in the case of tension, the presence of the corner quarter-circle crack affects the stress intensity factor along the semi-elliptical crack front. The present results clearly indicate that the effect of the corner quarter-circle crack on the surface semi-elliptical crack is weaker in the case of bending, when compared to the tension case. The largest percentage differences (5–22%) occur at the farthest crack tip of the semi-elliptical surface crack from the corner crack. The largest percent differences occur for a2/a1 < 1. In general, the deeper the semi-elliptical crack, the lower this percentage difference. When comparing maximal absolute values of the SIF, that normally occurs when b1/a1 = 1 for the cases undertaken. In general, the maximum values are found at the closest tip of the semi-elliptical crack to the corner quarter-circle crack. When b1/a1 is different from 1, then the maximum’s location can depend on the value of H/a2 and S/a2 irrespective of the ratio a2/a1. In these cases, the absolute maximum can occur in the vicinity of the deepest point or in the vicinity of the farthest crack tip of the semi-elliptical crack. As in the case of tension, in the case of bending the presence of the Corner Quarter-Circle Crack changes the stress intensity factor along the semi-elliptical crack front. The change reaches its maximum at the tip of the semi-elliptical crack closest to the corner crack, and it monotonically decreases moving away from this tip for the case b1/a1 = 1. For most of the cases b1/a1 < 1, the maximum occurs in the vicinity of the midpoint of the semi-elliptical crack and decreases monotonically in both directions from the midpoint.

During pipeline operation, internal cracks may occur. The severity around the crack tip can be quantified by the stress intensity factor (KI), which is a linear–elastic fracture mechanics parameter. For pressurized pipes featuring infinitely long internal surface cracks, KI can be interpolated from a function considering pressure, geometry, and crack size, as presented in API 579-1/ASME FFS-1. To enhance KI prediction accuracy, an artificial neural network (ANN) model was developed for such pressurized pipes. Predictions from the ANN model and API 579-1/ASME FFS-1 were compared with precise finite element analysis (FEA). The ANN model with an eight-neuron sub-layer outperformed others, displaying the lowest mean squared error (MSE) and minimal validation discrepancies. Nonlinear validation data improved both MSE and testing performance compared to uniform validation. The ANN model accurately predicted normalized KI, with differences of 2.2% or lower when compared to FEA results. Conversely, API 579-1/ASME FFS-1′s bilinear interpolation predicted inaccurately, exhibiting disparities of up to 4.3% within the linear zone and 24% within the nonlinearity zone. Additionally, the ANN model effectively forecasted the critical crack size (aC), differing by 0.59% from FEA, while API 579-1/ASME FFS-1′s bilinear interpolation underestimated aC by 4.13%. In summary, the developed ANN model offers accurate forecasts of normalized KI and critical crack size for pressurized pipes, providing valuable insights for structural assessments in critical engineering applications.

Analytic solutions have been obtained for the stress intensity factor for a propagating and arresting anti-plane strain edge crack in a semi-infinite plate. Kostrov's analysis for this system has been used, and solutions have been obtained up to the time it takes for a stress wave emitted from the crack tip at the start of crack propagation to be reflected first from the edge of the plate, then from the crack tip, from the edge of the plate again and to arrive again at the crack tip. To simplify the analysis a constant velocity of crack propagation has been assumed. If the crack arrests before the arrival at the crack tip of a stress wave emitted from the crack tip at the start of propagation and reflected from the edge of the plate, the stress intensity factor remains constant after arrest until this stress wave arrives. The stress intensity factor then increases until the time of arrival of a reflected stress wave emitted at the instant of arrest. After this the stress intensity factor decreases again. If the crack arrests after the arrival of the first reflected stress wave, the stress intensity factor increases after arrest under the influence of reflected stress waves emitted earlier during the propagation of the crack. The changes in the stress intensity factor are more pronounced at higher crack velocities.

The stress intensity factor solutions of weld toe surface flaws near the saddle point of an X-joint under brace tension and in-plane bending loads have been calculated by three-dimensional finite element analyses. For the analyses the warped-crack surfaces following the brace-chord intersection were modeled using the quarter point crack tip singular elements. The crack tip asymptotic displacement values sampled from the quarter point elements were converted into the appropriate stress intensity factor solutions. The solutions indicate that the crack tip material behavior is predominantly Mode I for the brace tension load case, and it is predominantly mixed mode for the in-plane bending case. Using the solutions of sixteen flaws with seven different depths and three different lengths, the trend of the fatigue crack driving force is analyzed. This analyzed trend of the stress intensity factors is consistent with the fatigue crack propagation behavior observed in laboratory tests on tubular joints. It also indicates that, for the brace tension case, the fatigue crack growth rate on the surface can be independent of the crack length at a certain flaw depth. INTRODUCTION For a fatigue analysis of a conventional tubular joint structure, a fracture mechanics method, which considers the crack geometry explicitly, is the most reliable. It is so because, in a welded component, welding-induced structural discontinuities behave as crack initiators, and thus, the fatigue life consists primarily of the crack propagation life. Although experimental fatigue studies are abundant, however, so far, no rigorous fracture mechanics fatigue crack growth analysis has been performed for surface flaws of offshore tubular joints because of the absence of the stress intensity factor solutions, which reflect explicitly both the flaw depth and length. Experimental fatigue crack growth rate data can be used to calculate a crack driving force parameter, which is expressed in a stress intensity factor equation in terms of the surface crack lengthl(or crack depth2) as the crack size parameter. Unless a problem is uncoupled mode (pure Mode I, II, or III), the crack driving force parameter cannot be considered as a stress intensity factor, which represents the magnitude of the crack tip stress singularity. The reason is that such a parameter incorporates all the stress intensity factors existing in the problem, as discussed later. Even for a single-mode problem, such an experimentally calibrated parameter cannot be general as the stress intensity factor of a surface flaw, if it ignores in its expression the other crack geometry parameter, viz., the crack depthl or the crack length.2 Even with today's sophisticated numerical analysis methods, the complexity in tubular joint structural geometries makes the evaluation of the stress intensity factors of a weld toe surface flaw a significant challenge. As can be seen in Figure 1, under fatigue loading, a weld toe surface flaw of a tubular joint tends to grow following the tubular intersection, and its growing path along the tubular-thickness direction also usually curves. The resulting crack surfaces are doubly warped, in general. For this type of complex flaw geometry, no solution procedure has been established for the stress intensity factors until recently.3

About the effect of plastic dissipation in heat at the crack tip on the stress intensity factor under cyclic loading

The elastic interaction between screw dislocation and the internal crack near a free surface has been investigated. The stress intensity factor at the crack tip, crack extension force, the image force on the dislocation are affected by the free surface. The number and nature of dislocations, m, inside the crack also play an important role in fracture. In order to understand the plastic zone, the zero-force points of dislocation along the x-axis are involved. The dislocation emitted from the right-hand crack tip is enhanced by positive m and reduced by negative m. On the other hand, if the internal crack is closer to the free surface, a dislocation generated from the right-hand crack tip is easier for negative m and more difficult for positive m. However, the role of m on the dislocation emission for the left-hand crack tip is opposite to that for the right-hand crack tip. Finally, three special cases can be obtained from our results. (1) The interaction between a dislocation and a surface crack; (2) the interaction between a dislocation and an internal crack; (3) the interaction between two dislocations.

The present work investigates the motion of a semi-infinite moving crack inside a semi-infinite half-space of orthotropic medium subjected to anti-plane shear wave. The crack is located at a finite depth from the surface of semi-infinite orthotropic medium. Our aim is to examine how such anisotropy and geometric parameters can be adjusted to reduce the magnitude of stress intensity factor (SIF) to control the crack propagation near the crack tip region. As mathematical tools, Fourier transformation and inverse Fourier transformation techniques are employed to convert the governing mixed boundary value problem to the well-known Weiner-Hopf equation with suitable boundary conditions. Some physical quantities such as SIF at the crack tip and crack opening displacement (COD) around the crack tip have been derived. Graphical exhibition has been carried out to show the impact of relevant parameters such as crack velocity, layer depth from the surface to crack and orthotropic material properties on SIF and COD. The numerical results show that SIF decay with crack depth from the layer. It is also observed that SIF decreases with an increase in crack velocity and finally tends to zero as crack velocity approches near SH-wave velocity. Also, the value of COD decays as we move along the damage near the crack tip along negative x-axis and finally tends to zero at the crack tip. This behavior of COD is consistent with the physical nature of the semi-infinite crack of the problem. The results are validated for isotropic material with some reported work and are well in agreement. The study of these physical quantities (SIF, COD) ensures the arrest of onset of crack expansion by monitoring geometric parameters and wave velocity to avoid fracture.

The weight function for cracks in piezoelectrics

Cyclic fatigue crack growth rate and crack resistance curve testing were undertaken on 6 different grades of Mg-PSZ. The width of the transformation zone at the flanks of the cracks was determined using Raman spectroscopy and, combined with R-curve toughening values, used to ascertain the level of crack-tip shielding during cyclic fatigue crack growth and hence the crack-tip stress intensity factor amplitude. By normalising the crack-tip stress intensity factor amplitude with the intrinsic toughness of the material, it was found that the cyclic fatigue threshold stress intensity factor was independent of the extent of crack-tip shielding and a function of the stress intensity factor at the crack tip. In situ SEM observations of cyclic fatigue revealed crack bridging by uncracked ligaments and the precipitate phase. Under cyclic loading the precipitate bridges were postulated to undergo frictional degradation at the precipitate/matrix interface with the degree of degradation determined by the cyclic amplitude. Acoustic emission testing revealed acoustic emissions at three distinct levels during the loading cycle: firstly, near the maximum applied stress intensity factor caused by crack propagation; secondly, at the mid-range of the applied stress intensity factor attributed to crack closure near the crack tip, presumably as a result of transformation induced dilation; and thirdly, intermittently near the base of the loading cycle as a result of fracture surface contact due to surface roughness at a significant distance behind the crack tip. Crack closure near the crack tip due to dilation is proposed to significantly reduce the crack tip stress intensity factor amplitude and hence the degree of cyclic fatigue.

The analysis of the interaction of a main crack with an array of microcracks, placed near the tip of the main crack, has been performed considering 2-D polycarbonate flat specimens. Analytical results, based on the elastic potential theory for the stress intensity factor K, have been well compared with experimental results obtained through the application of the caustic method. Two different specimen configurations have been analysed involving one and two microcracks. INTRODUCTION The problem of the interaction of a main crack with an array of microcracks placed near its tip, of strong relevance in the prediction of the reliability of structural materials, can be experimentally analyzed by applying the caustic method on 2-D specimens. In fact, the method of caustics can be applied to the determination of the stress intensity factors at crack tips approaching other crack tips or boundaries of the specimen. Several problems of interaction of cracks with other cracks or boundaries have been reported [1,2]. In these problems two special cases can be distinguished: in the first case, the crack tip at which the stress intensity factor is determined does not lie very near another crack or boundary [3]. In the second case the crack tip lies very near another crack tip (or boundary) and the shape of the caustic around this crack tip is influenced by the other crack (or boundary). On the other hand, the elastic interaction between both types of cracks can be analytically studied by applying a method based on the combination of the double layer potential technique and the Willis polynomial conservation theorem stating that the COD of a crack embedded into a polynomial stress field of degree N has the form (ellipse)x(polynomial) [4]. Following this approach and Transactions on Modelling and Simulation vol 10, © 1995 WIT Press, www.witpress.com, ISSN 1743-355X 620 Computational Methods and Experimental Measurements expressing the displacement field as well as the stress field by an integral of the double layer potential type, the problem can be reduced to the one of finding vectorial functions-microcrack CODs b(x) and the stress intensity factor K, at the macrocrack tip from the system obtained by solving the system of integral equations which express the boundary conditions on the crack faces. The microcrack CODs are represented in the form (ellipse)x(polynomial) where the first multiplier corresponds to the crack embedded into a uniform stress field and the second multiplier accounts for crack interactions. Thus, the system of singular integral equations is reduced to a system of linear algebraic equations which must be solved to obtain the polynomial coefficients. POTENTIAL REPRESENTATION THEORY The schematic representation of a main crack (-L, i) which can interact with one microcrack is shown in Fig. La and with two microcracks in Fig.l.b. In order to describe the elastic interaction between both cracks, plane stress conditions and Mode I are assumed. Moreover, the case where the microcrack is embedded into the stress field of the main crack tip is considered. Using the constant approximation the stress field within the microcrack line (c-l, c+l) may be approximated by a constant equal to the value of the field at the microcrack centre (x-c) and, following symmetry, a* = â fcj is the only stress component acting along the microcrack line. Fig.l .a: Main crack collinear with one microcrack. Transactions on Modelling and Simulation vol 10, © 1995 WIT Press, www.witpress.com, ISSN 1743-355X Computational Methods and Experimental Measurements 621 Thus, the overall stress field in the vicinity of the main crack tip is given by the superposition: