常用应力强度因子计算方法比较.

27th ICAF Symposium – Jerusalem, 5 – 7 June 2013The Pursuit of K:Reflections on the Current State of the Art inStress Intensity Factor Solutions forPractical Aerospace ApplicationsR. Craig McClung,1 Yi-Der Lee,1 Joseph W. Cardinal,1 and Yajun Guo2 1Southwest Research Institute, San Antonio, Texas, USA2Jacobs ESCG, Houston, Texas, USAAbstract: The stress intensity factor (K is the foundation offracture mechanics analysis for aircraft structures. This paperprovides several reflections on the current state of the art in Ksolution methods used for practical aerospace applications,including a brief historical perspective, descriptions of somerecent and ongoing advances, and comments on some remainingchallenges. Examples are selectively drawn from the recentliterature, from recent enhancements in the NASGRO andDARWIN software, and from new research, emphasizingintegrated approaches that combine different methods to createengineering tools for real-world analysis. Verification andvalidation challenges are highlighted.INTRODUCTIONThe stress intensity factor (commonly denoted K is the foundation of fracture mechanics (FM analysis for aircraft structures. This parameter describes the first-order effects of stress magnitude and distribution as well as the geometry of bothstructure/component and crack. Hence, the calculation of K is often the most significant step in fatigue crack growth (FCG life analysis. This paper provides several reflections on the current state of the art in K solution methods used for practical aerospace applications, including a brief historical perspective, descriptions of some recent and ongoing advances, and comments on some remaining challenges. No attempt is made to be exhaustive in this review—that would be a daunting task—but key citations are woven into the practical experiences of the authors.HISTORICAL SURVEYHandbooksThe early compilations of K solutions in handbooks by Tada, Paris, and Irwin [1] and Rooke and Cartwright [2] were invaluable contributions. Those compilers collected many different published K solutions available at the time while alsoR. C. McClung, Y.-D. Lee, J. W. Cardinal, and Y. Guo 2contributing new solutions from their own research. However, these handbooks had some practical limitations. First of all, many of the solutions were presented in graphical form based on complex numerical computations (without any corresponding closed-form equations, and so they could not be immediately incorporated into engineering software for production use. Second, many of the solutions were for configurations without immediate practical value: infinite bodies, point loads, out-of-plane loading modes, etc. While many of these solutions unquestionably provided essential foundations for laterwork, they were often not accessible to the everyday practitioner who needed to analyze real cracks in real structures, and do so quickly.Closed-Form Equations Derived From Finite Element ResultsNewman and Raju (NR made significant early contributions to practical structural analysis by developing closed-form K equations for surface and corner cracks in simplified finite geometries, often based on empirical fits of finite element (FE solutions. For example, a landmark Raju-Newman (RN paper [3] summarized the FE calculation of K values for semi-elliptical surface cracks in finite-thickness rectangular plates under uniform tension. Stretching the state of the art of that time, they employed FE models using up to 6900 degrees of freedom (!. Noting that previously published solutions for the same geometry exhibited large variations, they carefully verified their modeling techniques and compared their results against other reliable sources where available. Their key result was a summary table of correction factors for a simple matrix of normalized crack shapes and crack depths at various angular positions around the crack front. Newman-Raju [4] then used these discrete data to derive an empirical equation that could be used to calculate K quickly and accurately for any crack shape and normalized size (relative to plate dimensions within the scope of the original FE matrix, which now included both tension and bending. This simple equation has remained in widespread use to the present day.A later paper [5] summarized additional work by RN to develop K equations for elliptical embedded, surface, and corner cracks in plates, and surface and corner cracks at holes under uniform remote tension, again building on previously published tabulations of FE results. In some cases, the K equations included additional correction factors for finite geometry effects (finite width, one crack vs. two symmetric cracks based on theoretical considerations or other published work. NR later published slightly modified versions of these equations [6].The NR solutions began to be more widely disseminated and used after they were incorporated into early versions of the NASA/FLAGRO computer code, which was originally developed to support fracture control for the space shuttle orbiter and other space structures [7]. The FLAGRO team also began to expand and amend the original NR solutions to accommodate other loading modes and geometric variations, as well as to address some perceived accuracy issues. Other researchers and computer codes have subsequently contributed their own derivatives of theseThe Pursuit of K3 solutions. It is a remarkable testimonial to these solutions that they are still in widespread use over thirty years later, even though the original FE models were very coarse by current standards.Recent Finite Element MethodsComputational power has increased dramatically over the last thirty years, of course, and so the prospect of using this power to generate improved K solutions has grown more and more attractive. The ideal situation would be to be able to generate the exact K solution for each problem of interest using a faithful FE model of the actual configuration of interest (and to update the crack model as the crack grows. It is certainly possible to do this today, and in fact several commercial computer codes offer the capability. This can be an attractive option for solving very specific problems (such as a critical field cracking issue, but the resource requirements (including the computation time itself still render this approach impractical as a general design tool for complex structures with many fracture-critical locations.However, the increased computational power is being used to update the older engineering approaches (e.g., NR. At the least, the original matrices of FE solutions can be revisited with finer meshes, or expanded to wider geometry limits. Furthermore, the new power can be coupled with automated mesh construction and solution methods to generate much large numbers of solutions for a much wider range of parameters. Therecent leaders in this area have been Fawaz and Andersson (FA, who have used the p-version FE method. They began by revisiting the basic corner-crack-at-hole geometry, moving on to develop a large database of solutions for two unequal corner cracks at holes [8]. “Large” is something of an understatement in this case: their single (or symmetric corner-crack-at-hole database contained 7150 combinations of the non-dimensional ratios R/t, a/t, and a/c. The different combinations of diametrically-opposed unequal corner cracks (under tension, bending, or bearing load pushed the total number of solutions over five million (not to mention that each K solution was obtained at a large number of positions around the perimeter of the crack.This wealth of information could be used in several different ways. Initially, FA used it to evaluate the legacy RN solutions and NR equations. They found that the older solutions were remarkably good in many places and not so good in a few others. This prompted some efforts to develop empirical corrections (based on the new FE database to the original empirical fits to the original FE database. Unfortunately, the product of two necessarily inaccurate empirical fits is itself necessarily still inaccurate.However, the availability of such a dense matrix of reliable new FE solutions suggested that it might be preferable to use this matrix directly as a master table from which local interpolation could be performed, hence eliminating any inaccuracies of empirical multi-variable fits. Unfortunately, the size of the matricesR. C. McClung, Y.-D. Lee, J. W. Cardinal, and Y. Guo 4can result in very substantial penalties for computer memory (gigabytes of storage for only one family of crack geometries and, to a lesser extent, for processing time as well. FA have continued to generate even more solutions (literally millions and millions for other geometry families in work that is largely unpublished at this writing. While the promise of such bounty is exciting, it is not yet clear how the information could best be put to practical use, given the memory burden.This “FE database” approach to K solutions also retains two other disadvantages. First of all, even the dense matrices of automatically-generated FE solutions cannot capture all of the finite geometry effects, such as the influence of narrow plate width or hole offset / short edge distance (neither of which was addressed in the five million unequal corner crack solutions. Therefore, additional correction factors (of perhaps questionable accuracy or generality are still required for practical use. Second, the huge numbers of results make the task of verification difficult, if not impossible. How do we really know that all of those solutions are actually correct? The numerical method itself may claim that numerical convergence is a guarantee of success, but a careful inspection of the original Fawaz-Andersson database by the current authors found some zeros or even negative (! values where they should not occur. Further work is ongoing to address these concerns.Compounding MethodsCompounding approaches—multiplying together various geometry correction factors—are among the earliest building blocks of K solution development for more complex geometries. The old handbook K solutions were frequently used as compounding factors, and in fact this was exactly how the handbook solutions led to practical solutions for practical geometries in many cases. For example, this is how one of the classic early solutions for the corner crack at a hole was developed [9]. Even the FE-derived NR solutions themselves depended on additional compounding factors to address some finite geometry effects.The compounding approaches are attractive because of their low computational cost and conceptual robustness. In this sense they are the opposite of the brute-force FE methods. However, their accuracy is often in question, since the generality of the compounding is almost always limited to some extent. However, the compounding approaches can also be an intriguing complement to the numerical methods: as thegeneration of FE solutions becomes easier, it becomes easier to generate the necessary solutions to calibrate or verify the more general compounding approaches.In recent years, researchers at NRC–Canada have published some remarkable results using compounding approaches to generate accurate solutions for very difficult problems. For example, they have developed a very elaborate framework for multi-site damage (MSD—multiple unequally-sized cracks at multiple unequally-sized and unequally-spaced holes—and have employed advanced FEThe Pursuit of K5 methods to demonstrate the accuracy of the resulting solutions [10]. Even the building-block K solutions that provide the foundation for the MSD method are themselves interesting studies in the use of compounding approaches to solve multi-dimensional problems. For example, Bombardier and Liao [11] have recently published a powerful new K solution for unequal through cracks at an offset hole under tension, bend, or pin loading. This solution has been extended and refined by the current authors and recently implemented in the NASGRO® computer code. Figure 1 indicates the accuracy of the new compounding-based NASGRO TC23 solution for one or two cracks in comparison to K solutions from independent BE or FE solutions for specific geometries, over a range of geometry parameters for center and off-center holes under tension.This success prompted attempts to develop a similar compounding solution for the problem of unequal quarter-elliptical corner cracks at a hole, as an alternative to the brute-force FA database solution for the same geometry. The results to date are acceptable for some solutions but errors (relative to the benchmark Fawaz-Andersson solutions are greater than 10% for many other solutions. See Figure 2. This failure may indicate that further work is needed, or it may indicate that the problem contains too many interrelated geometry parameters to admit a simple compounding solution.Figure 1. Comparison of geometry factors for two unequal through cracks at a hole based on compounding methods versus independent BE or FE solutionsFigure 2. Comparison of geometry factors for two unequal corner cracks at a holeestimated from compounding versus Fawaz-Andersson FE solutionsWeight Function MethodsAll of the closed-form, tabulated numerical, and compounding solutions discussed so far (and many others like them share the characteristic of simple remote loading (uniform tension, bend, or pin load with simple load paths to the crack in a uniform geometry. However, many significant cracks in real-world components and structures occur in complex stress gradient fields that are not accurately approximated by uniform tension or bending loads. This is a job for weight function (WF methods, where the arbitrary stress distribution on the crack plane in the corresponding uncracked body (typically determined using FE methods is employed to determine K . The weight function method itself has been around since the earliest days of fracture mechanics analysis, but there have been a number of significant advances in the practical application of WF methods in recent years.Lee [12] combined the Glinka WF formulations with a large database of reference solutions for various finite geometries generated using the FADD3D boundary element (BE software to develop several new WF solutions that accommodate univariant stress gradients (stresses varying in one dimension only. Lee [13, 14] later adapted the Orynyak WF formulation to develop a new WF formulation for cracks in two-dimensional (bivariant stress fields. He again employed a large matrix of FADD3D reference solutions to develop new bivariant WF K solutions for surface, corner, and embedded cracks in plates and surface and corner cracks atcircular holes in plates. Due to the very large reference solution matrices, the geometry ranges of several of these new WF solutions are considerably wider than in the corresponding legacy NR solutions. For example, the surface-crack-in-plate solutions allow 0 ≤a/c≤ 8, 0 ≤a/t≤ 0.9, and off-center offsets up to 90%. Figure 3 demonstrates the accuracy of the bivariant WF solution for a corner crack at a hole at the two extreme tipsin comparison to independent 3D BE solutions. Figure 4 shows how the WF formulation can also be used to determine the K values around the entire crack front for a surface crack in a plate under various bivariant stress fields, with verification against independent FE and BE solutions.Figure 3. Comparison of normalized WF K solutions with independent FADD3D BE solutions at surface tip (top left and deepest tip (top right for quarter-elliptical corner crack at hole (bottom right under a bivariant stress gradient (bottom left and many different sets of geometry parameters.Figure 4. Evaluation of normalized WF K solutions around the perimeter of a semi-elliptical surface crack in a plate under three different bivariant stress fields in comparison to independent BE and FE solutionsWF solutions are much faster than FE or BE solutions, but can still be much slower than closed-form solutions, especially for bivariant solutions that require 2D numerical integration. Novel pre-integration and dynamic tabular methods [14] have been developed that substantially increase the speed of these advanced WF solutions. Portions of the numerical integration are performed in advance, so that the numerical integration step can be replaced with a simple series summation. In some cases, a limited matrix of K solutions is tabulated in advance so that the K calculation during crack growth is merely an interpolation from the table, and the tabulated matrix of solutions is dynamically updated as the crack grows beyond the bounds of the initial matrix.One of the advantages of WF K solutions is that they can be used to determine K for nearly any geometry by employing FE stress analysis results on the crack plane in the component of interest without the crack being explicitly modeled (the crack-free stress distribution, as long as the crack itself does not cause redistribution of the external structural load. The crack plane is typically identified as the plane of maximum principal stress at the crack origin. Combined loading modes can often be characterized in terms of the combined stress field, or the different K values arising from the different loading modes can be determined independently and then summed. The WF approach can also be used to determine K arising from residual stresses, either from a FE analysis of the residual stress field, or from directexperimental measurements of the residual stresses.Another great power of the basic WF methods is that they support solutions for an unlimited number of different stress profiles within a given geometric framework. This has recently been exploited to develop a large family of accurate K solutions for corner, surface, and through cracks at internal or external notches with very wide ranges of shapes, sizes, acuities, and offsets [the external notch solutions are illustrated in Figure 5]. Schjive [15] pointed out long ago that the K solution for a crack emanating from an arbitrary notch was an approximate function of the notch root radius and the peak stressat the notch root. Building on this insight, the curent authors developed and validated relationships to calculate the crack plane stresses at many different angled and elliptical notches (both internal and external. For external notches, the stress field ahead of the notch root was found to be a consistent function of the notch root radius and the total notch depth, irrespective of the notch angle or shape. Figure 6 shows six different edge notch shapes and Figure 7 shows the corresponding stress gradients for tension or bending. This relationship made it possible to estimate the stress field for any edge notch from a simple matrix of 2D FE reference solutions for various root radii and depth values. For internal slots or elliptical holes, some additional factors were derived to address theeffects of offset or shape on the stress field. For both cases, the stress gradient solutions were combined with the appropriate WF solutions for cracks in plates or cracks at holes to develop the new K solutions, which have all been recently incorporated into the NASGRO software.Figure 5. Geometries of arbitrary external notches supported bynew weight function stress intensity factor solutionsFigure 6. Different edge notch shapes (0°,15, 30, and 45 angles, and elliptical and keyhole profiles for notch stress field studyFigure 7. Comparison of near-tip stress fields for six edge notch shapes.Note that all six sets of lines are coincident.PRACTICAL IMPLEMENTATIONThe practical utility of advanced K methods, including WF and numerical methods, is greatly enhanced if the FM life analysis can be directly linked with digital models of the actual structure (e.g., FE stress analysis models. Two approaches to this linkage are described here.The first approach, which has been implemented in the DARWIN® software, directly interfaces the FM life analysis with the FE model of the uncracked component. Through a powerful graphical user interface, simplified FM life models (e.g., rectangular plate models can be constructed and visualized directly on the component FE model, as shown in Figure 8. The computer then automatically collects the geometry and stress gradient information needed to support the WF K solutions employed in the life calculation. These plate models can be sized and oriented manually by the user with the computer mouse. Alternatively, expert logic has been developed to automatically build (size/orient optimum simple geometry models at any arbitrary location in a component [16]. This algorithm can be applied to a large number of discrete locations (e. g, every node on a cross-section plane, performing the life calculation at each location from a common initial crack size to generate FCG life contour maps, all without user intervention. This paradigm has also been recently extended to the automatic calculation of fracture risk [17], considering variability in key input parameters such as initial crack size and also considering that the initial crack could occur randomly anywhere in the component (perhaps at an inherent material anomaly. The key outcome of all these developments is that engineering life analysis can be carried out with substantially improved speed (and reduced opportunities for user error without sacrificing accuracy [the advanced WF methods actually provide superior accuracy to the legacy K solutions].This framework can also be used to address comprehensively and efficiently the effects of bulk residual stresses that can arise (for example in large forged components. Figure 9 shows service stresses in a simple disk geometry with and without superimposed bulk residual stresses as calculated by manufacturing process simulation software, alongwith the corresponding FCG life contours [18]. Another new integrated approach currently under development links the component FE model and a 3D numerical fracture analysis built with the same component model to generate a table of K values at a specific location that can then be employed efficiently to perform the life calculation. This is not practical for a large number of locations but could be employed on a limited basis for engineering analysis. The interface is designed to minimize the required user intervention to build and execute the 3D numerical model, thereby streamlining the workflow and facilitating the use of the tool by non-experts.Figure 8. Geometry model for embedded crack in rectangular plate superimposed on axisymmetric FE model with hoop plane stress contoursService Stresses Only Service Stresses + Residual StressesStress ContoursFigure 9. Effect of bulk manufacturing residual stresses on FCG life contoursCONFUSION AND RESOLUTIONThis paper earlier reviewed the different approaches to the development of K solutions for practical engineering applications: simple-closed form equations, compounding, tabulated numerical solutions, and WF methods. Not surprisingly, each of these methods has been used to generate K solutions for the same common simplified geometries (e.g., corner crack at a hole in a plate. Furthermore, common methods have been used by different researchers (even by the same researchers! to generate multiple solutions. The end result has been a wealth of available solutions that ultimately produces confusion: which solution is correct? For example, a recent NASGRO version offered five different solutions for corner-crack-in-plate, each with its own unique developmenthistory, and each giving an answer that is different in at least part (if not all the solution space. Other computer codes offer still different solutions for the same simple geometry. Newer solutions usually employ newer and more powerful computer models (not always more accurate! and sometimes (not always! reveal inaccuracies in some of the older solutions. But mixing and matching these solutions is not simple, since the underlying formulations (or the ranges of geometric validity, or the types of admissible loading are often different.To make matters more complicated, there are also ambiguities about the proper application of a mathematical K solution for the calculation of FCG rates. Many researchers have shown that in order to predict correctly the growth of a semi-elliptical surface crack in a plate geometry using two degrees of freedom (a surface value and a maximum depth value of K around the crack front, the K value at the surface must somehow be adjusted downward, either multiplying the actual K value at the surface position by a correction factor on the order of 0.9 [19], or by taking the surface value from some angular position inside the surface. To make matters worse, most numerical methods exhibit some numerical instabilities when calculating K at the surface intersection of a part-through crack, and so the value at the surface itself is usually unavailable or conspicuously unreliable.This principle of adjusting the (near- surface K value seems applicable to other part-through crack geometries as well (e.g., corner cracks, or cracks at holes, but the validation is less straightforward and the supporting evidence often not available. The result is yet another dilemma: whether or not to apply a surface correction, or to use K values at some angle inside the surface, when calculating K corresponding to the surface location for some part-through crack geometry. Not surprisingly, different computer codes use different approaches for different geometries. The differences introduced are not huge—typically the K value is adjusted by about 10%--but this can still lead to a shiftof about 40% in the calculated life. This inconsistency makes it further difficult to sort out the differences in different K solutions as implemented in different software.The NASGRO team has recently been working to resolve some of the ambiguities associated with different legacy K solutions for the corner crack at hole by building a new hybrid solution with the best available technology from multiple sources. The new solution starts with the large FA database for a single/symmetric corner crack at a hole, purging anomalous results in the original database and replacing them with interpolated values from more reliable neighbors in the solution matrix. Since the FA database was generated using wide plate FE models with centered holes, correction factors are still needed for narrow plate widths and hole offsets. The legacy NR finite width (Fw correction factors were found to be inaccurate for some geometries when compared with the database of BE reference solutions generated previously by Lee to support his CC08 WF solution. The Lee database was then used to guide the development of improved Fw factors (see Figure 10. Note that the NR Fw correction was derived for the c-tip but is often used for the a-tip as well. The new Fw equations are different for the two tips. Comparisons of the AFGROW hole-offset factor with the Lee database indicated that it was acceptably accurate for B < W/2 (but not for B > W/2, so this correction factor was adopted as is, with that limit. Additional work is underway to compare the Lee WF solutions with the FA FE solutions and determine if any specific solutions need to be revisited or revised. The end result of this entire process will likely still include separate tension/bend/pin and WF solutions, since the geometry ranges of the FA solutions and the Lee WF solutions are considerably different, but the remaining solutions should be much more consistent. Studies are also underway to evaluate the use of surface correction factors for this geometry.。