常用应力强度因子计算方法比较

27th ICAF Symposium – Jerusalem, 5 – 7 June 2013

The Pursuit of K:

Reflections on the Current State of the Art in

Stress Intensity Factor Solutions for

Practical Aerospace Applications

R. Craig McClung,1 Yi-Der Lee,1 Joseph W. Cardinal,1 and Yajun Guo2 1Southwest Research Institute, San Antonio, Texas, USA

2Jacobs ESCG, Houston, Texas, USA

Abstract: The stress intensity factor (K) is the foundation of

fracture mechanics analysis for aircraft structures. 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. Examples are selectively drawn from the recent

literature, from recent enhancements in the NASGRO and

DARWIN software, and from new research, emphasizing

integrated approaches that combine different methods to create

engineering tools for real-world analysis. Verification and

validation challenges are highlighted.

INTRODUCTION

The 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 both structure/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 SURVEY

Handbooks

The 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 also

R. C. McClung, Y.-D. Lee, J. W. Cardinal, and Y. Guo 2

contributing 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 later work, 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 Results

Newman 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 these

The 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 Methods

Computational 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. The recent 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 matrices

R. C. McClung, Y.-D. Lee, J. W. Cardinal, and Y. Guo 4

can 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 Methods

Compounding 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 the generation 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 FE

The 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 solutions

R. C. McClung, Y.-D. Lee, J. W. Cardinal, and Y. Guo 6

Figure 2. Comparison of geometry factors for two unequal corner cracks at a hole estimated from compounding versus Fawaz-Andersson FE solutions

Weight Function Methods

All 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 at

The Pursuit of K7 circular 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 tips in 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.

R. C. McClung, Y.-D. Lee, J. W. Cardinal, and Y. Guo 8

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 solutions

WF 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 direct experimental measurements of the residual stresses.

The Pursuit of K9 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 stress at 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 the effects 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 by

new weight function stress intensity factor solutions

R. C. McClung, Y.-D. Lee, J. W. Cardinal, and Y. Guo

10

Figure 6. Different edge notch shapes (0°,15, 30, and 45 angles, and elliptical and

keyhole profiles) for notch stress field study

Figure 7. Comparison of near-tip stress fields for six edge notch shapes.

Note that all six sets of lines are coincident.

The Pursuit of K11

PRACTICAL IMPLEMENTATION

The 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, along with 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.

R. C. McClung, Y.-D. Lee, J. W. Cardinal, and Y. Guo 12

Figure 8. Geometry model for embedded crack in rectangular plate superimposed on axisymmetric FE model with hoop plane stress contours

Service Stresses Only Service Stresses + Residual Stresses

Stress Contours

Fatigue Crack Growth Life Contours

Figure 9. Effect of bulk manufacturing residual stresses on FCG life contours

The Pursuit of K13

CONFUSION AND RESOLUTION

This 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 development history, 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 shift of 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.

R. C. McClung, Y.-D. Lee, J. W. Cardinal, and Y. Guo 14

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.

Figure 10. Comparison of empirical finite width correction factors for corner crack at hole with correction factors derived from WF solution CC08

The Pursuit of K15 VERIFICATION AND VALIDATION

This discussion of confusion and resolution highlights a very fundamental issue implied by several earlier comments in the paper but not yet addressed directly: Exactly how does anyone know if a K solution is “correct”, and what does it mean for a solution to be correct? The rigorous technical terms here are verification and validation (V&V). There are recent trends in the solid mechanics community to address V&V much more rigorously, and this has led to the publication of some foundational guidelines by a special ASME committee [20]. It is useful now to consider how this new rigor can help the fracture mechanics community sort out the confusion and improve the overall level of credibility in K solutions.

The ASME Guide defines verification as “the process of determining that a computational model accurately represents the underlying mathematical model and its solution” and validation as “the process of determining the degree to which a model is an accurate representation of the real world from the perspective of the intended uses of the model.” In short, verification describes whether or not the computations themselves are being done accurately, and validation describes whether or not the model matches real-world behavior (typically by comparison with experimental data).

Unfortunately, K cannot be directly measured, and so validation is a challenge. The usefulness of K is ultimately determined by its ability to predict actual cracking behavior (e.g., FCG lifetime), but many other models (such as material models) will also influence the accuracy of a fracture mechanics based life prediction. The ASME Guide points out that validation of complex models (models comprising many submodels) must be performed hierarchically, one building block at a time. Consider, for example, the simple proposed hierarchy for FCG life models shown in Figure 11. The K solution (part of the crack driving force model) is not the only factor, or even the most sensitive factor, influencing the lifetime calculation. One implication of this reality is that the use of a particular K to correctly (or incorrectly) predict FCG life does not automatically indicate that the K model is correct (or incorrect), because the accuracy of the other submodels will also impact the predicted lifetime.

On the other hand, verification of K solutions is not so simple, either. Our instinct is that K does represent a rigorous mathematical quantity that has one and only one exact solution for a given geometry/stress combination within the well-defined framework of linear mechanics. However, exact analytical K solutions are available for only a few specialized geometry/stress combinations. These few exact solutions can be used to verify a general computational approach for K calculation, but this does not ensure that a computer model of K for a more complex geometry/stress combination (for which no exact solution is available) is adequately accurate. What to do? The current practical answer seems to be to compare the K calculated by one approach with the K calculated by a completely

R. C. McClung, Y.-D. Lee, J. W. Cardinal, and Y. Guo 16

independent approach. If they give answers that are adequately similar, this builds credibility. If they give answers that are not adequately similar, then perhaps we search for a third opinion as a referee. This is the approach that the present authors have taken in their own development of new K solutions. However, it must be noted that there are inherent uncertainties in any numerical method, and so these comparisons must be carried out with appropriate care.

Figure 11. Model hierarchy for calculation of fatigue crack growth lifetime

CURRENT AND FUTURE CHALLENGES

Many current and future needs and challenges remain. There are many other geometries and configurations for which engineering K solutions are not readily available. For example, many cracks in aircraft structures occur at countersunk holes. However, even this relatively simple configuration introduces several additional parameters compared to the simple hole-in-plate (countersink depth and angle, plus three potential corners at which a crack can form) that complicate the derivation of general solutions. Multiple-site damage is another configuration of great practical interest that is complicated by the very large number of geometry parameters (now including also different numbers of multiple dissimilar cracks, further changing as cracks link up or transition). Weldments appear to be good candidates for WF methods (addressing also the potential effects of residual stresses), but the stress analysis needed to feed the WF solution can be complicated by the effects of misalignment and other geometrical variations in the weld.

Sometimes even simple features of the idealized crack models present problems. Determination of K for a pin-loaded hole requires making assumptions about the nature of the stresses at the bearing surface that may differ from the real world (including the effects of fastener interference). Nearly all engineering K solutions are based on assumptions about simple crack shape—planar cracks with straight or elliptical crack fronts—that are sometimes compromised by reality. This

The Pursuit of K17 disconnect can become severe in some cases, such as complex residual stress fields at cold-worked holes. Engineering K solutions also depend on the externally applied loads, which can themselves sometimes be redistributed to other structural members as cracks grow long, resulting in diminished values of K.

Other needs and challenges were mentioned earlier in the paper. For example, how should K values at the intersection of a part-through crack with a free surface be adjusted due to local constraint loss (and should a similar constraint-loss correction be applied as a crack tip approaches a back surface, when the mathematics imply that the K value is rapidly increasing)?

It is tempting to think that all of these challenges could be resolved with numerical calculation of specific K solutions for specific configurations, accompanied by extensive crack growth rate testing to validate the resulting solutions. However, as noted earlier, both of these “solutions” can be expensive and impractical for real-world design applications. It appears that engineering methods (accompanied by good engineering judgment) will continue to be the backbone of engineering analysis for many years to come.

CONCLUDING REMARKS

Stress intensity factors have been available to support engineering analysis for fracture control for more than forty years, and many legacy solutions have been used for more than thirty years. The past decade has seen a resurgence of interest and activity in developing new and improved K solutions, supported by enhanced computer power, improved numerical methods, and clever new formulations. These new K solutions are also more widely available today (and easier to use) in sophisticated engineering software, so that a much larger number of engineers have access to the advanced technology. The continued contributions of the research community are needed to ensure that this technology growth continues and is able to address the significant number of remaining needs and challenges.

ACKNOWLEDGEMENTS

Portions of this work were suppported by the NASGRO Industrial Consortium, the National Aeronautics and Space Administration,and the Federal Aviation Administration. The assistance of V. Shivakumar (Jacobs ESCG) and V. Bhamidipati (SwRI) is gratefully acknowledged.

R. C. McClung, Y.-D. Lee, J. W. Cardinal, and Y. Guo 18

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[19] Newman, Jr., J.C., and Raju, I.S. (1984), In: Advances in Fracture Research

(Fracture 84), Sixth International Conference on Fracture, pp. 1597-1608. [20] Guide for Verification and Validation in Computational Solid Mechanics

(2006), ASME V&V 10-2006.

使用ABAQUS计算应力强度因子

------------------------------------------------------------------------------------------------------- 如何使用ABAQUS计算应力强度因子 Simwefanhj（fanhjhj@https://www.wendangku.net/doc/a74629530.html,） 2011.9.9 ------------------------------------------------------------------------------------------------------- 问题描述：以无限大平板含有一贯穿裂纹为例，裂纹长度为10mm（2a），在远场受双向均布拉应力σ＝100N/mm2。按解析解，此I型裂纹计算出的应力＝396.23（N.mm-3/2） 强度因子π σa K= I 以下为使用ABAQUS6.10的计算该问题的过程。 第一步：进入part模块 ①建立平板part（2D Planar;Deformation;shell），平板的尺寸相对于裂纹足够大，本例的尺寸为100×50（mm）。 ②使用Partation Face:sketch工具，将part分隔成如图1形式。 图1 第二步：进入property模块 ①建立弹性材料； ②截面选择平面问题的solid，homogeneous； ③赋予截面。

第三步：进入Assembly模块 不详述。需注意的是：实体的类型（instance type）选择independent。 第四步：进入mesh模块 除小圈内使用CPS6单元外，其它位置使用CPS8单元离散（图2）。裂纹尖端的奇异在interaction模块中（图4）考虑。 图2 第五步：进入interaction模块 ①指定裂纹special/creak/assign seam，选中示意图3中的黄色线，done！ ②生成裂纹crack 1，special/crack/create,name：crack 1，type: contour integral. 当提示选择裂纹前端时，选则示意图的红圈区域，当提示裂纹尖端区域时选择红圈的圆心，用向量q表示裂纹扩展方向（示意图3绿色箭头）。用同样的方法建立crack 2（示意图3中的蓝色区域）。 special/crack/edit，对两个裂纹进行应力奇异的设置，如图4所示。

计算应力强度因子

基于ANSYS的断裂参数的计算 本文介绍了断裂参数的计算理论，并使用ANSYS进行了实例计算。通过计算说明了ANSYS可以用于计算断裂问题并且可以取得很好的计算结果。 1 引言 断裂事故在重型机械中是比较常见的，我国每年因断裂造成的损失十分巨大。一方面，由于传统的设计是以完整构件的静强度和疲劳强度为依据，并给以较大的安全系数，但是含裂纹在役设备还是常有断裂事故发生。另一方面，对于一些关键设备，缺乏对不完整构件剩余强度的估算，让其提前退役，从而造成了不必要的浪费。因此，有必要对含裂纹构件的断裂参量进行评定，如应力强度因了和J积分。确定应力强度因了的方法较多，典型的有解析法、边界配位法、有限单元法等。对于工程上常见的受复杂载荷并包含不规则裂纹的构件，数值模拟分析是解决这些复杂问题的最有效方法。本文以某一锻件中取出的一维断裂试样为计算模型，介绍了利用有限元软件ANSYS计算应力强度因子。 2 断裂参量数值模拟的理论基础 对于线弹性材料裂纹尖端的应力场和应变场可以表述为： 其中K是应力强度因子，r和θ是极坐标参量，可参见图1，(1)式可以应用到三个断裂模型的任意一种。 图1 裂纹尖端的极坐标系

应力强度因子和能量释放率的关系： G=K/E" (3) 其中：G为能量释放率。 平面应变：E"=E/(1-v2) 平面应力：E=E" 3 求解断裂力学问题 断裂分析包括应力分析和计算断裂力学的参数。应力分析是标准的ANSYS线弹性或非线性弹性问题分析。因为在裂纹尖端存在高的应力梯度，所以包含裂纹的有限元模型要特别注意存在裂纹的区域。如图2所示，图中给出了二维和三维裂纹的术语和表示方法。 图2 二维和三维裂纹的结构示意图 3.1 裂纹尖端区域的建模 裂纹尖端的应力和变形场通常具有很高的梯度值。场值得精确度取决于材料，几何和其他因素。为了捕获到迅速变化的应力和变形场，在裂纹尖端区域需要网格细化。对于线弹性问题，裂纹尖端附近的位移场与成正比，其中r是到裂纹尖端的距离。在裂纹尖端应力和应变是奇异的，并且随1/变化而变化。为了产生裂纹尖端应力和应变的奇异性，裂纹尖端的划分网格应该具有以下特征： ·裂纹面一定要是一致的。 ·围绕裂纹尖端或裂纹前缘的单元一定是二次单元，并且他的中间节点在四分之一边处。这样的单元也称作为奇异单元。

abaqus计算应力强度因子

重庆大学 课题：Abaqus计算裂纹应力强度因子 学院： 专业： 学号： 姓名：

一、计算裂纹应力强度因子

问题描述：以无限大平板含有一单边裂纹为例，裂纹长度为a=10mm，平板宽度h=30，弹性模量E=210000Pa，泊松比v=0.33，在远场受双向均布拉应力。 使用Abaqus计算该问题： 1、进入part模块 建立平板part，平板的尺寸相对于裂纹足够大，本例尺寸为50x30 （mm）；使用Partation Face:sketch工具，将part分隔成如图1形式 图1 2、进入property模块 建立弹性材料；截面选择平面问题的solid，homogeneous；赋予截 面。 3、进入Assembly模块 实体的类型（instance type）选择independent。 4、进入mesh模块 划分单元格如图2所示。

图2 5、进入interaction模块 指定裂纹special/creak/assign seam；生成裂纹crack 1， special/crack/create；special/crack/edit，对两个裂纹进行应力奇异的 设置。 6、进入step模块 在initial步之后建立static，general步；在 output/history output requests/create/中创建输出变量。 7、进入load模块 定义位移和荷载边界，如图3所示。

图3 8、进入job模块，提交计算 Mises应力分布见图4，在.dat文件中（图5）查看应力强度因子。 图4

图5 计算解析解： 由公式F=1.12?0.23(a/h)+10.6(a/h)2?21.71(a/h)3+30.38(a/h)4 计算得解析解为k=1001 应力强度因子误差为0.09% 二、误差分析 改变板的长度，其他条件不变 1.当长度L=100时 误差为0.5% 2.当板长L=30

第二章应力强度因子的计算.

第二章 应力强度因子的计算 K --应力、位移场的度量?K 的计算很重要,计算K 值的几种方法: 1.数学分析法:复变函数法、积分变换； 2.近似计算法：边界配置法、有限元法； 3.实验标定法：柔度标定法； 4.实验应力分析法：光弹性法. §2-1 三种基本裂纹应力强度因子的计算 一、无限大板Ⅰ型裂纹应力强度因子的计算 K Z ξ→=→ⅠⅠ计算K 的基本公式，适用于Ⅱ、Ⅲ型裂纹. 1.在“无限大”平板中具有长度为2a 的穿透板厚的裂纹表面上，距离x b =±处各作用一对集中力p . Re Im x Z y Z σ'=-ⅠⅠ Re Im y Z y Z σ'=+ⅠⅠ Re xy y Z τ'=-Ⅰ 选取复变解析函数： 22 2() Z z b π=-边界条件： a.,0x y xy z σστ→∞===. b.,z a <出去z b =±处裂纹为自由表面上0,0y xy στ==。 c.如切出xy 坐标系内的第一象限的薄平板，在x 轴所在截面上内力总和为p 。 y '

以新坐标表示： Z= ?lim() K Z ξ ξ → == Ⅰ 2.在无限大平板中,具有长度为2a的穿透板厚的裂纹表面上，在距离 1 x a =±的范围内受均布载荷q作用. 利用叠加原理: 微段→集中力qdx →dK= Ⅰ ? K=? Ⅰ 令cos cos x a a θθ ==,cos dx a d θθ = ?111 sin() 1 cos 22( cos a a a a a K d a θ θ θ - - == Ⅰ 当整个表面受均布载荷时, 1 a a →. ?1 2()a a K- == Ⅰ 3.受二向均布拉力作用的无限大平板,在x轴上有一系列长度为2a,间距为2b 的裂纹.

ABAQUS计算裂纹尖端应力强度因子有效性的算例研究

ABAQUS计算裂纹尖端应力强度因子有效性的算例研究 发表时间：2018-09-11T11:34:12.223Z 来源：《新材料.新装饰》2018年3月下作者：汪波[导读] 在实际工程领域中，相当部分的脆性材料总是不可避免的存在着裂纹或是缺陷。在实际环境中材料的受力往往是相当复杂的。基于ABAQUS平台的裂纹仿真软件，它具有简单易用的特点。（成都理工大学工程技术学院，四川乐山 614000） 摘要：在实际工程领域中，相当部分的脆性材料总是不可避免的存在着裂纹或是缺陷。在实际环境中材料的受力往往是相当复杂的。基于ABAQUS平台的裂纹仿真软件，它具有简单易用的特点。通过算例分析验证表明，该软件的计算结果具有较高的精度，完全可以用于实际工程问题的计算，通过分析验证表明该软件的设计是成功的。此外，今后可以在它的基础上进行更多功能扩展，从而使它拥有分析更为复杂问题的能力。 关键词：裂纹；应力强度因子；断裂力学；ABAQUS 引言 材料在成型和加工过程中在其内部造成了很多缺陷，而其破坏正好均源于构件内部的微小裂纹，所以研究带裂纹的物体力学性能具有十分重要的意义。 图1存在于岩石和混凝土地面中的裂缝 1920年， Griffith[1-2]提出了在材料中存在裂纹的设想，而从Irwin[]3-4]在1957年提出了应力强度因子以及其后形成的断裂韧度的概念后，断裂力学理论出现了重大的突破，奠定了线弹性断裂力学的基础。 1基本原理 近年来以数值分析为基础的手段来解决断裂力学相关问题的技术得到了广泛的发展应用，并且不断的调整完善。该技术在一定程度上较好的克服了实验条件下的不足。对于线弹性断裂力学而言，裂尖区域的位移场、应力、应变场由应力强度因子决定，故而通过有限元计算的结果来得到具体的应力强度因子的值是线弹性断裂力学中用有限元法的基本要求。 1.1 ABAQUS求解裂纹尖端的应力强度因子 传统的有限元在计算裂纹尖端的应力强度因子的时候，无可避免地遇到裂尖复杂应力场和位移场的计算，J积分则可以完全避免这种复杂的处理过程。 为了计算二维情况下的J积分，ABAQUS定义了围绕裂纹尖端由单元组成的环形的积分域，如下图所示。 图2 ABAQUS中围线的定义 ABAQUS在计算围线积分时，采用的是先计算出围线上面所取的若干个离散点处J积分值，然后乘以每个点对应的加权值后，所有点相加来近似地求解出围线积分，即J积分的值和，进而得到复合裂纹的应力强度因子和。 2两条共线裂纹应力强度因子的算例分析 2.1共线双裂纹在压缩荷载作用下应力强度因子的解析解 有许多学者对含有裂纹的无限大板，裂纹尖端的应力强度因子进行了研究。Zhu Z M[5] 等从理论和实验两个方面都做了详细的研究与探讨。基于前人的研究结果，Zhu Z M 给出了共线裂纹的应力函数及其应力强度因子的基本公式，并就共线双裂纹问题进行了研究，给出了裂纹应力强度因子精确的解析解。 图3压缩载荷作用下的含有共线双裂纹的无限大板 2.2 ABAQUS计算共线裂纹应力强度因子

常用应力强度因子计算方法比较.

27th ICAF Symposium – Jerusalem, 5 – 7 June 2013 The Pursuit of K: Reflections on the Current State of the Art in Stress Intensity Factor Solutions for Practical Aerospace Applications R. Craig McClung,1 Yi-Der Lee,1 Joseph W. Cardinal,1 and Yajun Guo2 1Southwest Research Institute, San Antonio, Texas, USA 2Jacobs ESCG, Houston, Texas, USA Abstract: The stress intensity factor (K is the foundation of fracture mechanics analysis for aircraft structures. 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. Examples are selectively drawn from the recent literature, from recent enhancements in the NASGRO and DARWIN software, and from new research, emphasizing integrated approaches that combine different methods to create engineering tools for real-world analysis. Verification and

应力强度因子的求解方法的综述

应力强度因子的求解方法的综述 摘要：应力强度因子是结构断裂分析中的重要物理量，计算应力强度因子的方法主要有数学分析法、有限元法、边界配置法以及光弹性法。本文分别介绍了上述几种方法求解的原理和过程，并概述了近几年来求解应力强度因子的新方法，广义参数有限元法，利用G*积分理论求解，单元初始应力法，区间分析方法，扩展有限元法，蒙特卡罗方法，样条虚边界元法，无网格—直接位移法，半解析有限元法等。 关键词：断裂力学；应力强度因子；断裂损伤； Solution Methods for Stress Intensity Factor of Fracture Mechanics Shuanglin LU (HUANGSHI Power Survey&Design Ltd.) Abstract: The solution methods for stress intensity factor of fracture mechanics was reviewed, which include mathematical analysis method, finite element method, boundary collocation method and photo elastic method. The principles and processes of those methods were introduced, and the characteristics of each method were also simply analyzed in this paper. Key words: fracture mechanics; stress intensity factors 0 引言 断裂力学的基础理论最初起源于1920年Griffith的研究工作[1]。Griffith在研究玻璃、陶瓷等脆性材料的断裂现象时，认为裂纹的存在及传播是造成断裂的原因。裂纹的扩展过程,从能量的观点来看,存在着两种完全对抗的因素：一种是阻止裂纹扩展的因素，另一种是推动裂纹扩展的因素。Griffith由此建立了材料的脆性断裂判据[1]： (1) 在(1)式中：—断裂应力；E—材料的弹性模量；—材料的表面能；a—裂纹长度的一半。 Griffith判据并不能完全成功地应用于金属断裂问题。1949年, Orowan考虑到裂纹释放的应变能不仅转化成表面能，也同时转化成使裂纹顶附近材料发生塑性变形所需要的功。因此，Orowan对Griffith判据进行修正并得到了具有塑性变形的金属材料的断裂判据[1]：

应力强度因子

应力强度因子

断裂与损伤力学 应力强度因子 数值计算方法综述 2013年6月 第一章应力强度因子求解方法概述 含有裂纹的工程结构的断裂力学分析一直是一个重要问题，在断裂力学理论中应力强度因子是线弹性断裂力学中最重要的参量。它是由构件的尺寸、形状和所受的载荷形式而确定。由于裂尖应力场强度取决于应力强度因子，因此在计算各种构件或试件的应力强度因子是线弹性断裂力学的一项重要任务。 由于应力强度因子在裂纹体分析中的中心地位，它的求解自断裂力学问世以来就受到了高度的重视。迄今为止，已经产生了众多的理论和致值解法。70年代中期以前的有关工作在文献中已有相当全面的总结，近20年来，求解的方

法又得刭了明显的发展与完善。下文将穿透裂纹问题(二维)与部分穿透裂纹问题（三维）分开讨论。 第二章 二维裂纹问题 2.1 复变函数法 由Muskhelishvili 的复变函数法，应力函数为： _])()()([2/1)]()(Re[z z z z z z z z χψψχψ++=+=Φ 平面应变情况下的应力与位移为： )]('Re[42222z y x y x ?φφσσ=??+??=+ )]('')(''[22z z z i xy y x χ?τσσ+=+- )](')('[21)(243x z z z iv u χ?μ ?μμ+--=+ 可以证明，在裂纹尖端区域： )]('lim[220z z z iK K K I ?π-=-=∏ 由上式可见。由于k 仅与)(z φ有关，因此只需确定一个解析函数)(z φ，就能求得k I ，这一方法一般只能用来解无限体裂纹问题。对于含孔边裂纹的无限大板，通常可利用复变函数的保角映射原理来简化解题过程。如采用复变（解析）变分方法，则可求解具有复杂几何形状的含裂纹有限大板的应力强度因子。 2.2 积分方程法 弹性边值问题可以变为求解下列形式的积分方程： )() )(()().,(r f dt t b a t t P t r M -=--? 由积分方程解出沿裂纹的坐标的函数，便能直接求出应力强度因子k 。这个积分方程在有些特殊情况下可用普通的Gauss-Chebyshellr 积分或它的修正形式来求解。

应力强度因子计算

应力强度因子计算 FRANC3D使用M-积分来计算应力强度因子，M-积分又称为交互积分，与J-积分具有相似的数学表达形式，能考虑温度、裂纹面接触、裂纹面牵引及残余应力等因素的影响，并能实现多工况的应力强度因子的叠加。 FRANC3D对围绕裂纹尖端的两个单元环执行守恒积分计算，积分域包括一个15节点奇异楔形单元的内环和一个20节点六面体单元的外环。FRANC3D的自适应网格划分技术，还会在裂纹尖端周围布置第三个六面体单元环，但不参与积分计算。 M-积分在FRANC3D中的实现 利用M-积分可同时计算出三种断裂模式的应力强度因子（KI、KII和KIII），其中，KII 用来预测裂纹扭转角度以确定裂纹前缘的扩展方向。FRANC3D可计算各项同性和一般各向异性材料中的三种模式的应力强度因子，也是目前唯一一款可以计算一般各向异性材料中三种断裂模式应力强度因子的软件。同时，还能提供J-积分、T-Stress、Kink Angle等断裂力学参数的结果。 FRANC3D计算应力强度因子时可以考虑温度、裂纹面牵引、裂纹面接触以及它们的组合的影响，还提供多种选项来定义结构中的残余应力或初始条件，包括： ●恒定的裂纹面压强载荷 ●1维径向分布的残余应力 ●2维（轴向和径向）分布的残余应力 ●表面处理后的残余应力 ●基于网格的残余应力（将有限元应力分析结果映射到裂纹网格上，FRANC3D自动 计算并转换为裂纹面牵引力） FRANC3D还提供位移法（COD）来计算应力强度因子，也可使用VCCT技术来计算获得能量释放率（GI、GII、GIII）的结果。

计算应力强度因子 FRANC3D可以图形化和以列表形式显示应力强度因子的计算结果，能同时显示K I、K II、K III的结果，同时还能显示J-积分和T-应力的结果，并提供多种选项供用户输出想要的结果和数据格式。 结果显示和输出

在ANSYS中计算裂缝应力强度因子的技巧

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3．划分网格 为了方便裂纹尖端因子的计算，ansys软件专门提供了一个对裂纹尖端划分扇形单元的命令，即：“kscon”。其命令流如下： LESIZE, 2,,,15,,,,,1 ！对线指定单元个数 LESIZE, 4,,,15,0.3,,,,1 LESIZE, 3,,,12,,,,,1 KSCON,5,3.5/H,1,8 ！对裂纹尖端所在的位置划分扇形单元 ESIZE,3/H,0, AMESH,1 FINISH

4．加载和求解 ?]痏I囚_ _R /SOLU ！进入求解器 嶊?$~菐宅鷋_'?l|錑鈑 壓庢uK麡睽KK畵>Ou?__ 訽 DL,4,,SYMM閼 :!痱摋铪6鸰._@ SFL,3,PRES,-1 ！在3号线上施加布力倪猸 _湋繽丈\g颻湀}OUTPR,ALL }b畇__濠N鲭|FINISH 'b镫淖瑵_鲱v蠄瀯屋璅 甆€_鼍_恄7]僟濢Z嵹!_価 _dDO_N谶l

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《机械设计基础》课程单元教学设计 单元标题：轴向拉伸与压缩的应力 及强度计算条件 单元教学学时 2 在整体设计中的位置第10次 授课班级上课地点 教学目标 能力目标知识目标素质目标 1.能求轴向拉伸与压缩横截面 上应力； 2.能利用胡克定律求变形。 3.能利用强度计算条件解决三 类问题 1.理解应力的概念； 2.掌握拉压杆正应力计 算； 3.理解应变的概念； 4.掌握胡克定律的第一 第二表达式； 5.掌握强度计算条件 1.培养学生热爱本专业、爱 学、会学的思想意识。 2.培养学生应用理论知识分 析和解决实际问题的能力； 3.培养学生的团队合作意 识； 4.培养学生仔细、认真、严 谨的工作态度。 能力训 练任务及案例任务1：计算拉压杆的应力；任务2：计算拉压杆的变形; 教学材料1.教材; 2.使用多媒体辅助教学。

单元教学进度 步骤教学内容教学方法学生活动工具 手段 时间 分配 1复习、导 入复习：拉压杆的受力变形特点、截面法求轴 力直接法求轴力 导入：在求轴力时，我们已经知道轴力的大 小不能代表一个杆件的受力强弱，那谁能代 表呢？ 提问 讲授 讨论 回答 黑板 课件 视频 5 分钟 2提出任务如图(a)所示的三角形托架，P=75kN，AB杆 为圆形截面钢杆，其[σ1]=160MPa；BC杆为 正方形截面木杆，其[σ2]=10MPa，试确定 AB杆的直径d和BC杆的边长a。 情景教 问题探究 问题引领 听讲 思考 黑 板、 ppt 5 分钟 一.应力 应力：内力在截面上某点处的分布集 度，称为该点的应力。 在拉（压）杆横截面上，与轴力N相对 应的是正应力，一般用σ表示。 N A σ= 案例应用1： 一变截面圆钢杆ABCD如图5-6（a）所 示，已知F1=20kN，F2=35kN，F3=35kN， d1=12mm，d2=16mm，d3=24mm。试求： （1）各截面上的轴力，并作轴力图。 （2）杆的最大正应力。 15分 钟

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第二章 应力强度因子的计算 K --应力、位移场的度量?K 的计算很重要,计算K 值的几种方法: 1.数学分析法:复变函数法、积分变换； 2.近似计算法：边界配置法、有限元法； 3.实验标定法：柔度标定法； 4.实验应力分析法：光弹性法. §2-1 三种基本裂纹应力强度因子的计算 一、无限大板Ⅰ型裂纹应力强度因子的计算 K Z ξ→=→ⅠⅠ计算K 的基本公式，适用于Ⅱ、Ⅲ型裂纹. 1.在“无限大”平板中具有长度为2a 的穿透板厚的裂纹表面上，距离x b =±处各作用一对集中力p . Re Im x Z y Z σ'=-ⅠⅠ ！ Re Im y Z y Z σ' =+ⅠⅠ Re xy y Z τ'=-Ⅰ 选取复变解析函数： Z = 边界条件： a.,0x y xy z σστ→∞===. b.,z a <出去z b =±处裂纹为自由表面上0,0y xy στ==。 c.如切出xy 坐标系内的第一象限的薄平板，在x 轴所在截面上内力总和为p 。 / y '

以新坐标表示： Z= ?lim() K Z ξ ξ → == Ⅰ \ 2.在无限大平板中,具有长度为2a的穿透板厚的裂纹表面上，在距离 1 x a =±的范围内受均布载荷q作用. 利用叠加原理: 微段→集中力qdx →dK= Ⅰ ? a K=? Ⅰ 、 令cos cos x a a θθ ==,cos dx a d θθ = ?111 sin() 1 cos 22() cos a a a a a K d a θ θ θ - - == Ⅰ 当整个表面受均布载荷时, 1 a a →. ?1 2(a a K- == Ⅰ 3.受二向均布拉力作用的无限大平板,在x轴上有一系列长度为2a,间距为2b

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第二章应力强度因子的计算 K--应力、位移场的度量 K的计算很重要,计算K值的几种方法: 1.数学分析法:复变函数法、积分变换； 2.近似计算法：边界配置法、有限元法； 3.实验标定法：柔度标定法； 4.实验应力分析法：光弹性法. 论述实测应力强度因子的方法 应力强度因子是反映裂纹尖端弹性应力场强弱的物理量。它和裂纹大小、构件几何尺寸以及外应力有关。应力在裂纹尖端有奇异性，而应力强度因子在裂纹尖端为有限值。 网格法是以网格为制图单元，反映制图对象特征的一种地图表示方法。其制图精度取决于网眼大小，网眼越小，精度越高。网眼大小的确定，取决于制图目的、比例尺和掌握制图资料的详细程度等。网格法既可表示制图对象的数量特征，也可表示其质量特征。使用该法编图时，首先把制图区域按照一定原则，用规定的网眼尺寸画出格网，然后根据掌握的制图资料、野外考察得到的制图对象的分布特征，分别用每个网眼赋值。当表示数量差异时，填入分级级别；表示质量特征时，填入类型代码等。最后用色彩或面状网线符号区分它们。这种方法在计算机辅助制图、统计制图中得到广泛应用。实验应力分析方法的一种。网格法是在试件表面印制或刻划网格，则当试件受载而发生变形时，网格随之变形，通过测量网格因变形而引起的位移，以确定试件的位移场或应变场。它适用于测量5%以上的大应变,而用于测量较小的应变时，精度很低。此法于20世纪40年代开始应用，后来在较大程度上被云纹法所取代。 光弹性法应用光学原理研究弹性力学问题的一种[[实验应力分析]]方法。将具有双折射效应的透明塑料制成的结构模型置于偏振光场中，当给模型加上载荷时，即可看到模型上产生的干涉条纹图。测量此干涉条纹，通过计算，就能确定结构模型在受载情况下的应力状态。 20世纪初，E.G.科克尔和L.N.G.菲伦用光弹性法研究桥梁结构等的应力分布。40年代，M.M. 弗罗赫特对光弹性的基本原理、测量方法和模型制造等方面的问题，作了全面系统的总结，从而使光弹性法在工程上获得广泛的应用。 利用光弹性法，可以研究几何形状和载荷条件都比较复杂的工程构件的应力分布状态，特别是应力集中的区域和三维内部应力问题。对于断裂力学、岩石力学、生物力学、粘弹性理论、复合材料力学等，也可用光弹性法验证其所提出的新理论、新假设的合理性和有效性，为发展新理论提供科学依据。 在偏振光场中，各向同性的光弹性模型在载荷作用下会产生暂时双折射效应，其在光弹性实验中，通常出现两组干涉条纹：等差线和等倾线。 从光弹性实验可以直接获得主应力差和主应力方向。为了确定单独的应力分量，还须借助于其他实验方法或计算方法。 对于二维应力问题，确定主应力或正应力分量的实验方法，有侧向变形法、电比拟法、云纹法、光弹性斜射法、全息光弹性法和全息干涉法等。 三维应力分析大多数工程结构在载荷作用下常处于三维应力状态。应用三维光弹性实验法能有效地确定工程结构内部的三维应力状态。三维光弹性实验法包

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《ANSYS12.0结构分析工程应用实例解析第3版》连载14 发表时间：2012-5-16 作者: 张朝辉来源: 机械工业出版社 关键字: ANSYS 复合材料结构分析 本文是《ANSYS12.0结构分析工程应用实例解析第3版》连载。由机械工业出版社独家授权e-works转载，任何人不得复制、转载、摘编等任何方式进行使用。如需联系出版相关书籍，请联系机械工业出版社张淑谦先生，电话： 本书目录请点击优惠购买本书请点击 8.2 结构断裂分析实例详解——二维断裂问题 8.2.1 问题描述 图8.5所示为一断裂试样结构示意图，厚度为5mm，试计算其应力强度因子。 试样材料参数：弹性模量E=220GPa；泊松比n=0.25；载荷P=0.12MPa 8.2.2 问题分析 由于长度和宽度方向的尺寸远大于厚度方向的尺寸，且所承受的载荷位于长宽方向所构成的平面内，所以该问题满足平面应力问题的条件，可以简化为平面应力问题进行求解。 根据对称性，取整体模型的1/2建立几何模型；选择六节点三角形单元PLANE183模拟加载过程； 先进行普通结构分析求解，再采用特殊的后处理命令计算断裂参数。 8.2.3 求解步骤

1.定义工作文件名和工作标题 1）选择Utility Menu︱ Jobname命令，出现Change Jobname对话框，在［/FILNAM］ Enter new jobname 文本框中输入工作文件名EXERCISE1，单击OK按钮关闭该对话框。 2）选择Utility Menu︱ Title命令，出现Change Title对话框，在文本框中输入 ANALYSIS OF THE STRESS INTENSITY FACTOR，单击OK按钮关闭该对话框。 2.定义单元类型 1）选择Main Menu︱Preprocessor︱Element Type︱Add/Edit/Delete命令，出现Element Types对话框，单击Add按钮，出现Library of Element Types对话框。 2）在Library of Element Types列表框中选择Structural Solid︱Quad 8node 183，在 Element type reference number文本框中输入1，如图8.6所示，单击OK按钮关闭该对话框。 3）单击Element Types对话框上的Options按钮，出现PLANE183 element type options对话框，在Element behavior K3下拉列表中选择Plane strs w/thk，其余选项采用默认设置，如图8.7所示，单击OK 按钮关闭该对话框。 4）单击Element Types对话框上的Close按钮，关闭该对话框。 5）选择Main Menu︱Preprocessor︱Real Constants︱Add/Edit/Delete命令，出现Real Constants对话框，单击Add按钮，出现Element Type for Real Constants对话框，单击OK按钮，出现 Real Constants Set Number 2，for PLANE183对话框，在Real Constant Set No. 文本框中输入1，在Thickness THK文本框中输入5，如图8.8所示，单击OK按钮关闭该对话框。

工程力学 第九章 梁的应力及强度计算

课时授课计划 掌握弯曲应力基本概念； 掌握弯曲正应力及弯曲剪应力的计算；掌握弯曲正应力的强度计算； 掌握弯曲剪应力强度校核。

教学过程： 复习：1、复习刚架的组成及特点。 2、复习平面静定刚架内力图的绘制过程。 新课： 第九章梁的应力及强度计算 第一节纯弯曲梁横截面上的正应力 一、纯弯曲横梁截面上的正应力计算公式 平面弯曲时，如果某段梁的横截面上只有弯矩M，而无剪力Q = 0，这种弯曲称为纯弯曲。 1、矩形截面梁纯弯曲时的变形观察 现象： （1）变形后各横向线仍为直线，只是相对旋转了一个角度，且与变形后的梁轴曲线保持垂直，即小矩形格仍为直角； （2）梁表面的纵向直线均弯曲成弧线，而且，靠顶面的纵线缩短，靠底面的纵线拉长，而位于中间位置的纵线长度不变。 2、假设

（1）平面假设：梁变形后，横截面仍保持为平面，只是绕某一轴旋转了一个角度，且仍与变形后的梁轴曲线垂直。 中性层：梁纯弯曲变形后，在凸边的纤维伸长，凹边的纤维缩短，纤维层中必有一层既不伸长也不缩短，这一纤维层称为中性层。 中性轴：中性层与横截面的交线称为中性轴。 中性轴将横截面分为两个区域——拉伸区和压缩区。 注意：中性层是对整个梁而言的； 中性轴是对某个横截面而言的。 中性轴通过横截面的形心，是截面的形心主惯性轴。 （2）纵向纤维假设：梁是由许多纵向纤维组成的，且各纵向纤维之间无挤压。各纵向纤维只产生单向的拉伸或压缩。 3、推理 纯弯曲梁横截面上只存在正应力，不存在剪应力。 二、纯弯曲横梁截面上正应力分布规律 由于各纵向纤维只承受轴向拉伸或压缩，于是在正应力不超过比例极限时，由胡克定律可知 ρ εσy E E =?= 通过上式可知横截面上正应力的分布规律，即横截面上任意一点的正应力与该点到中性轴之间的距离成正比，也就是正应力沿截面高度呈线性分布，而中性轴上各点的正应力为零。

ANSYS中应力强度因子与J积分的计算

ANSYS中应力强度因子与J积分的计算 (2009-06-02 10:55:08) 转载 标签： 分类：软件学习资料 j积分 ansys 断裂力学 裂纹 塑性区 杂谈 裂缝应力强度因子用ANSYS中怎么求呀。另外，建模时，裂纹应该怎么处理呀，难道只有画出一条线吗？ 首先说一下裂纹怎么画，其实裂纹很简单啊。只要画出裂纹的上下表面（线）就可以了，即使是两个面（线）重合也一定要是两个面（线）；如果考虑道对称模型就更好办了，裂纹尖点左面用一个面（线），右边用另外一个面（线），加上对称边界约束。 再说一下裂尖点附近网格的划分。ansys提供了一个kscon的命令，主要是使得crack tip 的第一层单元变成奇异单元，用来模拟断裂奇异性(singularity)。当然这个步骤不是必须的，有的人说起用ansys算强度因子的时候就一定要用奇异单元，其实是误区（原因下面解释） 好了，回到强度因子的计算。其实只要学过一些断裂力学都知道，K的求法很多。就拿Mode I的KI来说吧，Ansys自己提供了一个办法（displacement extrapolation），中文可能翻译作“位移外推”法，其实就是根据解析解的位移公式来对计算数据进行fitting的。分3步走，如果你已经算完了： 第一步，先定义一个crack-tip的局部坐标系，这是ansys帮助文件中说的，其实如果你的裂纹尖端就是整体坐标原点的话，而且你的x-axis就顺着裂纹，就没有什么必要了。 第二步，定义一个始于crack-tip的path,什么什么？path怎么定义？？看看帮助吧，在索引里面查找fracture mechanics，找到怎么计算断裂强度因子。（my god,我这3步全是在copy 帮助中的东东啊）。 第三步，Nodal Calcs>Stress Int Factr ，别忘了，这是在后处理postproc中啊。