A simple method based for computing crack shapes

Engineering Failure Analysis 59 (2016) 41–56

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Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal

A simple method based for computing crack shapes D. Peng a,⁎, P. Huang a, R. Jones a, A. Bowler b, D. Edwards c a b c

Department of Mechanical and Aerospace Engineering, Monash University, P.O. Box 31, Monash University, Victoria 3800, Australia Maritime Systems Project Office (MPSPO), RAAF Base Edinburgh, Edinburgh, SA 5111, Australia Airbus Group Australia Pacific, Buildings 46, Corner of East Avenue and Explosives Road, Edinburgh, SA 5111, Australia

a r t i c l e

i n f o

Article history: Received 15 July 2015 Received in revised form 27 September 2015 Accepted 6 November 2015 Available online 10 November 2015 Keywords: Crack shapes Intergranular cracking Nastran Genetic algorithm

a b s t r a c t The growth of cracks from small naturally occurring material discontinuities plays a major role in the operational lifespan of aircraft structures. Calculating the life of an airframe structure requires the determination of the crack path(s) which for complex real life geometries can often be highly complex. This paper presents a simple method based on finite element analysis for estimating the crack growth path. The analysis is based on an element removal approach and uses the major principal stress as the failure criteria. Evolution strategies are derived from the biological process of evolution. Three examples are presented demonstrating the utility of the proposed technique. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction A knowledge of potential crack paths1/shapes is important both when calculating the lifespan of an airframe and when determining the appropriate inspection procedures. For many “real life” problems, the development of the shapes of life limiting cracks is often quite complex. A general analytical solution does not yet exist. A number of methods can be used for this class of problems. Of these methods: the finite element [1–14], boundary element [15], mesh free [16,17], extended finite element [18,19], virtual crack closure-integral [20] and s- and p-element [21,22] methods are perhaps the most widely used techniques for computing the stress intensity factors needed to simulate 3D crack growth. However, as explained in Appendix X3 of the ASTM fatigue crack growth standard E-647-13a, a significant proportion of the life of an operational structure is spent in the regime where the crack is small.2 For cracks growing in primary structural members the crack shape often has a complex three dimensional shape. However, in the absence of fractography data, it is not possible to know the crack shapes prior to the analysis. The analyst is therefore forced to make educated guesses of the shapes or compute new K (stress intensity factor) solutions after each increment in the crack growth. Both approaches have limitations. To determine the stress intensity factors and thereby the subsequent crack shape for small cracks requires a very fine local mesh. Such meshes result in a commensurate large CPU solution time. Furthermore, it is generally necessary to re-mesh and resolve after each increment of crack growth. This usually involves substantial computational effort and thereby often limits such approaches to simple problems. As such the challenge is to develop a simple standalone method whereby the crack shapes can be determined prior to any fatigue analysis. This would enable the stress intensity factors needed to life a structure to be determined prior to the fatigue analysis. To meet this challenge, the present paper investigates the use of the ‘Nibbling Algorithm’ approach described in [23,24]. Here attention is focused on the potential for determining the crack path/shapes associated with growth of small three dimensional cracks in realistic structural geometries. This ‘Nibbling Algorithm’ approach computes the crack path(s)/shapes by sequentially ⁎ Corresponding author. 1 In this paper the phrase “crack path prediction” is defined as predicting the successive position of the crack front. 2 Here the definition of “small” is as defined in ASTM E647.

http://dx.doi.org/10.1016/j.engfailanal.2015.11.016 1350-6307/© 2015 Elsevier Ltd. All rights reserved.

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removing those elements that have the highest maximum principal stress. In [23,24] it was shown to successfully simulate both the growth of squats in railway tracks as well as the growth of an inclined crack under constant amplitude loading. An advantage of this method is that it only uses an uncracked finite element mesh and does not require re-meshing every time the crack grows. To illustrate this approach, three problems are analysed and the computed crack shapes are compared with those seen in test specimens. Two problems involve crack growth from fastener holes. One “fastener hole” problem is related to cracking fastener holes in RAAF (Royal Australian Air Force) AP3C (Orion) aircraft. In this context, when assessing the fatigue life of RAAF AP3C (Orion) aircraft [28,29] attention has recently focused on understanding a new class of multi-site damage problems. This involves the growth of small sub mm cracks in 7075-T6 wing skins at fastener holes containing intergranular cracking. The other “fastener hole” problem involves cracks that grow from etch cracks [31,32] at fastener holes in a specimen test performed in support of RAAF F/A-18 aircraft. The other problem considered examines crack growth in the helicopter round robin test programme described in [25,27]. The results of these studies illustrate the ability of the ‘Nibbling Algorithm’ to generate complex three dimensional crack shapes that are in reasonable agreement with those seen experimentally. 2. Methodology This section describes a simple evolutionary procedure for computing crack growth under alternating loads. This technique is termed a ‘Nibbling Algorithm’. The ‘Nibbling Algorithm’ is a heuristic method. It works by removing elements from highly stressed regions. In this study, as in [23,24], a representative maximum principal stress for each element and an ultimate tensile strength are used in the element selection criterion [33]. The average maximum principal stress for each element is derived from the corresponding gauss point stresses and its value is used to determine if the element will be removed from the structure. In a given stage (iteration) the ith element is removed if: σ i ≥ ð1−SF Þ  Maxhσ 1 max jσ u i

ð1Þ

where σi is the representative average maximum principal stress for the ith element, σ1, max is the peak maximum principal stress for all the elements in the structure and σu is the ultimate tensile strength of material and SF is an elimination factor. The elimination factor plays an important role in controlling the iteration process. A high value will lead to a rapid convergence, but may cause instability. The instability may drive the solution away from a correct pathway of crack growth. In contrast, a very low value will require a large number of iterations and can dramatically increase the solution time. Finite element analysis is generally used for determining the structural response. Once an element has “failed”3 it is then removed from the structure. As outlined above the maximum principle stress at the centroid of each element is chosen as the element removal criteria and elements with highest maximum principle stress are eliminated at each evaluation/iteration. This is done by changing the Young's modulus of the structure to a very small value, typically 1/1000th of its previous value. This results in a new FE model. The updated (new) structure is then re-analysed. Depending on the response of the new structure the algorithm will again use the removal criteria to identify elements and eliminate them from the structure. This process is continued and the associated crack shapes determined until the resulting structure fails or the user terminates the iterative process. 3. Crack growth in a helicopter lift frame As the first example considered the problem of crack growth in the helicopter component described in [25]. This component was a flanged plate with a central lightning hole made of the 7010 alloy, see Fig. 1. The finite element model used in [26] to study crack growth in this geometry is shown in Figs. 2 and 3. Due to symmetry considerations only 1/4 of the structure was analysed, see Fig. 3. The resultant mesh had 17,451 elements and 78,608 nodes, see [26]. The material of the round-robin was taken to be an aluminium alloy 7010-T73651 which had a Young's modulus and a Poisson's ratio of E = 70,000 MPa, and ν = 0.3 respectively. The average room temperature tensile strength for this material is 502 MPa and the yield strength is 440 MPa. See Fig. 4 As per [25] it was assumed that there was a small initial corner defect on the inner edge of the large central hole, see Fig. 3. Two different elimination factors were used in this example, viz.: 0.01 (case 1) and 0.05 (case 2). The predicted crack growth shapes are shown in Figs. 2 and 5 for case 1 and case 2 respectively where we see minimal difference between the results obtained using these two elimination factors. Comparing the computed (Fig. 6) and measured (Fig. 7), there is good agreement between the computed and the experimental crack shapes [27]. 4. Crack tunnelling at a fastener hole with intergranular cracking Having established the potential of this approach to accurately capture crack shapes. Let us investigate a new class of multi-site damage problems, viz.: the interaction between intergranular cracks and cracks that grow from small naturally occurring material discontinuities at a fastener hole [28–31]. The specific example studied involves crack growth associated with a P-3 DNH (Dome Nut Hole) coupon [30] which has IGC (Intergranular Cracking) at the fastener hole, see Fig. 8. The unit used in this specimen 3

The term ‘failure’ means that the element is cracked or is not taking part or contributing to the overall performance of the structure.

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Fig. 1. Schematic diagram of the round-robin component (all dimensions are in mm), from [26].

drawing is millimetre. This coupon was part of a larger test programme [30] performed in order to study the interaction of IGC with cracks that grow from small corrosion pits at the bore of the hole in a specimen that was representative of a regions in RAAF AP3C (Orion) aircraft. The geometry of the test specimen is shown below in Fig. 8. The material was 7075-T6 aluminium alloy with a Young's modulus and a Poisson's ratio of E = 73,100 MPa, and ν = 0.33 respectively. The elimination factor used in this example was 0.001. The paper by Jones, Lo, Peng, Bowler, Dorman, Janardhana, and Iyyer [28], which studied cracking at a fastener hole where IGC divided the thickness of the structure into thirds, was the first to reveal the potential for cracking to be confined to a ligament between IGC events, see Fig. 9. We will term this phenomenon, i.e. whereby cracking is confined between IGC events, as “crack tunnelling”. The phenomenon of crack tunnelling was subsequently seen in the experimental tests reported in [30], see Fig. 10 which presents details associated with the failure surface. In this instance there were two IGC strikes, with lengths, as measured from the bore of the hole, of approximately 0.8 mm and 0.7 mm. This figure illustrates the complex nature of associated crack fronts which, as can be seen in Fig. 10, first turns the corner of the lower IGC and then proceed to grow back along the IGC and also down through the thickness of the specimen. Let us next evaluate the potential of this approach to compute crack growth once the crack has reached the end of the ligament between the two IGC events. Due to symmetry, only one quarter of the structure was analysed, see Fig. 11. The resultant mesh had

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Fig. 2. FEA model, from [26].

84,601 11-noded parabolic tetrahedron elements and 418,620 nodes, see Fig. 12. The results of the analysis are shown in Figs. 13 and 14. Comparing Figs. 10 and 14 we see qualitative agreement between the computed and experimental crack shapes. The computed results also show that the crack first turns the corner of the lower IGC, then grows back along the IGC, i.e. towards the bore of the hole, and down through the thickness of the specimen. Given the complexity of the IGC and recognizing the potential for differences between the IGC in the structure and the idealization adopted in the analysis, the similarities between the computed and measured shapes is particular pleasing. The results achieved were deemed sufficiently satisfactory to enable us to proceed to the next stage of the project namely a more detailed analysis of multi-layer large intergranular cracks with cracks at a fastener hole. 4.1. Crack growth from a fastener hole with multi-layer intergranular corrosion In this section, we concentrate on geometries that differ from those studied in [31] but are nevertheless representative of areas in RAAF AP3C aircraft. The particular problem analysed is that studied in [28], viz.: 3 mm deep multi-layer intergranular cracks (MLIC) that (in the vicinity of the fastener hole) divide the plate thickness into thirds interacting with cracks that emanate from the bore of the fastener hole in a 7075-T6 aluminium plate. As in [28,29] this study analysed crack growth from a 6 mm diameter hole in a 152.4 × 152.4 × 8 mm (thick) 7075-T6 plate with 2 equally spaced 3 mm deep intergranular cracks emanating from the bore of the hole. Each of the intergranular cracks were considered to have a circular plan form which allowing for the 6 mm diameter hole in the middle results in the width of the IGC being 12 mm. As explained in [28,29] these dimensions were

Fig. 3. A close up view of the stresses at the inner edge of the large central hole, from [26].

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Fig. 4. Crack progression associated with case 1 (elimination factor equal to 0.01).

chosen to approximate IGC seen in P3C panels provided by the RAAF Maritime Systems Project Office (MPSPO). For the purpose of the analysis the specimen was considered to be subjected to a remote uniaxial stress of 1 MPa. Due to symmetry, only one quarter of the plate was modelled, see Fig. 15, from [28]. The finite element model of this case contained 96,720 11-noded parabolic tetrahedron elements and 469,640 nodes. The accuracy/convergence of this mesh was established in [28]. Since intergranular cracking does not in general produce smooth planar surfaces, it was decided, as in [29,34], to investigate the case when the surfaces of the intergranular cracks were rough and therefore had the potential to (locally) contact each

Fig. 5. Crack progression associated with case 2 (elimination factor equal to 0.05).

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Fig. 6. Crack growth path of case 1.

other and transfer load. To investigate the possible effect that load transfer across the IGC has on fatigue crack growth, it was assumed that the contact between the upper and lower surfaces was random, see [29] for more details. Since the level of contact was unknown, it was decided to analyse four cases where the percentages of the surfaces contact were 0, 23.1, 58 and 81%. These percentages were chosen because finite element models for each of these four cases were developed in [29]. As reported in [29] the resultant stress fields and hence the KT's vary slightly depending on the level of contact. In the crack propagation analysis, two different initial crack locations were considered, viz.: Case A. Two small 0.05 mm deep quadrant cracks initiating at the intersection of the bore of the hole with the lower IGC, see Fig. 16; Case B. Four small 0.05 mm deep quadrant cracks that initiate at the intersection of the bore of the hole with both the lower and the upper IGCs, see Fig. 17. In each case the cracks were allowed to grow and there were no constraint placed on their shape. As previously the elimination factor for both cases was 0.01. The predicted crack growth shapes, for both Cases A and B, are shown in Figs. 18–24, which correspond to cases of surfaces contact were 0, 23.1, 58, 81% respectively. For the case of no contact between opposing surfaces of the IGC (i.e. 0% surface contact) we see that when the cracks are small there are differences in the crack shapes associated with the two and four initial cracks cases, i.e. Case A and Case B, as shown in Figs. 18 and 19 repsectively. The fatigue crack shapes have been indicated/presented by dash line. However, once the cracks have reached a distance of 3 mm, as measured from the bore of the fastener hole, the configurations are very much alike. In contracst when the level of surface contact was 81% it was found that the results associated with Case A and Case B were quite similar. Hence in this instance only the results associated with Case B, i.e. small cracks that initiate from corrosion pits at both the lower and the upper IGC's, are shown, see Fig. 24. For the remaining cases, which are associated with surface contact levels of 23.1 and 58%, when the crack depth is less than 3 mm the crack shapes associated with various surfaces contact levels are different. As such it would appear that the level of surface contact between opposing faces of the IGC has an effect on shape of the fatigue cracks. However, once the cracks reached a distance of 3 mm from the fastener hole the crack shapes have become similar.

Fig. 7. Schematic diagram of crack growth, from [27].

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Fig. 8. A drawing of the simulated IGC test specimen.

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Fig. 9. Crack tunnelling shown in [29].

Fig. 10. Crack tunnelling within a ligament of specimen containing IGC. The fatigue crack eventually grows longer than the IGC and continues to grow, from [30].

From Fig. 20 (Case A with a surface contact level of 23.1%) we see that crack shape is very complex. The lower crack grew at a faster rate than higher one and both cracks then continued to grow until a new crack initiated at the hole above upper (top) IGC event. These cracks then continued to grow subsequently link. The reason for this non-symmetrical crack growth is due to random nature of the contact, in the finite element model developed in [29], between the upper and lower surfaces of the IGC. It is interesting to note that several of these features are similar to those seen in the experimental test data discussed in [30]. The analysis also suggests that the precise shape of the crack path is heavily dependent on the level of contact seen between the opposing surfaces of the IGC. However, as first shown in [28,29] and substantiated in [30], despite the different crack paths the fatigue lives associated with cracks at fastener holes with intergranular cracking appear to be relatively unaffected by the presence of IGC. Although this analysis has been focused on a particular geometry and a particular IGC configuration it is hypothesised that it could be used to study any arbitrary geometries and IGC events. As such this approach has the potential to enable an

Fig. 11. Geometry of the simulated IGC test specimen.

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Fig. 12. FEA model.

understanding of the features associated with fatigue crack propagation at a fastener hole with intergranular cracks. It also has the potential to assist in the development of beta solutions for crack growth involved multi-layer intergranular corrosion (MLIC) in the vicinity of the fastener hole. 5. Crack growth from an etch pit The next example to be studied involves crack growth in the 7050-T7451 dog-bone specimens tested in [32]. The specimen geometry is shown in Fig. 25. The thickness and width of the specimen was 10 mm and 32 mm respectively and the specimen contains a centrally located 6.35 mm diameter hole. Due to symmetry, only 1/4 of this structure has been modelled, see Fig. 26. This mesh had 56,901 11-noded parabolic tetrahedron elements and 219,161 nodes. The Young's modulus and Poisson's ratio of material was taken to be E = 73,100 MPa, and ν = 0.33 respectively. The elimination factor used in this example was 0.01. Case 2. Nine small cracks that initiate from bore of the hole, see Fig. 28.

Fig. 13. Computed crack fronts.

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Fig. 14. Close up of the computed crack fronts.

Fig. 15. Schematic diagram of the section modelled, 11 noded element models with a maximum gap of 0.005 mm, from [28].

Experimental fracture surface [32] of the failed coupon KK1H348, which from the crack growth data and from the failure surface given in [32] appears to have multiple cracks (on each side of the fastener hole) that grow from day one and subsequently link, is shown in Fig. 27. A close examination of this specimen reveals what appeared to be approximately nine cracks on each side of the hole and what appears to be (to a first approximation) roughly similar growth on both sides of the hole. The results of the crack growth analysis for this specimen KK1H348, which as shown in Fig. 28 assumed nine equal semi-circular cracks with a radius of 0.05 mm on both sides of the hole, are shown in Figs. 29 and 30. Since it was difficult to accurately determine

Fig. 16. A close up view of the stresses at the hole in the base mode with intergranular cracking and the location of cracks that initiate on either side of the lower IGC.

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Fig. 17. A close up view of the stresses at the hole in the base mode with intergranular cracking with four initiating cracks, two on either side of both the upper and lower IGC events.

Fig. 18. Computed crack growth shapes for Case A, no surface contact.

the number of initial cracks the analysis was repeated for both five and seven equal initial semi-circular cracks each with a radius of 0.05 mm on both sides of the hole. The resultant crack shapes were similar in that as per Figs. 29 and 30 they resulted in the crack in the middle of the specimen growing the fastest with the other cracks growing in an almost identical (to each other but slower than the central crack) semi-elliptic shape and post link up having an almost identical semi elliptical shape to that shown in Figs. 29 and 30, see Figs. 31 and 32. Comparing the computed crack fronts shown in Figs. 29 and 30 with the measured crack growth shown in Fig. 27 and noting that as explained above the initial number of starting cracks makes little difference post link up we see that, allowing for differences in the precise location of the initial cracks, there is reasonably good quantitative agreement between the computed and the experimental shapes [32]. Given the potential for scatter that can be experienced in such nominally identical experimental tests [32] and the complex nature of the initiating cracks the results achieved were deemed sufficiently satisfactory.

Fig. 19. Computed crack growth shapes for Case B, no surface contact.

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Fig. 20. Computed crack growth model for Case A with 23.1% surface contact.

Fig. 21. Computed crack growth shapes for Case B with 23.1% surface contact.

6. Conclusion The analysis of crack growth in real life structures often involves cracks that evolve to have complex three dimensional shapes. A knowledge of the associated stress factors associated with each of these shapes is essential for determining the associated crack growth histories. However, in the absence of fractography data, it is not possible to know the crack shapes prior to the analysis. The analyst is therefore forced to make educated guesses of the shapes or compute new K solutions after each increment of the crack growth. Both approaches have limitations. An alternative approach could be to obtain engineering estimates for the crack shapes by using the simplistic model presented in this paper which as shown above and in [23,24] appears to yield reasonable engineering estimates for the crack shapes. The finite element solutions for the stress intensity factors and hence the beta solutions associated with these shapes could then be used to perform the subsequent fatigue life calculations.

Fig. 22. Computed crack growth shapes for Case A with 58% surface contact.

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Fig. 23. Computed crack growth shapes for Case B with 58% surface contact.

Fig. 24. Computed crack growth shapes for Case B with 81% surface contact.

To the best of the authors' knowledge this study is one of the first to attempt to model the evolution of 3D cracks associated with intergranular cracking at a fastener hole. Comparison with test data suggests that this approach appears to have the potential to capture the intrinsic features associated with the interaction between intergranular cracks and cracks that grow from small naturally occurring material discontinuities at a fastener hole. In this context it was mentioned in [28,29] that provided that

Fig. 25. A drawing of the specimen used in the crack growth analysis [32].

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Fig. 26. FEA model.

Fig. 27. Fracture surface of coupon KK1H 348, from [32].

Fig. 28. A close up view of the stresses at the hole in the base mode with nine small cracks that initiate from bore of the hole.

spalling does not occur the effect of IGC on fatigue life is expected to be small. In this context the approach presented in this paper has the potential to investigate the conditions under which this (spalling) may occur. However, it must be stressed that this is a preliminary study and that further work is required, particularly in respect to the effect of different mesh densities on convergence.

Fig. 29. Computed instantaneous crack shapes for the case of eleven initial cracks.

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Fig. 30. Computed crack growth for the 11 initial crack models.

Fig. 31. Computed crack growth for the 7 initial crack models.

Leaving aside the question of convergence for the moment it should be noted that one attractive feature of this approach is that it is easy to use and that the various model parameters can be easily modified and their effects evaluated. A further advantage is that remeshing is not necessary since cracks are not explicitly modelled. Although the present study was performed using NASTRAN the analysis procedure can be implemented into any general purpose finite element computer code.

Fig. 32. Computed crack growth for the 5 initial crack models.

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