EFFECT OF BODY GEOMETRY AND MATERIAL PROPERTIES ON RESIDUAL STRESS DISTRIBUTION ON ARRESTING CRACK HOLES. C. Rubio-González a, G. Mesmacque b, J. Santos-García b, A. Amrouche b a Centro de Ingeniería y Desarrollo Industrial, Pie de la Cuesta 7, Desarrollo San Pablo, Querétaro Qro. 763, México b Laboratoire de Méchanique de Lille, Université de Lille, Villeneuve d Ascq 5965, France Abstract Drilling a hole at the tip of fatigue cracks followed by a cold expansion has proven to be an efficient way to arrest crack propagation in transportation structures. Nucleation of new cracks is delayed, and a decrease of the subsequent growth has been observed due to the presence of favorable compressive residual stresses. In this paper an investigation is conducted to asses the influence of geometric characteristics such as hole diameter, ligament size and distance h on the residual stress field generated by the cold expansion. The effect of cold expansion percentage (CE) is analyzed as well. Residual stress distribution due to hole expansion is estimated considering different stress-strain behaviors. It is desirable to have a deep understanding of the variables affecting this arresting crack technique. This analysis is made by numerical simulation using the finite element method. It has been shown that some geometric variables of the cracked body have important influence in results. The cold expansion percentage (CE) is a parameter that controls the compressive residual tangential stress distribution in case that the crack length is small compared to other specimen geometric dimensions. It has been shown that hole diameter has no important influence in the residual stress distribution. If ligament size or h is reduced then a reduction in the zone of compressive residual stresses is observed as well. This means that for a successful application of this repairing technique it is necessary that crack length to be much smaller compared with specimen width and height. Key words: residual stress, arrest, fatigue crack, finite element. Introduction Different practical methods have been employed for repairing cracked components []. For example shallow fatigue cracks may be grinded away. If this method weak the structure a weld overlay is required after grinding. Other technique is to drill an arresting hole to remove the crack tip singularity. In this case, the crack has to be re-initiated before growth can be continued. The stop hole may be further cold expanded to introduce favorable residual stresses to delay crack re-initiation and subsequent growth. In literature the work is mainly oriented to analyze the effect of cold expansion of fasteners holes. This is motivated by the great fatigue life increase obtained with this process in fastened structures. Different techniques have been used: analytical, to study the plastic zone and residual stresses around the hole [,3]; experimental [4-7] and numerical techniques [8-9]. On the other hand, little work has been done to quantitatively analyze the effect of cold expansion in arresting crack holes, [,]. Cold expansion of fasteners holes and crack arresting holes have some similar features, in both cases cold expansion introduces residual compressive stresses to delay crack nucleation and propagation due to fatigue loading. However there are also some differences, in the fastener hole the plastic zone is symmetrical but this is not the case in the arresting crack hole. Ghfiri, Shi, Guo and Mesmacque [] provide experimental evidence that using this technique, fatigue life of cracked structures may be increased by a factor of 3. In addition fatigue crack growth rate is decreased. However there is no quantitative basis to analyze this kind of problems. It is known that the compressive residual stress field generated when the hole is cold expanded, is responsible of this fatigue life increase. In this work a detailed analysis of the residual stresses and plastic zone for the arresting crack hole is presented. The commercial finite element code ANSYS is used for this purpose. From the practical point of view, it is desirable to investigate the effect of cracked body geometry on the residual stress distribution produced by the cold expansion on the arresting crack hole. The aim of the paper is to analyze the influence Corresponding author, e-mail: crubio@cidesi.mx, fax: +5(4)9839 a 6 de Junho de 3 / June to 6 3 Rio de Janeiro - RJ - Brasil
of geometric characteristics such as hole diameter, ligament size and distance h on the residual stress field as well as the effect of the percentage of cold expansion (CE). It is desirable to have a good understanding of the variables affecting this arresting crack technique and the conditions under it may be applied effectively. A plastic zone and a residual stress field are also generated by overload. A comparison between the plastic zone induced by overload and that generated by hole expansion is made. Finally, the residual stress distribution is estimated considering specimens made of different structural materials.. The arresting crack hole Drilling a hole at the crack tip removes the stress singularity. Hole expansion may be carried out passing through the plate a hard ball or mandrel. During the cold expansion process a plastic zone is generated in the hole neighborhood. The tangential stress distribution S t is shown in Figure. Note that S t is positive at this stage; after getting a maximum it decays slowly, vanishing far away from the hole surface. After the expansion, Figure, the unloading process generates a zone of compressive residual stresses. From Figure one concludes that it is not appropriate to leave the ball inside the hole and apply fatigue loading to the cracked structure. The unloading stage should be reached before cyclic loading can be applied. 3. Numerical simulation Specimen Geometry Consider the edge notched specimen with dimensions h x b, shown in Figure. The crack length is a and the ligament length is l=b-a-d/. A hole of diameter d is made at the crack tip, hence the singular stress field around the crack tip is eliminated. Cold expansion is carried out by inserting a rigid mandrel of diameter D. Percentage of cold expansion, CE, is defined as CE = D d () d () Tangential Tangential Hole Hole Radial Radial Fig. Stress field during the expansion process, and after the expansion process.
The mechanical properties considered for the analysis are those for aluminum alloy 65A T6 which are summarized in Table. The material stress-strain curve was obtained with the standard tension test and it is shown in Figure 3. h d θ a b Fig. Schematic illustration of the specimen used in the numerical simulation. Table Mechanical properties used in the analysis. Yield stress σ y Tensile strength Elongation (%) Young s moludus (MPa) (MPa) (GPa) 65A T6 85 69.5 Solution procedure The commercial finite element code ANSYS was used in the non-linear analysis to simulate the coldworking expansion process. Two dimensional isoparametric elements with eight nodes were used in this task. A multilinear model of the stress-strain curve was considered along with the kinematic hardening rule. A typical mesh is shown in Figure 4. Because of symmetry, only the upper half part is considered. Plane stress conditions were assumed. The loading process consists of a uniform radial expansion of the nodes at the hole surface, next a switch from displacement control to force control is applied, finally the forces at the hole surface nodes are removed and the corresponding residual stress field is calculated. 3 Stress-strain curves 5 Stress (MPa) Stress (MPa) 5 Aluminum 65A T6 5...3.4.5.6.7.8.9 strain Fig. 3 Stress-strain curve for the 65A T6 aluminum alloy used in the analysis. 3
4. Results Comparison between fastener holes and arresting crack holes Consider a square plate with a central hole which is expanded by a percentage CE. The residual stress distribution is shown in Figure 5. Fig. 4 Finite element discretization of the half specimen. Figure 5 shows the residual von Mises stress distribution for a fastener hole and for an arresting crack hole. Equivalent von Mises stress is defined as [( s s ) + ( s s ) + ( s s ) + 6( s + s s )] / s e = xx yy yy zz xx zz xy yz + where s ij are the stress tensor components. As it was expected, on the first case the stress distribution is symmetrical around the hole center, while on the second case, there is no such symmetry. For this analysis CE=3.5 and d=6 mm. We can observe in Figure 6 that compressive residual tangential stress distribution is almost the same in both cases, however, there is a difference on the maximum tension residual tangential stress and furthermore, for the arresting crack hole, it decays to zero slower than in the fastener hole case. xz Fig. 5 Residual stress field (von Mises) after cold expansion of a fastener hole, and an arresting crack hole. 4
Figure 6 shows the residual von Mises stress distribution at the hole surface as a function of the angle θ defined in figure. In Figure 6 the normalization factor is the yielding stress σ y..6 Fastener hole and arresting crack hole Residual von Mises stress at hole surface.4 Arresting crack hole.9..8.7 St/σy St / σ y -. -.4 -.6 Fastener hole Se/σy / σ y.6.5.4.3 arresting crack hole fastener hole -.8 -.. -. 4 6 8 4 6 8 x / R 4 6 8 4 6 8 θ degrees Effect of diameter Varying the hole diameter but keeping CE constant it is shown in Figure 7 that residual tangential stress changes slightly, diameters considered were 4, 6 and 8mm. In their analysis, Ghfiri et al. [] have observed a fatigue life increase for high hole diameters. This fact is explained since the stress concentration factor is lower. Distance x from the hole surface is normalized with the respective hole radius. Effect of CE Fig. 6 Residual tangential stress distribution for a fastener hole and an arresting crack hole. Residual von Mises stress distribution at the hole surface. Figure 8 shows the influence of the cold expansion percentage (CE) on the residual stress distribution. Note that the greater CE value the bigger the compressive residual stress zone size, R c. Also note that the tensile peak stress increases with the CE percentage..6 Residual tangential stress, different diam Residual von Mises stress at hole surface.4.9..8.7.6 St/σy St / σ y -. -.4 -.6 Se/σy Se / σ y.5.4.3 -.8. - -. 4 6 8 4 6 8 x / R. 4 6 8 4 6 8 θ degrees Fig. 7 Influence of hole diameter on the residual stress distribution keeping CE constant. Tangential stress, von Mises stress at the hole surface. 5
.6.4. Residual tangential stress, different CE CE= CE=3.5 CE=5 St / σ y -. -.4 St/σy -.6 -.8-4 6 8 x / R Fig. 8 Influence of CE on the residual tangential stress distribution. Effect of ligament size The ligament size (l=b-a-d/) has a significant importance on the residual tangential stress distribution, as shown in Figure 9. First note that the zone of compressive stresses reduces as the ligament size is reduced. In addition, the tensile residual stresses increased as the ligament size is decreased. This suggests that for a safe application of this technique, the ligament size should be at least times the hole radius. This assures a reasonable zone of compressive residual tangential stress. In this case CE=3.5%, a=5mm and d=6mm. 3 Influence of ligament size 5 8.3.5 (b-a)/r = 5 St/σy / σ y -.5 - -.5 3 4 5 6 x / R Fig. 9 Influence of ligament size on the residual tangential stress distribution. 6
Influence of h Parameter h is the vertical distance from crack faces to the upper and bottom edges in specimen illustrated in Figure. It is desirable to know if this repairing technique is applicable in case that crack faces are very close to the edge. It is observed in Figure that as distance h is smaller the zone size of compressive residual stresses reduces, this means that if crack faces are close to the edge then there is no big advantage in using this arresting crack technique. It is shown by Figure that in order to have a zone of compressive stresses of one radius it is necessary for h to be at least 6 times the hole radius. In these results the parameters CE=3.5 and d=6mm were used. Plastic zone size Figure shows the plastic zone size, R p, as a function of the ligament length for different expansion percentages CE. The parameter used to define the appearance of plastic deformations and hence the plastic zone size is the equivalent plastic strain ε e = ( + ν ) p p p p p p 3 p p p ( ε xx ε xx ) + ( ε yy ε zz ) + ( ε xx ε zz ) + ( γ xy + γ yz + γ xz ) Note that as the ligament size increases the plastic zone tends to a stable value, that is, the plastic zone is unaffected with the ligament length for long ligaments. It is observed that for ligaments longer than o 5 times the hole radius, the plastic zone size is constant. Other wise there is effect of the finite specimen size. Influence of h.6 /.4 3. St / σ y St/σy -. -.4 -.6 -.8-3.3 5 6.6 h/r = 3.3.5.5.5 3 3.5 4 4.5 5 x / R Fig. Influence of h on the residual tangential stress distribution. 7
3.8 Plastic zone size 3.6 3.4 CE = 3.5 CE = 5. r p / R Rp/R 3. 3.8.6.4 5 5 5 3 ligament l/r Fig. Plastic zone size as a function of ligament length 5. Overload In the following, another method to generate a plastic zone and a residual stress field is described and analysed. Consider again the edge notched specimen used in previous analyses but in this case a remote tensile stress S is applied instead, see Figure. S h d a b Figure. Edge notched specimen used in the analysis. A remote stress S is applied. As the remote stress is increased, yielding will occur close to the hole and a plastic zone is generated. After removing the remote stress a residual stress field will be induced. In both cases, the hole expansion and overload, a plastic zone and a residual stress field are developed. However, plastic zone shape is different in each case. Figure 3 compares the plastic zone shape for theses cases. In Figure 3 the cold expansion percentage was CE= 5.5% and in Figure 3 the remote stress was S=8MPa. Recall that yield stress is MPa (form Table ). In both cases specimen dimensions were h= 8mm, d=6mm, a=5mm, b=5mm. These dimension were maintained for the following analyses. 8
R p R p Figure 3. Plastic zone shapes for hole expansion, CE=5.5%, and overload, S=8MPa. Continuing with the comparative analysis, it would be desirable to have a relationship between both methods to produce the plastic zone. Figure 4 makes this comparison. Figure 4 shows the plastic zone size (R p ) evolution varying the degree of expansion. The compressive residual stress zone size (R c ) is plotted as well. Note that at least, for R c there is a limiting value between 5 and 6mm. Figure 4 shows the same parameter but for the overload case. Note that neither R p nor R c has a limiting value, they always increase as the remote stress S is increased. Zone size with hole expansion Zone size with overload zone size (mm) 9 8 7 6 5 4 3 4 6 8 degree of expansion % plastic zone size comp. residual stresses zone size zone size (mm) 9 8 7 6 5 4 3 5 6 7 8 9 stress S (MPa) plastic zone size comp. residual stresses zone size Figure 4. Plastic zone size, Rp, and compressive residual stresses zone size, Rc, for a) Hole expansion and b) overload. Given specific values of the plastic zone size (R p ) or the compressive residual stresses zone size (R c ) it is possible to determine from Figure 4 and 4 the degree of expansion (CE) or remote stress (S) required to develop them. This fact is illustrated in Figure 5 where the tangential residual stress distribution induced by hole expansion and overload are plotted. In both cases R c is the same although the stress distribution is slightly different. These parameters are R c =3.7mm, CE=3% and S=8MPa. 9
Figure 5. Tangential compressive residual stress distribution for the hole expansion and overload cases with the same R c value. 6. Residual stress field for different materials Now consider the expansion process performed on specimens made of different materials. The materials and their properties used in the analysis are given in Table. Table Different materials and mechanical properties used in the analysis. Yield stress σ y (MPa) Young s moludus (GPa) Steel 355 45.4 9.3 Steel A49 565.3 4. Steel HLE 557.9 6.3 Aluminum 65A T6. 69.5 Aluminum 68 T6 67.5 68.3 Figure6 shows the tangential residual stress distribution for the different materials mentioned above. Two degrees of expansion are considered: % and 3.5%. In both cases hole diameter was d=6mm. Distance x from hole surface was normalized with the hole radius. Note that for steels the plots are very similar, in fact the compressive residual stress zone size is almost the same the only difference is the minimum residual stress. This is due to the different yield stress. However, for aluminium the behaviour is slightly different. It was observed that the compressive residual stress zone size, R c, was almost the same for the three steels, and for the two aluminums for a given CE value. It was observed that only the cold expansion percentage CE, had influence on R c. Young s modulus and Poisson ratio were very similar for the three steels considered and the set of aluminums. 7. Conclusions A numerical investigation of the residual stress field ahead of the arresting crack hole has been conducted using the finite element method. It has been shown that some geometric variables of the cracked body have important influence in results. The cold expansion percentage (CE) is a parameter that controls the compressive residual tangential stress distribution in case that the crack length is small compared to other specimen geometric dimensions. It has been shown that hole diameter has no important influence in the residual stress distribution. If ligament size or h is reduced then a reduction in the zone of compressive
residual stresses is observed as well. This means that for a successful application of this repairing technique it is necessary that crack length to be much smaller compared with specimen width and height. Figure 6. Tangential residual stress distribution for different materials and degrees of expansion, CE=%, CE=3.5%. Another way (called overload) to introduce a plastic zone and a compressive residual stress field ahead of the arresting crack hole was analyzed. In this way a remote stress is applied to open the crack faces followed by unloading. A comparative analysis of results from both techniques is made. The residual stress distribution from hole expansion was calculated when the specimen was made of different materials, three different steels and two different aluminums. It was observed that the compressive residual stress zone size, R c, was almost the same for the three steels, and the same for the two aluminums for a given CE value. It was observed that only the cold expansion percentage CE, had influence on R c. References. C.S. Shin, C.M. Wang and P.S. Song, Fatigue Damage Repair: A Comparison of Some Possible Methods. Int. J. of Fatigue, 996, v.8(8), pp.535-546.. G. Wanlin, Elastic Plastic Analysis of a Finite Sheet with a Cold Worked Hole. Engineering Fracture Mechanics, 993, v.46(3), pp.465-47. 3. Y.C. Hsu and R.G. Forman, Elastic Plastic Analysis of an Infinite Sheet Having Circular Hole Under Pressure. Journal of Applied Mechanics, 975, pp.347-35. 4. R. Herman, Three-dimensional Stress Distribution Around Cold Expanded Holes in Aluminum Alloys. Engineering Fracture Mechanics, 994, v.48(6), pp.89-835. 5. P.R. Arora, B. Dattaguru and H.S. Subramanya, The Fatigue Crack Growth Rate in L-7 Al Alloy Plate Specimens with Cold Worked Holes. Engineering Fracture Mechanics, 99, v.4(6), pp.989-. 6. A.T. Ozdemir and L. Edwards, Relaxation of Residual Stresses at Cold Worked Fasteners Holes Due to Fatigue Loading. Fatigue Fract. Engng. Mater. Struct, 997, v.(), pp.443-45. 7. D.L. Ball and D.R. Lowry, Experimental Investigation on the Effects of Cold Expansion of Fasteners Holes. Fatigue Fract. Engng. Mater. Struct. 998 v.7, pp.7-34. 8. M. Bernard, T. Bui-Quoc and M. Burlat, Effect of Re-cold Working on Fatigue Life Enhacement of a Fastener Hole. Fatigue Fract. Engng. Mater. Struct. 995, v.8(7/8), pp765-775. 9. M.J. Pavier, G.C. Pousard and D.J. Smith, Effect of Residual Stress Around Cold Worked Holes on Fracture under Superimposed Mechanical Load. Engineering Fracture Mechanics, 999, v.63, pp.75-773.. R. Ghfiri, H.J. Shi, R. Guo and G. Mesmacque, Effect of Expanded and non-expanded Hole on the Delay of Arresting Crack Propagation for Aluminum Alloys. Material Science and Engineering A,, v.86, pp.44-49.. N. Vulic, S. Stjepan and V. Grubisic, Validation of Crack Arrest Technique by Numerical Modeling. Int. J. of Fatigue, 997, v.9(4), pp.83-9.