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Experimental and numerical fatigue crack growth of an aluminium pipe repaired by composite patch
Engineering Structures 133 (2017) 24–32
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Experimental and numerical fatigue crack growth of an aluminium pipe repaired by composite patch H. Zarrinzadeh, M.Z. Kabir ⇑, A. Deylami Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, P.O. Box: 158754413, Iran
a r t i c l e
i n f o
Article history: Received 18 June 2016 Revised 8 December 2016 Accepted 10 December 2016
Keywords: Crack growth experiment XFEM crack growth Patch repaired pipe Stress intensity factors
a b s t r a c t In this study a cylindrical cracked aluminium pipe is considered. Fatigue crack growth behaviour of the pipe is observed through experimental tests. Stress intensity factors are computed for the pipe with an inclined crack under axial tensile load. Fatigue crack trajectory and also the crack growth curves versus number of cycles of load, are extracted. Validation of results is then achieved through the extended finite element method (XFEM). A stand-alone MATLAB programming package is developed to study such structures with 3D degenerated shell elements. The cracked pipe is finally repaired by glass/epoxy polymer composite and the effect of the patch is observed on the extension of fatigue life through experimental tests and the XFEM framework. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Crack initiation may occur in different structures such as metal or concrete elements. Fatigue is one of the most common kinds of loading which causes cracking initiation and propagation in structural elements. These structural parts can be found in building area or industry. For example, bridges which are utilized for providing passage of vehicles or trains, are subjected to fatigue loading that may cause damage in the bridge elements. Pipelines are another example of strategic infrastructures which are used for energy transferring. They may contain fluids such as gas, petroleum or water while pressure changes of the fluid would lead to internal fatigue loading in the pipe. Most of the aforementioned structures can be considered as 3D Shell elements. While this assumption takes into account the three dimensional geometry of the parts, the computational cost will decrease compared with 3D solid element assumption. Different methods are proposed to analyze shell elements which can be categorized as analytical, numerical and experimental studies. Erdogan et al. [1,2] studied on the cracked panels using an analytical formulation for the fracture parameters such as stress intensity factor. Other closed-form expressions for SIFs are presented by Zahoor [3], Sanders [4] and Forman [5] for cracked cylindrical pipes. Zárate et al. [6] presented a framework to update and predict crack length as a function of the number of cycles in structural elements subjected to fatigue. ⇑ Corresponding author. E-mail address: [email protected] (M.Z. Kabir). http://dx.doi.org/10.1016/j.engstruct.2016.12.011 0141-0296/Ó 2016 Elsevier Ltd. All rights reserved.
Analytical methods are mostly capable to solve just simple geometric problems with particular loading and boundary conditions. Of course, it should be mentioned that the analytical procedures are basis of the recent powerful numerical techniques. Finite element method is one of these techniques that can solve varieties of simple or complex engineering problems. Structures can be modeled and analyzed under arbitrary loading and boundary conditions through this framework. Lam et al. [7] studied on a cracked steel circular tube repaired by FRP patching through FEM method. Tong and Sun [8–11] have studied the effect of curvature existence in elements on the adhesive stress and fracture toughness. Pavlou et al. [12] proposed a new methodology to simulate the crack trajectories under mixed-mode fatigue loading through FEM method. The conventional finite element method remains simple until there is no discontinuity in the model. Discontinuities in elements would lead to singularities that make solving the problem more difficult. For instance, in problems containing cracked parts the generated mesh should be in a way that the crack body coincides the element edges. Singular elements should also be used as crack tip elements. This would lead to an irregular mashed part of the structure. The problem becomes more complex when crack propagation needs to be considered. In the case of crack propagation such as fatigue crack growth problems the crack body and crack tip will not be coincided with the element edges after growing the crack, so a new mesh generation is needed for the model in each step of solving the problem. This procedure is done in a work by Ghaffari and Hosseini-Toudeshky [13]. They studied on crack growth of a steel pipe under fatigue loading with and without FRP patching through FEM method. An automated re-meshing
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technique is performed through ANSYS Parametric Design Language (APDL) to detect the new geometry of the crack in each step of crack growth and assign adaptive mesh to the cracked panel. All the deficiencies of the conventional finite element method resulted to appear a new technique known as an extended finite element method (XFEM) in which a regular mesh generation can be used in even cracked parts. The elements in this method can be cut in arbitrary manners and just the formulation of the elements around the crack is modified in a way that the singularity of the problem is taken into account. Areias and Belytschko [14–16] worked on cracked shells by XFEM method considering the material nonlinearities without applying enrichment functions of crack tips. Bayesteh and Mohammadi [17] have extended Areias [14] works to study cracked shells applying Mindlin-Reissner theory, while they also considered the effect of the crack tip enrichment function. Wyart et al. [18] applied extended finite element method to analyze cracks in aircraft thin walled structures. Experimental testing is another way of studying problems. It is obvious that every structural part cannot be tested experimentally due to time and cost aspects, but the simplest experimental test can be utilized to validate the results of the numerical method. After a damage or crack is initiated in an element, affordable techniques of repairing are required for the damaged parts. Traditional techniques like using bolts or welding have some disadvantages such as vulnerability of welding to fatigue loading, high stress concentrations near bolts and increasing the total weight of the structure after using metal repairs. Among the new techniques of repairing, polymer composite materials are a proper alternatives which cover the aforementioned deficiencies of traditional retrofitting methods. Glass/epoxy polymer composite is used in this research as a repair patch. Gandhi et al. [19] studied on fatigue crack growth in stiffened steel tubular joints in seawater environment. Kabir and Nazari [20–23] experimentally studied on a cracked cylindrical steel pipe under compressive loading. The results were also compared with FEM models for the patched an un-patched pipe. In this study a cylindrical cracked aluminium pipe is studied by XFEM method and then validated by experimental tests. Fatigue crack growth analysis of the pipe is performed through the extended finite element method assuming 3D degenerated shell elements. The effect of glass/epoxy patch repair is then observed. The transparency of the patch makes it possible to trace the crack trajectory in the lower panel. Besides developing a stand-alone XFEM package in the MATLAB programming software, similar test specimens are provided to experimentally validate the numerical results. 2. Numerical formulation
at node i in the middle surface. fl3i tion cosine vector v 3i .
m3i
T
n3i g is related to direc-
2.2. XFEM formulation In XFEM formulation, the displacement field is divided into two parts
u ¼ uFE þ uENR in which u
uFE ðxÞ ¼
FE
ð2Þ
is the conventional finite element displacement.
n o ^i e2i ^ i Ma ^ i e1i þ M b Ni ðn; gÞ u
n X
ð3Þ
i¼1
Five degrees of freedom are considered in the conventional shell ^ i and two formulation which consist of three nodal displacements u ^i with respect to local orthonormal vectors e2i ^i; b local rotations a and e1i . uENR is related to the enriched part of approximation. To model weak or strong discontinuities in XFEM framework, one need to incorporate two types of enrichment functions into displacement approximation. The first type of enrichment function which is used to present the discontinuity across the crack, is the Heaviside step function. Dolbow introduced this function to simplify the representation of crack away from the tip. The Heaviside function is
hðxÞ ¼
abov e the crack
1
1 below the crack
ð4Þ
The Heaviside function comes in the degenerated shell formulation as below [17]
uHeav iside ðxÞ ¼
n n X Ni ðn; gÞ ðHð/ðxÞÞ Hð/ðxi ÞÞÞ i¼1
2.1. Degenerated shell element Using the so-called degenerated shell element formulation, computations are done similarly in a planar scheme, while the 3D properties of element are kept. Fig. 1 shows schematic of a degenerated shell element, nodal points and coordinate systems. The geometry of an element is described as
8 9 88 9 8 99 > > > = X <> < xi > = 1 < l3i > => = 8 y ¼ Ni ðn; gÞ yi þ M i ðfÞ m3i > > > > : > ; i¼1 :> : > ; 2 : ;> ; z zi n3i
Fig. 1. Schematic of a degenerated shell element, nodal points and coordinate systems.
ð1Þ
where n; g are the two curvilinear coordinates in the middle of plane of shell and f is a linear coordinate in the thickness direction. N i ðn; gÞ are shape functions in element plane directions, fxi yi zi gT is the Cartesian coordinate. Mi ðfÞ ¼ f t2i is the one dimensional shape function in f direction and t i is the thickness
o ai Maai e1i þ Mabi e2i
ð5Þ
where ai is the vector of enriched displacement degrees of freedom at the mid-surface of the shell and aai and abi are rotations with respect to e2 and e1 respectively. The second type of functions called crack-tip enrichment functions, is usually derived from the asymptotic analytical solution. They are used to represent the singularity of stress field near the crack tip. The crack tip enrichment functions are used in a similar formulation as shown in Eq. (5) for shell elements, which consist of in-plane and out-of-plane enrichment functions as below [17] The enrichment functions for in-plane degrees of freedom are pffiffiffi pffiffiffi h pffiffiffi h pffiffiffi h h Fðr; hÞ ¼ r sin ; r cos ; r sin sinðhÞ; r cos sinðhÞ 2 2 2 2 ð6Þ
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The above functions are also used to enrich the rotational degrees of freedom. pffiffiffi pffiffiffi h pffiffiffi h pffiffiffi h h ; r cos ; r sin sinðhÞ; r cos sinðhÞ Rðr; hÞ ¼ r sin 2 2 2 2 ð7Þ
the analysis computations for millions of cycles, after calculating SIFs for one step of loading, a predefined crack growth increment Da is assumed, then the number of cycles DN related to this Da is determined with respect to material constants c; m.
The other degree of freedom related to out-of-plane displacement u3 is enriched with the following function
3. Experiments
Gðr; hÞ ¼
pffiffiffi h r sin 2
3.1. Specimens’ geometry
ð8Þ
where r; h are the polar coordinates in the local crack tip coordinate system. The crack tip is the origin of the system and h ¼ 0 is parallel to the crack. Fig. 2 shows the enriched nodes with the Heaviside and cracktip enrichment functions around the crack body. Stress intensity factors (SIFs) in the current shell model are calculated through the use of J-Integral method [24]. To perform a fatigue crack growth analysis, the Paris law is used
Da ¼ cðDKÞm DN
ð9Þ
where Da is crack growth increment, DN the increment for number of cycles, DK the range of SIF due to maximum and minimum load value and c; m are material constants for Paris equation. To facilitate
A cylindrical pipe made of 6063 aluminium alloy with a through-the-thickness crack is considered. The pipe has a length of 400 mm with internal and external diameters of 86.5 mm and 90.2 mm, respectively. Maximum tensile stress of 60 Mpa with a load ratio R ¼ 0:1 applies to pipe. The initial crack is placed circumferentially in the middle of the pipe. As the wire-cut technique could not be used to make such a crack in the pipe, Electrical Discharge Machining (EDM) technique is utilized as an alternative method. In this technique pure copper electrode with thickness of 0.18 mm has been prepared to be used in the spark procedure. To maximize the accuracy of inserting a proper initial crack, the copper electrode was modeled in a computer designing program and then this shape was extracted from a copper sheet by a CNC laser cutting machine. Fig. 3 shows the copper electrode and the aluminium pipe in the EDM machine. The initial length of the crack
Fig. 2. Definition of enriched nodes and illustration of level-set functions.
Fig. 3. (a) Copper electrode used to insert crack in the spark procedure. (b) The aluminium pipe in the EDM machine.
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Fig. 4. (a) DARTEC 9600 testing machine with the installed fixture. (b) Schematic of the lower part of the designed fixture to apply tensile load on the pipe.
after applying a pre-cracking to provide a sharpened crack tip is a0 = 30 mm. 3.2. Test preparation The specimens are tested with a DARTEC 9600 fatigue testing machine. As there was no prefabricated fixture to perform a tensile fatigue loading test on a cylindrical pipe, a particular fixture shown in Fig. 4 has been designed and built. To apply a uniform tensile stress, inside of the pipe is filled with a solid circular steel filler at both ends. Outside surfaces of both ends are then gripped with two other pieces which hold the pipe edges fixed during the loading procedure. The whole pieces are connected to the machine with two wedges at both ends. 4. Results and discussion 4.1. Fatigue crack growth in a pipe with a circumferential crack To extract the Paris constants c and m, experimental results of the cracked pipe with known magnitudes of Da; DN and DK are required in different steps of fatigue crack growth. To extract values of DK for each step of crack growth, analytical formulations or numerical modeling techniques can be applied. Here the specimens are modeled in the prepared MATLAB code (called as Shell MXFEM), to simulate the fatigue crack growth through the XFEM framework. The results of SIFs are also verified by the analytical expression proposed by Forman [5] who proposed the following formulation to calculate the stress intensity factor for a cylindrical pipe subjected to a tension axial loading with a circumferential crack of length 2a and central angle of 2a for the crack
KI ¼
I
1=2
2pa
pffiffiffiffiffiffi
r pa
Considering r as the axial tension stress applied at the ends of the pipe. I is defined as
h pffiffiffii I ¼ e1 a2 gðaÞ þ pk1 C 2 2 2
in which k is a crack length parameter equal to ðk ¼ a=2eÞ and C is
8 p k2 0:0293k3 < 1 þ 16 C ¼ pffiffi 0:5
0:885 : 2 2k þ 0:179 k p The other parameters
e2
t 1=2 ½12ð1 m2 Þ ¼ R
0 pffiffiffi gðaÞ ¼ 2 2@1 þ
9 k 6 1= k > 1;
e and gðaÞ are defined as below
12 1 a cot a h pffiffiffi pffiffiffiiA 2a cot a þ 2a cot ðp aÞ= 2
in which t and R are the thickness and radius of the pipe, respectively. The comparison of results is presented in Fig. 5. By this work, besides validating the prepared XFEM program, values of DK in Paris equation are computed so the material constants c and m can be predicted by a regression technique like the least square method. The computed material constants are c ¼ 1:4125 109 ; m ¼ 2:3. Fig. 6 shows the crack path in the test specimen while the XFEM modeled crack growth in the pipe is depicted in Fig. 7. 4.2. Fatigue crack growth in a pipe with an inclined crack The cylindrical aluminium pipe with the same geometry described in the previous section is tested with an inclined crack
of angle a ¼ 45 and initial length of a0 ¼ 25:5 mm. The fatigue crack trajectory and also the a-N curves are extracted from the test specimen, while they are also validated with XFEM results. Fig. 8 shows the calculated values of equivalent stress intensity factors of the XFEM model. It should be noted that an inclined initial crack in the pipe would lead to happen a mixed-mode fracture condition, so both KI and KII values should be considered to calculate DK into Paris equation. The proposed criterion by Richard[25] has been used to extract the equivalent stress intensity factor as below:
K eq ¼
KI 1 þ 2 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 2I þ 4ðaK II Þ2
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Fig. 5. XFEM and analytical values of SIFs versus different crack lengths.
Fig. 6. Circumferential crack in the pipe (a) Initial crack shape. (b) Crack growth under tensile axial stress.
The same expression is suggested to be used by Ghaffari and Hosseini-Toudeshky [13] who modeled fatigue crack growth in a repaired pipe by FEM method. After Computing SIFs in each step of crack growth, the a-N curve from the test experiments can be compared with the XFEM output. Fig. 9 shows the half crack length versus the number of load cycles for the pipe with the inclined crack. Reconsidering the logged data file of applied load cycles for the test specimen, it is observed that the maximum applied load has randomly exceeded the input value due to a difficulty in the hydraulic system of the testing machine so the experimental results of this test may prone to errors. Such a difficulty would lead to faster crack growth in the experimental test so the numerical predicted life is above the experimental result. A crack trajectory in the pipe with an inclined initial flaw is another issue in the experimental test that can be compared with XFEM models. Fig. 10 depicts the crack path in the test specimen and the predicted crack trajectory by the numerical method is shown by a dashed line in the figure. The results of the XFEM model are in a very good agreement with the experiment. The
inclined crack in the pipe behaves as a mixed-mode fracture problem, so a sudden turning occurs in the crack growth path. This turning happens at the first step of crack growth both in experimental test and XFEM model as shown in Fig. 10. By increasing the crack, since the load is in the axial direction, the first fracture mode becomes dominant and the crack tends to grow in a perpendicular path to the loading direction. So a horizontal extension of the crack can be observed in this figure. 4.3. Fatigue crack growth in a patch repaired pipe The pipe with a circumferential crack is now repaired with glass/epoxy polymer composite. The transparency of the glass/ epoxy patch makes it possible to trace the crack trajectory in the aluminium pipe. The pipe is wrapped in a length of 100 mm with the patch. The surface of the pipe is prepared using the ASTM D2651 specification. It should be noted the surface is degreased by non-etching alkaline cleaner. The cracked aluminium pipe is then wrapped by 2 layers of unidirectional fiberglass using epoxy resin. The fibers are aligned in the longitudinal axis of the pipe
H. Zarrinzadeh et al. / Engineering Structures 133 (2017) 24–32
29
Fig. 7. Schematic of the XFEM model of the circumferentially cracked pipe.
Fig. 8. Values of equivalent stress intensity factor versus different crack lengths of the pipe with an inclined crack.
which is perpendicular to the crack direction. The specimen is finally cured under curing temperature of 50 °C for at least 15 h. Fig. 11 shows the installed repaired pipe in the testing machine. Mechanical properties of the polymer composites are measured according to the ASTM D7205 standard as below:
E11 ¼ 31 GPa; E22 ¼ 4 GPa; G12 ¼ 3:6 GPa;
m12 ¼ 0:26;
total thickness ¼ 0:6 mm Fig. 12 compares the values of the SIF for the circumferential cracked pipe with and without patch repair.
As shown in this figure SIF values are reduced after repairing the pipe with the polymer composite. This reduction is more significant for a longer length of crack which means that the presence of the patch would be more effective as more as the crack grows. A comparison between the results of crack length versus number of cycles for the cracked pipe with and without patch repair is presented in Fig. 13. A considerable fatigue life extension is observed for the repaired pipe which is due to the reduction in SIF values in each step of crack growth after using the patch repair. This figure also exhibits the results of XFEM model of the pipe. As
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Fig. 9. Experimental and XFEM results of crack growth versus number of cycles for a pipe with inclined crack.
Fig. 10. (a) Schematic of crack deformation in the pipe with inclined crack. (b) Experimental and XFEM crack growth path.
shown in the figure a discrepancy is observed between the experimental and XFEM results. This difference may be explained by debonding of the patch in the experimental test which is not considered in XFEM simulation. The numerical results have a good agreement with the experimental output until no debonding has occurred between the patch and the pipe. At the half crack length of about a = 20 mm the debonding effect can be observed in the trend of the experimental curve but in numerical method the patch is assumed to be tied to the pipe surface. In future works numerical models can be developed which consider the debonding effect. Such a model should apply cohesive elements between the patch and the pipe which leads to a nonlinear and iterative model. 5. Conclusion
Fig. 11. Cracked pipe repaired by glass/epoxy polymer composite installed in the testing machine.
In the current study experimental tests has been performed to study the behaviour of crack growth in a cracked aluminium pipe under tensile fatigue loading. As the wire-cut technique could not be used to make such a crack in the pipe, Electrical Discharge Machining (EDM) technique is utilized as an alternative method to insert the initial crack in the pipe. The specimens are tested with a DARTEC 9600 fatigue testing machine. As there was no prefabricated fixture to perform a tensile fatigue loading test on a cylindrical pipe, a particular fixture has been designed and built.
H. Zarrinzadeh et al. / Engineering Structures 133 (2017) 24–32
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Fig. 12. Comparison of stress intensity factors for repairs and unrepaired pipe.
Fig. 13. Experimental and XFEM results of crack growth versus number of cycles for a the repaired and unrepaired pipe.
First a circumferentially cracked aluminium specimen is tested to extract the material constants required in the Paris equation for fatigue crack growth modeling. The pipe with an inclined crack is then tested under mixed-mode fracture condition. The results of the crack growth versus the number of applied cycles of load and also the fatigue crack trajectories are extracted from the experimental tests. The results are also validated by a XFEM code developed in MATLAB software. 3D degenerated shell elements are utilized to model such a cracked structure. It is shown that the numerical method can model the crack growth behaviour of the pipe in a very good agreement with experimental results. To increase the strength of the structure, the cracked pipe is wrapped by glass/epoxy polymer composite. The transparency of the patch makes it possible to trace the crack trajectory in the lower aluminium panel. It is observed that the stress intensity factor decreases after repairing the pipe with polymer composites and the fatigue life of the repaired pipe is extended considerably.
References [1] Erdogan F, Sih G. On the crack extension in plates under plane loading and transverse shear. J Basic Eng 1963;85:519–25. [2] Erdogan F, Kibler J. Cylindrical and spherical shells with cracks. Int J FractMech 1969;5:229–37.
[3] Zahoor A. Closed form expressions for fracture mechanics analysis of cracked pipes. J Pressure Vessel Technol 1985;107:203–5. [4] Sanders JL. Circumferential through-cracks in cylindrical shells under tension. J Appl Mech 1982;49:103–7. [5] Forman R, Hickman J, Shivaskumar V. Stress intensity factors for circumferential through cracks in hollow cylinders subjected to combined tension and bending loads. Eng Fract Mech 1985;21:563–71. [6] Zárate BA, Caicedo JM, Yu J, Ziehl P. Bayesian model updating and prognosis of fatigue crack growth. Eng Struct 2012;45:53–61. [7] Lam C, Cheng J, Yam C. Finite element study of cracked steel circular tube repaired by FRP patching. Procedia Eng 2011;14:1106–13. [8] Tong L, Sun X. Shape optimization of bonded patch to cylindrical shell structures. Int J Numer Meth Eng 2003;58:793–820. [9] Tong L, Sun X. Optimization of ply drop-offs in bonded patch to cylindrical shell structures. J Compos Mater 2003;37:1289–314. [10] Tong L, Sun X. Nonlinear stress analysis for bonded patch to curved thinwalled structures. Int J Adhes Adhes 2003;23:349–64. [11] Sun X, Tong L. Fracture toughness analysis of inclined crack in cylindrical shell repaired with bonded composite patch. Compos Struct 2004;66:639–45. [12] Pavlou D, Labeas G, Vlachakis N, Pavlou F. Fatigue crack propagation trajectories under mixed-mode cyclic loading. Eng Struct 2003;25:869–75. [13] Ghaffari MA, Hosseini-Toudeshky H. Fatigue crack propagation analysis of repaired pipes with composite patch under cyclic pressure. J Pressure Vessel Technol 2013;135:031402. [14] Areias P, Belytschko T. Non-linear analysis of shells with arbitrary evolving cracks using XFEM. Int J Numer Meth Eng 2005;62:384–415. [15] Areias PM, Song J, Belytschko T. Analysis of fracture in thin shells by overlapping paired elements. Comput Methods Appl Mech Eng 2006;195: 5343–60. [16] Areias PM, Belytschko T. Analysis of finite strain anisotropic elastoplastic fracture in thin plates and shells. J Aerospace Eng 2006;19:259–70.
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[17] Bayesteh H, Mohammadi S. XFEM fracture analysis of shells: the effect of crack tip enrichments. Comput Mater Sci 2011;50:2793–813. [18] Wyart E, Coulon D, Pardoen T, Remacle J-F, Lani F. Application of the substructured finite element/extended finite element method (S-FE/XFE) to the analysis of cracks in aircraft thin walled structures. Eng Fract Mech 2009;76:44–58. [19] Gandhi P, Murthy DR, Raghava G, Rao AM. Fatigue crack growth in stiffened steel tubular joints in seawater environment. Eng Struct 2000;22:1390–401. [20] Kabir MZ, Nazari A. The study of ultimate strength in notched cylinders subjected to axial compression. J Constr Steel Res 2011;67:1442–52. [21] Kabir MZ, Nazari A. Numerical Study on Reinforcing of Thin Walled Cracked Metal Cylindrical Columns Using FRP Patch. Sci Iran Trans A, Civ Eng 2010;17:407.
[22] Kabir MZ, Nazari A. Experimental and numerical study on the nonlinear response of notched cylinders under compressive loading. Sci Iran 2012;19:355–65. [23] Kabir MZ, Nazari A. Enhancing ultimate compressive strength of notch embedded steel cylinders using overwrap CFRP patch. Appl Compos Mater 2012;19:723–38. [24] Chau-Dinh T, Zi G, Lee P-S, Rabczuk T, Song J-H. Phantom-node method for shell models with arbitrary cracks. Comput Struct 2012;92:242–56. [25] Richard HA, Buchholz FG, Kullmer G, Schöllmann M. 2D-and 3D-mixed mode fracture criteria. Key Engineering Materials. Trans Tech Publ; 2003. p. 251–60.
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Experimental and numerical fatigue crack growth of an aluminium pipe repaired by composite patch H. Zarrinzadeh, M.Z. Kabir ⇑, A. Deylami Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, P.O. Box: 158754413, Iran
a r t i c l e
i n f o
Article history: Received 18 June 2016 Revised 8 December 2016 Accepted 10 December 2016
Keywords: Crack growth experiment XFEM crack growth Patch repaired pipe Stress intensity factors
a b s t r a c t In this study a cylindrical cracked aluminium pipe is considered. Fatigue crack growth behaviour of the pipe is observed through experimental tests. Stress intensity factors are computed for the pipe with an inclined crack under axial tensile load. Fatigue crack trajectory and also the crack growth curves versus number of cycles of load, are extracted. Validation of results is then achieved through the extended finite element method (XFEM). A stand-alone MATLAB programming package is developed to study such structures with 3D degenerated shell elements. The cracked pipe is finally repaired by glass/epoxy polymer composite and the effect of the patch is observed on the extension of fatigue life through experimental tests and the XFEM framework. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Crack initiation may occur in different structures such as metal or concrete elements. Fatigue is one of the most common kinds of loading which causes cracking initiation and propagation in structural elements. These structural parts can be found in building area or industry. For example, bridges which are utilized for providing passage of vehicles or trains, are subjected to fatigue loading that may cause damage in the bridge elements. Pipelines are another example of strategic infrastructures which are used for energy transferring. They may contain fluids such as gas, petroleum or water while pressure changes of the fluid would lead to internal fatigue loading in the pipe. Most of the aforementioned structures can be considered as 3D Shell elements. While this assumption takes into account the three dimensional geometry of the parts, the computational cost will decrease compared with 3D solid element assumption. Different methods are proposed to analyze shell elements which can be categorized as analytical, numerical and experimental studies. Erdogan et al. [1,2] studied on the cracked panels using an analytical formulation for the fracture parameters such as stress intensity factor. Other closed-form expressions for SIFs are presented by Zahoor [3], Sanders [4] and Forman [5] for cracked cylindrical pipes. Zárate et al. [6] presented a framework to update and predict crack length as a function of the number of cycles in structural elements subjected to fatigue. ⇑ Corresponding author. E-mail address: [email protected] (M.Z. Kabir). http://dx.doi.org/10.1016/j.engstruct.2016.12.011 0141-0296/Ó 2016 Elsevier Ltd. All rights reserved.
Analytical methods are mostly capable to solve just simple geometric problems with particular loading and boundary conditions. Of course, it should be mentioned that the analytical procedures are basis of the recent powerful numerical techniques. Finite element method is one of these techniques that can solve varieties of simple or complex engineering problems. Structures can be modeled and analyzed under arbitrary loading and boundary conditions through this framework. Lam et al. [7] studied on a cracked steel circular tube repaired by FRP patching through FEM method. Tong and Sun [8–11] have studied the effect of curvature existence in elements on the adhesive stress and fracture toughness. Pavlou et al. [12] proposed a new methodology to simulate the crack trajectories under mixed-mode fatigue loading through FEM method. The conventional finite element method remains simple until there is no discontinuity in the model. Discontinuities in elements would lead to singularities that make solving the problem more difficult. For instance, in problems containing cracked parts the generated mesh should be in a way that the crack body coincides the element edges. Singular elements should also be used as crack tip elements. This would lead to an irregular mashed part of the structure. The problem becomes more complex when crack propagation needs to be considered. In the case of crack propagation such as fatigue crack growth problems the crack body and crack tip will not be coincided with the element edges after growing the crack, so a new mesh generation is needed for the model in each step of solving the problem. This procedure is done in a work by Ghaffari and Hosseini-Toudeshky [13]. They studied on crack growth of a steel pipe under fatigue loading with and without FRP patching through FEM method. An automated re-meshing
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technique is performed through ANSYS Parametric Design Language (APDL) to detect the new geometry of the crack in each step of crack growth and assign adaptive mesh to the cracked panel. All the deficiencies of the conventional finite element method resulted to appear a new technique known as an extended finite element method (XFEM) in which a regular mesh generation can be used in even cracked parts. The elements in this method can be cut in arbitrary manners and just the formulation of the elements around the crack is modified in a way that the singularity of the problem is taken into account. Areias and Belytschko [14–16] worked on cracked shells by XFEM method considering the material nonlinearities without applying enrichment functions of crack tips. Bayesteh and Mohammadi [17] have extended Areias [14] works to study cracked shells applying Mindlin-Reissner theory, while they also considered the effect of the crack tip enrichment function. Wyart et al. [18] applied extended finite element method to analyze cracks in aircraft thin walled structures. Experimental testing is another way of studying problems. It is obvious that every structural part cannot be tested experimentally due to time and cost aspects, but the simplest experimental test can be utilized to validate the results of the numerical method. After a damage or crack is initiated in an element, affordable techniques of repairing are required for the damaged parts. Traditional techniques like using bolts or welding have some disadvantages such as vulnerability of welding to fatigue loading, high stress concentrations near bolts and increasing the total weight of the structure after using metal repairs. Among the new techniques of repairing, polymer composite materials are a proper alternatives which cover the aforementioned deficiencies of traditional retrofitting methods. Glass/epoxy polymer composite is used in this research as a repair patch. Gandhi et al. [19] studied on fatigue crack growth in stiffened steel tubular joints in seawater environment. Kabir and Nazari [20–23] experimentally studied on a cracked cylindrical steel pipe under compressive loading. The results were also compared with FEM models for the patched an un-patched pipe. In this study a cylindrical cracked aluminium pipe is studied by XFEM method and then validated by experimental tests. Fatigue crack growth analysis of the pipe is performed through the extended finite element method assuming 3D degenerated shell elements. The effect of glass/epoxy patch repair is then observed. The transparency of the patch makes it possible to trace the crack trajectory in the lower panel. Besides developing a stand-alone XFEM package in the MATLAB programming software, similar test specimens are provided to experimentally validate the numerical results. 2. Numerical formulation
at node i in the middle surface. fl3i tion cosine vector v 3i .
m3i
T
n3i g is related to direc-
2.2. XFEM formulation In XFEM formulation, the displacement field is divided into two parts
u ¼ uFE þ uENR in which u
uFE ðxÞ ¼
FE
ð2Þ
is the conventional finite element displacement.
n o ^i e2i ^ i Ma ^ i e1i þ M b Ni ðn; gÞ u
n X
ð3Þ
i¼1
Five degrees of freedom are considered in the conventional shell ^ i and two formulation which consist of three nodal displacements u ^i with respect to local orthonormal vectors e2i ^i; b local rotations a and e1i . uENR is related to the enriched part of approximation. To model weak or strong discontinuities in XFEM framework, one need to incorporate two types of enrichment functions into displacement approximation. The first type of enrichment function which is used to present the discontinuity across the crack, is the Heaviside step function. Dolbow introduced this function to simplify the representation of crack away from the tip. The Heaviside function is
hðxÞ ¼
abov e the crack
1
1 below the crack
ð4Þ
The Heaviside function comes in the degenerated shell formulation as below [17]
uHeav iside ðxÞ ¼
n n X Ni ðn; gÞ ðHð/ðxÞÞ Hð/ðxi ÞÞÞ i¼1
2.1. Degenerated shell element Using the so-called degenerated shell element formulation, computations are done similarly in a planar scheme, while the 3D properties of element are kept. Fig. 1 shows schematic of a degenerated shell element, nodal points and coordinate systems. The geometry of an element is described as
8 9 88 9 8 99 > > >
Fig. 1. Schematic of a degenerated shell element, nodal points and coordinate systems.
ð1Þ
where n; g are the two curvilinear coordinates in the middle of plane of shell and f is a linear coordinate in the thickness direction. N i ðn; gÞ are shape functions in element plane directions, fxi yi zi gT is the Cartesian coordinate. Mi ðfÞ ¼ f t2i is the one dimensional shape function in f direction and t i is the thickness
o ai Maai e1i þ Mabi e2i
ð5Þ
where ai is the vector of enriched displacement degrees of freedom at the mid-surface of the shell and aai and abi are rotations with respect to e2 and e1 respectively. The second type of functions called crack-tip enrichment functions, is usually derived from the asymptotic analytical solution. They are used to represent the singularity of stress field near the crack tip. The crack tip enrichment functions are used in a similar formulation as shown in Eq. (5) for shell elements, which consist of in-plane and out-of-plane enrichment functions as below [17] The enrichment functions for in-plane degrees of freedom are pffiffiffi pffiffiffi h pffiffiffi h pffiffiffi h h Fðr; hÞ ¼ r sin ; r cos ; r sin sinðhÞ; r cos sinðhÞ 2 2 2 2 ð6Þ
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The above functions are also used to enrich the rotational degrees of freedom. pffiffiffi pffiffiffi h pffiffiffi h pffiffiffi h h ; r cos ; r sin sinðhÞ; r cos sinðhÞ Rðr; hÞ ¼ r sin 2 2 2 2 ð7Þ
the analysis computations for millions of cycles, after calculating SIFs for one step of loading, a predefined crack growth increment Da is assumed, then the number of cycles DN related to this Da is determined with respect to material constants c; m.
The other degree of freedom related to out-of-plane displacement u3 is enriched with the following function
3. Experiments
Gðr; hÞ ¼
pffiffiffi h r sin 2
3.1. Specimens’ geometry
ð8Þ
where r; h are the polar coordinates in the local crack tip coordinate system. The crack tip is the origin of the system and h ¼ 0 is parallel to the crack. Fig. 2 shows the enriched nodes with the Heaviside and cracktip enrichment functions around the crack body. Stress intensity factors (SIFs) in the current shell model are calculated through the use of J-Integral method [24]. To perform a fatigue crack growth analysis, the Paris law is used
Da ¼ cðDKÞm DN
ð9Þ
where Da is crack growth increment, DN the increment for number of cycles, DK the range of SIF due to maximum and minimum load value and c; m are material constants for Paris equation. To facilitate
A cylindrical pipe made of 6063 aluminium alloy with a through-the-thickness crack is considered. The pipe has a length of 400 mm with internal and external diameters of 86.5 mm and 90.2 mm, respectively. Maximum tensile stress of 60 Mpa with a load ratio R ¼ 0:1 applies to pipe. The initial crack is placed circumferentially in the middle of the pipe. As the wire-cut technique could not be used to make such a crack in the pipe, Electrical Discharge Machining (EDM) technique is utilized as an alternative method. In this technique pure copper electrode with thickness of 0.18 mm has been prepared to be used in the spark procedure. To maximize the accuracy of inserting a proper initial crack, the copper electrode was modeled in a computer designing program and then this shape was extracted from a copper sheet by a CNC laser cutting machine. Fig. 3 shows the copper electrode and the aluminium pipe in the EDM machine. The initial length of the crack
Fig. 2. Definition of enriched nodes and illustration of level-set functions.
Fig. 3. (a) Copper electrode used to insert crack in the spark procedure. (b) The aluminium pipe in the EDM machine.
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Fig. 4. (a) DARTEC 9600 testing machine with the installed fixture. (b) Schematic of the lower part of the designed fixture to apply tensile load on the pipe.
after applying a pre-cracking to provide a sharpened crack tip is a0 = 30 mm. 3.2. Test preparation The specimens are tested with a DARTEC 9600 fatigue testing machine. As there was no prefabricated fixture to perform a tensile fatigue loading test on a cylindrical pipe, a particular fixture shown in Fig. 4 has been designed and built. To apply a uniform tensile stress, inside of the pipe is filled with a solid circular steel filler at both ends. Outside surfaces of both ends are then gripped with two other pieces which hold the pipe edges fixed during the loading procedure. The whole pieces are connected to the machine with two wedges at both ends. 4. Results and discussion 4.1. Fatigue crack growth in a pipe with a circumferential crack To extract the Paris constants c and m, experimental results of the cracked pipe with known magnitudes of Da; DN and DK are required in different steps of fatigue crack growth. To extract values of DK for each step of crack growth, analytical formulations or numerical modeling techniques can be applied. Here the specimens are modeled in the prepared MATLAB code (called as Shell MXFEM), to simulate the fatigue crack growth through the XFEM framework. The results of SIFs are also verified by the analytical expression proposed by Forman [5] who proposed the following formulation to calculate the stress intensity factor for a cylindrical pipe subjected to a tension axial loading with a circumferential crack of length 2a and central angle of 2a for the crack
KI ¼
I
1=2
2pa
pffiffiffiffiffiffi
r pa
Considering r as the axial tension stress applied at the ends of the pipe. I is defined as
h pffiffiffii I ¼ e1 a2 gðaÞ þ pk1 C 2 2 2
in which k is a crack length parameter equal to ðk ¼ a=2eÞ and C is
8 p k2 0:0293k3 < 1 þ 16 C ¼ pffiffi 0:5
0:885 : 2 2k þ 0:179 k p The other parameters
e2
t 1=2 ½12ð1 m2 Þ ¼ R
0 pffiffiffi gðaÞ ¼ 2 2@1 þ
9 k 6 1= k > 1;
e and gðaÞ are defined as below
12 1 a cot a h pffiffiffi pffiffiffiiA 2a cot a þ 2a cot ðp aÞ= 2
in which t and R are the thickness and radius of the pipe, respectively. The comparison of results is presented in Fig. 5. By this work, besides validating the prepared XFEM program, values of DK in Paris equation are computed so the material constants c and m can be predicted by a regression technique like the least square method. The computed material constants are c ¼ 1:4125 109 ; m ¼ 2:3. Fig. 6 shows the crack path in the test specimen while the XFEM modeled crack growth in the pipe is depicted in Fig. 7. 4.2. Fatigue crack growth in a pipe with an inclined crack The cylindrical aluminium pipe with the same geometry described in the previous section is tested with an inclined crack
of angle a ¼ 45 and initial length of a0 ¼ 25:5 mm. The fatigue crack trajectory and also the a-N curves are extracted from the test specimen, while they are also validated with XFEM results. Fig. 8 shows the calculated values of equivalent stress intensity factors of the XFEM model. It should be noted that an inclined initial crack in the pipe would lead to happen a mixed-mode fracture condition, so both KI and KII values should be considered to calculate DK into Paris equation. The proposed criterion by Richard[25] has been used to extract the equivalent stress intensity factor as below:
K eq ¼
KI 1 þ 2 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 2I þ 4ðaK II Þ2
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Fig. 5. XFEM and analytical values of SIFs versus different crack lengths.
Fig. 6. Circumferential crack in the pipe (a) Initial crack shape. (b) Crack growth under tensile axial stress.
The same expression is suggested to be used by Ghaffari and Hosseini-Toudeshky [13] who modeled fatigue crack growth in a repaired pipe by FEM method. After Computing SIFs in each step of crack growth, the a-N curve from the test experiments can be compared with the XFEM output. Fig. 9 shows the half crack length versus the number of load cycles for the pipe with the inclined crack. Reconsidering the logged data file of applied load cycles for the test specimen, it is observed that the maximum applied load has randomly exceeded the input value due to a difficulty in the hydraulic system of the testing machine so the experimental results of this test may prone to errors. Such a difficulty would lead to faster crack growth in the experimental test so the numerical predicted life is above the experimental result. A crack trajectory in the pipe with an inclined initial flaw is another issue in the experimental test that can be compared with XFEM models. Fig. 10 depicts the crack path in the test specimen and the predicted crack trajectory by the numerical method is shown by a dashed line in the figure. The results of the XFEM model are in a very good agreement with the experiment. The
inclined crack in the pipe behaves as a mixed-mode fracture problem, so a sudden turning occurs in the crack growth path. This turning happens at the first step of crack growth both in experimental test and XFEM model as shown in Fig. 10. By increasing the crack, since the load is in the axial direction, the first fracture mode becomes dominant and the crack tends to grow in a perpendicular path to the loading direction. So a horizontal extension of the crack can be observed in this figure. 4.3. Fatigue crack growth in a patch repaired pipe The pipe with a circumferential crack is now repaired with glass/epoxy polymer composite. The transparency of the glass/ epoxy patch makes it possible to trace the crack trajectory in the aluminium pipe. The pipe is wrapped in a length of 100 mm with the patch. The surface of the pipe is prepared using the ASTM D2651 specification. It should be noted the surface is degreased by non-etching alkaline cleaner. The cracked aluminium pipe is then wrapped by 2 layers of unidirectional fiberglass using epoxy resin. The fibers are aligned in the longitudinal axis of the pipe
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Fig. 7. Schematic of the XFEM model of the circumferentially cracked pipe.
Fig. 8. Values of equivalent stress intensity factor versus different crack lengths of the pipe with an inclined crack.
which is perpendicular to the crack direction. The specimen is finally cured under curing temperature of 50 °C for at least 15 h. Fig. 11 shows the installed repaired pipe in the testing machine. Mechanical properties of the polymer composites are measured according to the ASTM D7205 standard as below:
E11 ¼ 31 GPa; E22 ¼ 4 GPa; G12 ¼ 3:6 GPa;
m12 ¼ 0:26;
total thickness ¼ 0:6 mm Fig. 12 compares the values of the SIF for the circumferential cracked pipe with and without patch repair.
As shown in this figure SIF values are reduced after repairing the pipe with the polymer composite. This reduction is more significant for a longer length of crack which means that the presence of the patch would be more effective as more as the crack grows. A comparison between the results of crack length versus number of cycles for the cracked pipe with and without patch repair is presented in Fig. 13. A considerable fatigue life extension is observed for the repaired pipe which is due to the reduction in SIF values in each step of crack growth after using the patch repair. This figure also exhibits the results of XFEM model of the pipe. As
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Fig. 9. Experimental and XFEM results of crack growth versus number of cycles for a pipe with inclined crack.
Fig. 10. (a) Schematic of crack deformation in the pipe with inclined crack. (b) Experimental and XFEM crack growth path.
shown in the figure a discrepancy is observed between the experimental and XFEM results. This difference may be explained by debonding of the patch in the experimental test which is not considered in XFEM simulation. The numerical results have a good agreement with the experimental output until no debonding has occurred between the patch and the pipe. At the half crack length of about a = 20 mm the debonding effect can be observed in the trend of the experimental curve but in numerical method the patch is assumed to be tied to the pipe surface. In future works numerical models can be developed which consider the debonding effect. Such a model should apply cohesive elements between the patch and the pipe which leads to a nonlinear and iterative model. 5. Conclusion
Fig. 11. Cracked pipe repaired by glass/epoxy polymer composite installed in the testing machine.
In the current study experimental tests has been performed to study the behaviour of crack growth in a cracked aluminium pipe under tensile fatigue loading. As the wire-cut technique could not be used to make such a crack in the pipe, Electrical Discharge Machining (EDM) technique is utilized as an alternative method to insert the initial crack in the pipe. The specimens are tested with a DARTEC 9600 fatigue testing machine. As there was no prefabricated fixture to perform a tensile fatigue loading test on a cylindrical pipe, a particular fixture has been designed and built.
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Fig. 12. Comparison of stress intensity factors for repairs and unrepaired pipe.
Fig. 13. Experimental and XFEM results of crack growth versus number of cycles for a the repaired and unrepaired pipe.
First a circumferentially cracked aluminium specimen is tested to extract the material constants required in the Paris equation for fatigue crack growth modeling. The pipe with an inclined crack is then tested under mixed-mode fracture condition. The results of the crack growth versus the number of applied cycles of load and also the fatigue crack trajectories are extracted from the experimental tests. The results are also validated by a XFEM code developed in MATLAB software. 3D degenerated shell elements are utilized to model such a cracked structure. It is shown that the numerical method can model the crack growth behaviour of the pipe in a very good agreement with experimental results. To increase the strength of the structure, the cracked pipe is wrapped by glass/epoxy polymer composite. The transparency of the patch makes it possible to trace the crack trajectory in the lower aluminium panel. It is observed that the stress intensity factor decreases after repairing the pipe with polymer composites and the fatigue life of the repaired pipe is extended considerably.
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