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Stress intensity factor of semi-elliptical surface crack in a cylinder with hoop wrapped composite layer
International Journal of Pressure Vessels and Piping xxx (2013) 1e5
Contents lists available at SciVerse ScienceDirect
International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
Stress intensity factor of semi-elliptical surface crack in a cylinder with hoop wrapped composite layer Jianjun Chen*, Hongliang Pan Mechanical and Power Engineering Department, East China University of Science and Technology, 200237, China
a b s t r a c t Keywords: Stress intensity factor Cylinder Axial crack Finite element methods Hoop wrapped composite
In this paper the fracture behavior of the compressed natural gas (CNG) cylinder with hoop wrapped composite layer is investigated for the axial crack at the inner surface. By the aid of the threedimensional finite element method, the stress intensity factors along the crack front are obtained for different crack profiles. The effects of the cylinder geometry, hoop wrapped layer thickness and the property distributions of the composite layer on the stress intensity factor are discussed in detail. The numerical results show that the hoop wrapped composite cylinder can lower the stress intensity factor value greatly and ensure the safe use of pressure vessels containing defects in service. The composite property distribution owns a distinct effect on the crack behavior in different directions that provides a clear guide to the maintenance of CNG cylinder. Finally an approximate formula with high precision is proposed to evaluate the stress intensity factor along the axial crack front in the hoop wrapped cylinder with composite layer. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction In the transportation and energy industrials, steel liner hoop wrapped cylinders are often used to store industrial gases such as oxygen, nitrogen and the compressed natural gas (CNG). As a clean, reliable and competitive source of energy, natural gas owns vast reserves and relatively lower price compared to other energy resources. Because the CNG is stored in the cylinder with a high pressure, the safety problem becomes to be a key factor for its further usage and improvement. Although the design and use of CNG are strictly abide by certain safety requirements and standards [1], there are still explosion reports of CNG cylinder in recent years. The accidents reports showed that some of the explosions are caused by hydrogen embrittlement and others are due to the inappropriate choice of the liner material. Because the hydrogen is very sensitive to the surface defects, even a tiny notch would accelerate hydrogen corrosion rate and leads to final cylinder explosion. To avoid the catastrophic explosion it should pay more attentions to the surface quality of the CNG cylinder. On the other hand, the engineers and researchers must gain more fracture knowledge of CNG cylinder for the better design and maintenance work. In
* Corresponding author. E-mail address: [email protected] (J. Chen).
recent years, considerable effort has been devoted to the calculation of fracture responses of the pressurized cylinders [2e6]. However, the research work on the surface crack in the hoopwrapped composite cylinders is still limited in the open literature. The purpose of this paper is to systematically analyze the fracture behavior of the axial elliptical crack in a hoop-wrapped composite CNG cylinder. The stress intensity factors along the crack front are calculated by three dimensional finite element method for a wide range of crack geometries, cylinder dimensions, composite layer thicknesses and property distributions of the composite hoop-wrapped layer. An approximate formula for the stress intensity factor evaluation is also proposed and compared with the finite element results. 2. Stress intensity factor of the hoop-wrapped cylinder 2.1. Geometry and properties Consider a cylinder with a finite elliptical axial surface crack, subject to the internal pressure pi, as depicted in Fig. 1. The cylinder can be considered to be made by two parts: the steel liner and the hoop-wrapped composite layer. The inner and outer radius of the steel liner is ri and ro respectively. The thickness of the liner is t (¼ro ri) and the thickness of hoop-wrapped layer is tc. The depth and the semi-length of the elliptical crack are depicted by ‘a’ and ‘c’ respectively. The different node position along the crack front will
0308-0161/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijpvp.2013.04.026
Please cite this article in press as: Chen J, Pan H, Stress intensity factor of semi-elliptical surface crack in a cylinder with hoop wrapped composite layer, International Journal of Pressure Vessels and Piping (2013), http://dx.doi.org/10.1016/j.ijpvp.2013.04.026
2
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Fig. 1. Schematic view of composite hoop-wrapped cylinder with semi-elliptical crack at the inner surface.
crack front the local mesh is refined enough to achieve the stress and strain distribution accurately. In order to obtain the square root singularity at the crack tip, the element is meshed by shifting the mid-side nodes to the quarter point location in the region around the crack front. The nodes in the crack front elements are collapsed as illustrated in Fig. 2(b). A local cylindrical coordinate system is established for the better understanding of radial, circumferential and axial stress distribution inside the hoop-wrapped cylinder. In this new local coordinate system the r- and q-axis locate in the crack plane and the z-axis lies along the length direction of the cylinder. The stress intensity factors of the CNG cylinder are extracted from the stress-strain field using a domain integral procedure provided by ABAQUS. By the domain integral method, a crack-tip contour integral can be expressed as a volume integral over a finite domain surrounding the crack tip. The process of recasting the contour integral into a volume integral is advantageous for numerical purposes. By the domain integral method the fracture parameters can be accurately obtained even for a coarse mesh. A more general discussion on crack-tip contour integrals and their associated domain integral method can be found from Moran and Shih’s classic work [8]. 3. Results and discussion
be denoted with the central angle of the semi-elliptical crack with symbol q. Notation for relevant dimensions is shown in Fig. 1 as well. In this work CreMo steel is selected as the metallic liner material and its Young’s modulus and Poisson’s ratio are 200 GPa and 0.28 respectively. Different composite materials can be chosen to cover the steel liner for very different purposes. Here the E-glass fiber/epoxy composite, the commonest choice in the real application, is chosen as the hoop-wrapped layer material in the study case. The steel liner and the hoop-wrapped composite layer are modeled as a linear isotropic and orthotropic material in the FE model respectively. The mechanical properties of the E-glass fiber/ epoxy, are listed in the Table 1.
The finite element results can provide a high accurate strainstress field around the crack front by the refined meshes. Fig. 3 shows the distribution of the von Mises stress near the crack front. It can be seen that the stress level is changed along the crack front due to the cylinder geometry and the constraint. To give a better understanding to the fracture behavior of the hoop-wrapped composite cylinder, in the following analyses the calculated stress intensity factors are normalized by the internal stress pi. Because the numerical simulations are limited within the elastic field, the normalization data can provide a clear role of different influencing factors excluding the internal pressure.
2.2. Finite element model
Fig. 4(a) shows the distribution of the stress intensity factor of the axial crack in the hoop-wrapped composite cylinder. The relative crack depth a/t is 0.5 and the crack semi-length c is changed from a/2 to 4a. The distribution of the stress intensity factor along the axial crack front is varied according to different crack profiles. For a slender crack (c/a > 1) the stress intensity factor reaches its largest values at the deepest points (2q/p ¼ 1) and the lowest value near the inner surface. However for the deep crack (c/a < 1) the maximum stress intensity factor occurs at the corner point (2q/ p ¼ 0) of the inner surface. This means that a probable crack propagation of the slender crack may start from the deepest point; while the starting point of the deep crack is at the corner end for the axial crack in CNG cylinder. The variation of the maximum intensity factor value as a function of crack profile c/a is illustrated in Fig. 4(b). It can be seen that the maximum stress intensity factor increases monotonically with the increasing of c/a. As for the cylinder structure, the axial crack appears more dangerous than the circumferential crack because the hoop stress is much larger than other stress components. So in the CNG cylinder the axial crack plays a more dangerous role than the circumferential one even when they have the same dimensions.
For a pressurized cylinder with hoop wrapped composite layer, its stress intensity factor K may be expressed as
K ¼
pi ri pffiffiffiffiffiffi ri a c tc paF ; ; ; ; q t t t a t
(1)
where F represents the shape function for the stress intensity factor as a function of ri/t, a/t, c/a, tc/t and q. Values of F for different crack profile can be obtained from the numerical calculations. The general-purpose finite element code ABAQUS [7] is used in this study to obtain the stress fields of the hoop-wrapped composite cylinder and the stress intensity factor along the semi-elliptical crack front. As a result of the symmetry, only one-eighth of the cylinder is considered in the modeling. The twenty-node reduced element C3D20R is used to construct the model. The finite element mesh for the axial crack is shown in Fig. 2(a). The whole FE-model has nearly 40,000 elements and 200,000 nodes for the hoopwrapped cylinder with different crack dimensions. Around the
Table 1 Mechanical properties of steel liner and composite layer materials.
3.1. Crack profile
3.2. Cylinder geometry
Composite material
E1 (GPa)
E2 (GPa)
G12 (GPa)
n12
n23
E-glass fiber/epoxy
78
26
13
0.25
0.25
The effect of the cylinder radius on the SIF of hoop wrapped cylinder is plotted in Fig. 5(a). It can be found that the SIF curve of
Please cite this article in press as: Chen J, Pan H, Stress intensity factor of semi-elliptical surface crack in a cylinder with hoop wrapped composite layer, International Journal of Pressure Vessels and Piping (2013), http://dx.doi.org/10.1016/j.ijpvp.2013.04.026
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Fig. 2. (a) Whole mesh of a 1/8 hoop-wrapped cylinder with axial crack (b) 20-node brick element and collapsed singularity element at the crack front.
the axial crack isn’t convergent at the corner point (2q/p ¼ 0). The change of the cylinder geometry makes the SIF curves move in parallel. Fig. 5(b) shows the variation of the maximum stress intensity as a function of the cylinder radius. It can be seen that the stress intensity factor vary greatly with the increase of the cylinder radius ri. It means that for the same crack dimensions the larger radius the cylinder has, the more dangerous the cylinder becomes. 3.3. Thickness of the composite layer Fig. 6(a) shows the distribution of the stress intensity factor against the different hoop-wrapped layer thicknesses. The variation of the maximum stress intensity factor along the crack front is plotted in Fig. 6(b) for the case of composite layer thickness tc varying from 0 to t. Again the maximum SIF value decreases with the increasing of the layer thickness. 3.4. Effect of property distribution The effects of the composite property distribution on the stress intensity factor of the axial crack are illustrated in Fig. 7(a) and Fig. 7,(b). It can be found that the SIF curve declines distinctively with the increasing of E1, but the curves for different E2 are almost overlapped. It implies that the change of elastic modulus along the circumferential direction has a large effect on the stress intensity factor value, while the property change along the axial direction shows little contribute to the SIF curves. Therefore, to extend the service life of a CNG cylinder containing defect at the inner surface, different hoop wrapped materials should be chosen according to variant internal crack orientation. Different property distributions of the composite layers have diverse functions on the crack behavior. A composite owns large E1 Young’s modulus is suitable to the axial crack. If the cylinder contains both axial and circumferential crack, the composite with a large E1 Young’s modulus is preferred since the axial crack has a much higher SIF value than the circumferential one when they have the same crack dimensions.
4. Estimation of SIF of axial crack Because the axial crack appears more dangerous in the cylinder structure as mentioned above, developing a simple approach to predict the stress intensity factor along the elliptical crack front becomes to be very important for the safe use of CNG. However, the existing SIF formula for the cylinder surface crack cannot be used to evaluate the fracture parameter of hoop wrapped cylinder directly. To obtain an appropriate expression for the elliptic crack in the CNG cylinder, a modified coefficient f is introduced and defined as
f ¼ Khwc =Kc
(2)
here Khwc is the stress intensity factor for the hoop wrapped cylinder and Kc is the stress intensity factor for the pure cylinder. The modified coefficient can be achieved by deducing the finite element analyses results. It can be found that the modified coefficient f almost keep constant under the constant value of tc/t and E1/Ec. Moreover the relative deviation is less than 5% along the crack front. So the modified coefficient f can be regarded as a function only containing the elastic modulus ratio (E1/Ec) and thickness ratio (tc/t). A more detailed distribution of f can be derived as:
f ¼
1þ
Kc ¼ savg
tc t
0:11:1E1 =Ec þ0:25ðE1 =Ec Þ2
rffiffiffiffiffiffiffiffi a a a r p F ; ; i; q Q c t t
(3)
(4)
in which
Q ¼ 1:464ða=cÞ1:65
(5)
savg is the average hoop stress along the wall thickness and is given by
Fig. 3. Mises stress distribution near the crack front.
Please cite this article in press as: Chen J, Pan H, Stress intensity factor of semi-elliptical surface crack in a cylinder with hoop wrapped composite layer, International Journal of Pressure Vessels and Piping (2013), http://dx.doi.org/10.1016/j.ijpvp.2013.04.026
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Fig. 4. (a) Variation of stress intensity factor along the crack front with different crack shape c/a (b) Maximum stress intensity factor for different crack shapes (c/a).
Fig. 5. (a) Variation of stress intensity factor along the crack front with different cylinder radii (b) Variation of maximum stress intensity value as function of cylinder radii.
Zro
savg ¼
ri
pi ri2 ro2 ri2
! r2 1 þ o2 dr ri
ro ri
Kc ¼ f
¼
pi ri pr ¼ i i ro ri t
(6)
The expression of F for the cylinder surface is introduced in Ref. [9] and the detailed expression can be found in the Appendix as well. Thus a complete stress intensity factor formula for the hoop wrapped cylinder can be written as
pi ri t
rffiffiffiffiffiffiffiffi a a a r p F ; ; i; q Q c t t
(7)
To verify above equation, a specific case is calculated for the hoop wrapped cylinder with an axial crack. The cylinder’s geometry dimensions are ri ¼ 0.33 m, ro ¼ 0.34 m, a ¼ 0.5 mm, c ¼ 1.5 mm and tc ¼ 10 mm. The elastic modulus ratio is E1/Ec ¼ 78/200. The K1 obtained from above formula is plotted in Fig. 8 as well as the results obtained by the 3-D finite element analysis. The two SIF curves are coincident and the maximum deviation is within 5 percent. This shows the feasibility and accuracy of our proposed formula for the stress intensity factor evaluation of the hoop wrapped cylinder.
Fig. 6. (a) Variation of stress intensity factor along the crack front with different composite layer thickness (b) variation of maximum stress intensity value as function of hoopwrapped layer thickness.
Please cite this article in press as: Chen J, Pan H, Stress intensity factor of semi-elliptical surface crack in a cylinder with hoop wrapped composite layer, International Journal of Pressure Vessels and Piping (2013), http://dx.doi.org/10.1016/j.ijpvp.2013.04.026
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Fig. 7. (a) Effect of E1 on the stress intensity factor distribution (b) Effect of E2 for the stress intensity factor distribution.
a2 a4 i h gf4 fc F ¼ 0:97 M1 þ M2 þ M3 t t
(A.1)
where
M1 ¼ 1:13 0:09 M2 ¼ 0:54 þ
M3 ¼ 0:5
Fig. 8. Value of the stress intensity factor from finite element analysis and proposed formula.
5. Conclusion In the present paper the three-dimensional finite element calculations are performed to investigate the axial surface crack inside the CNG cylinder with hoop wrapped composite layer. The effects of the crack profile and the cylinder geometry as well as composite thickness and the composite property distribution on the stress intensity factor along the crack front are determined. Moreover an empirical formula for the stress intensity factor of an axial crack in the pressurized hoop wrapped cylinder is deduced. The KI result from this equation has a high accuracy and covers a wide range of configuration parameters within 5 percent error band. It is believed the results of this analysis will provide the engineers with a valuable knowledge that may be used in a more realistic assessment of CNG cylinders containing flaws.
Acknowledgment The authors are grateful to the financial support by Natural Science Foundation of China (51105143), the Fundamental Research Funds for the Central Universities (1114036) and Shanghai University Young Teachers Training Funds (YG0142129).
Appendix A The expression of function F can be drawn as:
a c
(A.2)
0:89 0:2 þ
a c
a24 a þ 14 1 c 0:65 þ c 1
(A.3)
(A.4)
2 h a2 i 1 sin2 4 g ¼ 1 þ 0:1 þ 0:35 t
(A.5)
h a2 i1=4 cos2 4 f4 ¼ sin2 4 þ c
(A.6)
fc ¼
rffiffiffi! a t þ 1 0:5 2 2 t ri ro þ ri ro2 ri2
(A.7)
And the parameters f and q obeys
tanq ¼
a tan4 c
(A.8)
References [1] E/ECE/TRANS/505. Specific components of motor vehicles using compressed natural gas (CNG) in their propulsion system 2001. Rev. 2, Add. 109, Reg. 110. [2] Newman Jr JC, Raju IS. Stress intensity factor for internal and external surface cracks in cylindrical vessels. J Press Vess Tech 1982;104:293e8. [3] Delale F, Erdogan F. Application of line-spring model to a cylindrical shell containing a circumferential or axial part-trough crack. J Appl Mech 1982;49: 97e102. [4] Lin XB, Smith RA. Fatigue growth prediction of internal surface cracks in pressure vessels. J Press Vess Tech 1998;120:17e23. [5] Kim Y-J, Kim J-S, Park Y-J, Kim Y-J. Elasticeplastic fracture mechanics method for finite internal axial surface cracks in cylinders. Eng Fract Mech 2004;71: 925e44. [6] Perl M, Levy C, Wang J. Interaction effects in combined arrays of radial and longitudinal semi-elliptical surface cracks in pressurized thick-walled cylinder. J Press Vess Tech 1997;119:167e74. [7] ABAQUS Version 6.5: user’s manual. RI: ABAQUS, Inc; 2005. [8] Moran B, Shih CF. Crack tip and associated domain integrals from momentum and energy balance. Eng Fract Mech 1987;27:615e42. [9] Newman JC, Raju IS. Stress-intensity factors for internal surface cracks in cylindrical pressure vessels. J Press Vess Tech 1980;102:342e6.
Please cite this article in press as: Chen J, Pan H, Stress intensity factor of semi-elliptical surface crack in a cylinder with hoop wrapped composite layer, International Journal of Pressure Vessels and Piping (2013), http://dx.doi.org/10.1016/j.ijpvp.2013.04.026
Contents lists available at SciVerse ScienceDirect
International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
Stress intensity factor of semi-elliptical surface crack in a cylinder with hoop wrapped composite layer Jianjun Chen*, Hongliang Pan Mechanical and Power Engineering Department, East China University of Science and Technology, 200237, China
a b s t r a c t Keywords: Stress intensity factor Cylinder Axial crack Finite element methods Hoop wrapped composite
In this paper the fracture behavior of the compressed natural gas (CNG) cylinder with hoop wrapped composite layer is investigated for the axial crack at the inner surface. By the aid of the threedimensional finite element method, the stress intensity factors along the crack front are obtained for different crack profiles. The effects of the cylinder geometry, hoop wrapped layer thickness and the property distributions of the composite layer on the stress intensity factor are discussed in detail. The numerical results show that the hoop wrapped composite cylinder can lower the stress intensity factor value greatly and ensure the safe use of pressure vessels containing defects in service. The composite property distribution owns a distinct effect on the crack behavior in different directions that provides a clear guide to the maintenance of CNG cylinder. Finally an approximate formula with high precision is proposed to evaluate the stress intensity factor along the axial crack front in the hoop wrapped cylinder with composite layer. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction In the transportation and energy industrials, steel liner hoop wrapped cylinders are often used to store industrial gases such as oxygen, nitrogen and the compressed natural gas (CNG). As a clean, reliable and competitive source of energy, natural gas owns vast reserves and relatively lower price compared to other energy resources. Because the CNG is stored in the cylinder with a high pressure, the safety problem becomes to be a key factor for its further usage and improvement. Although the design and use of CNG are strictly abide by certain safety requirements and standards [1], there are still explosion reports of CNG cylinder in recent years. The accidents reports showed that some of the explosions are caused by hydrogen embrittlement and others are due to the inappropriate choice of the liner material. Because the hydrogen is very sensitive to the surface defects, even a tiny notch would accelerate hydrogen corrosion rate and leads to final cylinder explosion. To avoid the catastrophic explosion it should pay more attentions to the surface quality of the CNG cylinder. On the other hand, the engineers and researchers must gain more fracture knowledge of CNG cylinder for the better design and maintenance work. In
* Corresponding author. E-mail address: [email protected] (J. Chen).
recent years, considerable effort has been devoted to the calculation of fracture responses of the pressurized cylinders [2e6]. However, the research work on the surface crack in the hoopwrapped composite cylinders is still limited in the open literature. The purpose of this paper is to systematically analyze the fracture behavior of the axial elliptical crack in a hoop-wrapped composite CNG cylinder. The stress intensity factors along the crack front are calculated by three dimensional finite element method for a wide range of crack geometries, cylinder dimensions, composite layer thicknesses and property distributions of the composite hoop-wrapped layer. An approximate formula for the stress intensity factor evaluation is also proposed and compared with the finite element results. 2. Stress intensity factor of the hoop-wrapped cylinder 2.1. Geometry and properties Consider a cylinder with a finite elliptical axial surface crack, subject to the internal pressure pi, as depicted in Fig. 1. The cylinder can be considered to be made by two parts: the steel liner and the hoop-wrapped composite layer. The inner and outer radius of the steel liner is ri and ro respectively. The thickness of the liner is t (¼ro ri) and the thickness of hoop-wrapped layer is tc. The depth and the semi-length of the elliptical crack are depicted by ‘a’ and ‘c’ respectively. The different node position along the crack front will
0308-0161/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijpvp.2013.04.026
Please cite this article in press as: Chen J, Pan H, Stress intensity factor of semi-elliptical surface crack in a cylinder with hoop wrapped composite layer, International Journal of Pressure Vessels and Piping (2013), http://dx.doi.org/10.1016/j.ijpvp.2013.04.026
2
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Fig. 1. Schematic view of composite hoop-wrapped cylinder with semi-elliptical crack at the inner surface.
crack front the local mesh is refined enough to achieve the stress and strain distribution accurately. In order to obtain the square root singularity at the crack tip, the element is meshed by shifting the mid-side nodes to the quarter point location in the region around the crack front. The nodes in the crack front elements are collapsed as illustrated in Fig. 2(b). A local cylindrical coordinate system is established for the better understanding of radial, circumferential and axial stress distribution inside the hoop-wrapped cylinder. In this new local coordinate system the r- and q-axis locate in the crack plane and the z-axis lies along the length direction of the cylinder. The stress intensity factors of the CNG cylinder are extracted from the stress-strain field using a domain integral procedure provided by ABAQUS. By the domain integral method, a crack-tip contour integral can be expressed as a volume integral over a finite domain surrounding the crack tip. The process of recasting the contour integral into a volume integral is advantageous for numerical purposes. By the domain integral method the fracture parameters can be accurately obtained even for a coarse mesh. A more general discussion on crack-tip contour integrals and their associated domain integral method can be found from Moran and Shih’s classic work [8]. 3. Results and discussion
be denoted with the central angle of the semi-elliptical crack with symbol q. Notation for relevant dimensions is shown in Fig. 1 as well. In this work CreMo steel is selected as the metallic liner material and its Young’s modulus and Poisson’s ratio are 200 GPa and 0.28 respectively. Different composite materials can be chosen to cover the steel liner for very different purposes. Here the E-glass fiber/epoxy composite, the commonest choice in the real application, is chosen as the hoop-wrapped layer material in the study case. The steel liner and the hoop-wrapped composite layer are modeled as a linear isotropic and orthotropic material in the FE model respectively. The mechanical properties of the E-glass fiber/ epoxy, are listed in the Table 1.
The finite element results can provide a high accurate strainstress field around the crack front by the refined meshes. Fig. 3 shows the distribution of the von Mises stress near the crack front. It can be seen that the stress level is changed along the crack front due to the cylinder geometry and the constraint. To give a better understanding to the fracture behavior of the hoop-wrapped composite cylinder, in the following analyses the calculated stress intensity factors are normalized by the internal stress pi. Because the numerical simulations are limited within the elastic field, the normalization data can provide a clear role of different influencing factors excluding the internal pressure.
2.2. Finite element model
Fig. 4(a) shows the distribution of the stress intensity factor of the axial crack in the hoop-wrapped composite cylinder. The relative crack depth a/t is 0.5 and the crack semi-length c is changed from a/2 to 4a. The distribution of the stress intensity factor along the axial crack front is varied according to different crack profiles. For a slender crack (c/a > 1) the stress intensity factor reaches its largest values at the deepest points (2q/p ¼ 1) and the lowest value near the inner surface. However for the deep crack (c/a < 1) the maximum stress intensity factor occurs at the corner point (2q/ p ¼ 0) of the inner surface. This means that a probable crack propagation of the slender crack may start from the deepest point; while the starting point of the deep crack is at the corner end for the axial crack in CNG cylinder. The variation of the maximum intensity factor value as a function of crack profile c/a is illustrated in Fig. 4(b). It can be seen that the maximum stress intensity factor increases monotonically with the increasing of c/a. As for the cylinder structure, the axial crack appears more dangerous than the circumferential crack because the hoop stress is much larger than other stress components. So in the CNG cylinder the axial crack plays a more dangerous role than the circumferential one even when they have the same dimensions.
For a pressurized cylinder with hoop wrapped composite layer, its stress intensity factor K may be expressed as
K ¼
pi ri pffiffiffiffiffiffi ri a c tc paF ; ; ; ; q t t t a t
(1)
where F represents the shape function for the stress intensity factor as a function of ri/t, a/t, c/a, tc/t and q. Values of F for different crack profile can be obtained from the numerical calculations. The general-purpose finite element code ABAQUS [7] is used in this study to obtain the stress fields of the hoop-wrapped composite cylinder and the stress intensity factor along the semi-elliptical crack front. As a result of the symmetry, only one-eighth of the cylinder is considered in the modeling. The twenty-node reduced element C3D20R is used to construct the model. The finite element mesh for the axial crack is shown in Fig. 2(a). The whole FE-model has nearly 40,000 elements and 200,000 nodes for the hoopwrapped cylinder with different crack dimensions. Around the
Table 1 Mechanical properties of steel liner and composite layer materials.
3.1. Crack profile
3.2. Cylinder geometry
Composite material
E1 (GPa)
E2 (GPa)
G12 (GPa)
n12
n23
E-glass fiber/epoxy
78
26
13
0.25
0.25
The effect of the cylinder radius on the SIF of hoop wrapped cylinder is plotted in Fig. 5(a). It can be found that the SIF curve of
Please cite this article in press as: Chen J, Pan H, Stress intensity factor of semi-elliptical surface crack in a cylinder with hoop wrapped composite layer, International Journal of Pressure Vessels and Piping (2013), http://dx.doi.org/10.1016/j.ijpvp.2013.04.026
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Fig. 2. (a) Whole mesh of a 1/8 hoop-wrapped cylinder with axial crack (b) 20-node brick element and collapsed singularity element at the crack front.
the axial crack isn’t convergent at the corner point (2q/p ¼ 0). The change of the cylinder geometry makes the SIF curves move in parallel. Fig. 5(b) shows the variation of the maximum stress intensity as a function of the cylinder radius. It can be seen that the stress intensity factor vary greatly with the increase of the cylinder radius ri. It means that for the same crack dimensions the larger radius the cylinder has, the more dangerous the cylinder becomes. 3.3. Thickness of the composite layer Fig. 6(a) shows the distribution of the stress intensity factor against the different hoop-wrapped layer thicknesses. The variation of the maximum stress intensity factor along the crack front is plotted in Fig. 6(b) for the case of composite layer thickness tc varying from 0 to t. Again the maximum SIF value decreases with the increasing of the layer thickness. 3.4. Effect of property distribution The effects of the composite property distribution on the stress intensity factor of the axial crack are illustrated in Fig. 7(a) and Fig. 7,(b). It can be found that the SIF curve declines distinctively with the increasing of E1, but the curves for different E2 are almost overlapped. It implies that the change of elastic modulus along the circumferential direction has a large effect on the stress intensity factor value, while the property change along the axial direction shows little contribute to the SIF curves. Therefore, to extend the service life of a CNG cylinder containing defect at the inner surface, different hoop wrapped materials should be chosen according to variant internal crack orientation. Different property distributions of the composite layers have diverse functions on the crack behavior. A composite owns large E1 Young’s modulus is suitable to the axial crack. If the cylinder contains both axial and circumferential crack, the composite with a large E1 Young’s modulus is preferred since the axial crack has a much higher SIF value than the circumferential one when they have the same crack dimensions.
4. Estimation of SIF of axial crack Because the axial crack appears more dangerous in the cylinder structure as mentioned above, developing a simple approach to predict the stress intensity factor along the elliptical crack front becomes to be very important for the safe use of CNG. However, the existing SIF formula for the cylinder surface crack cannot be used to evaluate the fracture parameter of hoop wrapped cylinder directly. To obtain an appropriate expression for the elliptic crack in the CNG cylinder, a modified coefficient f is introduced and defined as
f ¼ Khwc =Kc
(2)
here Khwc is the stress intensity factor for the hoop wrapped cylinder and Kc is the stress intensity factor for the pure cylinder. The modified coefficient can be achieved by deducing the finite element analyses results. It can be found that the modified coefficient f almost keep constant under the constant value of tc/t and E1/Ec. Moreover the relative deviation is less than 5% along the crack front. So the modified coefficient f can be regarded as a function only containing the elastic modulus ratio (E1/Ec) and thickness ratio (tc/t). A more detailed distribution of f can be derived as:
f ¼
1þ
Kc ¼ savg
tc t
0:11:1E1 =Ec þ0:25ðE1 =Ec Þ2
rffiffiffiffiffiffiffiffi a a a r p F ; ; i; q Q c t t
(3)
(4)
in which
Q ¼ 1:464ða=cÞ1:65
(5)
savg is the average hoop stress along the wall thickness and is given by
Fig. 3. Mises stress distribution near the crack front.
Please cite this article in press as: Chen J, Pan H, Stress intensity factor of semi-elliptical surface crack in a cylinder with hoop wrapped composite layer, International Journal of Pressure Vessels and Piping (2013), http://dx.doi.org/10.1016/j.ijpvp.2013.04.026
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J. Chen, H. Pan / International Journal of Pressure Vessels and Piping xxx (2013) 1e5
Fig. 4. (a) Variation of stress intensity factor along the crack front with different crack shape c/a (b) Maximum stress intensity factor for different crack shapes (c/a).
Fig. 5. (a) Variation of stress intensity factor along the crack front with different cylinder radii (b) Variation of maximum stress intensity value as function of cylinder radii.
Zro
savg ¼
ri
pi ri2 ro2 ri2
! r2 1 þ o2 dr ri
ro ri
Kc ¼ f
¼
pi ri pr ¼ i i ro ri t
(6)
The expression of F for the cylinder surface is introduced in Ref. [9] and the detailed expression can be found in the Appendix as well. Thus a complete stress intensity factor formula for the hoop wrapped cylinder can be written as
pi ri t
rffiffiffiffiffiffiffiffi a a a r p F ; ; i; q Q c t t
(7)
To verify above equation, a specific case is calculated for the hoop wrapped cylinder with an axial crack. The cylinder’s geometry dimensions are ri ¼ 0.33 m, ro ¼ 0.34 m, a ¼ 0.5 mm, c ¼ 1.5 mm and tc ¼ 10 mm. The elastic modulus ratio is E1/Ec ¼ 78/200. The K1 obtained from above formula is plotted in Fig. 8 as well as the results obtained by the 3-D finite element analysis. The two SIF curves are coincident and the maximum deviation is within 5 percent. This shows the feasibility and accuracy of our proposed formula for the stress intensity factor evaluation of the hoop wrapped cylinder.
Fig. 6. (a) Variation of stress intensity factor along the crack front with different composite layer thickness (b) variation of maximum stress intensity value as function of hoopwrapped layer thickness.
Please cite this article in press as: Chen J, Pan H, Stress intensity factor of semi-elliptical surface crack in a cylinder with hoop wrapped composite layer, International Journal of Pressure Vessels and Piping (2013), http://dx.doi.org/10.1016/j.ijpvp.2013.04.026
J. Chen, H. Pan / International Journal of Pressure Vessels and Piping xxx (2013) 1e5
5
Fig. 7. (a) Effect of E1 on the stress intensity factor distribution (b) Effect of E2 for the stress intensity factor distribution.
a2 a4 i h gf4 fc F ¼ 0:97 M1 þ M2 þ M3 t t
(A.1)
where
M1 ¼ 1:13 0:09 M2 ¼ 0:54 þ
M3 ¼ 0:5
Fig. 8. Value of the stress intensity factor from finite element analysis and proposed formula.
5. Conclusion In the present paper the three-dimensional finite element calculations are performed to investigate the axial surface crack inside the CNG cylinder with hoop wrapped composite layer. The effects of the crack profile and the cylinder geometry as well as composite thickness and the composite property distribution on the stress intensity factor along the crack front are determined. Moreover an empirical formula for the stress intensity factor of an axial crack in the pressurized hoop wrapped cylinder is deduced. The KI result from this equation has a high accuracy and covers a wide range of configuration parameters within 5 percent error band. It is believed the results of this analysis will provide the engineers with a valuable knowledge that may be used in a more realistic assessment of CNG cylinders containing flaws.
Acknowledgment The authors are grateful to the financial support by Natural Science Foundation of China (51105143), the Fundamental Research Funds for the Central Universities (1114036) and Shanghai University Young Teachers Training Funds (YG0142129).
Appendix A The expression of function F can be drawn as:
a c
(A.2)
0:89 0:2 þ
a c
a24 a þ 14 1 c 0:65 þ c 1
(A.3)
(A.4)
2 h a2 i 1 sin2 4 g ¼ 1 þ 0:1 þ 0:35 t
(A.5)
h a2 i1=4 cos2 4 f4 ¼ sin2 4 þ c
(A.6)
fc ¼
rffiffiffi! a t þ 1 0:5 2 2 t ri ro þ ri ro2 ri2
(A.7)
And the parameters f and q obeys
tanq ¼
a tan4 c
(A.8)
References [1] E/ECE/TRANS/505. Specific components of motor vehicles using compressed natural gas (CNG) in their propulsion system 2001. Rev. 2, Add. 109, Reg. 110. [2] Newman Jr JC, Raju IS. Stress intensity factor for internal and external surface cracks in cylindrical vessels. J Press Vess Tech 1982;104:293e8. [3] Delale F, Erdogan F. Application of line-spring model to a cylindrical shell containing a circumferential or axial part-trough crack. J Appl Mech 1982;49: 97e102. [4] Lin XB, Smith RA. Fatigue growth prediction of internal surface cracks in pressure vessels. J Press Vess Tech 1998;120:17e23. [5] Kim Y-J, Kim J-S, Park Y-J, Kim Y-J. Elasticeplastic fracture mechanics method for finite internal axial surface cracks in cylinders. Eng Fract Mech 2004;71: 925e44. [6] Perl M, Levy C, Wang J. Interaction effects in combined arrays of radial and longitudinal semi-elliptical surface cracks in pressurized thick-walled cylinder. J Press Vess Tech 1997;119:167e74. [7] ABAQUS Version 6.5: user’s manual. RI: ABAQUS, Inc; 2005. [8] Moran B, Shih CF. Crack tip and associated domain integrals from momentum and energy balance. Eng Fract Mech 1987;27:615e42. [9] Newman JC, Raju IS. Stress-intensity factors for internal surface cracks in cylindrical pressure vessels. J Press Vess Tech 1980;102:342e6.
Please cite this article in press as: Chen J, Pan H, Stress intensity factor of semi-elliptical surface crack in a cylinder with hoop wrapped composite layer, International Journal of Pressure Vessels and Piping (2013), http://dx.doi.org/10.1016/j.ijpvp.2013.04.026