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Analytical and numerical fracture analysis of pressure vessel containing wall crack and reinforcement with CFRP laminates
Thin-Walled Structures 127 (2018) 210–220
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Full length article
Analytical and numerical fracture analysis of pressure vessel containing wall crack and reinforcement with CFRP laminates E. Alizadeh, M. Dehestani
T
⁎
Faculty of Civil Engineering, Babol Noshirvani University of Technology, Postal Box: 484, Babol 47148-71167, Islamic Republic of Iran
A R T I C L E I N F O
A B S T R A C T
Keywords: Pressure vessel Crack Fracture mechanic Extended finite element method (X-FEM) FRP
This paper concentrates on the analytical and numerical calculation of the critical internal load for a pressure vessel containing a longitudinal edge crack or cracks. Initially, the vessel’s capacity is analyzed based on the theoretical fracture methods for seven material properties, different crack lengths, and vessel’s wall thickness. Theses analyses are conducted using an extended finite element method (XFEM) to observe its accuracy and applicability. In problems with complex configuration of crack, it’s difficult to use theoretical method for analyzing the structure. Therefore, after verifying the XFEM with excellent accuracy, several analyses are made for different cases. By employing the XFEM, the effects of having multiple cracks along the vessel’s circumference, crack width along the vessel’s wall, crack location on the internal or external edge of the vessel, and applying the FRP laminates to reinforce the vessel are investigated. Besides, the effect of mode II (sliding mode) on behavior of vessel and the elastic-plastic analysis are analytically studied. Results show that the critical internal pressure for a single cracked and a multiple cracked vessel are the same unless two cracks be very close to each other. Increasing the crack width decreases the critical pressure meaningfully. It is also shown that the vessel is more vulnerable to fail by external crack than an internal crack with similar length and width. Also, the cracked bodies are reinforced with FRP laminates which proves that laminates with higher modulus of elasticity have more effect on the critical internal pressure. Moreover, the elastic-plastic analysis does not have significant influence on critical pressure load due to small plastic zone.
1. Introduction Pressure vessels are widely used in industries and cracks can usually be observed in this type of structures. In addition to manufacturing procedures, environmental effect like corrosion can cause the vessels to be cracked. Several researches have been conducted in order to investigate the behavior of pressure vessels and pipes containing cracks. Shariati et al. [1] proposed a novel numerical model to define the behavior of semi-elliptical crack growth in thick-walled pressure vessels. They also experimentally investigated the influence of fatigue crack growth in the vessels. The numerical results of fatigue loading on the specimen are compared with the experimental results. They concluded that numerical and experimental methods have shown good agreement. Viehrig et al. [2] experimentally investigated the hardness, tensile, and fracture toughness of several reactor pressure vessels. They concluded that the distribution of the crack initiation region is not necessarily correlated to the structure of the different welding beads along the crack front. Furthermore, it was found that the fracture toughness
⁎
values at cleavage failure, Kjc , determined with T-S (crack extension through the thickness) specimens is significantly higher than in case of the T-L (crack extension in welding direction) specimens. Shlyannikov et al. [3] studied the structural behavior of power plant pipes under creep and fatigue conditions. The effect of internal pressure and shape defect of a pipe bend on stress-strain redistributions and creep-fatigue crack growth rates was investigated using FE-analysis and experiments. It was found that the creep-fatigue crack growth rate are depend on the critical zone positions in the considered pipe bend. Furthermore, creep stress intensity factor was used as a new parameter for characterization of the crack growth resistance for power plant materials and structures under elevated temperature. Ruggieri et al. [4] numerically investigated the crack front region and effects of crack-tip constraint in conventional fracture specimens with transverse delamination cracks. They proposed important features of 3-D crack front fields in fracture specimens and concluded that these features had a direct effect on toughness of an isotropic materials. Mirzaei et al. [5] experimentally and numerically evaluated the failure of a compressed natural gas (CNG) fuel tank. Several transient-
Corresponding author. E-mail addresses: [email protected] (E. Alizadeh), [email protected] (M. Dehestani).
https://doi.org/10.1016/j.tws.2018.02.009 Received 12 May 2017; Received in revised form 12 January 2018; Accepted 8 February 2018 0263-8231/ © 2018 Elsevier Ltd. All rights reserved.
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dynamic elastic-plastic finite element (FE) analyses was conducted to simulate the structural response of the tank under dynamic pressure. It was found that the failure characteristics, like the overall asymmetric deformation and fracture patterns, initiation and partial growth of parallel cracks at the same section, multiple cracking at the neck, and the self-similar growth of the main axial crack were all initiated by traveling sonic pressure wave from the neck towards the bottom of the tank. Mirzaei et al. [6] studied dynamic ductile rupture of steel pipes under high-speed internal moving pressures. Formation of special fracture surface patterns due to cyclic crack growth was experimentally identified. The overall transient dynamic response of the pipe to explosion loading, the detonation-driven crack growth, the cyclic bulging of the crack flaps, and the resultant crack branching were simulated numerically. Their study demonstrated that the propagation of the initial axial cracks in the pipe was the incremental cyclic growth governed by the structural waves. Oikonomidis et al. [7] proposed a specimen for evaluation the high strain rate flow and fracture properties of pipe material and tuning a strain rate dependent damage model (SRDD). They used the SRDD model to simulate axial crack propagation and arrest in natural gas pipelines. The result reveals that the model correctly predicted the crack initiation pressure, the crack speed and the crack arrest length. Fracture tests of carbon steel piping with a single and two circumferential flaws on the inner surface were conducted by Ogawa et al. [8] to investigate a method for fracture assessment of ferritic steel piping with multiple flaws. The results showed that fracture assessment based on the twice elastic slope method and the plastic collapse mechanism reached inadequate results for a large single flaw. Furthermore, two kinds of elastic-plastic fracture assessment method, one using the Z-factor in the JSME FFS code and the other by ductile instability analysis, showed conservative estimates of fracture strength even when the structural factor SF was not considered. Śnieżek et al. [9] experimentally studied the propagation of the semi-elliptical cracks in austenitic steel which usually used for construction of the industrial pipelines and pressure vessels. The method of electrical potential drop (EPD) is used to investigate the fatigue crack growth. In order to confirm accuracy of the electrical potential drop measurement method, two other experimental methods were used in comparative studies. It was found that Elaborated EPD measurement methodology enables a continuous recording of the dimensions of propagating semi-elliptical cracks, both under variable tension and under constant bending, which was confirmed during the tests. Moshayedi et al. [10] experimentally and numerically investigated the influence of welding residual stresses on brittle fracture in a pipe containing internal surface crack. They assumed an internal circumferential crack at the weld line. Their results indicate that, in the welded pipe, the fracture toughness decreased dramatically in comparison with the pipe without welding residual stresses, this is due to the influence of tensile welding on the crack tip stress state, which cause a decrease in the opening mode stresses at the near crack tip. Mehmanparast [11] experimentally and analytically studied the behavior of creep crack growth in 316H stainless steel for several specimens. The behavior of two material states including as-received and pre-compressed conditions have been experimentally evaluated and the experimental results were compared with the predicted creep crack growth which obtained analytically. It was shown that the predicted creep crack growth had a good agreement with experimental results. Mehmanparast et al. [12] experimentally investigated the effects of the material pre-straining level on the tensile, creep deformation, creep crack initiation and crack growth behavior of 316H stainless steel. It was found that creep ductility and the time of rupture processes was decreased with an increase in pre-strain levels. McCready et al. [13] examined the retrofitting of damaged pipes with composite layers. The specimens were subjected to a combined
loading of internal pressure and external bending. They concluded that composite repairs significantly increased the flexural stiffness of the damaged pipe. Furthermore, post-flexural, hydro-static testing indicated that the composite repairs were able to sustain significant damage while retaining their capability to support the pressure capacity of the pipe. Mazurkiewicz et al. [14] numerically and experimentally evaluated the mechanical behavior of steel pipe containing part-wall defect reinforced with GFRP layers. The analyses shown that local reduction of pipe wall thickness due to corrosion defect can reduce the pressure resistance significantly. Besides, pipes which were repaired by a GFRP layers could sustain more pressure in comparison to an original steel pipe considering burst pressure. Jeong et al. [15] proposed the plastic influence functions, for estimates of J -integral of a pipe containing a complex crack based on the systematic 3-D elastic-plastic finite element (FE) analyses by using Ramberg-Osgood (R-O) relation. In their function, global bending moment, axial tension and internal pressure were considered as loading conditions. They concluded that the proposed engineering J prediction method can be utilized to assess instability of a complex crack in pipes for R-O material. Arafah et al. [16] presented analytical model for calculating plates with semi-elliptical surface cracks subjected to tension, bending, and combined tension-bending and biaxial tension. The comparison between the predicted critical loads and experimental burst testfailure loads demonstrated satisfying agreement with each other. Schonberg et al. [17] proposed a model to predict the eventuality of crack formation and propagation under an impact crater in a thin plate. This model was used to examine the influence of penetration depth on crack formation and whether or not the crack might grow through the tank wall thickness. The predictions of the model were compared to experimental data with encouraging results. Sowards et al. [18] evaluated the fatigue crack propagation in three steel specimens in a simulated fuel-grade ethanol environment. A fracture mechanics testing approach was used to determine crack propagation rates as a function of the stress-intensity-factor amplitude. The experimental and fracture analysis results indicate that fatigue damage in fuel-grade ethanol environments is increased for all three materials. Based on these results, they proposed a model for determining crack growth rates in ethanol fuel. Yokozeki et al. [19] experimentally studied the gas leakage through a composite laminates containing multilayer matrix cracks. They used helium leak gas detector at room temperature to examine the helium gas leak rates through the damaged laminates subjected to uniaxial loadings. They also evaluated the effect of angles of matrix cracks on gas leakage. Richardson et al. [20] presented a method for simulating quasistatic crack propagation in 2-D which combines the extended finite element method (XFEM) with a general algorithm for cutting triangulated domains, and introduce a simple and flexible quadrature rule based on the same geometric algorithm. They concluded that the cutting algorithm provides a flexible and systematic way of determining material connectivity, which is required by the XFEM enrichment functions. Also, their integration pattern is accurate, without requiring a triangulation that incorporates the new crack edges. The application of this cutting algorithm and integration rule is for geometrically complicated domains and complex crack patterns. Giner et al. [21] proposed an implementation of the extended finite element method for fracture problems within the finite element software ABAQUS. They provided details on the data input format together with the subroutine elements, however, pre-processing tools that are necessary for an X-FEM method, but not directly related to ABAQUS, are not provided. They finally presented several numerical examples in fracture mechanics to reveal the benefits of the proposed implementation. Wang et al. [22] numerically studied the ultimate load capacity and 211
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the failure of composite vessel. The material property degradation and cohesive element methods were used to model the intralaminar failure and the initiation of delamination at the interface of composite laminates, respectively. The numerical model is compared with experimental results and they were compatible. Gutman et al. [23] used the thin elastic cylindrical shell model to investigate the stability of high-pressure vessel subjected to inside corrosion. Estekanchi and Vafai [24] utilized the finite element model to study the buckling of cylindrical shell with cracks. Besides, Akrami and Erfani [25] analytically and numerically investigated the buckling load of cracked cylindrical shell. They studied the effect of crack location, crack severity and length of cylinder on the load capacity of cracked body. Gato [26] proposed the mesh-free model to study the dynamic fracture of pressure vessel. This method can model the fracture of thinwalled structure in a simple way. Although, analytical methods for studying cracked bodies are usually exact and easy to use, but analytical results are available for only limited problems with simple configurations and loadings. Therefore, for many practical complicated engineering problems like pressure vessels, FEM is applied. The main purpose of current article is to use both analytical and finite element method to study a pressure vessel containing a crack or multiple cracks subjected an internal pressure. Also, the effect of some retrofitting methods with FRP laminates are investigated.
Fig. 1. Configuration of cracked pressure vessel (all dimensions are in millimeter).
shown in Eq. (5), where σz and σy are longitudinal and yield stresses, respectively [27]. In this case, the critical pressure (Pcr ) can be computed from Eq. (6).
2. Theoretical analysis
σ 2θ − σθ σz − σ 2 z = σ 2 y
The investigation of the failure behavior of supposed pressure vessel is conducted based on fracture mechanic theories. Fracture mechanics is focuses on the analyzing the structures containing initial cracks that can affect the load‐carrying capacity of an engineering structures. Defects are initiated in the material by manufacturing procedures or can be created during the service life, by fatigue, environmental effects or creep. In the theoretical section of current paper, it is assumed that there is an initial crack in the internal side of the thickness of the supposed pressure vessel and a characteristic quantity which defines the propensity of the crack to extend is determined. The characteristic quantity depends on the particular failure criterion used, that for this paper, the stress intensity factor is applied. The failure criterion is expressed by Eq. (1), in which KI is stress intensity factor at the crack tip which depends on the applied load, the initial crack length and the geometrical configurations of the cracked plate and K c is fracture toughness that depends on the material parameter which can be determined experimentally [27].
Pcr =
PR t
Kc t 1.12R πa
(5)
2σy t 3R
(6)
3.1. Constitutive modelling and elements In current study, the pressure vessels made of seven metals are designed according to ASMEVIII- Division1 (Rules for Construction of Pressure Vessels) [29]. The minimum required thickness of vessels under internal pressure should not be less than that computed by Eq. (7), where E is the efficiency of appropriate joint in cylindrical or spherical vessels, or the efficiency of ligaments between openings, whichever is less, that is considered 0.85 in this paper. P is internal pressure, R is the radius of the vessel, S is the maximum allowable stress value and tmin is minimum required thickness of shell [29].
(2)
tmin =
(3)
PR SE −0.6P
(7)
The outside diameter of pressure vessel is considered 2000 mm. The wall thickness (t ) is designed according to Eq. (7). A crack with initial length (a) and width (b) is considered in current studies. Configuration of pressure vessel and its dimensions are demonstrated in Fig. 1. The location of initial crack in cylindrical part of the vessel and its position, which could be on the internal or external side of the vessel, are shown in Fig. 2.
By combining the Eqs. (1)–(3), the critical pressure (Pcr ) is calculated as Eq. (4).
Pcr =
PR 2t
A three-dimensional finite element analysis is carried out to evaluate the behavior of the pressure vessel. The numerical analysis is performed by finite element software ABAQUS [28]. The procedure is summarized as follows.
The amount of σ can be calculated from Eq. (3), where σθ is circumferential stress, P is internal pressure of the vessel and R and t are the radius and the thickness of the vessel, respectively.
σ = σθ =
σz =
3. Finite element analysis
The stress intensity factor at the crack tip (KI) of the single-edgecracked plate under uniform far field tension can be obtained from Eq. (2), where σ is the applied stress on the plate and a is the crack length perpendicular to load of σ [27].
KI = 1.12σ πa
PR , t
The diagram of critical pressure Pcr versus crack length a , can be plotted by using Eqs. (4) and (6) for a pressure vessel which is designed safely against both yielding and fracture due to initial crack. The critical pressure of a vessels with seven different materials are calculated and illustrated in Figs. 3–5. The mechanical properties of considered materials are given in Table 1.
(1)
KI = K c
σθ =
(4)
In addition, the vessel must be designed safely against failure due to yielding (according to the Von Mises yield criterion, for example) as 212
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Fig. 2. (a) Crack location in cylindrical part of vessel, (b) location of internal crack in a sector of vessel, (c) location of external crack in a sector of vessel.
Fig. 3. Mesh configuration around the crack region.
Solid elements are used to model the pressure vessel. The type of the element is chosen as 3D8R that includes eight nodes and three translational degrees of freedom at each node. In order to make computational time minimum and achieving the highest accuracy simultaneously, based on comprehensive mesh convergence studies, mesh size is adopted with a maximum element size of 10 mm in the region around the crack tip and 80 mm for other parts of the specimen which are far from the crack domain, as shown in Fig. 3. An extended finite element method (XFEM) is used for modelling the crack. In comparison with the classical finite element method or linear elastic fracture mechanics (LEFM), the XFEM provides significant benefits in the numerical modelling of crack propagation. In the traditional formulation of the FEM, the existence of a crack is modeled by requiring the crack to follow element edges. In contrast, the crack geometry in the XFEM need not be aligned with the element edges, which provides more flexibility and versatility in modelling. The method is based on the enrichment of the FE model with additional degrees of freedom (DOFs) that are tied to the nodes of the elements intersected by the crack [21]. In this manner, the discontinuity is included in the numerical model without modifying the discretization, as the mesh is generated without taking into account the presence of the crack. Therefore, only a single mesh is needed for any crack length and orientation. In addition, nodes surrounding the crack-tip are enriched
Fig. 4. Critical pressure (Pcr ) versus crack length (a ) for pressure vessel that made of steel material.
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Fig. 5. Critical pressure (Pcr ) versus crack length (a ) for pressure vessel that made of titanium material.
Fig. 6. Critical pressure (Pcr ) versus crack length (a ) for pressure vessel that made of aluminum material.
Table 1 Mechanical properties of materials [27]. Material
σy (MPa)
K c (MPa m )
Steel 4340 Steel 4335 Steel 350 maraging
860 1300 1550
100 70 55
Ti 6A1-4V Ti 4A1-4Mo-2Sn-0.5Si
1103 945
38.5 70.3
AL 2014-T4 Al 7075-T651
448 545
28.6 29.7
In first part of verification, the pressure vessels with different initial crack length (a) , constant crack width (b = 10 mm), and wall thickness of 20 mm, are modeled for seven material properties. The critical tolerable pressure for each vessel with a certain initial crack is investigated. The analytical and numerical results of critical pressure are plotted versus crack length for steel, titanium and aluminum vessels in Figs. 4–6, respectively. As illustrated in Figs. 4–6, the diagram of failure pressure against crack length (a ) is composed of two pieces; one linear piece that shows the failure due to the yielding of the steel, and a curved piece that shows the fracture of the vessel due to crack growth. Figs. 4–6 prove that results obtained from the numerical analysis by XFEM employed here coincide with the theoretical answers in both yielding and fracture part of the diagram. The numerical critical stress (σcr ) at crack region versus crack length (a ) for three type of steel material is presented in Fig. 7. Besides, Fig. 8 demonstrates the numerical circumferential stress distribution and crack propagation process of crack length 10 mm in steel 4340 pressure vessel for two different internal pressure P = 6.68 MPa and P = 10.3 MPa.
with DOFs associated with functions that reproduce the asymptotic LEFM fields. This enables the modelling of the crack discontinuity within the crack-tip element and substantially increases the accuracy in the computation of the stress intensity factors (SIFs) [21]. In the ABAQUS the maximum principal stress damage (Maxps Damage) is defined as the material behavior of the region in which the crack could growth. The crack propagation occurs when the critical energy release rate (Gc ) reaches a critical value. The connection between the energy release rate (Gc ) , which is a global quantity, and the stress intensity factor (K c ) , which expresses the strength of the local elastic stress field in the neighborhood of the crack tip, is very important. The equivalence between critical energy release rate and critical stress intensity factor for plane stress material is shown in Eq. (8) [27]. The critical energy release rate is calculated for each material in Table 1 and it is used for defining the material behavior in ABAQUS.
Gc =
Kc2 E
(8)
In addition to defining Gc , the critical stress of crack must be introduced to the program which is computed as Eq. (9) and is used for each initial crack length at a certain material type in the ABAQUS.
σc =
Kc 1.12 πa
(9)
3.2. Validation of numerical model The verification studies are conducted for various values of internal crack length (a) , various different wall thickness of the vessel, and for seven different material properties (steel, titanium, and aluminum).
Fig. 7. Numerical critical stress (σcr ) versus crack length (a ) for pressure vessel.
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Fig. 8. Circumferential stress distribution and crack propagation of crack initial length 10 mm for two different internal pressure (a ) P = 6.68
MPa (b)P = 10.3 MPa.
4.1. Multiple cracks
In the second part of the verification, vessels with crack length of a = 20 mm, crack width b = 10 mm, and three material properties with various vessel’s wall thicknesses are considered. For such vessels, results of theoretical analysis which are calculated based on Eq. (4), and XFEM analysis are compared (Fig. 9). The studies in this section, indicate that the results of numerical method are in excellent agreement with analytical method. Thus, FEM is going to be utilized to study the more complex problem of cracks.
The possible effects of simultaneous cracks on a body is numerically analyzed in two configuration types. In the first one, the effect of angular distance between two cracks on the vessel’s body is analyzed. As shown in Fig. 10(a), two cracks are located in the internal circumference of the vessel with the angle of θ between them (θ = 5, 10, 20, 30, 60, 90, 120, 150, 180 ). In the second type, n equally-spaced cracks (n = 1 − 9) are located in the vessel’s internal circumference while the angle between them are 360/n ° (Fig. 10(b)). As predicted, in both types above, no significant differences are observed when one compares the critical pressure of a single cracked body with a multiple cracked one. This is because each crack affects only on a very tiny limited area around the crack tip. Therefore, two
4. Parametric study The analytical and numerical investigations are proposed in this section.
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Fig. 11. Critical pressure (Pcr ) versus crack width (b ) for steel 4340 pressure vessel.
Fig. 9. Critical pressure (Pcr ) versus wall thickness (t ) for pressure vessel with 20 mm initial crack.
across the vessel’s thickness. In order to evaluate the effects of crack location on the critical load, the numerical analyses are conducted for a pressure steel 4340 vessel containing internal or external crack (As shown in Fig. 2). In order to have a better comprehension, the analyses are made for three thickness of the vessel 20, 30 and 40 mm. As shown in Fig. 12, for specimens with small crack length, by changing the crack location from internal to the external edge, the critical pressure capacity is meaningfully decreased, but for longer crack length the difference between two crack locations is decreased until the two curves are converged to each other where the crack length is equal to the thickness of the vessel. In other words, cracks on the outer surfaces of the vessel are more critical than cracks on the inner one, besides, the external crack are usually more probable due to external impact load, scratches, or environmental effects.
neighbor cracks have very little effects on each other and on the overall capacity of the vessel. Two neighbor cracks should be very close to each other to have effects on the overall capacity. For example in this case, when two cracks are placed at the distance of 5 mm along the circumference, the critical pressure decreases less than 15%. 4.2. Effects of crack width (b ) on critical pressure (Pcr ) Each crack is simulated as rectangular discontinuity in the vessel’s wall thickness. In previous sections of this paper, the width of cracks are considered 10 mm, but in current section, the crack width is changed to four different value of 5, 10 , 15, and 20 mm. The effect of crack width (b) on critical pressure of steel 4340 vessel is numerically investigated. As shown in Fig. 11, the critical pressure is reduced for all crack length by increasing the crack width. The effect of crack width is more significant for crack with smaller crack length, and the influence of crack width is decreased by increasing the length of crack. For example, the curve of a crack with 5 mm length is more sensitive to increasing the b , when compared to the curve of crack length 20 mm.
4.4. Reinforcement with FRP laminates In order to strengthening the specimens containing cracks, different materials could be used as a reinforcing layer. But it is reasonable to use materials with modulus of elasticity close to steel or higher than that. Therefore, among different types of FRP laminates (Carbon, Glass, and Aramid), CFRP types are considered due to its high value of modulus of elasticity. It is noticeable that several investigations are conducted to study the load capacity and failure of carbon composite vessels [22]. In this study, CFRP laminate with thickness of 7 mm and 400 mm width is applied in a region around the crack tip (Fig. 13). This
4.3. Effect of crack location along vessel’s wall thickness In all previous sections, cracks are located on the internal edge of the circumference, while cracks’ location influences the stress field
Fig. 10. (a) Configuration of two cracks located in the internal circumference of the vessel, (b) Configuration of multiple cracks located in the internal circumference of the vessel.
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Fig. 12. Critical pressure (Pcr ) versus crack length (a ) for steel 4340 pressure vessel with internal and external crack location.
Fig. 14. Critical pressure (Pcr ) versus crack length (a ) for steel 4340 pressure vessel reinforced with CFRP laminates.
Fig. 13. Configuration of CFRP layer around the crack region.
Fig. 15. Circumferential stress (σ ) versus crack length (a ) for steel 4340 pressure vessel with and without CFRP laminate.
numerical study is done by three different types of CFRPs with different mechanical properties that are given in Table 2. In the FEM program the CFRP layer is modeled as orthotropic material via “shell” elements. This types of element consists of four nodes and six degrees of freedom per node, three translational and three rotational. The Fig. 14 demonstrates the effects of reinforcement with CFRP laminates, on the critical pressure for a cracked body. The linear piece of all diagrams in Fig. 14 (smaller values for a ) shows the limit where the crack has no effect on the critical internal pressure. Adding a CFRP layer expands this limits for crack length and stretches the linear piece of the diagram. Also, as predicted, adding a CFRP layer, increases the capacity of the vessel. These two effects are more pronounced for CFRPs with higher modulus of elasticity. Furthermore, the circumferential stress (σ ) versus crack length (a )
for steel 4340 vessel with and without CFRP laminate is shown in Fig. 15. The stress in vessel without CFRP layer is demonstrated at critical pressure load for each crack length. Besides, the stress in vessel with CFRP layer is shown at a same load as vessel without CFRP layer for each crack length. The results reveal that the stress in vessel with CFRP layer is lower than the other vessel at a same load, so utilizing CFRP layer at the crack region can enhance the load bearing capacity of vessel. 4.5. The effect of combined modes on the vessel An analytical investigation is conducted to study the effect of combined opening (mode I) and sliding (mode II) modes on crack
Table 2 Mechanical properties of CFRP laminates. Material
Product name
Type
Ply orientation
Modulus (GPa)
G12 (GPa)
Longitudinal
Transverse
ϑ12
Strength (MPa) Longitudinal
CFRP
SK-CPS 0512
High strength carbon
0°/90°
165
13.75
7.27
0.218
3000
CFRP
SK-CPI 0614
Intermediate modulus carbon
0°/90°
210
17.5
8.42
0.214
2400
CFRP
SK-CPH 0514
High modulus carbon
0°/90°
300
25
8.76
0.212
1300
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Fig. 16. (a) Inclined crack in the pressure vessel (b) the opening and sliding modes of loading in rotated plate.
instability. Thus, the orientation of crack is rotated from vertical status and the mode II of loading appears in the inclined crack. A shown in Fig. 16, the rotation angle between θ and θ´ axes is presented by β and σL and σθ are longitudinal and circumferential stresses, respectively. Fig. 16(a), demonstrates the inclined crack in the pressure vessel. In order to calculate the stress intensity factor of inclined crack, the plate is rotated and the opening and sliding modes of loading are shown in Fig. 16(b). ′ are estimated By stress transformation relations, the σθ′, σz′ and τθz and given in Eq. (10) [27].
σθ +σL σ σ 3σ + θ − L cos2β = + 2 2 2 σθ +σL σθ −σL 3σ cos2β = = − − 2 2 2 σθ −σL σ sin 2β = − sin 2β = − 2 2
σ ´θ = σ´z τ ´θz
KII = −
1 Pcr R b P R b sin2(30) π ( ) = −0.2165 cr π( ) 2 2t 2 t 2
KI K + II = 1 KIc KIIc 0.625
π (2 )
0.2165 −
100
0.00625
σ cos2β 2 σ cos2β 2
Pcr =
(10)
π (2 )
108.65
=1
Pcr R b π( ) = 1 t 2
t 0.004257×R ×
As shown in Eq. (11), the mode I and II stress intensity factor (KI , KII ) are obtained by using the transformed stresses (Eq. (10)).
b
Pcr R t
Pcr R b P R b π ( ) − 0.00199 cr π( ) = 1 t 2 t 2
0.004257
b
π (2)
=
20 b
7.5434× ( 2 )
=
2.6513 b
(2)
According to the above equations, the critical pressure (Pcr ) is calculated for different rotation angle (β = 30° , 45° , 60° , 75° and 90°) and then the different critical pressure (Pcr ) versus crack length (b ) diagrams are plotted and presented in Fig. 17.
b b 3 1 KI = σz′ π ⎛ ⎞ = ⎡ − cos 2β⎤ σ π ⎛ ⎞ 2 2 2 2 ⎣ ⎦ ⎝ ⎠ ⎝ ⎠ b σ b ′ π ⎛ ⎞ = − sin 2β π ⎛ ⎞ KII = τθz 2 ⎝2⎠ ⎝2⎠
b
Pcr R t
(11)
Likewise, the critical pressure (Pcr) can be calculated via Eqs. (11) and (12), in which the fracture toughness of mode I and II are presented by KIc and KIIc , respectively [30].
KI K + II = 1, KIc KIIc σ=
b 3 1 P R P R π( ) [ − cos2β] cr (− cr4t Pcr R 2 2 2t 2 → + 2t KIc
b
sin2β π ( 2 )) KIIc
=1 (12) Furthermore, the calculation procedure of critical pressure for crack with rotation angle of 30° is shown below. It is noticeable that the steel 4340 material and the wall thickness of 20 mm are used in this section. Besides, KIc is considered 100 MPa m and KIIc is also assumed 108.65 MPa m [30].
KI = [
3 1 P R b P R b π ( ) = 0.625 cr π( ) − cos2(30)] cr 2 2 2t 2 t 2
Fig. 17. Critical pressure (Pcr ) versus crack length (b ) for steel 4340 pressure vessel with different crack orientations.
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As illustrated in Fig. 17, the most critical pressure load is related to the initial crack (the crack that was investigated in previous sections) with vertical status and rotation angle of 90°. It should be noted that the vessel must be designed safely against failures by yielding and fracture. Thus, as stated before, the failure pressure of the vessel is studied due to yielding based on Eq. (6). In each crack length, the lower load calculated with Eqs. (6) and (12) is considered as a critical pressure. As an example, in vertical crack with crack lengths lower than 6.5 mm, the yield pressure is lower than the fracture one, thus, the failure will occur due to yielding. It means that, although the vessel could withstand the pressure Pcr = 32 MPa for crack length 2.5 mm, but it fails at Pcr = 19.8 by yielding. Otherwise, the failure will happen because of fracture for vertical cracks with more than 6.5 mm length. Besides, the fracture load for crack with rotation angle of 30° is higher than yield pressure, hence, the failure of vessel will be caused by yielding of material in this case. 4.6. The elastic-plastic analysis In this section, the elastic-plastic analysis has been conducted via analytical calculation. The inelasticity in the form of plasticity, creep or phase changes can be identified in the neighborhood of crack tip. As shown in Fig. 18, the inelastic region is sufficiently small compared to k-dominant zone [27]. The occurrence of plasticity in crack tip, makes the crack behave as if it was longer than its real and physical size. Therefore, the plate performs as if it contained a crack with slightly larger length. Furthermore, the effective crack size, aeff , can be calculated by a +rp , where a is the crack length and rp is the radius of plastic zone near the crack tip. The principal stresses shown in Eq. (13) can be utilized in von mises criterion to calculate the radius of plastic zone [27].
σ1 = σ2 = σ3 =
Fig. 19. Critical pressure (Pcr ) versus crack length (a ) for elastic and elastic-plastic analysis (steel 350 maraging).
In addition, the stress intensity factor corresponding to the effective crack, called effective stress intensity factor (K eff ) for plane strain status, has been computed by using Eq. (16) and is shown in Eq. (17). Consequently, the effective pressure (Peff ) is also estimated as given in Eq. (18) [27].
K = 1.12σθ π (a + rp ) = 1.12×
1−[
⎟
1.12σ (1 − 2ϑ) 2 ] 2 σy
(17)
2σy
=
K2 3 2 [ sin θ+(1 − 2ϑ)2 (1+cosθ)] 4πσY 2 2
K2 (1
Kc t 1.12R πa +
(14)
1.12RK c (1 − 2ϑ) 2 σy
(18)
According to Eq. (18), the Critical effective pressure (Pcr ) versus
The radius of plastic zone (rp) is calculated based on principal stresses and von mises criterion and it is shown in Eq. (15). Besides, rp is estimated for θ = 0° at crack tip (Eq. (16)).
K2 [2(1 − 2ϑ)2] = 4πσ 2Y
K2 (1 − 2ϑ)2 ⎞ → K eff 2πσY 2 ⎠
K 2 (1 − 2ϑ)2 PR 2 ) ×[πa + c ] → P2 eff t 2σY 2 K c 2t 2 = → Peff K 2 (1 − 2ϑ)2 2 2 1.12 R [πa + c ] 2
(13)
(σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2 = 2σ 2y
rP =
⎜
K c 2 = 1.122 × (
The von mises criterion is presented in Eq. (14), in which σy demonstrates the yield stress of material [27].
rP (θ) =
π ⎛a + ⎝
1.12σ πa
=
θ θ K1 cos (1 + sin ) 2 2 2πr θ θ K1 cos (1 − sin ) 2 2 2πr θ 2ϑK1 cos 2 2πr
PR × t
(15)
2ϑ)2
− 2πσ 2Y
(16)
Fig. 20. Critical pressure (Pcr ) versus crack length (a ) for elastic and elastic-plastic analysis (steel 4335).
Fig. 18. The inelastic and k-dominant regions in the crack tip.
219
Thin-Walled Structures 127 (2018) 210–220
E. Alizadeh, M. Dehestani
crack length (a ) for 350 maraging and 4335 steel is computed and is compared with elastic analysis in Figs. 19 and 20, respectively. It is noticeable that the plastic zone near crack tip is considered in elasticplastic analysis. Thus, the crack length is slightly larger than its real length and the critical pressure is also slightly lower than elastic analysis. As shown in Figs. 19 and 20, the difference between elastic and elastic-plastic analysis is not significant due to small plastic zone.
[6]
[7]
[8]
5. Summary and conclusion [9]
In current article, the fracture analysis of a pressure vessel is considered for different configurations of cracks with seven metal material properties. Initially, some analytical studies are conducted to obtain the critical internal pressure of the vessel. Analyzing these problems by XFEM illustrates that numerical methods could be beneficial in studying the cracked structures, especially for cases in which theoretical analyses are difficult to achieve. The analytical and numerical result are presented below:
[10]
[11]
[12]
[13]
• Inducing • • • • •
multiple cracks has no effects on the critical internal pressure practically except for cracks which are very close to each other. Greater crack width causes lower critical pressures; this effect is more observable for smaller crack lengths. By changing the location of the crack on the vessel’s wall thickness, it is found that an external crack has more influence on the vessel’s capacity than an internal crack of the same configurations. Reinforcing the cracked region of the vessel reveals that laminates with higher modulus of elasticity could be more useful for strengthening a defected structure. The effect of sliding mode is investigated due to rotating the crack orientation and the load capacity of vertical crack is the most critical one among others. The difference between elastic-plastic and elastic analyses is not significant due to the small plastic zone in the crack tip.
[14]
[15]
[16]
[17]
[18]
[19]
[20]
Acknowledgment This paper presents findings of a research project which was supported by Babol Noshirvani University of Technology in Iran via faculty member grants (No. 10162).
[21] [22]
References
[23] [24]
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Full length article
Analytical and numerical fracture analysis of pressure vessel containing wall crack and reinforcement with CFRP laminates E. Alizadeh, M. Dehestani
T
⁎
Faculty of Civil Engineering, Babol Noshirvani University of Technology, Postal Box: 484, Babol 47148-71167, Islamic Republic of Iran
A R T I C L E I N F O
A B S T R A C T
Keywords: Pressure vessel Crack Fracture mechanic Extended finite element method (X-FEM) FRP
This paper concentrates on the analytical and numerical calculation of the critical internal load for a pressure vessel containing a longitudinal edge crack or cracks. Initially, the vessel’s capacity is analyzed based on the theoretical fracture methods for seven material properties, different crack lengths, and vessel’s wall thickness. Theses analyses are conducted using an extended finite element method (XFEM) to observe its accuracy and applicability. In problems with complex configuration of crack, it’s difficult to use theoretical method for analyzing the structure. Therefore, after verifying the XFEM with excellent accuracy, several analyses are made for different cases. By employing the XFEM, the effects of having multiple cracks along the vessel’s circumference, crack width along the vessel’s wall, crack location on the internal or external edge of the vessel, and applying the FRP laminates to reinforce the vessel are investigated. Besides, the effect of mode II (sliding mode) on behavior of vessel and the elastic-plastic analysis are analytically studied. Results show that the critical internal pressure for a single cracked and a multiple cracked vessel are the same unless two cracks be very close to each other. Increasing the crack width decreases the critical pressure meaningfully. It is also shown that the vessel is more vulnerable to fail by external crack than an internal crack with similar length and width. Also, the cracked bodies are reinforced with FRP laminates which proves that laminates with higher modulus of elasticity have more effect on the critical internal pressure. Moreover, the elastic-plastic analysis does not have significant influence on critical pressure load due to small plastic zone.
1. Introduction Pressure vessels are widely used in industries and cracks can usually be observed in this type of structures. In addition to manufacturing procedures, environmental effect like corrosion can cause the vessels to be cracked. Several researches have been conducted in order to investigate the behavior of pressure vessels and pipes containing cracks. Shariati et al. [1] proposed a novel numerical model to define the behavior of semi-elliptical crack growth in thick-walled pressure vessels. They also experimentally investigated the influence of fatigue crack growth in the vessels. The numerical results of fatigue loading on the specimen are compared with the experimental results. They concluded that numerical and experimental methods have shown good agreement. Viehrig et al. [2] experimentally investigated the hardness, tensile, and fracture toughness of several reactor pressure vessels. They concluded that the distribution of the crack initiation region is not necessarily correlated to the structure of the different welding beads along the crack front. Furthermore, it was found that the fracture toughness
⁎
values at cleavage failure, Kjc , determined with T-S (crack extension through the thickness) specimens is significantly higher than in case of the T-L (crack extension in welding direction) specimens. Shlyannikov et al. [3] studied the structural behavior of power plant pipes under creep and fatigue conditions. The effect of internal pressure and shape defect of a pipe bend on stress-strain redistributions and creep-fatigue crack growth rates was investigated using FE-analysis and experiments. It was found that the creep-fatigue crack growth rate are depend on the critical zone positions in the considered pipe bend. Furthermore, creep stress intensity factor was used as a new parameter for characterization of the crack growth resistance for power plant materials and structures under elevated temperature. Ruggieri et al. [4] numerically investigated the crack front region and effects of crack-tip constraint in conventional fracture specimens with transverse delamination cracks. They proposed important features of 3-D crack front fields in fracture specimens and concluded that these features had a direct effect on toughness of an isotropic materials. Mirzaei et al. [5] experimentally and numerically evaluated the failure of a compressed natural gas (CNG) fuel tank. Several transient-
Corresponding author. E-mail addresses: [email protected] (E. Alizadeh), [email protected] (M. Dehestani).
https://doi.org/10.1016/j.tws.2018.02.009 Received 12 May 2017; Received in revised form 12 January 2018; Accepted 8 February 2018 0263-8231/ © 2018 Elsevier Ltd. All rights reserved.
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E. Alizadeh, M. Dehestani
dynamic elastic-plastic finite element (FE) analyses was conducted to simulate the structural response of the tank under dynamic pressure. It was found that the failure characteristics, like the overall asymmetric deformation and fracture patterns, initiation and partial growth of parallel cracks at the same section, multiple cracking at the neck, and the self-similar growth of the main axial crack were all initiated by traveling sonic pressure wave from the neck towards the bottom of the tank. Mirzaei et al. [6] studied dynamic ductile rupture of steel pipes under high-speed internal moving pressures. Formation of special fracture surface patterns due to cyclic crack growth was experimentally identified. The overall transient dynamic response of the pipe to explosion loading, the detonation-driven crack growth, the cyclic bulging of the crack flaps, and the resultant crack branching were simulated numerically. Their study demonstrated that the propagation of the initial axial cracks in the pipe was the incremental cyclic growth governed by the structural waves. Oikonomidis et al. [7] proposed a specimen for evaluation the high strain rate flow and fracture properties of pipe material and tuning a strain rate dependent damage model (SRDD). They used the SRDD model to simulate axial crack propagation and arrest in natural gas pipelines. The result reveals that the model correctly predicted the crack initiation pressure, the crack speed and the crack arrest length. Fracture tests of carbon steel piping with a single and two circumferential flaws on the inner surface were conducted by Ogawa et al. [8] to investigate a method for fracture assessment of ferritic steel piping with multiple flaws. The results showed that fracture assessment based on the twice elastic slope method and the plastic collapse mechanism reached inadequate results for a large single flaw. Furthermore, two kinds of elastic-plastic fracture assessment method, one using the Z-factor in the JSME FFS code and the other by ductile instability analysis, showed conservative estimates of fracture strength even when the structural factor SF was not considered. Śnieżek et al. [9] experimentally studied the propagation of the semi-elliptical cracks in austenitic steel which usually used for construction of the industrial pipelines and pressure vessels. The method of electrical potential drop (EPD) is used to investigate the fatigue crack growth. In order to confirm accuracy of the electrical potential drop measurement method, two other experimental methods were used in comparative studies. It was found that Elaborated EPD measurement methodology enables a continuous recording of the dimensions of propagating semi-elliptical cracks, both under variable tension and under constant bending, which was confirmed during the tests. Moshayedi et al. [10] experimentally and numerically investigated the influence of welding residual stresses on brittle fracture in a pipe containing internal surface crack. They assumed an internal circumferential crack at the weld line. Their results indicate that, in the welded pipe, the fracture toughness decreased dramatically in comparison with the pipe without welding residual stresses, this is due to the influence of tensile welding on the crack tip stress state, which cause a decrease in the opening mode stresses at the near crack tip. Mehmanparast [11] experimentally and analytically studied the behavior of creep crack growth in 316H stainless steel for several specimens. The behavior of two material states including as-received and pre-compressed conditions have been experimentally evaluated and the experimental results were compared with the predicted creep crack growth which obtained analytically. It was shown that the predicted creep crack growth had a good agreement with experimental results. Mehmanparast et al. [12] experimentally investigated the effects of the material pre-straining level on the tensile, creep deformation, creep crack initiation and crack growth behavior of 316H stainless steel. It was found that creep ductility and the time of rupture processes was decreased with an increase in pre-strain levels. McCready et al. [13] examined the retrofitting of damaged pipes with composite layers. The specimens were subjected to a combined
loading of internal pressure and external bending. They concluded that composite repairs significantly increased the flexural stiffness of the damaged pipe. Furthermore, post-flexural, hydro-static testing indicated that the composite repairs were able to sustain significant damage while retaining their capability to support the pressure capacity of the pipe. Mazurkiewicz et al. [14] numerically and experimentally evaluated the mechanical behavior of steel pipe containing part-wall defect reinforced with GFRP layers. The analyses shown that local reduction of pipe wall thickness due to corrosion defect can reduce the pressure resistance significantly. Besides, pipes which were repaired by a GFRP layers could sustain more pressure in comparison to an original steel pipe considering burst pressure. Jeong et al. [15] proposed the plastic influence functions, for estimates of J -integral of a pipe containing a complex crack based on the systematic 3-D elastic-plastic finite element (FE) analyses by using Ramberg-Osgood (R-O) relation. In their function, global bending moment, axial tension and internal pressure were considered as loading conditions. They concluded that the proposed engineering J prediction method can be utilized to assess instability of a complex crack in pipes for R-O material. Arafah et al. [16] presented analytical model for calculating plates with semi-elliptical surface cracks subjected to tension, bending, and combined tension-bending and biaxial tension. The comparison between the predicted critical loads and experimental burst testfailure loads demonstrated satisfying agreement with each other. Schonberg et al. [17] proposed a model to predict the eventuality of crack formation and propagation under an impact crater in a thin plate. This model was used to examine the influence of penetration depth on crack formation and whether or not the crack might grow through the tank wall thickness. The predictions of the model were compared to experimental data with encouraging results. Sowards et al. [18] evaluated the fatigue crack propagation in three steel specimens in a simulated fuel-grade ethanol environment. A fracture mechanics testing approach was used to determine crack propagation rates as a function of the stress-intensity-factor amplitude. The experimental and fracture analysis results indicate that fatigue damage in fuel-grade ethanol environments is increased for all three materials. Based on these results, they proposed a model for determining crack growth rates in ethanol fuel. Yokozeki et al. [19] experimentally studied the gas leakage through a composite laminates containing multilayer matrix cracks. They used helium leak gas detector at room temperature to examine the helium gas leak rates through the damaged laminates subjected to uniaxial loadings. They also evaluated the effect of angles of matrix cracks on gas leakage. Richardson et al. [20] presented a method for simulating quasistatic crack propagation in 2-D which combines the extended finite element method (XFEM) with a general algorithm for cutting triangulated domains, and introduce a simple and flexible quadrature rule based on the same geometric algorithm. They concluded that the cutting algorithm provides a flexible and systematic way of determining material connectivity, which is required by the XFEM enrichment functions. Also, their integration pattern is accurate, without requiring a triangulation that incorporates the new crack edges. The application of this cutting algorithm and integration rule is for geometrically complicated domains and complex crack patterns. Giner et al. [21] proposed an implementation of the extended finite element method for fracture problems within the finite element software ABAQUS. They provided details on the data input format together with the subroutine elements, however, pre-processing tools that are necessary for an X-FEM method, but not directly related to ABAQUS, are not provided. They finally presented several numerical examples in fracture mechanics to reveal the benefits of the proposed implementation. Wang et al. [22] numerically studied the ultimate load capacity and 211
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E. Alizadeh, M. Dehestani
the failure of composite vessel. The material property degradation and cohesive element methods were used to model the intralaminar failure and the initiation of delamination at the interface of composite laminates, respectively. The numerical model is compared with experimental results and they were compatible. Gutman et al. [23] used the thin elastic cylindrical shell model to investigate the stability of high-pressure vessel subjected to inside corrosion. Estekanchi and Vafai [24] utilized the finite element model to study the buckling of cylindrical shell with cracks. Besides, Akrami and Erfani [25] analytically and numerically investigated the buckling load of cracked cylindrical shell. They studied the effect of crack location, crack severity and length of cylinder on the load capacity of cracked body. Gato [26] proposed the mesh-free model to study the dynamic fracture of pressure vessel. This method can model the fracture of thinwalled structure in a simple way. Although, analytical methods for studying cracked bodies are usually exact and easy to use, but analytical results are available for only limited problems with simple configurations and loadings. Therefore, for many practical complicated engineering problems like pressure vessels, FEM is applied. The main purpose of current article is to use both analytical and finite element method to study a pressure vessel containing a crack or multiple cracks subjected an internal pressure. Also, the effect of some retrofitting methods with FRP laminates are investigated.
Fig. 1. Configuration of cracked pressure vessel (all dimensions are in millimeter).
shown in Eq. (5), where σz and σy are longitudinal and yield stresses, respectively [27]. In this case, the critical pressure (Pcr ) can be computed from Eq. (6).
2. Theoretical analysis
σ 2θ − σθ σz − σ 2 z = σ 2 y
The investigation of the failure behavior of supposed pressure vessel is conducted based on fracture mechanic theories. Fracture mechanics is focuses on the analyzing the structures containing initial cracks that can affect the load‐carrying capacity of an engineering structures. Defects are initiated in the material by manufacturing procedures or can be created during the service life, by fatigue, environmental effects or creep. In the theoretical section of current paper, it is assumed that there is an initial crack in the internal side of the thickness of the supposed pressure vessel and a characteristic quantity which defines the propensity of the crack to extend is determined. The characteristic quantity depends on the particular failure criterion used, that for this paper, the stress intensity factor is applied. The failure criterion is expressed by Eq. (1), in which KI is stress intensity factor at the crack tip which depends on the applied load, the initial crack length and the geometrical configurations of the cracked plate and K c is fracture toughness that depends on the material parameter which can be determined experimentally [27].
Pcr =
PR t
Kc t 1.12R πa
(5)
2σy t 3R
(6)
3.1. Constitutive modelling and elements In current study, the pressure vessels made of seven metals are designed according to ASMEVIII- Division1 (Rules for Construction of Pressure Vessels) [29]. The minimum required thickness of vessels under internal pressure should not be less than that computed by Eq. (7), where E is the efficiency of appropriate joint in cylindrical or spherical vessels, or the efficiency of ligaments between openings, whichever is less, that is considered 0.85 in this paper. P is internal pressure, R is the radius of the vessel, S is the maximum allowable stress value and tmin is minimum required thickness of shell [29].
(2)
tmin =
(3)
PR SE −0.6P
(7)
The outside diameter of pressure vessel is considered 2000 mm. The wall thickness (t ) is designed according to Eq. (7). A crack with initial length (a) and width (b) is considered in current studies. Configuration of pressure vessel and its dimensions are demonstrated in Fig. 1. The location of initial crack in cylindrical part of the vessel and its position, which could be on the internal or external side of the vessel, are shown in Fig. 2.
By combining the Eqs. (1)–(3), the critical pressure (Pcr ) is calculated as Eq. (4).
Pcr =
PR 2t
A three-dimensional finite element analysis is carried out to evaluate the behavior of the pressure vessel. The numerical analysis is performed by finite element software ABAQUS [28]. The procedure is summarized as follows.
The amount of σ can be calculated from Eq. (3), where σθ is circumferential stress, P is internal pressure of the vessel and R and t are the radius and the thickness of the vessel, respectively.
σ = σθ =
σz =
3. Finite element analysis
The stress intensity factor at the crack tip (KI) of the single-edgecracked plate under uniform far field tension can be obtained from Eq. (2), where σ is the applied stress on the plate and a is the crack length perpendicular to load of σ [27].
KI = 1.12σ πa
PR , t
The diagram of critical pressure Pcr versus crack length a , can be plotted by using Eqs. (4) and (6) for a pressure vessel which is designed safely against both yielding and fracture due to initial crack. The critical pressure of a vessels with seven different materials are calculated and illustrated in Figs. 3–5. The mechanical properties of considered materials are given in Table 1.
(1)
KI = K c
σθ =
(4)
In addition, the vessel must be designed safely against failure due to yielding (according to the Von Mises yield criterion, for example) as 212
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E. Alizadeh, M. Dehestani
Fig. 2. (a) Crack location in cylindrical part of vessel, (b) location of internal crack in a sector of vessel, (c) location of external crack in a sector of vessel.
Fig. 3. Mesh configuration around the crack region.
Solid elements are used to model the pressure vessel. The type of the element is chosen as 3D8R that includes eight nodes and three translational degrees of freedom at each node. In order to make computational time minimum and achieving the highest accuracy simultaneously, based on comprehensive mesh convergence studies, mesh size is adopted with a maximum element size of 10 mm in the region around the crack tip and 80 mm for other parts of the specimen which are far from the crack domain, as shown in Fig. 3. An extended finite element method (XFEM) is used for modelling the crack. In comparison with the classical finite element method or linear elastic fracture mechanics (LEFM), the XFEM provides significant benefits in the numerical modelling of crack propagation. In the traditional formulation of the FEM, the existence of a crack is modeled by requiring the crack to follow element edges. In contrast, the crack geometry in the XFEM need not be aligned with the element edges, which provides more flexibility and versatility in modelling. The method is based on the enrichment of the FE model with additional degrees of freedom (DOFs) that are tied to the nodes of the elements intersected by the crack [21]. In this manner, the discontinuity is included in the numerical model without modifying the discretization, as the mesh is generated without taking into account the presence of the crack. Therefore, only a single mesh is needed for any crack length and orientation. In addition, nodes surrounding the crack-tip are enriched
Fig. 4. Critical pressure (Pcr ) versus crack length (a ) for pressure vessel that made of steel material.
213
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E. Alizadeh, M. Dehestani
Fig. 5. Critical pressure (Pcr ) versus crack length (a ) for pressure vessel that made of titanium material.
Fig. 6. Critical pressure (Pcr ) versus crack length (a ) for pressure vessel that made of aluminum material.
Table 1 Mechanical properties of materials [27]. Material
σy (MPa)
K c (MPa m )
Steel 4340 Steel 4335 Steel 350 maraging
860 1300 1550
100 70 55
Ti 6A1-4V Ti 4A1-4Mo-2Sn-0.5Si
1103 945
38.5 70.3
AL 2014-T4 Al 7075-T651
448 545
28.6 29.7
In first part of verification, the pressure vessels with different initial crack length (a) , constant crack width (b = 10 mm), and wall thickness of 20 mm, are modeled for seven material properties. The critical tolerable pressure for each vessel with a certain initial crack is investigated. The analytical and numerical results of critical pressure are plotted versus crack length for steel, titanium and aluminum vessels in Figs. 4–6, respectively. As illustrated in Figs. 4–6, the diagram of failure pressure against crack length (a ) is composed of two pieces; one linear piece that shows the failure due to the yielding of the steel, and a curved piece that shows the fracture of the vessel due to crack growth. Figs. 4–6 prove that results obtained from the numerical analysis by XFEM employed here coincide with the theoretical answers in both yielding and fracture part of the diagram. The numerical critical stress (σcr ) at crack region versus crack length (a ) for three type of steel material is presented in Fig. 7. Besides, Fig. 8 demonstrates the numerical circumferential stress distribution and crack propagation process of crack length 10 mm in steel 4340 pressure vessel for two different internal pressure P = 6.68 MPa and P = 10.3 MPa.
with DOFs associated with functions that reproduce the asymptotic LEFM fields. This enables the modelling of the crack discontinuity within the crack-tip element and substantially increases the accuracy in the computation of the stress intensity factors (SIFs) [21]. In the ABAQUS the maximum principal stress damage (Maxps Damage) is defined as the material behavior of the region in which the crack could growth. The crack propagation occurs when the critical energy release rate (Gc ) reaches a critical value. The connection between the energy release rate (Gc ) , which is a global quantity, and the stress intensity factor (K c ) , which expresses the strength of the local elastic stress field in the neighborhood of the crack tip, is very important. The equivalence between critical energy release rate and critical stress intensity factor for plane stress material is shown in Eq. (8) [27]. The critical energy release rate is calculated for each material in Table 1 and it is used for defining the material behavior in ABAQUS.
Gc =
Kc2 E
(8)
In addition to defining Gc , the critical stress of crack must be introduced to the program which is computed as Eq. (9) and is used for each initial crack length at a certain material type in the ABAQUS.
σc =
Kc 1.12 πa
(9)
3.2. Validation of numerical model The verification studies are conducted for various values of internal crack length (a) , various different wall thickness of the vessel, and for seven different material properties (steel, titanium, and aluminum).
Fig. 7. Numerical critical stress (σcr ) versus crack length (a ) for pressure vessel.
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Fig. 8. Circumferential stress distribution and crack propagation of crack initial length 10 mm for two different internal pressure (a ) P = 6.68
MPa (b)P = 10.3 MPa.
4.1. Multiple cracks
In the second part of the verification, vessels with crack length of a = 20 mm, crack width b = 10 mm, and three material properties with various vessel’s wall thicknesses are considered. For such vessels, results of theoretical analysis which are calculated based on Eq. (4), and XFEM analysis are compared (Fig. 9). The studies in this section, indicate that the results of numerical method are in excellent agreement with analytical method. Thus, FEM is going to be utilized to study the more complex problem of cracks.
The possible effects of simultaneous cracks on a body is numerically analyzed in two configuration types. In the first one, the effect of angular distance between two cracks on the vessel’s body is analyzed. As shown in Fig. 10(a), two cracks are located in the internal circumference of the vessel with the angle of θ between them (θ = 5, 10, 20, 30, 60, 90, 120, 150, 180 ). In the second type, n equally-spaced cracks (n = 1 − 9) are located in the vessel’s internal circumference while the angle between them are 360/n ° (Fig. 10(b)). As predicted, in both types above, no significant differences are observed when one compares the critical pressure of a single cracked body with a multiple cracked one. This is because each crack affects only on a very tiny limited area around the crack tip. Therefore, two
4. Parametric study The analytical and numerical investigations are proposed in this section.
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Fig. 11. Critical pressure (Pcr ) versus crack width (b ) for steel 4340 pressure vessel.
Fig. 9. Critical pressure (Pcr ) versus wall thickness (t ) for pressure vessel with 20 mm initial crack.
across the vessel’s thickness. In order to evaluate the effects of crack location on the critical load, the numerical analyses are conducted for a pressure steel 4340 vessel containing internal or external crack (As shown in Fig. 2). In order to have a better comprehension, the analyses are made for three thickness of the vessel 20, 30 and 40 mm. As shown in Fig. 12, for specimens with small crack length, by changing the crack location from internal to the external edge, the critical pressure capacity is meaningfully decreased, but for longer crack length the difference between two crack locations is decreased until the two curves are converged to each other where the crack length is equal to the thickness of the vessel. In other words, cracks on the outer surfaces of the vessel are more critical than cracks on the inner one, besides, the external crack are usually more probable due to external impact load, scratches, or environmental effects.
neighbor cracks have very little effects on each other and on the overall capacity of the vessel. Two neighbor cracks should be very close to each other to have effects on the overall capacity. For example in this case, when two cracks are placed at the distance of 5 mm along the circumference, the critical pressure decreases less than 15%. 4.2. Effects of crack width (b ) on critical pressure (Pcr ) Each crack is simulated as rectangular discontinuity in the vessel’s wall thickness. In previous sections of this paper, the width of cracks are considered 10 mm, but in current section, the crack width is changed to four different value of 5, 10 , 15, and 20 mm. The effect of crack width (b) on critical pressure of steel 4340 vessel is numerically investigated. As shown in Fig. 11, the critical pressure is reduced for all crack length by increasing the crack width. The effect of crack width is more significant for crack with smaller crack length, and the influence of crack width is decreased by increasing the length of crack. For example, the curve of a crack with 5 mm length is more sensitive to increasing the b , when compared to the curve of crack length 20 mm.
4.4. Reinforcement with FRP laminates In order to strengthening the specimens containing cracks, different materials could be used as a reinforcing layer. But it is reasonable to use materials with modulus of elasticity close to steel or higher than that. Therefore, among different types of FRP laminates (Carbon, Glass, and Aramid), CFRP types are considered due to its high value of modulus of elasticity. It is noticeable that several investigations are conducted to study the load capacity and failure of carbon composite vessels [22]. In this study, CFRP laminate with thickness of 7 mm and 400 mm width is applied in a region around the crack tip (Fig. 13). This
4.3. Effect of crack location along vessel’s wall thickness In all previous sections, cracks are located on the internal edge of the circumference, while cracks’ location influences the stress field
Fig. 10. (a) Configuration of two cracks located in the internal circumference of the vessel, (b) Configuration of multiple cracks located in the internal circumference of the vessel.
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Fig. 12. Critical pressure (Pcr ) versus crack length (a ) for steel 4340 pressure vessel with internal and external crack location.
Fig. 14. Critical pressure (Pcr ) versus crack length (a ) for steel 4340 pressure vessel reinforced with CFRP laminates.
Fig. 13. Configuration of CFRP layer around the crack region.
Fig. 15. Circumferential stress (σ ) versus crack length (a ) for steel 4340 pressure vessel with and without CFRP laminate.
numerical study is done by three different types of CFRPs with different mechanical properties that are given in Table 2. In the FEM program the CFRP layer is modeled as orthotropic material via “shell” elements. This types of element consists of four nodes and six degrees of freedom per node, three translational and three rotational. The Fig. 14 demonstrates the effects of reinforcement with CFRP laminates, on the critical pressure for a cracked body. The linear piece of all diagrams in Fig. 14 (smaller values for a ) shows the limit where the crack has no effect on the critical internal pressure. Adding a CFRP layer expands this limits for crack length and stretches the linear piece of the diagram. Also, as predicted, adding a CFRP layer, increases the capacity of the vessel. These two effects are more pronounced for CFRPs with higher modulus of elasticity. Furthermore, the circumferential stress (σ ) versus crack length (a )
for steel 4340 vessel with and without CFRP laminate is shown in Fig. 15. The stress in vessel without CFRP layer is demonstrated at critical pressure load for each crack length. Besides, the stress in vessel with CFRP layer is shown at a same load as vessel without CFRP layer for each crack length. The results reveal that the stress in vessel with CFRP layer is lower than the other vessel at a same load, so utilizing CFRP layer at the crack region can enhance the load bearing capacity of vessel. 4.5. The effect of combined modes on the vessel An analytical investigation is conducted to study the effect of combined opening (mode I) and sliding (mode II) modes on crack
Table 2 Mechanical properties of CFRP laminates. Material
Product name
Type
Ply orientation
Modulus (GPa)
G12 (GPa)
Longitudinal
Transverse
ϑ12
Strength (MPa) Longitudinal
CFRP
SK-CPS 0512
High strength carbon
0°/90°
165
13.75
7.27
0.218
3000
CFRP
SK-CPI 0614
Intermediate modulus carbon
0°/90°
210
17.5
8.42
0.214
2400
CFRP
SK-CPH 0514
High modulus carbon
0°/90°
300
25
8.76
0.212
1300
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Fig. 16. (a) Inclined crack in the pressure vessel (b) the opening and sliding modes of loading in rotated plate.
instability. Thus, the orientation of crack is rotated from vertical status and the mode II of loading appears in the inclined crack. A shown in Fig. 16, the rotation angle between θ and θ´ axes is presented by β and σL and σθ are longitudinal and circumferential stresses, respectively. Fig. 16(a), demonstrates the inclined crack in the pressure vessel. In order to calculate the stress intensity factor of inclined crack, the plate is rotated and the opening and sliding modes of loading are shown in Fig. 16(b). ′ are estimated By stress transformation relations, the σθ′, σz′ and τθz and given in Eq. (10) [27].
σθ +σL σ σ 3σ + θ − L cos2β = + 2 2 2 σθ +σL σθ −σL 3σ cos2β = = − − 2 2 2 σθ −σL σ sin 2β = − sin 2β = − 2 2
σ ´θ = σ´z τ ´θz
KII = −
1 Pcr R b P R b sin2(30) π ( ) = −0.2165 cr π( ) 2 2t 2 t 2
KI K + II = 1 KIc KIIc 0.625
π (2 )
0.2165 −
100
0.00625
σ cos2β 2 σ cos2β 2
Pcr =
(10)
π (2 )
108.65
=1
Pcr R b π( ) = 1 t 2
t 0.004257×R ×
As shown in Eq. (11), the mode I and II stress intensity factor (KI , KII ) are obtained by using the transformed stresses (Eq. (10)).
b
Pcr R t
Pcr R b P R b π ( ) − 0.00199 cr π( ) = 1 t 2 t 2
0.004257
b
π (2)
=
20 b
7.5434× ( 2 )
=
2.6513 b
(2)
According to the above equations, the critical pressure (Pcr ) is calculated for different rotation angle (β = 30° , 45° , 60° , 75° and 90°) and then the different critical pressure (Pcr ) versus crack length (b ) diagrams are plotted and presented in Fig. 17.
b b 3 1 KI = σz′ π ⎛ ⎞ = ⎡ − cos 2β⎤ σ π ⎛ ⎞ 2 2 2 2 ⎣ ⎦ ⎝ ⎠ ⎝ ⎠ b σ b ′ π ⎛ ⎞ = − sin 2β π ⎛ ⎞ KII = τθz 2 ⎝2⎠ ⎝2⎠
b
Pcr R t
(11)
Likewise, the critical pressure (Pcr) can be calculated via Eqs. (11) and (12), in which the fracture toughness of mode I and II are presented by KIc and KIIc , respectively [30].
KI K + II = 1, KIc KIIc σ=
b 3 1 P R P R π( ) [ − cos2β] cr (− cr4t Pcr R 2 2 2t 2 → + 2t KIc
b
sin2β π ( 2 )) KIIc
=1 (12) Furthermore, the calculation procedure of critical pressure for crack with rotation angle of 30° is shown below. It is noticeable that the steel 4340 material and the wall thickness of 20 mm are used in this section. Besides, KIc is considered 100 MPa m and KIIc is also assumed 108.65 MPa m [30].
KI = [
3 1 P R b P R b π ( ) = 0.625 cr π( ) − cos2(30)] cr 2 2 2t 2 t 2
Fig. 17. Critical pressure (Pcr ) versus crack length (b ) for steel 4340 pressure vessel with different crack orientations.
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As illustrated in Fig. 17, the most critical pressure load is related to the initial crack (the crack that was investigated in previous sections) with vertical status and rotation angle of 90°. It should be noted that the vessel must be designed safely against failures by yielding and fracture. Thus, as stated before, the failure pressure of the vessel is studied due to yielding based on Eq. (6). In each crack length, the lower load calculated with Eqs. (6) and (12) is considered as a critical pressure. As an example, in vertical crack with crack lengths lower than 6.5 mm, the yield pressure is lower than the fracture one, thus, the failure will occur due to yielding. It means that, although the vessel could withstand the pressure Pcr = 32 MPa for crack length 2.5 mm, but it fails at Pcr = 19.8 by yielding. Otherwise, the failure will happen because of fracture for vertical cracks with more than 6.5 mm length. Besides, the fracture load for crack with rotation angle of 30° is higher than yield pressure, hence, the failure of vessel will be caused by yielding of material in this case. 4.6. The elastic-plastic analysis In this section, the elastic-plastic analysis has been conducted via analytical calculation. The inelasticity in the form of plasticity, creep or phase changes can be identified in the neighborhood of crack tip. As shown in Fig. 18, the inelastic region is sufficiently small compared to k-dominant zone [27]. The occurrence of plasticity in crack tip, makes the crack behave as if it was longer than its real and physical size. Therefore, the plate performs as if it contained a crack with slightly larger length. Furthermore, the effective crack size, aeff , can be calculated by a +rp , where a is the crack length and rp is the radius of plastic zone near the crack tip. The principal stresses shown in Eq. (13) can be utilized in von mises criterion to calculate the radius of plastic zone [27].
σ1 = σ2 = σ3 =
Fig. 19. Critical pressure (Pcr ) versus crack length (a ) for elastic and elastic-plastic analysis (steel 350 maraging).
In addition, the stress intensity factor corresponding to the effective crack, called effective stress intensity factor (K eff ) for plane strain status, has been computed by using Eq. (16) and is shown in Eq. (17). Consequently, the effective pressure (Peff ) is also estimated as given in Eq. (18) [27].
K = 1.12σθ π (a + rp ) = 1.12×
1−[
⎟
1.12σ (1 − 2ϑ) 2 ] 2 σy
(17)
2σy
=
K2 3 2 [ sin θ+(1 − 2ϑ)2 (1+cosθ)] 4πσY 2 2
K2 (1
Kc t 1.12R πa +
(14)
1.12RK c (1 − 2ϑ) 2 σy
(18)
According to Eq. (18), the Critical effective pressure (Pcr ) versus
The radius of plastic zone (rp) is calculated based on principal stresses and von mises criterion and it is shown in Eq. (15). Besides, rp is estimated for θ = 0° at crack tip (Eq. (16)).
K2 [2(1 − 2ϑ)2] = 4πσ 2Y
K2 (1 − 2ϑ)2 ⎞ → K eff 2πσY 2 ⎠
K 2 (1 − 2ϑ)2 PR 2 ) ×[πa + c ] → P2 eff t 2σY 2 K c 2t 2 = → Peff K 2 (1 − 2ϑ)2 2 2 1.12 R [πa + c ] 2
(13)
(σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2 = 2σ 2y
rP =
⎜
K c 2 = 1.122 × (
The von mises criterion is presented in Eq. (14), in which σy demonstrates the yield stress of material [27].
rP (θ) =
π ⎛a + ⎝
1.12σ πa
=
θ θ K1 cos (1 + sin ) 2 2 2πr θ θ K1 cos (1 − sin ) 2 2 2πr θ 2ϑK1 cos 2 2πr
PR × t
(15)
2ϑ)2
− 2πσ 2Y
(16)
Fig. 20. Critical pressure (Pcr ) versus crack length (a ) for elastic and elastic-plastic analysis (steel 4335).
Fig. 18. The inelastic and k-dominant regions in the crack tip.
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crack length (a ) for 350 maraging and 4335 steel is computed and is compared with elastic analysis in Figs. 19 and 20, respectively. It is noticeable that the plastic zone near crack tip is considered in elasticplastic analysis. Thus, the crack length is slightly larger than its real length and the critical pressure is also slightly lower than elastic analysis. As shown in Figs. 19 and 20, the difference between elastic and elastic-plastic analysis is not significant due to small plastic zone.
[6]
[7]
[8]
5. Summary and conclusion [9]
In current article, the fracture analysis of a pressure vessel is considered for different configurations of cracks with seven metal material properties. Initially, some analytical studies are conducted to obtain the critical internal pressure of the vessel. Analyzing these problems by XFEM illustrates that numerical methods could be beneficial in studying the cracked structures, especially for cases in which theoretical analyses are difficult to achieve. The analytical and numerical result are presented below:
[10]
[11]
[12]
[13]
• Inducing • • • • •
multiple cracks has no effects on the critical internal pressure practically except for cracks which are very close to each other. Greater crack width causes lower critical pressures; this effect is more observable for smaller crack lengths. By changing the location of the crack on the vessel’s wall thickness, it is found that an external crack has more influence on the vessel’s capacity than an internal crack of the same configurations. Reinforcing the cracked region of the vessel reveals that laminates with higher modulus of elasticity could be more useful for strengthening a defected structure. The effect of sliding mode is investigated due to rotating the crack orientation and the load capacity of vertical crack is the most critical one among others. The difference between elastic-plastic and elastic analyses is not significant due to the small plastic zone in the crack tip.
[14]
[15]
[16]
[17]
[18]
[19]
[20]
Acknowledgment This paper presents findings of a research project which was supported by Babol Noshirvani University of Technology in Iran via faculty member grants (No. 10162).
[21] [22]
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