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Numerical study of elliptical cracks in cylinders with a thickness transition Abdelkader Saffih a,*, Saı¨d Hariri b a b

Laboratoire GMMS, Universite´ de Reims, Moulin de la Housse, B.P. 1039, 51687 Reims Cedex 2, France Ecole des Mines de Douai, De´partement MCM, 941, rue Charles Bourseul, F-59508 Douai Cedex, France Received 14 April 2005; received in revised form 3 October 2005; accepted 5 October 2005

Abstract The paper deals with elliptical cracks in a cylinder with a thickness transition. This structure is an assembly of two cylinders of thickness t and t2 (t!t2). In the transition zone, the thickness varies linearly. The purpose is to check whether tabulated data used for SIF calculations in a cylinder of uniform thickness t can be used for the cylinder with a thickness transition. A comparative study is made on the effect of a crack in a cylinder with a thickness transition and in a uniform thickness cylinder. Loads considered are pure tensile stress and bending moment. A numerical analysis was performed considering elastic behaviour at the crack tip. A crack mesh was designed and validated for 3D models. The results show that SIF calculations in the transition assuming a uniform thickness cylinder are conservative but not precise. The comparative study shows that the cylinder with a thickness transition is more vulnerable to a defect. q 2005 Elsevier Ltd. All rights reserved. Keywords: Thickness transition; Elliptical cracks; Stress intensity factor

1. Introduction The harmfulness of cracks can be measured with global parameters such as the stress intensity factor (SIF) K or the J integral [1,2]. Evaluation of these parameters is often done by finite element (FE) calculations [3–9]. The duration of calculations is very sensitive to the geometrical complexity and the material non-linearity. However, simplified methods for the evaluation of K and J exist. In the case of elastic materials, the superposition principle is applicable and a resolution of stresses in polynomial form combined with the use of the SIF parameters, allows quick calculation of K for current geometries subjected to various loadings [6,7,10–12]. For elastic–plastic material, the use of R6 and A16 rules leads to approximate evaluation of J [10]. For complex geometries, it is also of use in some cases to simplify the problem by substituting simpler geometries for which calculations are less expensive [8,9,13]. The substitute geometry must always lead to a structure that ensures great safety, i.e. it must be more

* Corresponding author. Tel.: C33 3 26 91 88 13; fax: C33 3 26 91 39 78. E-mail addresses: [email protected] (A. Saffih), [email protected] ensm-douai.fr (S. Hariri).

0308-0161/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2005.10.002

vulnerable to the defect. The choice of the substitute geometry is however not always obvious as, for example, in the case of thickness transitions. The French utility EDF and CEA1 jointly launched a study on cylinders comprising a thickness transition [13]. These structures are assemblies of two cylinders of the same inner radius but with different thickness t and t2 (t!t2). They generally correspond to connecting zones of pipes to thicker elements (tank pipe, valves, etc.). During service, circumferential cracks can appear at the base of the thickness transition. To characterize these cracks, and to be on the safe side, common industrial practice is to model these defects with similar defects located in a cylinder of uniform thickness t subjected to the loads calculated in the thickness transition zone. This approach has not been fully justified, and only a few studies, mainly focused on axisymmetric cracks subjected to tensile stress and/or thermal shocks, have been performed. In [13], it was shown that, for elastic material, the use of SIF parameters of a uniform thickness cylinder for K calculations in the thickness transition gave conservative and precise results. For elastic–plastic materials, simplified rules for J calculations, developed for the uniform thickness cylinder, are also applicable to defects in the thickness transition [13]. 1

Commissariat a` L’Energie Atomique (France).

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The case of elliptical cracks was however not treated. Also, taking account of bending moment was required to complete the work in [13]. To extend the previous study to the threedimensional case, external circumferential elliptical cracks are considered in this paper. The purpose is to check whether or not the conclusions reached previously for axisymmetric defects remain valid for elliptical cracks. A numerical study was performed considering pure tensile stress and global bending loads. First, in Sections 2–4, we calculate the SIF of an elliptical crack in the thickness transition in two ways: (i) directly and (ii) by using the K parameters of the uniform thickness cylinder, to evaluate the precision of this latter method. Then, in Section 5, a comparison of crack harmfulness in both the uniform thickness cylinder and the cylinder with a transition is made. 2. Geometry Thickness transitions met in industry are classified into two types: transitions with a single slope and transitions with a double slope [10,13]. The present study deals with the case of a single slope transition (Fig. 1). The thin part of the structure is a tube of thickness t and internal radius Ri. It is connected to a thick part (t2, Ri) by a transition zone where the thickness varies linearly from t to t2. The most severe case encountered in industrial facilities is a transition with a slope of 308, and a ratio t2/tZ1.5 [13]. Elliptical cracks are supposed to be located at the base of the transition on the thin cylinder side (Fig. 2). The various geometries of cylinder considered are defined by the non-dimensional ratios a/c, a/t and t/Ri, where a and c are, respectively, the depth and the semi-length of the crack. Three values of t/Ri are considered: 1/2, 1/10 and 1/100. These correspond, respectively, to three types of cylinder: thick, of average thickness and thin. The ratio t2/t is set equal to 1.5. The aspect ratio a/c, takes values 1/2, 1/4, and 1/8. Lastly, the ratio a/t varies between 10 and 80%. This gives a set of 45 geometries. 3. Calculation of K In elasticity, a crack is characterized by its stress intensity factor K: pﬃﬃﬃﬃﬃﬃ K Z Cs pa (1) where C is a constant related to the loading, crack size and geometry, s is the tensile stress applied on the crack surfaces and a is the depth of the defect. For the same geometry, the value of K depends on the distribution of stress, s. Rather than

Fig. 1. Geometry of a cylinder with a single slope thickness transition.

Fig. 2. Cracked cylinder with a thickness transition (elliptical circumferential external crack).

performing K calculations for each loading case, by applying the superposition principle, we can resolve the distribution of s into elementary loadings. The SIF may then be written as the sum of weighted contributions of the elementary loadings, each depending only on the geometry. For cylinders subjected to axisymmetric loading, a decomposition of stresses in a polynomial form of third or fourth degree is usually used [6,7,10–12]. By adopting a polynomial form of the third degree, the stresses at a point M at a radius R can be written as follows: s Z s0 C s1

u t

C s2

u 2 t

C s3

u 3 t

(2)

where ReZRiCt and uZReKR. s0, s1, s2, s3 are the coefficients of the polynomial form. By combining Eqs. (1) and (2), the SIF can be written in the form: K Z K0 ðs0 Þ C K1 ðs1 Þ C K2 ðs2 Þ C K3 ðs3 Þ

(3)

where Kj ðsj Þ Z sj ij

a j pﬃﬃﬃﬃﬃﬃ pa t

j Z 0; 1; 2; 3

If the cylinder is subjected to a bending moment: pﬃﬃﬃﬃﬃﬃ K Z Kgb ðsgb Þ Z sgb igb pa

(4)

(5)

where sgb is the maximum bending stress on the crack surface. The coefficients ij (jZ0, 1, 2, 3, gb) are the SIF parameters or influence functions. They are only functions of the geometry. For a uniform thickness cylinder subjected to pure tensile stress, KZK0. If the cylinder is only subjected to a global bending moment, KZKgb For a cylinder with a thickness transition subjected to tensile stress, the SIF will be calculated with the general formula of Eq. (3), since the stress distribution in the crack plane is modified by the thickness transition, which acts as a stress raiser. The values of the parameters ij (jZ0, 1, 2, 3) are obtained by calculating KEF (K value from a FE calculation) for a unit load function of (u/t)j applied on the crack surface, and using

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Fig. 3. Point location on the crack front; definition of angle f.

the relation: KjEF ij Z j pﬃﬃﬃﬃﬃﬃ u pa t

(6)

igb is obtained by applying a bending moment to the cylinder, so that the maximum bendingpﬃﬃﬃﬃﬃﬃ stress is equal to unity, and by using the relation: igb Z KjEF = pa. The software Castem2000 [14] was used for modelling and calculations. This integrates the G_theta method widely used to calculate the J integral along the crack front. For elastic behaviour at the crack tip, JZG where G is the elastic energy release rate [1,2]. Then, K can be deduced from the relation: G Z KI2 =E 0

(7)

where E 0 ZE in plane stress condition and E 0 ZE/(1Kn2) in plane strain [1,2,11,12]. The position of a point on the crack front is defined by the angle f (Fig. 3). f takes the value of 908 at the deepest point and 08 at the surface point. FE models are built with solid brick and prismatic quadratic elements. In this case, the load discretization on the edges of an element assigns 4/6 of the load magnitude to the medium node and 1/6 to each external node [11,12]. For elements with an edge on the crack front, the J value is calculated in the centre of the edge as follows: 1 Jelem Z ðJe1 C 4Jint C Je2 Þ 6

(8)

where Je1 and Je2 represent J values at the external nodes and Jint the value of J at the medium node. At the surface point, the Je2 value records a jump and will be ignored [11,12]. J will be calculated in this particular case by: 1 Jelem Z ðJe1 C 4Jint Þ 5

Fig. 4. Crack mesh.

(9)

The material considered in the calculations is a steel with a Young’s modulus EZ200,000 MPa and a Poisson’s ratio nZ0.3. 4. Meshing For 3D models of cracked cylinders, a procedure was developed for meshing elliptical cracks. Only the parameters a/c, a/t and t/Ri are required to construct the FE models. The procedure generates a block, of approximately 3400 elements, which can be inserted in planar or curved structures after geometrical transformation (Fig. 4). Fig. 5 presents the mesh of

Fig. 5. Meshing of a half-cylinder with a thickness transition and an elliptical crack.

a half-cylinder with a thickness transition and an external elliptical crack. A study of the mesh sensitivity was performed focusing on selected parameters related to the number and size of elements on, and in the neighbourhood of, the crack front and crack surface. Calibration of the model has been performed on the basis of SIF calculations for elliptical cracks in a uniform thickness cylinder. Data furnished by CEA were used p asﬃﬃﬃﬃﬃﬃ reference. Fig. 6 shows sample calculations results of KI =ðs paÞ along the crack front of an external circumferential elliptical crack using our model and values obtained by CEA. The mean of the absolute value of the relative difference between FE results and corresponding values of the CEA is 0.46% with a standard deviation equal to 0.52% and a maximum value of 3.67%. Our results are in very good agreement with the values of [11,12]. 5. Results and discussion We calculated the SIF for an elliptical crack in a cylinder with a thickness transition in two ways: (i) directly,

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Fig. 6. Calibration of the FE model; Sample results.

and (ii) by using the relation (3) and the SIF parameters for the uniform cylinder of thickness t. This allows an evaluation of the precision of SIF calculation by method (ii). We also compared, from the point of view of propagation risk, the harmfulness of a defect in a cylinder comprising a thickness transition with that for a defect in a uniform thickness cylinder. The geometries considered are those defined in Section 2. 5.1. Calculation of K for a crack in a cylinder with a thickness transition subjected to pure tension For a crack emerging at the cylinder surface, there are two particular points: the deepest point and the point where the crack emerges, indicated as the surface point. In general, the analysis of propagation at these two points is enough to judge the severity of the defect. For this reason, we restricted the analysis of our results to these two points. We define a relative difference DKT between KT, the value of the SIF obtained by direct FE calculation, and KT0 , the SIF value calculated with the SIF parameters of the uniform thickness cylinder: DKT Z 100 !

KT KK 0 T KT

(10)

DKT allows the precision of KT calculated with the SIF parameters of the uniform thickness cylinder to be judged. Fig. 7(a)–(c) represents the evolution of DKT as function of a/t at the deepest and the surface points of a crack for cylinders with ratios t/RiZ1/2, 1/4 and 1/100 subjected to pure tensile stress. At the deepest point, except for the case of shallow cracks in thin cylinders (t/RiZ1/100), DKT is negative for all ratios of a/t and t/Ri. Estimating KT with KT0 is thus conservative. Absolute values of DKT for a/t!0.4, are lower than 15%. This may be considered as good precision of the KT evaluation with the SIF parameters of the uniform thickness cylinder. However, for deep and elongated cracks, the difference between KT and KT0 increases, exceeding 20%. It reaches 25% for a/cZ1/8 and aZ0.8t. The accuracy is thus poor for these defect sizes.

Fig. 7. DKT values at the deepest point for tensile stress: (a) t/RiZ1/2. (b) t/RiZ 1/10. (c) t/RiZ1/100.

At the surface point, the absolute value of DKT remains low in general, less than 15%. Moreover, except for cracks with a/cZ1/8, DKT is negative, and thus indicates an over-estimate of KT by KT0 . The evaluation of KT with the SIF parameters of the uniform thickness cylinder is hence generally conservative and relatively precise at the surface point. 5.2. Calculation of K for a crack in a cylinder with thickness transition subjected to a bending moment In the case of global bending, the relation (5) is used. Comparing direct KT calculations in the thickness transition with calculations using the parameter (igb)d of the uniform thickness cylinder, is equivalent to comparing the values of (igb)T and (igb)d (subscripts T and d refer to the cylinder with a

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Fig. 9. K values at the deepest point for tensile stress (t/RiZ1/10).

5.3. Analysis of the harmfulness of the defects

Fig. 8. (a) Digb values at the deepest and the surface points in bending (t/RiZ 1/10). (b) DKT values at the deepest and the surface points in bending (t/RiZ 1/10) (using decomposition of stresses in polynomial form).

thickness transition and the uniform thickness cylinder, respectively, and (igb)T is given by relation (5)). We then define: Digb Z 100

ðigb ÞT Kðigb Þd ðigb ÞT

(11)

Fig. 8(a) shows the evolution of Digb as function of a/t for various ratios t/Ri. Digb varies over a wide range and reaches a maximum of 60%. We deduce that calculations of KT with the parameter (igb)d lead to significant undervaluation of KT On the other hand, since bending loading may be sometimes assessed locally as tensile loading, we used relation (2) to resolve the bending stresses in the thickness transition into a polynomial form, and relation (3) for the calculation of KT. This method gives a good estimate of the SIF at the deepest point. However, the estimate is still significantly in error at the surface point (Fig. 8(b)). We can deduce from these results that the calculation of KT is, in general, conservative under a tensile stress, with good precision at the surface point, but worse precision at the deepest point, particularly for long defects. In bending, the use of the SIF parameters of the uniform thickness cylinder gives erroneous estimation of KT. Hence, specific calculations must be done to determine K in the thickness transition. These results are not totally in accordance with corresponding results obtained for axisymmetric cracks [13], where the precision was found to be very good.

The SIF characterizes the crack tip loading of a defect in a structure for elastic response. The harmfulness of an elliptical crack in a structure may be related to the evolution of K values. In order to quantify the severity of a defect in the cylinders considered, we compared the evolution of K, as functions of a/t, in both the cylinders at the deepest and at the surface points of the crack. Fig. 9 depicts K variations for two cracks (a/cZ1/8 and a/ cZ1/2) in cylinders of ratio t/RiZ1/10 under a pure tensile stress. The SIF is an increasing function of a/t. K values are higher in the thickness transition at the surface point, for all a/t ratios, and at the deepest point for low and average crack depths. For the ratio a/t exceeding 0.4, the SIF of the uniform thickness cylinder becomes greater at the deepest point. This fact is due to the thickness transition, which acts as a stress raiser. Indeed, stresses are amplified near the surface, which gives greater K values at the surface point and at the deepest point for shallow cracks in the thickness transition. Far from the surface, normal stresses diminish to become lower than the constant stress in the uniform thickness cylinder. As a result, the SIF value at the deepest point is lower than the value in the uniform thickness cylinder. In the case of bending, the evolution of K is typically similar to that described for tensile stress. However, the increasing rate of K values at the surface point is less important than the previous case since stresses are a function of the distance to the neutral axis, and hence diminish when c rises. The interpretation provided here holds also for the other geometries analysed. Further, we introduce the rate of change of G, the energy release rate, defined by:

xðGÞ Z

ðKÞða=tÞCDða=tÞ KðKÞða=tÞ 2 DG DK Z Km Z Km E Dða=tÞ Dða=tÞ Dða=tÞ (12)

where Km is the mean value of K for two cracks of relative depths a/t and a/tCD(a/t). DG/D(a/t) is the increase of G corresponding to an increase of the crack relative depth by an

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Fig. 10. x(G) values for tensile stress (t/RiZ1/10): (a) at the deepest point; (b) at the surface point.

Fig. 11. x(G) values in bending (t/RiZ1/10): (a) at the deepest point; (b) at the surface point.

amount equal to D(a/t). x(G) provides information about the ability of the crack to develop. Fig. 10(a) shows that, for pure tensile stress, x(G) values obtained for the thickness transition at the deepest point, are higher than those for the uniform thickness cylinder for very small depths. They become lower than those for the uniform thickness cylinder once the crack depth rises slightly. At the surface point x(G) is always greater in the thickness transition (Fig. 10(b)). This is easily comprehended when we take into account the stress concentration effect explained above. Value of x(G) suggest that, compared to a defect in the uniform thickness cylinder, a shallow crack in the thickness transition, would first grow quickly at the deepest point, then slow down, while it would continue quicker growth at the surface point. Hence, during propagation, crack shape in the transition becomes more elongated compared to the same initial defect in the uniform thickness cylinder. This is also true for bending loads (Fig. 11(a) and (b)), and this tendency is the same whatever the values of t/Ri and a/t. We may conclude that cracks in the thickness transition present a greater risk than similar cracks in a uniform thickness cylinder. Indeed, if at the deepest point the behaviour of the two defects is almost the same, the crack in the thickness transition will propagate more quickly leaving a smaller resistant section, and other factors may accelerate the breaking of the tube with a thickness transition before the uniform thickness transition.

6. Conclusions In this work, we analysed elliptical cracks in a cylinder with a thickness transition. Two points were explored: (a) the calculation of the stress intensity factor using SIF parameters of a similar crack located in a cylinder of uniform thickness, and (b) a comparison, from the point of view of harmfulness, of this crack with a similar defect in a uniform thickness cylinder. We considered two types of loading, pure tension and bending. For the calculation of K, we concluded that the use of the elastic stress distribution in the cylinder with a thickness transition combined with the SIF parameters of the uniform thickness cylinder gives conservative results in pure tension. The precision at the deepest point is however relatively low. In bending, the use of the SIF value of the uniform thickness cylinder to characterize the elliptical crack at the thickness transition is not adequate, and specific calculations are necessary for the thickness transition. On the other hand, the analysis showed that cracks in the thickness transition present a greater risk of propagation at the surface of the cylinder but this risk is almost the same at the deepest point. This study extends the conclusions in preceding work in the literature on axisymmetric cracks in thickness transition zones. It considers bending loading, which were not treated before.

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