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FE validation of R6 elastic–plastic J estimation for circumferentially cracked pipes under mechanical and thermal loadings
Engineering Fracture Mechanics xxx (2014) xxx–xxx
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
FE validation of R6 elastic–plastic J estimation for circumferentially cracked pipes under mechanical and thermal loadings Chang-Young Oh a, Hyun-Suk Nam a, Yun-Jae Kim a,⇑, Robert A. Ainsworth b, Peter J. Budden c a
Korea University, Department of Mechanical Engineering, Anam-Dong, Sungbuk-Ku, Seoul 136-701, Republic of Korea The University of Manchester, Manchester M13 9PL, UK c Assessment Technology Group, EDF Energy, Barnwood, Gloucester GL4 3RS, UK b
a r t i c l e
i n f o
Article history: Received 21 October 2013 Received in revised form 12 March 2014 Accepted 18 March 2014 Available online xxxx Keywords: Combined mechanical and thermal loads Elastic–plastic J FE analysis Loading sequence effect V-factor
a b s t r a c t Objective: This paper provides validation of R6 elastic-plastic J-estimation methods, via elastic-plastic finite element analyses for circumferentially cracked pipes under combined mechanical and thermal loads. Materials method: In the FE analyses, the relative magnitudes of the mechanical and thermal loads, types of thermal loads, the crack sizes and material hardening are systematically varied. Loading sequence effects for the mechanical and thermal loads are also considered. The FE elastic-plastic J values are compared with those derived using R6 elastic-plastic J-estimation methods. Results: It is shown that there are significant load order effects on elastic-plastic J for large secondary stress cases, but these are conservatively estimated using newly proposed methods in R6. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Quantification of secondary stress effects on elastic–plastic fracture is important for the assessment of crack-like defects. Based on initial work by Ainsworth [1,2], practical approaches for component assessment under combined primary and secondary stresses have been incorporated in the R6 procedure [3] and other codes [4,5], where a parameter V [6–8] was introduced to accommodate the interaction between the primary and secondary loads. Within R6 [3], several procedures to estimate V are given and has been validated against numerous finite element (FE) and experimental data. Some experimental validation results can be found in Chapter V: Validation and worked examples of R6. Validation results suggest that existing procedures are overall conservative but can be unduly conservative. Recently, some attempts have been made to reduce over-conservatism embedded in the current V-factor approach using FE data [9–11]. For instance, James et al. [11] performed a series of FE analyses and indeed showed that current V-factor estimations in R6 can be quite conservative. The present authors also obtained a large number of detailed FE solutions for elastic–plastic J for circumferentially cracked pipes under combined mechanical and thermal loads [9,10]. They also showed that the current V-factor estimation in R6 can be quite conservative in general, but found that it can be non-conservative for cases of large elastic follow-up [12]. Based on FE results, the authors proposed a new approximation for the V-factor. Motivated by FE results, Ainsworth [13] made a new proposal based on an analytical approach by implementing relaxation equations to allow for reduction of the secondary loading due to plastic straining. ⇑ Corresponding author. Tel.: +82 2 3290 3372; fax: +82 2 926 9290. E-mail address: [email protected] (Y.-J. Kim). http://dx.doi.org/10.1016/j.engfracmech.2014.03.015 0013-7944/Ó 2014 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Oh C-Y et al. FE validation of R6 elastic–plastic J estimation for circumferentially cracked pipes under mechanical and thermal loadings. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.03.015
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Nomenclature a E E0 F, N, P FL f(Lr) J,Je JFE K Kmat Kr Lr NOR, POR r t V Vo a, n b DT v e, eref r, ry, rref ra, rh, rr h
w n
crack depth Young’s modulus E/(1m2) for plane strain; E for plane stress generalised load, axial tension, internal pressure limit load failure assessment curve J-integral, elastic value of J J from FE analysis linear elastic stress intensity factor fracture toughness proximity parameter for fracture in R6 proximity parameter for plastic collapse in R6 optimised reference axial tension and internal pressure mean pipe radius pipe thickness multiplying factor to incorporate the secondary stress effect on elastic–plastic J value of V for the secondary stress only coefficient and strain hardening index in the Ramberg–Osgood model parameter associated with the relative magnitude of the secondary stress temperature difference Poisson’s ratio strain and reference strain stress, yield (0.2% proof) strength and reference stress axial, hoop and radial stress components half circumferential angle of a circumferential crack biaxiality parameter R6 detailed estimate of V/V0
Superscripts p referring to the primary stress s referring to the secondary stress p+s referring to combined primary and secondary stresses
Fig. 1. (a) Pipes under combined pressure P and axial tension (or compression) N, and (b) circumferential part-through surface cracks. Directions for contour integral calculations are also shown.
Please cite this article in press as: Oh C-Y et al. FE validation of R6 elastic–plastic J estimation for circumferentially cracked pipes under mechanical and thermal loadings. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.03.015
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However, the FE data in Refs. [9–12] have been obtained for the case of an increasing primary (mechanical) load applied after an initial secondary (thermal) load. In actual plant conditions, however, thermal loads can be applied after an initial mechanical load, such as in thermal shock conditions. As the loading sequence of primary and secondary loads can affect elastic–plastic J, it is important to validate existing elastic–plastic J estimation procedures for combined primary and secondary stresses against FE data generated for various loading sequences, which is subject of this paper. In this paper, variations of elastic–plastic J with load magnitude under combined mechanical and thermal loading are assessed by FE analyses. Relative mechanical/thermal load magnitudes, crack sizes and material hardening are systematically varied. The loading sequence effect of mechanical and thermal loading is also investigated. The FE results are presented on the R6 failure assessment diagram by calculating the two parameters Kr and Lr using various methods for estimating the interaction parameter V. Section 2 briefly summarises the treatment of combined primary and secondary stresses in fracture assessment. Section 3 describes the FE analysis performed in this paper. Results are presented in Section 4. The presented work is concluded in Section 5. 2. R6 fracture assessment for combined primary and secondary stresses Within R6, fracture assessment is performed using the failure assessment diagram where two parameters, Kr and Lr, are calculated and failure is conceded when Kr = f(Lr), where f(Lr) is the failure assessment curve. A factor V is used to account for interactions between primary and secondary stress due to crack tip plasticity by defining the Kr parameter at the assessed load condition as:
Kr ¼
ðK p þ VK s Þ K mat
ð1Þ
t
x a
r
ΔT
(a)
(b) ΔT
(c) Fig. 2. Three types of thermal loading: (a) radial temperature gradient, (b) axial temperature gradient and (c) sectional temperature gradient.
Please cite this article in press as: Oh C-Y et al. FE validation of R6 elastic–plastic J estimation for circumferentially cracked pipes under mechanical and thermal loadings. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.03.015
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Fig. 3. Through-thickness uncracked cylinder variations of thermal stresses, resulting from elastic FE analysis: (a) axial temperature gradient, (b) radial temperature gradient and (c) sectional temperature gradient. Axial, radial and hoop stress components are denoted as ra, rr and rh, respectively.
Fig. 4. Typical FE mesh for a part-through surface crack.
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where Kp and Ks are the stress intensity factor for primary and secondary stresses, respectively, and Kmat is the fracture toughness. Eq. (1) is equivalent to an estimate of J for the combined loading given by 2
J¼
ðK p þ VK s Þ 2 f ðLr Þ E0
ð2Þ
0
where E = E/(1 m2) with E being Young’s modulus and m Poisson’s ratio. The failure assessment curve function f(Lr) depends on the proximity parameter for plastic collapse, Lr, defined at the assessed load condition by
Fig. 5. Variations of Kr with Lr for the axial temperature gradient (hp = 0.125, a/t = 0.3, n = 5) under three different loading sequences: (a and c) b = 0.5 and (d and f) b = 5.0.
Please cite this article in press as: Oh C-Y et al. FE validation of R6 elastic–plastic J estimation for circumferentially cracked pipes under mechanical and thermal loadings. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.03.015
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C.-Y. Oh et al. / Engineering Fracture Mechanics xxx (2014) xxx–xxx p
Lr ¼
rref F ¼ F L ða; ry Þ ry
ð3Þ
where F is the applied primary load and FL is the corresponding plastic limit load for the structure with a crack of size a for a rigid plastic material of yield stress, ry. Note that the reference stress for the primary loads, rpref , follows from Eq. (3) as rpref ¼ Lr ry . There are different Options in R6 and other codes for the failure assessment curve f(Lr). The R6 Option 1 curve is a function of Lr only and defined by
1=2 h i f ðLr Þ ¼ 1 þ 0:5L2r 0:3 þ 0:7 exp 0:6L6r
ð4Þ
The Option 2 curve depends on material and is defined by
f ðLr Þ ¼
" p Eeref
rpref
p
1 rref þ L2r p 2 Eeref
#1=2 ð5Þ
where epref is the strain on the stress–strain curve at the stress level rpref . Various options of estimating the V factor to be used in Eqs. (1) and (2) are also given in R6. In a simplified procedure, V is assumed to be a function of b and Lr, and given by
8 > < 1 þ 0:2Lr þ 0:02bð1 þ 2Lr Þ for Lr < Lr V ¼ 3:1 2Lr for Lr < Lr < 1:05 > : 1 for Lr > 1:05
ð6Þ
Fig. 6. Variations of Kr with Lr for the axial temperature gradient (hp = 0.125, a/t = 0.3, n = 5) under two extreme loading sequences: (a and b) b = 0.5 and (c and d) b = 5.0.
Please cite this article in press as: Oh C-Y et al. FE validation of R6 elastic–plastic J estimation for circumferentially cracked pipes under mechanical and thermal loadings. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.03.015
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where
b¼
Ks K p =Lr
ð7Þ
is a normalised secondary stress and Lr is defined in terms of b by equating the first two expressions in Eq. (6). It should be noted that, in the simplified procedure, V P 1 so that no benefit is taken for any plastic relaxation of secondary stresses, and its application is limited to b < 4. A more generally applicable method is the detailed procedure in R6, where it is necessary to first calculate the factor Vo, which is the value of V for secondary loading acting alone. Among a number of methods to calculate Vo suggested in R6, the most accurate one is to calculate Vo from Eq. (8) using an inelastic analysis of the cracked body.
Vo ¼
pffiffiffiffiffiffiffi E0 J s Ks
ð8Þ
where Js denotes the elastic–plastic J due to thermal loading only. Once Vo has been calculated, the value of V is obtained from
V ¼ nV o
ð9Þ
where n is a function of b and Lr, and is tabulated in R6 [3]. At low Lr, V is often close to unity indicating elastic response; at intermediate Lr (Lr < 1), V > 1, reflecting interaction between primary and secondary stress; at high Lr (Lr > 1), V < 1, indicating mechanical stress relief of the secondary stresses. Recently a new equation for evaluation of V based on a relaxation equation has been proposed by Ainsworth [13]:
V ¼ f ðLr Þ þ 0:42Lr ð0:72 þ Lr Þ½f ðLr Þ2 Vo
ð10Þ
Eq. (10) is considered to be an upper bound to V for typical plastic relaxation with limited elastic follow-up. However, Eq. (10) can be still non-conservative when large elastic follow-up is present [13].
Fig. 7. Variations of Kr with Lr for the radial temperature gradient (hp = 0.125, a/t = 0.3, n = 5) under two extreme loading sequences: (a and b) b = 0.5 and (c and d) b = 5.0.
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3. Finite element analysis 3.1. Geometry Consider a pipe with the mean radius, r, and thickness, t. Only one value of pipe geometry, r/t = 10, is considered. The half length of pipe was chosen to be ten times the pipe radius to avoid the end effect. Circumferential partthrough surface cracks are considered, as depicted in Fig. 1, characterised by the relative crack length, hp (where h denotes the half crack angle), and the relative crack depth, a/t. The surface crack is taken to have a rectangular shape, i.e., constant depth. Two values of hp are considered, hp = 0.125 and 1.0, with two relative crack depths, a/t = 0.3 and 0.5. 3.2. Mechanical and thermal loadings For the mechanical (primary) loadings, axial tension, pressure, and combined tension and pressure were applied as shown in Fig. 1 to generate various bi-axial stress states. To generate secondary stresses, three different types of thermal loads were applied. The first type has a temperature gradient through the pipe thickness (Fig. 2a, ‘‘radial temperature gradient’’). The second type has a gradient along the longitudinal direction (Fig. 2b, ‘‘axial temperature gradient’’). The third type has a temperature gradient across the cross-section, producing global bending stresses (Fig. 2c, ‘‘sectional temperature gradient’’). The magnitude of the temperature difference, DT, was adjusted to produce different levels of thermal stress. These three types of thermal gradient produce various bi-axial stress states, as shown in Fig. 3 where through-thickness variations of thermal stresses in the uncracked cylinder at the mid-plane, from elastic FE analyses, are plotted.
Fig. 8. Variations of Kr with Lr for the axial temperature gradient (hp = 1.0, a/t = 0.3, n = 5) under two extreme loading sequences: (a and b) b = 0.5 and (c and d) b = 5.0.
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3.3. Material For elastic analyses, an isotropic material was assumed with Young’s modulus E = 200 GPa and Poisson’s ratio m = 0.3. For elastic–plastic analyses, the material was assumed to follow the Ramberg–Osgood (R–O) relationship:
r r r e¼ þa y ry E E
n ð11Þ
where a and n are R–O parameters. The value of ry was chosen as ry = 300 MPa, but this choice does not affect the main results of this work, as these are presented in normalised form. To be consistent with the definition of ry as the 0.2% proof strength, a was chosen as
a¼
0:002E
ry
4 ¼ 3
ð12Þ
Finally to quantify the effect of the strain hardening index, two different values of n were considered; n = 5 and 10. 3.4. FE analysis A series of elastic and elastic–plastic FE analyses of circumferentially cracked pipes was performed using ABAQUS [14]. For axi-symmetric cases (hp = 1.0), eight-node reduced integration elements (CAX8R) were used. For efficient computations of three-dimensional cases, a quarter model was used allowing for symmetry. Twenty-node iso-parametric quadratic brick elements with reduced integration (C3D20R within ABAQUS) were used. Fig. 4 depicts a typical FE mesh for a surface cracked pipe. The crack-tip was designed with collapsed elements, and a ring of wedge-shaped elements was used in the crack-tip region. A total of eleven or twelve elements were used through the thickness. The numbers of elements and nodes in the FE
Fig. 9. Effect of strain hardening (n = 10) on variations of Kr with Lr for the axial temperature gradient (hp = 0.125, a/t = 0.3) under two extreme loading sequences: (a)–(b) b = 0.5 and (c)–(d) b = 5.0.
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meshes were about 3000 and 15,000, respectively. Note that, for modelling the part-through crack, a small rounded corner was introduced, which does not affect J calculations at the centre of the crack. The deformation plasticity option in ABAQUS was adopted in the FE analyses. Elastic FE analyses were performed first to determine the relative magnitude of the thermal stresses, which is quantified by Eq. (7). A value b = 1 corresponds to a secondary stress which produces a stress intensity factor equal to that for a primary load for which the primary load is equal to the limit load (or Lr = 1). Based on elastic FE analysis, the magnitude of the thermal loading was chosen to give two different values of b, b = 0.5 and 5.0. Note that the value b = 5 represents a very large thermal stress: a primary load equal to five times the collapse load (Lr = 5) would be required to generate a stress intensity factor equal to that produced by the secondary stress. For each magnitude of the thermal loading (each value of b), three sets of elastic–plastic FE analyses were performed; (i) mechanical loading only, (ii) thermal loading only and (iii) combined mechanical and thermal loading. To evaluate the loading sequence effect, FE analyses were performed where the sequence of application of the mechanical and thermal loadings was varied. Three different sequences were considered: (i) thermal loading followed by mechanical loading (where the thermal loading simulates residual stress, for instance); (ii) mechanical and thermal loading applied together in a proportional manner (simulating start-up conditions, for example); and (iii) mechanical loading followed by thermal loading (simulating pressurised thermal shock conditions, for instance). For thermal shock conditions, the value of b can be up to five or more. Values of the J-integral were extracted as a function of the applied load from the FE results using the domain integral method. The FE J values, JFE, were taken in the centre of the crack along the crack growth direction, as depicted in Fig. 1. Numerical J estimates using ABAQUS are generally known to be robust for monotonic loading, and the present meshes, as shown in Fig. 4, are judged to be sufficiently fine to produce accurate results. It is also found that the numerical J values are almost path-independent over six contours around the crack tip for the results presented in this paper. Averaged values of J (over six contours) are reported. Using the FE results, two parameters Kr and Lr for the failure assessment diagram are calculated. The finite element analyses results are used to evaluate Eq. (13):
Fig. 10. Variations of Kr with Lr for the radial temperature gradient (hp = 0.125, a/t = 0.3, n = 5) under two extreme loading sequences of pressure and thermal loading: (a)–(b) b = 0.5 and (c)–(d) b = 5.0.
Please cite this article in press as: Oh C-Y et al. FE validation of R6 elastic–plastic J estimation for circumferentially cracked pipes under mechanical and thermal loadings. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.03.015
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K p þ VK s K r ¼ qffiffiffiffiffiffiffiffiffiffiffiffi E0 J pþs FE
11
ð13Þ
with the detailed estimate of V from Eqs. (9) and (10). Values of Kp, Ks and JFE were taken from the FE results. Note that Eq. (13) represents the Option 3 failure assessment curve obtained using FE elastic–plastic J within R6 [3]. The curve is specific to a material and geometry, and is supposed to be more accurate than the Option 2 curve. The J-estimation approaches in R6 are then conservative if Kr from Eq. (13) lies outside the appropriate failure assessment curve, Kr = f(Lr). The R6 simplified estimates of Eq. (6) for estimating V are not assessed in this paper, as generalised comparisons with other methods have been given in Ref. [9] and they are expected to be very conservative at high values of Lr. 4. Results 4.1. Effects of b, a/t and hp Fig. 5 shows failure assessment diagrams for axial temperature gradients with b = 0.5 and b = 5.0 for the case n = 3, a/t = 0.3 and h/p = 0.125. Note that, in Fig. 5 as well as in subsequent figures, the optimised reference load (NOR for axial tension and POR for internal pressure) is used instead of the limit load to define Lr in Eq. (3). This is to remove conservatism in the estimation of J due to primary stress alone and thus to investigate conservatism solely due to the V-factor. More detailed explanation on the optimised reference load is given in our previous paper (see Eq. (18), Section 4.2 and Table 2 in Ref. [9]). Results are shown in Fig. 5 for three different loading sequences. In applying Eq. (13) to evaluate Kr from the FE analyses, V has been estimated in three ways. Firstly, the existing R6 detailed approach of Eq. (9) has been applied leading to the open squares. The second and third methods for estimating V have used Eq. (10) with f(Lr) defined by Eqs. (4) and (5) leading to the solid circles and the open triangles, respectively. The solid circles, with V estimated using the Option 1 failure
Fig. 11. Variations of Kr with Lr for the radial temperature gradient (hp = 0.125, a/t = 0.3) under combined tension and internal pressure: (a)–(b) b = 0.5 and (c)–(d) b = 5.0.
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assessment function, should be compared with the Option 1 failure assessment curve (solid line) whereas the open triangles should be compared with the Option 2 curve (dashed line). If the point lies (further) outside the failure assessment curve, the approach is (more) conservative. When secondary stresses are small, Fig. 5a–c, there is no significant loading sequence effect. However for large secondary stresses, b = 5.0, Fig. 5d–f, loading sequence effects are more significant and Fig. 5f shows values of Kr lower than those in Fig. 5d, indicating a higher value of J. The results in Fig. 5 suggest that the sequence of mechanical loading followed by thermal loading is the most severe loading case. The least severe loading sequence is the opposite one, thermal loading followed by mechanical loading. When the two loadings are applied proportionally, the J values are in between those from the above two loading sequences. The points for the existing R6 approach of Eq. (9) generally lie outside the failure assessment curve indicating conservatism in the approach. Also the results are less conservative if used with an Option 2 failure assessment curve. However, Fig. 5f shows that there is a small non-conservatism in the current R6 detailed approach if used with an Option 2 curve. In this respect, it is noted that the tabulated n values in R6 [3] were calculated using an Option 1 curve but are not particularly sensitive to the choice of f(Lr). The points using Eq. (10) (both sets), on the other hand, lie outside the corresponding failure assessment curves and are always conservative. Fig. 6 shows the results for a deeper crack depth (a/t = 0.5) with n = 5 and h/p = 0.125. Corresponding results for the radial thermal gradient with a/t = 0.3 are shown in Fig. 7. Fig. 8 finally shows the results for the axial temperature gradient but for a greater crack length (hp = 1.0) with a/t = 0.3 and n = 5. Only the two loading sequences producing the most and least severe cases are shown in Figs. 6–8 for clarity. Similar trends to those in Fig. 5 can be seen. 4.2. Effects of strain hardening In Fig. 9, the influence of strain hardening is demonstrated by taking a strain hardening index n = 10 as opposed to n = 5 in the earlier figures, again with a/t = 0.3 and h/p = 0.125. For small secondary loads (b = 0.5), Fig. 9a and b, all methods are
Fig. 12. Variations of Kr with Lr for the axial or radial temperature gradient (hp = 0.125, a/t = 0.3, n = 5) with large b(=10) under two extreme loading sequences: (a)–(b) for a fixed mechanical load magnitude, and (c)–(d) for a fixed thermal load magnitude.
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highly accurate. For high secondary loads (b = 5), Fig. 9c and d, the methods are significantly conservative for the mechanical load applied after the thermal load, Fig. 9c, but this conservatism is required to accurately predict the case of thermal load applied after mechanical loading, Fig. 9d. Another point is that the use of Eq. (10) is more accurate for the lower hardening material. 4.3. Bi-axial stress effects The results in the previous two sub-sections were for axial tension, producing uni-axial mechanical stress states in the cylinder. In this sub-section, a bi-axial mechanical stress effect is investigated by applying internal pressure or combined pressure and tension. The stress bi-axiality is defined in terms of the hoop and axial stresses, rh and ra, of an un-cracked body using a non-dimensional variable, w
w¼
rh 2ra
ð14Þ
For axial tension, w = 0 and for internal pressure, w = 1, based on the thin cylinder approximation. Results for internal pressure (w = 1.0) are shown in Fig. 10, and those for combined pressure and axial tension (w = 0.5) in Fig. 11, in both cases for n = 5, a/t = 0.3 and h/p = 0.125. It can be seen that even under bi-axial stress states, similar trends are observed to the earlier cases. 4.4. Results for large b cases There can be some practical cases when the b value from Eq. (7) is quite large. Two types of calculations were performed to produce b values close to 10. Firstly, a very large secondary stress Ks was increased for a fixed Kp/Lr by increasing the
Fig. 13. Variations of Kr with Lr for the sectional temperature gradient (hp = 0.125, a/t = 0.3) with small b(=0.5): (a)–(b) n = 5 and (c)–(d) n = 10.
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temperature difference DT. Secondly, Ks was fixed and Kp/Lr was decreased by applying compressive axial force (by increasing w to w = 3.0, see Eq. (14), by applying combined pressure and axial compression leading to a small net positive axial stress). In Fig. 12, the cases with large b (b = 10) are illustrated for n = 5 with a/t = 0.3 and h/p = 0.125 and so can be compared with the corresponding cases with smaller b in Fig. 5. The R6 detailed evaluations are not shown, as they cannot be applied for b > 5 due to the limits in the tabulated values of n in R6. The results in Fig. 12a and b are obtained for a fixed Kp/Lr (under axial tension) by increasing the temperature difference DT. When compared with the results in Fig. 5, it can be seen that Eq. (10) is more accurate for large b. In particular, the methods are accurate for the thermal load applied after the mechanical load. The results in Fig. 12c and d are obtained for the fixed Ks by decreasing Kp/Lr. It can be seen that the methods tend to be more conservative, compared to the cases in Fig. 12a and b. 4.5. Large elastic follow-up case In our previous work [12], it was shown that existing methods can be non-conservative when large elastic follow-up behaviour is present. Large elastic follow-up can occur when small secondary stresses producing a region of uniform secondary stress are present under contained yielding for low strain hardening materials. As shown in Fig. 3, the sectional temperature gradient produces uniform secondary stress through the thickness. Fig. 13 shows the results for sectional temperature gradients with b = 0.5, a/t = 0.3 and h/p = 0.125. The results for n = 5 are shown in Fig. 13a and b, whereas those for n = 10 are plotted in Fig. 13c and d. Due to the small secondary stresses (b = 0.5), loading sequence effects are not so significant. For n = 5, slight non-conservatism is observed at small Lr. For n = 10, the degree of non-conservatism increases. This suggests that caution in the application of Eq. (10) is needed when assessing a crack in low hardening materials under small secondary stresses that produce a uniform stress through the thickness. In [12], methods for allowing for such effects through an explicit dependence on an elastic follow-up factor are proposed, but these are not investigated in this paper.
Fig. 14. Effect of the use of a conservative reference stress on variations of Kr with Lr for an axial temperature gradient applied after mechanical loading: (a)–(b) n = 5 and (c)–(d) n = 10.
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4.6. Effect of conservative estimate of reference stress In presenting results in the previous sub-sections, the optimised reference load was used instead of the limit load to define Lr in Eq. (3), as explained in Section 4.1. The reason was to remove conservatism in estimates of J under primary stress alone and thus to investigate any conservatism solely due to the V-factor. In practical assessments, a lower reference stress (or Lr) is often used to ensure conservatism in estimating J. Fig. 14 shows the effect of the use of a conservative reference stress on variations of Kr with Lr for an axial temperature gradient. For the loading sequence, only the most severe case (thermal loading after mechanical loading) is considered for two different hardening exponents, n = 5 and 10; in Fig. 14a and b for n = 5 and in Fig. 14(c–d) for n = 10. For each case, a/t = 0.3 and h/p = 0.125 is assumed. In calculating Lr, the reference load is assumed to be ten percent smaller (and thus the reference stress is roughly ten percent higher). The reason to choose the 10% is rather arbitrary, but previous studies showed that the limit load in R6 for circumferential surface cracked pipes R6 reference stress is within 10% of the optimised reference load producing accurate J [9,10]. Due to the change in Lr, the b values are also changed slightly, as indicated in the figures. When compared with the results in Figs. 5 and 9, it can be seen that the points tend to move further from the corresponding failure assessment lines, suggesting that the assessment is more conservative. The assessment results are also more conservative for lower hardening materials. 5. Conclusions Variations of elastic–plastic J with load magnitude have been evaluated by finite element analyses for circumferentially surface cracked pipes under combined mechanical and thermal loads. For the mechanical loading, axial tension, pressure and combined tension and pressure have been considered. For the thermal loading, three different types of thermal gradient have been considered. Relative mechanical/thermal load magnitudes, crack sizes and material hardening have all been systematically varied. The loading sequence effect of mechanical and thermal loading have also been investigated. The FE results have been presented on the R6 failure assessment diagram by calculating the two parameters Kr and Lr using various methods for estimating the interaction parameter V. The V factor has been estimated using the current detailed R6 method and recent proposals dependent on the shape of the failure assessment curve. From the presented results, the following conclusions can be drawn. Significant loading sequence effects have been observed at large secondary stresses. The case when primary loading is applied before secondary loading, such as for pressurised thermal shock loading, is found to be a more severe loading case than that of secondary loading applied before primary loading. The current detailed R6 method is generally conservative and the conservatism increases with increasing relative magnitude of the thermal stresses. However, it can be slightly non-conservative if used with the Option 2 curve for high hardening materials for the case when a large thermal stress is applied after primary loading. Recent proposals for estimating V, on the other hand, are conservative even in such load cases. The recent proposals are shown to be more accurate for the case when the relative magnitude of the thermal stress is quite high. However, when large elastic follow-up is present, all methods are non-conservative in a small region at low values of Lr (<1.0). Acknowledgement This research is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (Nos. 2007-0056094, 2013M2B2A9A03051295, 2012M2A7A1051939). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
Ainsworth RA. The treatment of thermal and residual stresses in fracture assessments. Engng Fract Mech 1986;24:65–76. Ainsworth RA. The assessment of defects in structures of strain hardening material. Engng Fract Mech 1984;19:633–42. R6, Revision 4: Assessment of the integrity of structures containing defects. Gloucester, UK: EDF Energy Nuclear Generation Ltd.; 2013. BS7910: Guide on methods for assessing the acceptability of flaws in metallic structures. London: British standards Institution; 2000. API 579-1/ASME FFS-1. Fitness-for-service for pressure vessels, piping and storage tanks, 2007. American Petroleum Institute/American Society of Mechanical Engineers. Ainsworth RA, Smith SD, Wiesner CS. Treatment of thermal and residual stresses in defect assessment. British Energy Generation Limited Report EPD/ GEN/REP/0423/99, Gloucester; 1999. Ainsworth RA, Sharples JK, Smith SD. Effects of residual stresses on fracture behaviour – experimental results and assessment methods. J Strain Anal 2000;35:307–16. Stacey A, Barthelemy JY, Leggatt RH, Ainsworth RA. Incorporation of residual stresses into the SINTAP defect assessment procedure. Engng Fract Mech 2000;67:573–612. Song T-K, Kim Y-J, Nikbin K, Ainsworth RA. Approximate J estimates for circumferential cracked pipes under primary and secondary stresses. Engng Fract Mech 2009;76:2109–25. Oh C-Y, Kim Y-J, Budden PJ, Ainsworth RA. Biaxial stress effects on estimating J under combined mechanical and thermal stresses. Int J Pres Ves Piping 2011;88:365–74. James PM, Hooton DG, Higham LA, Madew CJ, Sharples JK, Watson CT. Continuing development of a simplified method to account for the interaction of primary and secondary Stresses. In: Proceedings of the 2008 ASME Pressure Vessels and Piping Conference, Paper PVP2008-61040; 2008.
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[12] Song T-K, Oh C-Y, Kim Y-J, Ainsworth RA, Nikbin K. Approximate J estimates for combined primary and secondary stresses with large elastic follow-up. Int J Press Vess Piping (electronic version available, see). [13] Ainsworth RA. Consideration of elastic follow-up in the treatment of combined primary and secondary stresses in fracture assessments. Engng Fract Mech 2012;96:558–69. [14] ABAQUS, ABAQUS Standard/User’s manual, Version 6.11.1. Dassaults Systemes Inc; 2011.
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Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
FE validation of R6 elastic–plastic J estimation for circumferentially cracked pipes under mechanical and thermal loadings Chang-Young Oh a, Hyun-Suk Nam a, Yun-Jae Kim a,⇑, Robert A. Ainsworth b, Peter J. Budden c a
Korea University, Department of Mechanical Engineering, Anam-Dong, Sungbuk-Ku, Seoul 136-701, Republic of Korea The University of Manchester, Manchester M13 9PL, UK c Assessment Technology Group, EDF Energy, Barnwood, Gloucester GL4 3RS, UK b
a r t i c l e
i n f o
Article history: Received 21 October 2013 Received in revised form 12 March 2014 Accepted 18 March 2014 Available online xxxx Keywords: Combined mechanical and thermal loads Elastic–plastic J FE analysis Loading sequence effect V-factor
a b s t r a c t Objective: This paper provides validation of R6 elastic-plastic J-estimation methods, via elastic-plastic finite element analyses for circumferentially cracked pipes under combined mechanical and thermal loads. Materials method: In the FE analyses, the relative magnitudes of the mechanical and thermal loads, types of thermal loads, the crack sizes and material hardening are systematically varied. Loading sequence effects for the mechanical and thermal loads are also considered. The FE elastic-plastic J values are compared with those derived using R6 elastic-plastic J-estimation methods. Results: It is shown that there are significant load order effects on elastic-plastic J for large secondary stress cases, but these are conservatively estimated using newly proposed methods in R6. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Quantification of secondary stress effects on elastic–plastic fracture is important for the assessment of crack-like defects. Based on initial work by Ainsworth [1,2], practical approaches for component assessment under combined primary and secondary stresses have been incorporated in the R6 procedure [3] and other codes [4,5], where a parameter V [6–8] was introduced to accommodate the interaction between the primary and secondary loads. Within R6 [3], several procedures to estimate V are given and has been validated against numerous finite element (FE) and experimental data. Some experimental validation results can be found in Chapter V: Validation and worked examples of R6. Validation results suggest that existing procedures are overall conservative but can be unduly conservative. Recently, some attempts have been made to reduce over-conservatism embedded in the current V-factor approach using FE data [9–11]. For instance, James et al. [11] performed a series of FE analyses and indeed showed that current V-factor estimations in R6 can be quite conservative. The present authors also obtained a large number of detailed FE solutions for elastic–plastic J for circumferentially cracked pipes under combined mechanical and thermal loads [9,10]. They also showed that the current V-factor estimation in R6 can be quite conservative in general, but found that it can be non-conservative for cases of large elastic follow-up [12]. Based on FE results, the authors proposed a new approximation for the V-factor. Motivated by FE results, Ainsworth [13] made a new proposal based on an analytical approach by implementing relaxation equations to allow for reduction of the secondary loading due to plastic straining. ⇑ Corresponding author. Tel.: +82 2 3290 3372; fax: +82 2 926 9290. E-mail address: [email protected] (Y.-J. Kim). http://dx.doi.org/10.1016/j.engfracmech.2014.03.015 0013-7944/Ó 2014 Elsevier Ltd. All rights reserved.
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Nomenclature a E E0 F, N, P FL f(Lr) J,Je JFE K Kmat Kr Lr NOR, POR r t V Vo a, n b DT v e, eref r, ry, rref ra, rh, rr h
w n
crack depth Young’s modulus E/(1m2) for plane strain; E for plane stress generalised load, axial tension, internal pressure limit load failure assessment curve J-integral, elastic value of J J from FE analysis linear elastic stress intensity factor fracture toughness proximity parameter for fracture in R6 proximity parameter for plastic collapse in R6 optimised reference axial tension and internal pressure mean pipe radius pipe thickness multiplying factor to incorporate the secondary stress effect on elastic–plastic J value of V for the secondary stress only coefficient and strain hardening index in the Ramberg–Osgood model parameter associated with the relative magnitude of the secondary stress temperature difference Poisson’s ratio strain and reference strain stress, yield (0.2% proof) strength and reference stress axial, hoop and radial stress components half circumferential angle of a circumferential crack biaxiality parameter R6 detailed estimate of V/V0
Superscripts p referring to the primary stress s referring to the secondary stress p+s referring to combined primary and secondary stresses
Fig. 1. (a) Pipes under combined pressure P and axial tension (or compression) N, and (b) circumferential part-through surface cracks. Directions for contour integral calculations are also shown.
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However, the FE data in Refs. [9–12] have been obtained for the case of an increasing primary (mechanical) load applied after an initial secondary (thermal) load. In actual plant conditions, however, thermal loads can be applied after an initial mechanical load, such as in thermal shock conditions. As the loading sequence of primary and secondary loads can affect elastic–plastic J, it is important to validate existing elastic–plastic J estimation procedures for combined primary and secondary stresses against FE data generated for various loading sequences, which is subject of this paper. In this paper, variations of elastic–plastic J with load magnitude under combined mechanical and thermal loading are assessed by FE analyses. Relative mechanical/thermal load magnitudes, crack sizes and material hardening are systematically varied. The loading sequence effect of mechanical and thermal loading is also investigated. The FE results are presented on the R6 failure assessment diagram by calculating the two parameters Kr and Lr using various methods for estimating the interaction parameter V. Section 2 briefly summarises the treatment of combined primary and secondary stresses in fracture assessment. Section 3 describes the FE analysis performed in this paper. Results are presented in Section 4. The presented work is concluded in Section 5. 2. R6 fracture assessment for combined primary and secondary stresses Within R6, fracture assessment is performed using the failure assessment diagram where two parameters, Kr and Lr, are calculated and failure is conceded when Kr = f(Lr), where f(Lr) is the failure assessment curve. A factor V is used to account for interactions between primary and secondary stress due to crack tip plasticity by defining the Kr parameter at the assessed load condition as:
Kr ¼
ðK p þ VK s Þ K mat
ð1Þ
t
x a
r
ΔT
(a)
(b) ΔT
(c) Fig. 2. Three types of thermal loading: (a) radial temperature gradient, (b) axial temperature gradient and (c) sectional temperature gradient.
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Fig. 3. Through-thickness uncracked cylinder variations of thermal stresses, resulting from elastic FE analysis: (a) axial temperature gradient, (b) radial temperature gradient and (c) sectional temperature gradient. Axial, radial and hoop stress components are denoted as ra, rr and rh, respectively.
Fig. 4. Typical FE mesh for a part-through surface crack.
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where Kp and Ks are the stress intensity factor for primary and secondary stresses, respectively, and Kmat is the fracture toughness. Eq. (1) is equivalent to an estimate of J for the combined loading given by 2
J¼
ðK p þ VK s Þ 2 f ðLr Þ E0
ð2Þ
0
where E = E/(1 m2) with E being Young’s modulus and m Poisson’s ratio. The failure assessment curve function f(Lr) depends on the proximity parameter for plastic collapse, Lr, defined at the assessed load condition by
Fig. 5. Variations of Kr with Lr for the axial temperature gradient (hp = 0.125, a/t = 0.3, n = 5) under three different loading sequences: (a and c) b = 0.5 and (d and f) b = 5.0.
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C.-Y. Oh et al. / Engineering Fracture Mechanics xxx (2014) xxx–xxx p
Lr ¼
rref F ¼ F L ða; ry Þ ry
ð3Þ
where F is the applied primary load and FL is the corresponding plastic limit load for the structure with a crack of size a for a rigid plastic material of yield stress, ry. Note that the reference stress for the primary loads, rpref , follows from Eq. (3) as rpref ¼ Lr ry . There are different Options in R6 and other codes for the failure assessment curve f(Lr). The R6 Option 1 curve is a function of Lr only and defined by
1=2 h i f ðLr Þ ¼ 1 þ 0:5L2r 0:3 þ 0:7 exp 0:6L6r
ð4Þ
The Option 2 curve depends on material and is defined by
f ðLr Þ ¼
" p Eeref
rpref
p
1 rref þ L2r p 2 Eeref
#1=2 ð5Þ
where epref is the strain on the stress–strain curve at the stress level rpref . Various options of estimating the V factor to be used in Eqs. (1) and (2) are also given in R6. In a simplified procedure, V is assumed to be a function of b and Lr, and given by
8 > < 1 þ 0:2Lr þ 0:02bð1 þ 2Lr Þ for Lr < Lr V ¼ 3:1 2Lr for Lr < Lr < 1:05 > : 1 for Lr > 1:05
ð6Þ
Fig. 6. Variations of Kr with Lr for the axial temperature gradient (hp = 0.125, a/t = 0.3, n = 5) under two extreme loading sequences: (a and b) b = 0.5 and (c and d) b = 5.0.
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where
b¼
Ks K p =Lr
ð7Þ
is a normalised secondary stress and Lr is defined in terms of b by equating the first two expressions in Eq. (6). It should be noted that, in the simplified procedure, V P 1 so that no benefit is taken for any plastic relaxation of secondary stresses, and its application is limited to b < 4. A more generally applicable method is the detailed procedure in R6, where it is necessary to first calculate the factor Vo, which is the value of V for secondary loading acting alone. Among a number of methods to calculate Vo suggested in R6, the most accurate one is to calculate Vo from Eq. (8) using an inelastic analysis of the cracked body.
Vo ¼
pffiffiffiffiffiffiffi E0 J s Ks
ð8Þ
where Js denotes the elastic–plastic J due to thermal loading only. Once Vo has been calculated, the value of V is obtained from
V ¼ nV o
ð9Þ
where n is a function of b and Lr, and is tabulated in R6 [3]. At low Lr, V is often close to unity indicating elastic response; at intermediate Lr (Lr < 1), V > 1, reflecting interaction between primary and secondary stress; at high Lr (Lr > 1), V < 1, indicating mechanical stress relief of the secondary stresses. Recently a new equation for evaluation of V based on a relaxation equation has been proposed by Ainsworth [13]:
V ¼ f ðLr Þ þ 0:42Lr ð0:72 þ Lr Þ½f ðLr Þ2 Vo
ð10Þ
Eq. (10) is considered to be an upper bound to V for typical plastic relaxation with limited elastic follow-up. However, Eq. (10) can be still non-conservative when large elastic follow-up is present [13].
Fig. 7. Variations of Kr with Lr for the radial temperature gradient (hp = 0.125, a/t = 0.3, n = 5) under two extreme loading sequences: (a and b) b = 0.5 and (c and d) b = 5.0.
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3. Finite element analysis 3.1. Geometry Consider a pipe with the mean radius, r, and thickness, t. Only one value of pipe geometry, r/t = 10, is considered. The half length of pipe was chosen to be ten times the pipe radius to avoid the end effect. Circumferential partthrough surface cracks are considered, as depicted in Fig. 1, characterised by the relative crack length, hp (where h denotes the half crack angle), and the relative crack depth, a/t. The surface crack is taken to have a rectangular shape, i.e., constant depth. Two values of hp are considered, hp = 0.125 and 1.0, with two relative crack depths, a/t = 0.3 and 0.5. 3.2. Mechanical and thermal loadings For the mechanical (primary) loadings, axial tension, pressure, and combined tension and pressure were applied as shown in Fig. 1 to generate various bi-axial stress states. To generate secondary stresses, three different types of thermal loads were applied. The first type has a temperature gradient through the pipe thickness (Fig. 2a, ‘‘radial temperature gradient’’). The second type has a gradient along the longitudinal direction (Fig. 2b, ‘‘axial temperature gradient’’). The third type has a temperature gradient across the cross-section, producing global bending stresses (Fig. 2c, ‘‘sectional temperature gradient’’). The magnitude of the temperature difference, DT, was adjusted to produce different levels of thermal stress. These three types of thermal gradient produce various bi-axial stress states, as shown in Fig. 3 where through-thickness variations of thermal stresses in the uncracked cylinder at the mid-plane, from elastic FE analyses, are plotted.
Fig. 8. Variations of Kr with Lr for the axial temperature gradient (hp = 1.0, a/t = 0.3, n = 5) under two extreme loading sequences: (a and b) b = 0.5 and (c and d) b = 5.0.
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3.3. Material For elastic analyses, an isotropic material was assumed with Young’s modulus E = 200 GPa and Poisson’s ratio m = 0.3. For elastic–plastic analyses, the material was assumed to follow the Ramberg–Osgood (R–O) relationship:
r r r e¼ þa y ry E E
n ð11Þ
where a and n are R–O parameters. The value of ry was chosen as ry = 300 MPa, but this choice does not affect the main results of this work, as these are presented in normalised form. To be consistent with the definition of ry as the 0.2% proof strength, a was chosen as
a¼
0:002E
ry
4 ¼ 3
ð12Þ
Finally to quantify the effect of the strain hardening index, two different values of n were considered; n = 5 and 10. 3.4. FE analysis A series of elastic and elastic–plastic FE analyses of circumferentially cracked pipes was performed using ABAQUS [14]. For axi-symmetric cases (hp = 1.0), eight-node reduced integration elements (CAX8R) were used. For efficient computations of three-dimensional cases, a quarter model was used allowing for symmetry. Twenty-node iso-parametric quadratic brick elements with reduced integration (C3D20R within ABAQUS) were used. Fig. 4 depicts a typical FE mesh for a surface cracked pipe. The crack-tip was designed with collapsed elements, and a ring of wedge-shaped elements was used in the crack-tip region. A total of eleven or twelve elements were used through the thickness. The numbers of elements and nodes in the FE
Fig. 9. Effect of strain hardening (n = 10) on variations of Kr with Lr for the axial temperature gradient (hp = 0.125, a/t = 0.3) under two extreme loading sequences: (a)–(b) b = 0.5 and (c)–(d) b = 5.0.
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meshes were about 3000 and 15,000, respectively. Note that, for modelling the part-through crack, a small rounded corner was introduced, which does not affect J calculations at the centre of the crack. The deformation plasticity option in ABAQUS was adopted in the FE analyses. Elastic FE analyses were performed first to determine the relative magnitude of the thermal stresses, which is quantified by Eq. (7). A value b = 1 corresponds to a secondary stress which produces a stress intensity factor equal to that for a primary load for which the primary load is equal to the limit load (or Lr = 1). Based on elastic FE analysis, the magnitude of the thermal loading was chosen to give two different values of b, b = 0.5 and 5.0. Note that the value b = 5 represents a very large thermal stress: a primary load equal to five times the collapse load (Lr = 5) would be required to generate a stress intensity factor equal to that produced by the secondary stress. For each magnitude of the thermal loading (each value of b), three sets of elastic–plastic FE analyses were performed; (i) mechanical loading only, (ii) thermal loading only and (iii) combined mechanical and thermal loading. To evaluate the loading sequence effect, FE analyses were performed where the sequence of application of the mechanical and thermal loadings was varied. Three different sequences were considered: (i) thermal loading followed by mechanical loading (where the thermal loading simulates residual stress, for instance); (ii) mechanical and thermal loading applied together in a proportional manner (simulating start-up conditions, for example); and (iii) mechanical loading followed by thermal loading (simulating pressurised thermal shock conditions, for instance). For thermal shock conditions, the value of b can be up to five or more. Values of the J-integral were extracted as a function of the applied load from the FE results using the domain integral method. The FE J values, JFE, were taken in the centre of the crack along the crack growth direction, as depicted in Fig. 1. Numerical J estimates using ABAQUS are generally known to be robust for monotonic loading, and the present meshes, as shown in Fig. 4, are judged to be sufficiently fine to produce accurate results. It is also found that the numerical J values are almost path-independent over six contours around the crack tip for the results presented in this paper. Averaged values of J (over six contours) are reported. Using the FE results, two parameters Kr and Lr for the failure assessment diagram are calculated. The finite element analyses results are used to evaluate Eq. (13):
Fig. 10. Variations of Kr with Lr for the radial temperature gradient (hp = 0.125, a/t = 0.3, n = 5) under two extreme loading sequences of pressure and thermal loading: (a)–(b) b = 0.5 and (c)–(d) b = 5.0.
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C.-Y. Oh et al. / Engineering Fracture Mechanics xxx (2014) xxx–xxx
K p þ VK s K r ¼ qffiffiffiffiffiffiffiffiffiffiffiffi E0 J pþs FE
11
ð13Þ
with the detailed estimate of V from Eqs. (9) and (10). Values of Kp, Ks and JFE were taken from the FE results. Note that Eq. (13) represents the Option 3 failure assessment curve obtained using FE elastic–plastic J within R6 [3]. The curve is specific to a material and geometry, and is supposed to be more accurate than the Option 2 curve. The J-estimation approaches in R6 are then conservative if Kr from Eq. (13) lies outside the appropriate failure assessment curve, Kr = f(Lr). The R6 simplified estimates of Eq. (6) for estimating V are not assessed in this paper, as generalised comparisons with other methods have been given in Ref. [9] and they are expected to be very conservative at high values of Lr. 4. Results 4.1. Effects of b, a/t and hp Fig. 5 shows failure assessment diagrams for axial temperature gradients with b = 0.5 and b = 5.0 for the case n = 3, a/t = 0.3 and h/p = 0.125. Note that, in Fig. 5 as well as in subsequent figures, the optimised reference load (NOR for axial tension and POR for internal pressure) is used instead of the limit load to define Lr in Eq. (3). This is to remove conservatism in the estimation of J due to primary stress alone and thus to investigate conservatism solely due to the V-factor. More detailed explanation on the optimised reference load is given in our previous paper (see Eq. (18), Section 4.2 and Table 2 in Ref. [9]). Results are shown in Fig. 5 for three different loading sequences. In applying Eq. (13) to evaluate Kr from the FE analyses, V has been estimated in three ways. Firstly, the existing R6 detailed approach of Eq. (9) has been applied leading to the open squares. The second and third methods for estimating V have used Eq. (10) with f(Lr) defined by Eqs. (4) and (5) leading to the solid circles and the open triangles, respectively. The solid circles, with V estimated using the Option 1 failure
Fig. 11. Variations of Kr with Lr for the radial temperature gradient (hp = 0.125, a/t = 0.3) under combined tension and internal pressure: (a)–(b) b = 0.5 and (c)–(d) b = 5.0.
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assessment function, should be compared with the Option 1 failure assessment curve (solid line) whereas the open triangles should be compared with the Option 2 curve (dashed line). If the point lies (further) outside the failure assessment curve, the approach is (more) conservative. When secondary stresses are small, Fig. 5a–c, there is no significant loading sequence effect. However for large secondary stresses, b = 5.0, Fig. 5d–f, loading sequence effects are more significant and Fig. 5f shows values of Kr lower than those in Fig. 5d, indicating a higher value of J. The results in Fig. 5 suggest that the sequence of mechanical loading followed by thermal loading is the most severe loading case. The least severe loading sequence is the opposite one, thermal loading followed by mechanical loading. When the two loadings are applied proportionally, the J values are in between those from the above two loading sequences. The points for the existing R6 approach of Eq. (9) generally lie outside the failure assessment curve indicating conservatism in the approach. Also the results are less conservative if used with an Option 2 failure assessment curve. However, Fig. 5f shows that there is a small non-conservatism in the current R6 detailed approach if used with an Option 2 curve. In this respect, it is noted that the tabulated n values in R6 [3] were calculated using an Option 1 curve but are not particularly sensitive to the choice of f(Lr). The points using Eq. (10) (both sets), on the other hand, lie outside the corresponding failure assessment curves and are always conservative. Fig. 6 shows the results for a deeper crack depth (a/t = 0.5) with n = 5 and h/p = 0.125. Corresponding results for the radial thermal gradient with a/t = 0.3 are shown in Fig. 7. Fig. 8 finally shows the results for the axial temperature gradient but for a greater crack length (hp = 1.0) with a/t = 0.3 and n = 5. Only the two loading sequences producing the most and least severe cases are shown in Figs. 6–8 for clarity. Similar trends to those in Fig. 5 can be seen. 4.2. Effects of strain hardening In Fig. 9, the influence of strain hardening is demonstrated by taking a strain hardening index n = 10 as opposed to n = 5 in the earlier figures, again with a/t = 0.3 and h/p = 0.125. For small secondary loads (b = 0.5), Fig. 9a and b, all methods are
Fig. 12. Variations of Kr with Lr for the axial or radial temperature gradient (hp = 0.125, a/t = 0.3, n = 5) with large b(=10) under two extreme loading sequences: (a)–(b) for a fixed mechanical load magnitude, and (c)–(d) for a fixed thermal load magnitude.
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highly accurate. For high secondary loads (b = 5), Fig. 9c and d, the methods are significantly conservative for the mechanical load applied after the thermal load, Fig. 9c, but this conservatism is required to accurately predict the case of thermal load applied after mechanical loading, Fig. 9d. Another point is that the use of Eq. (10) is more accurate for the lower hardening material. 4.3. Bi-axial stress effects The results in the previous two sub-sections were for axial tension, producing uni-axial mechanical stress states in the cylinder. In this sub-section, a bi-axial mechanical stress effect is investigated by applying internal pressure or combined pressure and tension. The stress bi-axiality is defined in terms of the hoop and axial stresses, rh and ra, of an un-cracked body using a non-dimensional variable, w
w¼
rh 2ra
ð14Þ
For axial tension, w = 0 and for internal pressure, w = 1, based on the thin cylinder approximation. Results for internal pressure (w = 1.0) are shown in Fig. 10, and those for combined pressure and axial tension (w = 0.5) in Fig. 11, in both cases for n = 5, a/t = 0.3 and h/p = 0.125. It can be seen that even under bi-axial stress states, similar trends are observed to the earlier cases. 4.4. Results for large b cases There can be some practical cases when the b value from Eq. (7) is quite large. Two types of calculations were performed to produce b values close to 10. Firstly, a very large secondary stress Ks was increased for a fixed Kp/Lr by increasing the
Fig. 13. Variations of Kr with Lr for the sectional temperature gradient (hp = 0.125, a/t = 0.3) with small b(=0.5): (a)–(b) n = 5 and (c)–(d) n = 10.
Please cite this article in press as: Oh C-Y et al. FE validation of R6 elastic–plastic J estimation for circumferentially cracked pipes under mechanical and thermal loadings. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.03.015
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temperature difference DT. Secondly, Ks was fixed and Kp/Lr was decreased by applying compressive axial force (by increasing w to w = 3.0, see Eq. (14), by applying combined pressure and axial compression leading to a small net positive axial stress). In Fig. 12, the cases with large b (b = 10) are illustrated for n = 5 with a/t = 0.3 and h/p = 0.125 and so can be compared with the corresponding cases with smaller b in Fig. 5. The R6 detailed evaluations are not shown, as they cannot be applied for b > 5 due to the limits in the tabulated values of n in R6. The results in Fig. 12a and b are obtained for a fixed Kp/Lr (under axial tension) by increasing the temperature difference DT. When compared with the results in Fig. 5, it can be seen that Eq. (10) is more accurate for large b. In particular, the methods are accurate for the thermal load applied after the mechanical load. The results in Fig. 12c and d are obtained for the fixed Ks by decreasing Kp/Lr. It can be seen that the methods tend to be more conservative, compared to the cases in Fig. 12a and b. 4.5. Large elastic follow-up case In our previous work [12], it was shown that existing methods can be non-conservative when large elastic follow-up behaviour is present. Large elastic follow-up can occur when small secondary stresses producing a region of uniform secondary stress are present under contained yielding for low strain hardening materials. As shown in Fig. 3, the sectional temperature gradient produces uniform secondary stress through the thickness. Fig. 13 shows the results for sectional temperature gradients with b = 0.5, a/t = 0.3 and h/p = 0.125. The results for n = 5 are shown in Fig. 13a and b, whereas those for n = 10 are plotted in Fig. 13c and d. Due to the small secondary stresses (b = 0.5), loading sequence effects are not so significant. For n = 5, slight non-conservatism is observed at small Lr. For n = 10, the degree of non-conservatism increases. This suggests that caution in the application of Eq. (10) is needed when assessing a crack in low hardening materials under small secondary stresses that produce a uniform stress through the thickness. In [12], methods for allowing for such effects through an explicit dependence on an elastic follow-up factor are proposed, but these are not investigated in this paper.
Fig. 14. Effect of the use of a conservative reference stress on variations of Kr with Lr for an axial temperature gradient applied after mechanical loading: (a)–(b) n = 5 and (c)–(d) n = 10.
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4.6. Effect of conservative estimate of reference stress In presenting results in the previous sub-sections, the optimised reference load was used instead of the limit load to define Lr in Eq. (3), as explained in Section 4.1. The reason was to remove conservatism in estimates of J under primary stress alone and thus to investigate any conservatism solely due to the V-factor. In practical assessments, a lower reference stress (or Lr) is often used to ensure conservatism in estimating J. Fig. 14 shows the effect of the use of a conservative reference stress on variations of Kr with Lr for an axial temperature gradient. For the loading sequence, only the most severe case (thermal loading after mechanical loading) is considered for two different hardening exponents, n = 5 and 10; in Fig. 14a and b for n = 5 and in Fig. 14(c–d) for n = 10. For each case, a/t = 0.3 and h/p = 0.125 is assumed. In calculating Lr, the reference load is assumed to be ten percent smaller (and thus the reference stress is roughly ten percent higher). The reason to choose the 10% is rather arbitrary, but previous studies showed that the limit load in R6 for circumferential surface cracked pipes R6 reference stress is within 10% of the optimised reference load producing accurate J [9,10]. Due to the change in Lr, the b values are also changed slightly, as indicated in the figures. When compared with the results in Figs. 5 and 9, it can be seen that the points tend to move further from the corresponding failure assessment lines, suggesting that the assessment is more conservative. The assessment results are also more conservative for lower hardening materials. 5. Conclusions Variations of elastic–plastic J with load magnitude have been evaluated by finite element analyses for circumferentially surface cracked pipes under combined mechanical and thermal loads. For the mechanical loading, axial tension, pressure and combined tension and pressure have been considered. For the thermal loading, three different types of thermal gradient have been considered. Relative mechanical/thermal load magnitudes, crack sizes and material hardening have all been systematically varied. The loading sequence effect of mechanical and thermal loading have also been investigated. The FE results have been presented on the R6 failure assessment diagram by calculating the two parameters Kr and Lr using various methods for estimating the interaction parameter V. The V factor has been estimated using the current detailed R6 method and recent proposals dependent on the shape of the failure assessment curve. From the presented results, the following conclusions can be drawn. Significant loading sequence effects have been observed at large secondary stresses. The case when primary loading is applied before secondary loading, such as for pressurised thermal shock loading, is found to be a more severe loading case than that of secondary loading applied before primary loading. The current detailed R6 method is generally conservative and the conservatism increases with increasing relative magnitude of the thermal stresses. However, it can be slightly non-conservative if used with the Option 2 curve for high hardening materials for the case when a large thermal stress is applied after primary loading. Recent proposals for estimating V, on the other hand, are conservative even in such load cases. The recent proposals are shown to be more accurate for the case when the relative magnitude of the thermal stress is quite high. However, when large elastic follow-up is present, all methods are non-conservative in a small region at low values of Lr (<1.0). Acknowledgement This research is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (Nos. 2007-0056094, 2013M2B2A9A03051295, 2012M2A7A1051939). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
Ainsworth RA. The treatment of thermal and residual stresses in fracture assessments. Engng Fract Mech 1986;24:65–76. Ainsworth RA. The assessment of defects in structures of strain hardening material. Engng Fract Mech 1984;19:633–42. R6, Revision 4: Assessment of the integrity of structures containing defects. Gloucester, UK: EDF Energy Nuclear Generation Ltd.; 2013. BS7910: Guide on methods for assessing the acceptability of flaws in metallic structures. London: British standards Institution; 2000. API 579-1/ASME FFS-1. Fitness-for-service for pressure vessels, piping and storage tanks, 2007. American Petroleum Institute/American Society of Mechanical Engineers. Ainsworth RA, Smith SD, Wiesner CS. Treatment of thermal and residual stresses in defect assessment. British Energy Generation Limited Report EPD/ GEN/REP/0423/99, Gloucester; 1999. Ainsworth RA, Sharples JK, Smith SD. Effects of residual stresses on fracture behaviour – experimental results and assessment methods. J Strain Anal 2000;35:307–16. Stacey A, Barthelemy JY, Leggatt RH, Ainsworth RA. Incorporation of residual stresses into the SINTAP defect assessment procedure. Engng Fract Mech 2000;67:573–612. Song T-K, Kim Y-J, Nikbin K, Ainsworth RA. Approximate J estimates for circumferential cracked pipes under primary and secondary stresses. Engng Fract Mech 2009;76:2109–25. Oh C-Y, Kim Y-J, Budden PJ, Ainsworth RA. Biaxial stress effects on estimating J under combined mechanical and thermal stresses. Int J Pres Ves Piping 2011;88:365–74. James PM, Hooton DG, Higham LA, Madew CJ, Sharples JK, Watson CT. Continuing development of a simplified method to account for the interaction of primary and secondary Stresses. In: Proceedings of the 2008 ASME Pressure Vessels and Piping Conference, Paper PVP2008-61040; 2008.
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[12] Song T-K, Oh C-Y, Kim Y-J, Ainsworth RA, Nikbin K. Approximate J estimates for combined primary and secondary stresses with large elastic follow-up. Int J Press Vess Piping (electronic version available, see
Please cite this article in press as: Oh C-Y et al. FE validation of R6 elastic–plastic J estimation for circumferentially cracked pipes under mechanical and thermal loadings. Engng Fract Mech (2014), http://dx.doi.org/10.1016/j.engfracmech.2014.03.015