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Derivation of J-resistance curve for through wall cracked pipes from crack mouth opening displacement
ARTICLE IN PRESS
International Journal of Pressure Vessels and Piping 83 (2006) 686–699 www.elsevier.com/locate/ijpvp
Derivation of J-resistance curve for through wall cracked pipes from crack mouth opening displacement Suneel K. Guptaa,, Vivek Bhasina, K.K. Vazea, A.K. Ghosha, H.S. Kushwahaa, S. Chapuliotb, S. Marieb, I. Kayserb a Bhabha Atomic Research Center, Reactor Safety Division, Mumbai 400 085, India Structural Integrity and Normalisation Laboratory, CEA/DEN/DM25/SEMT/LISN, CEA-Saclay, France
b
Received 30 July 2005; received in revised form 15 May 2006; accepted 20 May 2006
Abstract In this paper, factors for calculating the J-resistance curve from the measured crack mouth opening displacement (CMOD) record are derived for circumferential through wall cracked straight pipes subjected to pure bending. Rigorous elastic plastic finite element analyses of around 125 circumferential through wall cracked straight pipes subjected to four point bending have been carried out to validate the proposed equations. The factors have also been validated with a few fracture experimental results on straight pipes. For these tests the J–R curve has been evaluated using the conventional as well as using the proposed factors along with the moment-CMOD record. r 2006 Elsevier Ltd. All rights reserved. Keywords: J-integral; J–R curve; Z factor; g factor; Pipe; Crack; Rotation; CMOD
1. Introduction The J-integral is an important and established parameter to characterise the fracture behaviour of cracked structures made of ductile material. Since the introduction of the Jintegral by Rice [1], Hutchinson [2] and Rice and Rosengren [3], many researcher, Rice et al. [4], Hutchinson and Paris [5], Ernst et al. [6,7], Zahoor and Kanninen [8], Wilkowski et al. [9], Roos et al. [10] and Chattopadhayay et al. [11], have contributed to establishing procedures for experimental evaluation of the J-integral from the area under the load versus plastic part of load line displacement record. The J-integral evaluation procedure is based on Z and g factors. The plastic part of the J-integral is calculated by multiplying the Zpl factor by the area under the load versus plastic load line displacement curve obtained by conducting experiments on the cracked structure. However, for a growing crack, the above J-integral is modified using the gpl factor. The existence of these scaling factors relies upon the hypothesis of load separation, i.e. the load Corresponding author. Tel.: +91 22 559 15 28; fax: +91 22 50 51 51.
E-mail address: [email protected] (S.K. Gupta). 0308-0161/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2006.05.004
imposed on the cracked structure is equal to the product of two functions, one represents the cracked geometry and is independent of material and the other represents the deformation behaviour and depends on material only. Nuclear power plant (NPP) piping is normally made of ductile materials. Their rigorous fracture assessment is essential for demonstration of leak-before-break (LBB) capability and safety margins against fracture failure, under design basis accidents such as safe shutdown earthquake (SSE) loading. In general J-tearing analysis is performed for fracture assessment which in turn requires knowledge of material fracture properties i.e. J–R curve. The J-integral calculation is generally made from the test measured load and load line displacement (LLD) data. Hence, accuracy of the J-integral relies on the accurate measurement of the load and LLD, on a cracked specimen or component. Unlike small specimen testing, in large size component testing, such as a through wall cracked straight pipe under four point bending, the accuracy of the LLD measurement is influenced to a greater extent by various experimental factors such as machine/set up compliance, friction and denting at the loading and support points etc. As a result there is some difference between measured and
ARTICLE IN PRESS S.K. Gupta et al. / International Journal of Pressure Vessels and Piping 83 (2006) 686–699
Nomenclature A crack area E Young’s modulus J, Jel, Jpl total, elastic and plastic J-integral M total applied moment Mo limit moment n Ramberg–Osgood equation parameter P total applied load R, Ri, Ro mean, inside and outside radius of pipe crosssection t wall thickness of pipe Upl plastic part of strain energy Uf,pl plastic strain energy i.e. area under the moment vs. plastic load-point-rotation curve Un,pl area under the moment vs. plastic crack mouth opening displacement curve i.e. an indirect measure of plastic strain energy Greek letters a y
Ramberg–Osgood equation parameter half-circumferential crack angle
actual LLD. The testing machine and associated structure, which transfers load to testing component, may undergo significant deformation. Hence, in such testing, it becomes necessary to correct the measured LLD for machine compliance by subtracting the displacements caused by deformation of testing machine members and associated structure. In pipe fracture testing under bending load, the total LLD contains significant contribution of un-cracked component geometry deformation. J-calculation from load-LLD data requires that displacement of un-cracked component geometry be subtracted from the total LLD. Many times displacement due to un-cracked component geometry and machine compliance may not be readily available, which need to be evaluated with additional tests or calculations. In contrast, J-estimation from load and crack mouth opening displacement (CMOD) data is direct and there is no need to subtract any un-cracked or machine compliance CMOD. The CMOD does not include un-cracked deformation contribution and represents the crack-system deformation behaviour only. CMOD is independent from machine compliance since it is directly measured on specimen/component. The CMOD is normally measured using a clip gauge, which gives precise measurement of the deformation behaviour of the component during the experiment. In this study, the J-integral is evaluated using load versus CMOD data. The only difference in the evaluation procedure as compared to the procedure based on the moment-load-point-rotation (M–fpl) curve is the Z and g factors. Here new Zn,pl and gn,pl factors are derived which
687
D total load-line-displacement Del, Dpl elastic and plastic load-line-displacement due to crack-system f total load-point rotation fel, fpl elastic and plastic load-point rotation due the crack n, nel, npl total, elastic and plastic crack mouth opening displacement (CMOD) Zpl a function to multiply the area under the load vs. plastic load-point- deflection curve to get the plastic component of the J-integral gpl a function to correct the plastic J-integral evaluated by Zpl function in crack growth situations Zf,pl Zpl factor based on M–fpl record gf,pl gpl factor based on M–fpl record Zn,pl Zpl factor based on M–npl record gn,pl gpl factor based on M–npl record s stress so reference stress ¼ yield stress e strain corresponding to s eo reference strain (strain corresponding to so) i.e. so/E
are based on the moment-CMOD (M–npl) curve. The derivation of these factors is discussed in Section 3. To validate the proposed Zn,pl and gn,pl factors around 40 elastic and 125 elastic plastic finite element analyses have been performed on circumferential through wall cracked straight pipes subjected to four point bending and discussed in Section 5. Initially the Zf,pl factor has been evaluated from the M, fpl and Jpl curves obtained from finite element analyses. The Zf,pl factor as evaluated using FEM have been validated with equations in the literature. The objective of this exercise was to qualify the analyses and calculation procedures of the Zpl factor from FE data. Then the Zpl based on plastic CMOD (npl) is evaluated. Finally new Zn,pl and gn,pl factors validation with fracture test results on straight pipes have been discussed in Section 6. For these tests the J-integral and modified J integral have been evaluated using conventional as well as the new Z and g factors. 2. Theoretical background for gpl and cpl The plastic part of the J integral is defined as J pl ¼
dU pl , dA
(1)
where Upl is the plastic part of the strain energy and A is crack surface area. For a pipe of mean radius R, thickness t and having a through wall crack subtending an angle 2y at the centre (Fig. 1), the average Jpl given by Eq. (1) can be
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∆
P
R t Through wall Crack
2θ
L Z
Fig. 1. Schematic diagram of the circumferential through wall cracked straight pipe subjected to four point bend loading.
expressed as J pl ¼
1 dU pl . 2Rt dy
(2)
The plastic strain energy Upl can be obtained from the area under the moment versus plastic rotation (M–fpl) curve or load versus plastic LLD (P–Dpl) curve. Then, Upl is denoted as Uf,pl and is calculated by Z fpl Z Dpl U f;pl ¼ M dfpl ¼ P dDpl . (3) 0
0
Zahoor and Kanninen [8] have derived an expression for determination of Jpl from M–fpl or P–Dpl curves obtained from pipe fracture tests. It is based on dimensional analysis and on an alternative but equivalent definition of the J integral given by Rice et al. [4]. The resulting expression for evaluating Jpl for a stationary i.e. non-growing crack is Z fpl Z Dpl J pl ¼ Zf;pl M dfpl ¼ Zf;pl P dDpl . (4) 0
0
In the derivation of the Zf,pl, Zahoor and Kanninen [8] assumed that all the plasticity due to the presence of the crack is confined to the crack section. The assumption relies on the hypothesis that the relation between moment (M), crack angle (y) and plastic crack rotation (fpl) can be expressed in variable separable form. This form is given as M ¼ hðyÞgðfpl Þ,
(5)
where h(y) is a function of y alone and does not depend on material properties. The function g(fpl) represents the deformation behaviour and depends on material only. Based on this the resulting expression for Zf,pl as derived by Zahoor and Kanninen [8] is Zf;pl ¼
1 h0 ðyÞ , 2Rt hðyÞ
(6)
where h0 (y) is the derivative of h(y) with respect to y. For the circumferentially through wall cracked pipe geometry, the h(y) function is hðyÞ ¼ cosðy=2Þ 0:5 sinðyÞ.
(7)
The deformation function g(fpl) is material dependent and varies from one material to another. The general equation to evaluate Jpl from experimental data (M–fpl curve) when there is crack growth is as follows [8]: Z fpl Z y Zf;pl M dfpl þ gf;pl J pl dy, (8) J pl ¼ 0
yo
where gn,pl is gf;pl ¼
h00 ðyÞ . h0 ðyÞ
(9)
Here h00 (y) is the second derivative of h(y) with respect to y. The deformations fpl and Dpl are the plastic parts of the crack-system deformations of the pipe under four point bending. The crack-system deformation (i.e. the increased deformation due to the presence of the crack) is evaluated after subtracting the non-crack system deformation from the total deformation. 3. Derivation of gn,pl (CMOD-based) For a particular pipe geometry and material, Kumar et al. [12] have shown that Jpl, fpl, and npl are functions of crack size y and moment M expressed as J pl ¼ J pl ðM; yÞ,
(10)
fpl ¼ fpl ðM; yÞ,
(11)
npl ¼ npl ðM; yÞ.
(12)
From Eqs. (10) and (12), the Jpl can be represented as a function of y and npl J pl ¼ J pl ðnpl ; yÞ.
(13)
Similarly, from Eqs. (11) and (12), the npl can be represented as a function of y and fpl: npl ¼ npl ðfpl ; yÞ.
(14)
For a through wall circumferentially cracked pipe under pure bending, Kumar et al. [12] and Zahoor [13] have given expressions for fpl and npl. For a particular material, the dependence of the plastic rotation fpl and the plastic
ARTICLE IN PRESS S.K. Gupta et al. / International Journal of Pressure Vessels and Piping 83 (2006) 686–699
CMOD npl on loading is given as M n fpl / f f ðR; t; yÞ , Mo M n npl / f n ðR; t; yÞ , Mo
(15)
(16)
where ff and fn are functions of pipe size and crack size while Mo is a reference moment, which is usually taken as the limit load for the cracked pipe. Here n is strainhardening parameter of Ramberg–Osgood (R–O) equation representing the material’s stress–strain behaviour. Eqs. (15) and (16) show that the ratio npl/fpl is independent of applied loading and function of only pipe and crack size. Based on this fact following expression is postulated npl ¼ f ðR; t; yÞfpl .
(17)
Then Eq. (4), the expression for Jpl for a non-growing crack, can be rewritten in term of npl, as Z fpl Z npl dfpl M dfpl ¼ Zf;pl M (18) dnpl . J pl ¼ Zf;pl dnpl 0 0 For a stationary crack, dfpl/dnpl, from Eq. (17), is expressed as dfpl 1 . ¼ f ðR; t; yÞ dnpl
(19)
Therefore, Jpl, for a non-growing crack case can be expressed as Z npl M dnpl , (20) J pl ¼ Zn;pl
presence of the crack). A similar hypothesis was used by Smith [14,15] in which fc was related to the crack tip opening displacement CTOD (dpl) about the shifted neutral axis at the cracked section. dpl ¼ fc ðdistance of crack tip from neutral axisÞ, dpl ¼ fc Rðcos y þ cos bÞ.
Eq. (20) gives a linear relation between Jpl and the area under the M–npl curve. The area under the M–npl curve will be denoted by Un,pl. Un,pl does not directly represents the plastic work input but is given by Z npl M dnpl . (22) U n;pl ¼
ð23Þ
In the above equation, the angle b gives the neutral axis position as shown in Fig. 2. For elastic–perfectly plastic material, a free body analysis of the cracked section shows that the point of stress inversion i.e. the angle b is given by py . (24) 2 The logic given above can be extended to CMOD (npl), and Eq. (23), can also be written
b¼
npl ¼ fpl ðdistance of middle of the crack mouth from shifted neutral axisÞ. From Fig. 2, the distance of middle of the crack mouth has been calculated as ‘Ro+R cos b’, then npl ¼ fpl ðRo þ R cos bÞ, y npl ¼ fpl Ro þ R sin . 2
ð25Þ
On comparing Eqs. (17) and (25) the function f(R, t, y) is given as R y sin . (26) f ðR; t; yÞ ¼ Ro 1 þ Ro 2 From Eqs. (6) and (21), the Zn,pl function is given as
0
where Zn,pl is a CMOD based function and is expressed in term of Zf,pl using Eqs. (18)–(20). Zf;pl . (21) Zn;pl ¼ f ðR; t; yÞ
689
Zn;pl ¼
1 h0 ðyÞ 1 . 2Rt hðyÞ f ðR; t; yÞ
(27)
In the above equation the function h(y) is given by Eq. (7) and the function f(R, t, y) is given by Eq. (26). It may be noted that the above hypothesis is perfectly valid only if the material is elastic perfectly plastic and the gross state of stress is uniaxial. If the material is strain Point of CMOD measurement
0
It has been shown that fpl and npl as well as Zf,pl and Zn,pl are linearly related by a function f(R, t, y). This relationship can be appreciated further using the assumption that the deformation discontinuity at the cracked section is entirely in the form of rotation. The rotation of the cracked section fc is a pure manifestation of the applied load point rotation (crack-system) in the pipe subjected to pure bending (four-point bending) which implies that the crack system rotations about the neutral axis will remain constant between the cracked section and the load point. Thus, fc (i.e. the plastic part of the cracked section rotation about the shifted neutral axis) will be equal to fpl (i.e. the increased plastic rotation at the load point due to the
R+0.5t+RCosβ t 2θ R(Cosθ + Cosβ)
R C
C
Pipe Axis β
N
Neutral Axis
N
Fig. 2. Schematic diagram of the cracked section of the circumferential through wall cracked straight pipe.
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hardening in nature or if the gross state of stress is highly triaxial then the position of the neutral axis is not exactly the same as calculated above. However, the difference is not likely to be high. The finite element verification of such an assumption will be discussed in Section 6.
For a growing crack the J-integral expression is modified using the g factor. Prior to the initiation of crack growth, the plastic contribution of the J-integral is calculated from Eq. (20). After crack initiation and during propagation, the crack angle y is no longer constant. So it is necessary to know the evolution of the crack length and this also changes the plastic J-integral. The differential increment of Jpl can be derived from Eq. (13) as qJ pl qJ pl dnpl þ dy ¼ dJ pl;1 þ dJ pl;2 . (28) dJ pl ¼ qnpl y qy npl The increment dJpl,1 is further solved along with Eq. (20) and given as qJ pl dnpl dJ pl;1 ¼ qnpl y Z npl q ¼ Z M dnpl dnpl ¼ Zn;pl M dnpl . ð29Þ qnpl n;pl 0 y In the above equation M is expressed as product of h(y) and g(fpl) as mentioned in Eq. (5). The increment dJpl,2 is Z npl qJ pl q Z dJ pl;2 ¼ dy ¼ M dnpl dy. qy n;pl 0 qy npl npl From Eq. (5), the M is replaced in the above equation leading to Z npl q dJ pl;2 ¼ Zn;pl hðyÞ gðfpl Þ dnpl dy. (30) qy 0 npl As discussed earlier (in Section 3) the function g(fpl) does not depend on crack angle y and purely represents the material deformation behaviour. In the above equation Zn,pl and h(y) are functions of y. Then Eq. (31) can be rewritten as Z npl q ðhðyÞZn;pl Þ gðfpl Þ dnpl dy dJ pl;2 ¼ qy 0 leading to Z npl qZn;pl qhðyÞ þ Zn;pl gðfpl Þ dnpl dy dJ pl;2 ¼ hðyÞ qy qy 0
dJ pl;2 ¼
! Z npl 1 qZn;pl 1 qhðyÞ þ M dnpl dy. Zn;pl Zn;pl qy hðyÞ qy 0
The above equation is rewritten dJ pl;2 ¼ gn;pl J pl dy,
gn;pl ¼
1 qZn;pl 1 qhðyÞ . þ Zn;pl qy hðyÞ qy
(32)
From Eqs. (29) and (31), Eq. (28) is solved and leads to dJ pl ¼ dJ pl;1 þ dJ pl;2 ¼ Zn;pl M dnpl þ gn;pl J pl dy.
4. Derivation of cn,pl (CMOD-based)
and
where gn,pl is defined as
(31)
By integrating the above equation, an equation for the plastic J-integral after crack initiation is obtained and is given as Z npl Z y Zn;pl M dnpl þ gn;pl J pl dy. (33) J pl ¼ 0
yo
By differencing Eq. (27) we get 1 qZn;pl q2 hðyÞ=qy2 1 qhðyÞ 1 qf ðyÞ , ¼ Zn;pl qy hðyÞ qy f ðyÞ qy qhðyÞ=qy i.e. 1 qZn;pl h00 h0 f 0 ¼ 0 . Zn;pl qy h h f
(34)
From Eqs. (9), (32) and (34) the simplified equation for gn,pl can be expressed as gn;pl ¼
h00 f 0 f0 ¼ gf;pl , 0 h f f
(35)
where h0 and h00 are the first and second derivatives of h(y) with respect to y and f0 is the first derivative of f(R, t, y) with respect to y. 5. Finite element validations The finite element (FE) validation of the equations derived in Sections 3 and 4 will be discussed here. For FE validation, six pipe cases were considered and the geometry, material and crack sizes details are given in Table 1. The pipe-1, -2 and -3 geometries represent the pipes used in the primary heat transport (PHT) piping system of Indian pressurized heavy water reactors (PHWRs). These pipes are made of C–Mn steel conforming to material grade SA333Gr.6. In addition their geometries (R/t etc.) are similar, although the nominal pipe size and thickness differ. The material and geometry for pipe-1 and -3 are the same as the monotonic fracture experiments carried out by Bhabha Atomic Research Center (BARC), i.e. Chattopadhay et al. [16] while the pipe-2 geometry and material is the same as the cyclic tearing tests carried out by Gupta et al. [17,18]. The pipe-4, -5 and -6 were selected such to cover different pipe geometries (R/t: 5–10) and material stress–strain relation (strain hardening exponent n: 3–7). The materials were assumed to follow widely used and the well-known R–O constitutive law describing material’s
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Table 1 Pipe material and geometry details used for the finite element analysis Parameter
Outer diameter 2Ro (mm) Thickness t (mm) Radius R (mm) R/t Half-crack angle y (deg.) Ramberg–Osgood n Ramberg–Osgood a Yield stress so (MPa) Outer span Z (mm) Inner span L (mm)
Pipe no. Pipe-1
Pipe-2
Pipe-3
Pipe-4
Pipe-5
Pipe-6
219
219
406
219
219
300
15.18 101.91 6.71 30, 32, 34y100
15.5 101.75 6.56 30, 32, 34y100
32.3 186.85 5.78 40, 42, 34y110
10 104.50 10.45 20, 40y100
20 99.5 4.98 20, 40y100
25 137.50 5.50 20, 40y100
s–e curve given in Fig. 3a
s–e curve given in Fig. 3a
s–e in Fig. 3b
288 4000 1480
288 2500 860
312 5820 1480
3 20 250 4000 1480
5 10 250 4000 1480
7 5 250 4000 1480
Fig. 3. Stress–strain curve for (a) pipes 1 and 2, and (b) pipe 3. (c) Finite element mesh used for analysis of the straight pipe with circumferential through wall crack.
post-yield stress–strain (s–e) relation and is given as n s s ¼ , (36) þa o so so
where so is a reference stress, which can be arbitrary, but typically assumed to be the yield stress of the material, o ¼ so =E is the associated reference strain, and a and n are parameters of above power-law model (R–O equation)
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usually chosen from a best fit of the experimental data. Variability in stress–strain curve was considered by considering different a and n values for pipe-4 through pipe-6. From symmetry considerations one-quarter of the through wall circumferentially cracked pipe has been modelled using 20 noded three-dimensional (3D) isoparametric brick and prismatic elements. The thickness was modelled using three elements. The finite element code ADINA [19] has been used for modelling and analysis. The virtual crack extension method was used for J integral calculations. At the cut surfaces, symmetric boundary
conditions were applied. A typical finite element mesh used, is shown in Fig. 3. A spider type mesh pattern was used to mesh the crack tip zone, as shown in Fig. 3. In these analyses, the Load, LLD, J-integral, CMOD, rotations (f) at loading points, etc. have been evaluated. The data have been further processed and plastic parts of various fracture parameters have been calculated. The crack-system plastic parts of LLD, and f have been evaluated as Dpl ¼ D Del;crack Del;non_crack Dpl;non_crack , fpl ¼ f fel;crack fel;non_crack fpl;non_crack .
8000 θ = 30° : Pipe-1 θ = 40° : Pipe-1 θ = 50° : Pipe-1
7000
θ = 70°
θ = 80°
θ = 60°
θ = 60° : Pipe-1 θ = 70° : Pipe-1 θ = 80° : Pipe-1
6000
Jpl (N/mm)
θ = 90°
θ = 100°
θ = 50°
θ = 40°
θ = 90° : Pipe-1
5000
θ = 100° : Pipe-1
4000
θ = 30°
3000 2000 1000 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Uφ = ∫Mdφpl (N.mm)
2 x 107
pl
Fig. 4. Jpl (from FEM) versus Uf,pl (from FEM) for pipe 1.
14000 θ = 40° : PIPE3
12000
θ = 70° : PIPE3 θ = 80° : PIPE3
Jpl (N/mm)
10000
θ = 90° : PIPE3 θ = 100° : PIPE3 θ = 110° : PIPE3
8000
θ = 80°
θ = 90°
θ = 50° : PIPE3 θ = 60° : PIPE3
θ = 70° θ = 60° θ = 50°
θ = 100°
θ = 110° θ = 40°
6000
4000
2000
0
0
5
10 Uφ, pl = ∫Mdφpl (N.mm2)
Fig. 5. Jpl (from FEM) versus Uf,pl (from FEM) for pipe 3.
15 x 107
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The plastic parts of CMOD and J-integral have been evaluated as
It should be noted here that in certain cases the non-crack system rotations constitute a significant part of total rotation in case of shallow cracked geometry where load can go to a high level. This is due to the significant contributions from non-crack plastic LLD and rotation (Dpl,non_crack and fpl,non_crack) for smaller crack angles (generally y/po0.25). However for deeply cracked geometry, the non-crack rotation or LLD constitute an insignificant part of the total LLD and its effect may be small.
5000 4500
J pl ¼ J J el , npl ¼ n nel . In the above calculations, all the elastic crack, elastic non-crack and plastic non-crack terms have been evaluated from the finite element results.
θ = 80°
θ = 100° Pipe-4
693
θ = 60° θ = 100°
θ = 80° Pipe-5
θ = 40°
θ = 60°
4000 3500 Jpl (N/mm)
θ = 40°
3000 2500
θ = 20° Pipe-4
θ = 20° θ = 40° θ = 60° θ = 80°
2000 θ = 20° Pipe-5
1500
: PIPE4 : PIPE4 : PIPE4 : PIPE4
θ = 100° : PIPE4 θ = 20° : PIPE5 θ = 40° : PIPE5 θ = 60° : PIPE5 θ = 80° : PIPE5
1000 500
θ = 100° : PIPE5
0 2
0
4
6
8
10
12
14
Uφ, pl = ∫Mdφpl (N.mm2)
16 x 106
Fig. 6. Jpl (from FEM) versus Uf,pl (from FEM) for pipes 4 and 5.
1
x 10-3 PIPE1 PIPE2 PIPE3 PIPE4
0.8
PIPE5 PIPE6
ηφ, pl
0.6
0.4
0.2
0 20
30
40
50 60 70 80 Half Crack Angle θ (Degree)
90
100
Fig. 7. The value of Zf,pl (FEM) versus half-crack angle y for all pipes.
110
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5.1. Calculation of Zf,pl from FE analyses The value of Uf,pl, i.e. the integral given in Eq. (3), has been calculated using the trapezoidal rule from the finite element results for all six pipe cases. The value of Jpl has been plotted against the calculated Uf,pl for all the pipes. Sample plot for pipe-1, -3, -4 and -5 (which cover all the geometry and crack sizes) are shown in Figs. 4–6. These figures show that the curves are linear except for initial small Uf,pl values. In other words the slopes of these curves are constant and independent of load. This further proves the load separability discussed in Eq. (5).
The calculated value of Zf,pl (from the slope of the Jpl vs. Uf,pl curve) have been plotted in Fig. 7. The Zf,pl is also calculated using Eq. (6) for all cases. Fig. 8 compares RtZf,pl calculated using Eq. (6) (i.e., RtZf;pl ¼ 0:5 h0 ðyÞ= hðyÞ) and using the finite element results, against half-crack angle y. This figure shows that Zf,pl evaluated from the finite element analysis is in good agreement with Zf,pl calculated from the limit load based Eq. (6). After this satisfactory benchmarking of the Zf,pl evaluation procedure, the value of Zn,pl (based on M–npl curve) was calculated using a similar procedure as discussed next.
1.2 FEM : PIPE1 FEM : PIPE2 FEM : PIPE3 FEM : PIPE4 FEM : PIPE5 FEM : PIPE6 EQUATION No. 6
1.1 1
R*t*ηφ, pl
0.9 0.8 0.7 0.6 0.5 0.4 0.3 20
30
40
50 60 70 80 Half Crack Angle 'θ' (Degree)
90
100
110
Fig. 8. The comparison of RtZf,pl versus half-crack angle y calculated from FEM and using the literature Eq. (5) for all pipes.
8000
θ = 30° : Pipe-1 θ = 40° : Pipe-1
7000
θ = 100°
θ = 50° : Pipe-1 θ = 60° : Pipe-1 θ = 70° : Pipe-1 θ = 80° : Pipe-1 θ = 90° : Pipe-1
6000
Jpl (N/mm)
5000
θ = 100° : Pipe-1
4000
θ = 30°
3000 2000 1000 0
0
0.5
1
1.5 Uν = ∫Mdνpl pl
2 (N.mm2)
Fig. 9. Jpl (from FEM) versus Un,pl (from FEM) for pipe 1.
2.5
3 x 109
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6000
695
θ = 100° Pipe-4
θ = 40° Pipe-4
5000
θ = 100° Pipe-5
Jpl (N/mm)
4000
3000 θ = 20° θ = 40° θ = 60° θ = 80°
2000
: PIPE4 : PIPE4 : PIPE4
θ = 100° : PIPE4 θ = 20° : PIPE5 θ = 40° : PIPE5
θ = 20° Pipe-5
1000
0
: PIPE4
θ = 60° : PIPE5 θ = 80° : PIPE5 θ = 100° : PIPE5
0
0.5
1
1.5
2
2.5
Uν, pl = ∫Mdνpl (N.mm3)
x 109
Fig. 10. Jpl (from FEM) versus Un,pl (from FEM) for pipes 4 and 5.
12000 θ = 40° Pipe-6
Jpl (N/mm)
10000
θ = 100° Pipe-6
θ = 110° Pipe-3
8000
θ = 40° Pipe-3
6000
θ = 40° : PIPE3 θ = 50° : PIPE3 θ = 60° : PIPE3 θ = 70° : PIPE3
4000
θ =80° : PIPE3 θ = 90° : PIPE3 θ = 100° : PIPE3 θ = 110° : PIPE3 θ = 20° : PIPE6 θ = 40° : PIPE6 θ = 60° : PIPE6 θ = 80° : PIPE6
2000
0 0
θ = 100° : PIPE6
0.5
1
1.5
2
2.5
Uν, pl = ∫Mdνpl (N.mm3)
3
3.5 x 1010
Fig. 11. Jpl (from FEM) versus Un,pl (from FEM) for pipes 3 and 6.
5.2. Calculation of Zn,pl from FE analyses To verify the relation between Jpl and npl, Un,pl, i.e. Eq. (22), has been evaluated. Figs. 9–11 plot the Jpl versus Un,pl for the pipe-1 to -6 geometries for the crack angles analysed. Figs. 9–11 clearly verify the linear relation between the plastic J-integral and the area under the M–npl curve, i.e. given by Eq. (20). The value of Zn,pl are plotted against the half-crack angle y in Fig. 12 for all six pipes having different crack angles. Eq. (27) has been validated by comparing the term RoRtZn,pl as evaluated directly from
the FE results and by the equation Ro RtZn;pl ¼
1 1 h0 ðyÞ R y 1þ sin . 2 hðyÞ Ro 2
The value of RoRtZn,pl using Eq. (27) as given above, was evaluated for two bounding R/t values, viz, R=t ¼ 5 and 10. The evaluated values are plotted against y in Fig. 13. The figure demonstrates good agreement between RoRtZn,pl evaluated from finite element analyses and Eq. (27). This also indirectly validates the function f.
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6
x 10-6
PIPE1 PIPE2 PIPE3 PIPE4 PIPE5 PIPE6
5
ην, pl
4
3
2
1
0 20
30
40
50 60 70 80 Half Crack Angle θ (Degree)
90
100
110
Fig. 12. The value of Zn,pl (FEM) versus half-crack angle y for all pipes.
0.7 0.65 FEM : PIPE1 FEM : PIPE2 FEM : PIPE3 FEM : PIPE4 FEM : PIPE5 FEM : PIPE6 EQUATION NO. 27 (R/t=10) EQUATION NO. 27 (R/t=5)
0.6
RO*R*t*ην, pl
0.55 0.5 0.45 0.4 0.35 0.3 0.25 20
30
40
50 60 70 80 Half Crack Angle 'θ' (Degree)
90
100
110
Fig. 13. The comparison of RoRtZn,pl versus half-crack angle y calculated from FEM and using Eq. (27) for all pipes.
6. Validations using pipe fracture tests In this section, validation have been made with respect to three-pipe fracture tests, carried out on circumferential through wall cracked pipes of 200 mm nominal diameter and schedule 100, by BARC as a part of a component integrity testing programme [16]. These tests were conducted under four point bending. The test specimens were made of SA333Gr.6 carbon steel that is the same as the primary heat transport (PHT) piping material of the Indian pressurized heavy water reactor (PHWR). The pipe
geometry for all three tests is nearly the same and represented by pipe-1 (Table 1) in finite element analyses. The geometry and material details of these tests are given in Table 2. The load and CMOD for pipe-1 have been calculated, corresponding to the experimentally known data of LLD (corrected for machine/set up compliance) and crack size, from the finite element analyses results. Then the load and CMOD evaluated from FEM and the load and CMOD obtained from the tests are plotted against the LLD in Figs. 14 and 15. These figures show reasonable agreement
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Table 2 Pipe material and geometry details for monotonic fracture experiments Test name
Outer radius Ro (mm)
Thickness t (mm)
Outer span Z (mm)
Inner span L (mm)
Crack angles (2y1) after fatigue precracking
Young’s modulus E (N/mm2)
Material’s s–e curve
Yield stress so
SPBMTWC81 SPBMTWC82 SPBMTWC83
109.5 109.5 109.5
15.15 15.10 15.29
4000 4000 4000
1480 1480 1480
65.6 93.9 126.4
203,000 203,000 203,000
Given in Fig. 3a
288 288 288
2.5
x 105
LOAD 'P' (N)
2
1.5
1 TEST DATA : SPBMTWC81 FEM : Load determined corresponding to SPBMTWC81 test data (∆, θ) TEST DATA : SPBMTWC82 FEM : Load determined corresponding to SPBMTWC82 test data (∆, θ) TEST DATA : SPBMTWC83 FEM : Load determined corresponding to SPBMTWC83 test data (∆, θ)
0.5
0
0
20
40
60
80 LLD (mm)
100
120
140 150
Fig. 14. Comparison of FEM predicted (using stationary crack FEM results) and experimental load versus load line displacement.
40 TEST DATA : SPBMTWC81 FEM : CMOD determined corresponding to SPBMTWC81 test data (∆, θ) TEST DATA : SPBMTWC82 FEM : CMOD determined corresponding to SPBMTWC82 test data (∆, θ) TEST DATA : SPBMTWC83 FEM : CMOD determined corresponding to SPBMTWC83 test data (∆, θ)
35
CMOD 'ν' (mm)
30 25 20 15 10 5 0
0
20
40
60
80 LLD (mm)
100
120
140 150
Fig. 15. Comparison of FEM predicted (using stationary crack FEM results) and experimental CMOD versus load line displacement.
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698
8000
8000 Test : SPBMTWC81 Test : SPBMTWC82 Test : SPBMTWC83
7000
6000 Jr (νpl) (N/mm)
Jr (φpl) (N/mm)
6000 5000 4000 3000
5000 4000 3000
2000
2000
1000
1000
0 (a)
Test : SPBMTWC81 Test : SPBMTWC82 Test : SPBMTWC83
7000
0
10 20 30 40 Half Crack Extension ∆a (mm)
0
50 (b)
0
10 20 30 40 Half Crack Extension ∆a (mm)
50
Fig. 16. Comparison of J-integral versus crack extension Da, evaluated from M–npl record and from M–fpl record for tests on 219 mm outer diameter pipes. (a) J–R curve evaluated from M–fpl record; (b) J–R curve evaluated from M–npl record.
8000 From M-φ
: SPBMTWC81
From M-φ
: SPBMTWC82
From M-φ
: SPBMTWC83
pl
7000
pl pl
From M-ν pl : SPBMTWC81
6000
From M-ν pl : SPBMTWC82
Jm,pl (N/mm)
From M-ν pl : SPBMTWC83
5000 4000 3000 2000 1000 0
0
5
10
15 20 25 30 Half Crack Extension ∆a (mm)
35
40
45
Fig. 17. Jm,pl-integral versus crack extension Da for tests on 219 mm outer diameter pipes.
between the finite element evaluated and experimental values. This indirectly validates the proposed equations for Zn,pl. Further the J–R curve has been evaluated using the proposed value of Zn,pl, gn,pl and the area under the M–npl curve in Eq. (33) and has been plotted in Fig. 16a. The J–R curve has also been evaluated using the conventional Zf,pl, gf,pl and the area under the M–fpl curve in Eq. (8) and has been plotted in Fig. 16b. It can be observed that for SPBMTWC83 test the LLD based J–R curve and the CMOD based J–R curve are in good agreement while for the other two tests i.e. SPBMTWC81 and SPBMTWC82 the LLD based J–R curve is slightly higher than the CMOD based J–R curve. Moreover, the CMOD based
J–R curve has less scatter. Further Ernst’s [20] modified version of the J-integral has been evaluated by both procedures. The modified plastic J integral is defined as Z y qJ pl J m;pl ¼ J pl dy, yo qy Dpl Z y ) J m;pl ¼ J pl gpl J pl dy. yo
The value of Jm,pl evaluated from the M–npl curve and from the M–fpl curve for all three tests are plotted in Fig. 17. This figure shows that all the points lie in a narrow band and are in agreement.
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7. Conclusion
Acknowledgements
The following conclusions have been drawn from the present work.
The authors gratefully acknowledge Mr. J. Chattopadhaya and Mr. T.V. Pavankumar of RSD, BARC, for providing digital data of the pipe fracture experiments for the validation of the work.
1. It is possible to evaluate the J–R curve from CMOD data obtained in pipe fracture tests. The following equation is solved iteratively along with the proposed Zn,pl and gn,pl factors for evaluation of the J–R curve. Z
Z
npl
J pl ¼
y
Zn;pl M dnpl þ 0
gn;pl J pl dy, yo
where Zn;pl ¼
Zf;pl ; f ðR; t; yÞ
gn;pl ¼ gf;pl
f0 ; f
f ¼ Ro þ R sin
2.
3.
4.
5.
Zf;pl ¼
gf;pl ¼
y ; 2
1 h0 ðyÞ , 2Rt hðyÞ
h00 ðyÞ , h0 ðyÞ
hðyÞ ¼ cosðy=2Þ 0:5 sinðyÞ.
J-calculation from load-LLD or moment-rotation data requires that displacements or rotation of noncrack geometry and machine compliance be subtracted from total LLD or rotation. In contrast, formulation in terms of CMOD helps in the direct evaluation from measured moment-CMOD data with no need to subtract any non-crack or machine compliance deformations. The proposed Zn,pl and gn,pl factors have been rigorously validated by finite element analysis which cover a wide range of parameters such as crack angle y (20–1101), Ramberg–Osgood parameters n (3–7), a (5–20), radius to thickness ratio R/t (4.98–10.45), thickness t (10–32.3 mm), outer diameter (219–406) and different inner and outer spans. The parameters i.e., R/t and n cover the wide range of pipe sizes and materials used in NPP piping. For one case i.e. pipe-1, the load versus LLD and CMOD versus LLD from finite element analyses have been validated with three monotonic fracture experiments on a similar pipe geometry. This is an indirect validation of the Zn,pl and gn,pl factors. The calculation procedure for Z has been validated with established conventional equation, which is based on the M–fpl record. The J–R curve and modified plastic J-integral ‘Jm,pl’ for these tests have been evaluated using the proposed equation i.e., Zn,pl and gn,pl factor with M–npl curve and compared with the J–R curve based on M–fpl curve.
References [1] Rice JR. A path-independent integral and the approximate analysis of strain concentrations by notches and cracks. J Appl Mech 1968;35:379–86. [2] Hutchinson JW. Singular behaviour at the end of tensile crack in a hardening material. J Mech Phys Solid 1968;16(1):13–31. [3] Rice JR, Rosengren CF. Plain strain deformation near a crack tip in a power law hardening material. J Mech Phys Solid 1968;16(1):1–12. [4] Rice JR, Paris PC, Merkle JG. Some further resulta of J-integral analysis and estimates. In: Progress in flaw growth and fracture toughness testing, ASTM STP-536, 1973. p. 231–45. [5] Hutchinson JW, Paris PC. Stability analysis of J controlled crack growth. In: Elastic-Plastic Fracture, ASTM STP-66, 1979. p. 37–64. [6] Ernst HA, Paris PC, Rossow M, Hutchinson JW. Analysis of load displacement relation to determine J–R curve and tearing instability material properties. In: Smith CW, editor. Fracture Mechanics, ASTM STP-677, 1979. p. 581–99. [7] Ernst HA, Paris PC, landes JD. Estimation of J-integral and tearing modulus T from a single specimen test record. In: Robert R, editor. Fracture mechanics, ASTM STP-743, 1981. p. 476–502. [8] Zahoor A, Kanninen MF. A plastic fracture mechanics prediction of fracture instability in a circumferentially cracked pipe in bending—Part I: J-integral analysis. J Press Vessel Technol Trans ASME 1981;103:352–8. [9] Wilkowski GM, Zahoor A, Kanninen MF. A plastic fracture mechanics prediction of fracture instability in a circumferentially cracked pipe in bending—Part II: Experimental verification on a type 304 stainless steel pipe. J Press Vessel Technol Trans ASME 1981;103:359–65. [10] Roos E, Eisele U, Silcher H. ‘A procedure for the experimental assessment of J-integral by mean of specimen of different geometries. Int J Pres Ves Piping 1986;23:81–93. [11] Chattopadhaya J, Dutta BK, Kushwaha HS. Derivation of ‘g’ parameter from limit load expression of cracked component to evaluate J–R curve. Int J Pres Ves Piping 2001;78:401–27. [12] Kumar V, German MD, Shih CF. An engineering approach for elastic–plastic fracture analysis. EPRI-NP-1931, Project 1287-1, Topical Report, Electric Power Research Institute, Palo Alto, CA, 1981. [13] Zahoor A. Ductile fracture handbook. Vol. I: Circumferential through wall cracks’ Report No. EPRI NP-6301-D, Electric Power Research Institute, Palo Alto, CA, 1989. [14] Smith E. The effect of axial forces on the conservatism of the netsection stress criterion for the failure of cracked stainless steel piping. In: Proceedings of SmiRT—12, paper G05/1, 1993. [15] Smith E. Effect of inertial loadings on the criterion for stable growth of a crack in a piping system. Int J Pres Ves Piping 1996;67:119–25. [16] Chattopadhaya J, Dutta BK, Kushwaha HS. Experimental and analytical study of three point bend specimens and through wall circumferentially cracked straight pipe. Int J Pres Ves Piping 2000;77:455–71. [17] Gupta SK, Bhasin V, Vaze KK, Kushwaha HS. Fracture tests of through wall cracked straight pipe under cyclic loading and comparison with monotonic results. 28 MPA-Seminar, 10 and 11 October 2002. [18] Gupta SK, Bhasin V, Vaze KK, Kushwaha HS. Effect of earthquake loads on LBB assessment of high energy piping. In: First National Conference on Nuclear Reactor Safety, November 25–27, 2002, Mumbai, India. [19] ADINA, ADINA Version 8.0, September 2002. [20] Ernst HA. Further developments on modified J, JM. Nonlinear Fracture Mechanics, vol. II. ASTM STP 995, Philadelphia, 1989.
International Journal of Pressure Vessels and Piping 83 (2006) 686–699 www.elsevier.com/locate/ijpvp
Derivation of J-resistance curve for through wall cracked pipes from crack mouth opening displacement Suneel K. Guptaa,, Vivek Bhasina, K.K. Vazea, A.K. Ghosha, H.S. Kushwahaa, S. Chapuliotb, S. Marieb, I. Kayserb a Bhabha Atomic Research Center, Reactor Safety Division, Mumbai 400 085, India Structural Integrity and Normalisation Laboratory, CEA/DEN/DM25/SEMT/LISN, CEA-Saclay, France
b
Received 30 July 2005; received in revised form 15 May 2006; accepted 20 May 2006
Abstract In this paper, factors for calculating the J-resistance curve from the measured crack mouth opening displacement (CMOD) record are derived for circumferential through wall cracked straight pipes subjected to pure bending. Rigorous elastic plastic finite element analyses of around 125 circumferential through wall cracked straight pipes subjected to four point bending have been carried out to validate the proposed equations. The factors have also been validated with a few fracture experimental results on straight pipes. For these tests the J–R curve has been evaluated using the conventional as well as using the proposed factors along with the moment-CMOD record. r 2006 Elsevier Ltd. All rights reserved. Keywords: J-integral; J–R curve; Z factor; g factor; Pipe; Crack; Rotation; CMOD
1. Introduction The J-integral is an important and established parameter to characterise the fracture behaviour of cracked structures made of ductile material. Since the introduction of the Jintegral by Rice [1], Hutchinson [2] and Rice and Rosengren [3], many researcher, Rice et al. [4], Hutchinson and Paris [5], Ernst et al. [6,7], Zahoor and Kanninen [8], Wilkowski et al. [9], Roos et al. [10] and Chattopadhayay et al. [11], have contributed to establishing procedures for experimental evaluation of the J-integral from the area under the load versus plastic part of load line displacement record. The J-integral evaluation procedure is based on Z and g factors. The plastic part of the J-integral is calculated by multiplying the Zpl factor by the area under the load versus plastic load line displacement curve obtained by conducting experiments on the cracked structure. However, for a growing crack, the above J-integral is modified using the gpl factor. The existence of these scaling factors relies upon the hypothesis of load separation, i.e. the load Corresponding author. Tel.: +91 22 559 15 28; fax: +91 22 50 51 51.
E-mail address: [email protected] (S.K. Gupta). 0308-0161/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2006.05.004
imposed on the cracked structure is equal to the product of two functions, one represents the cracked geometry and is independent of material and the other represents the deformation behaviour and depends on material only. Nuclear power plant (NPP) piping is normally made of ductile materials. Their rigorous fracture assessment is essential for demonstration of leak-before-break (LBB) capability and safety margins against fracture failure, under design basis accidents such as safe shutdown earthquake (SSE) loading. In general J-tearing analysis is performed for fracture assessment which in turn requires knowledge of material fracture properties i.e. J–R curve. The J-integral calculation is generally made from the test measured load and load line displacement (LLD) data. Hence, accuracy of the J-integral relies on the accurate measurement of the load and LLD, on a cracked specimen or component. Unlike small specimen testing, in large size component testing, such as a through wall cracked straight pipe under four point bending, the accuracy of the LLD measurement is influenced to a greater extent by various experimental factors such as machine/set up compliance, friction and denting at the loading and support points etc. As a result there is some difference between measured and
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Nomenclature A crack area E Young’s modulus J, Jel, Jpl total, elastic and plastic J-integral M total applied moment Mo limit moment n Ramberg–Osgood equation parameter P total applied load R, Ri, Ro mean, inside and outside radius of pipe crosssection t wall thickness of pipe Upl plastic part of strain energy Uf,pl plastic strain energy i.e. area under the moment vs. plastic load-point-rotation curve Un,pl area under the moment vs. plastic crack mouth opening displacement curve i.e. an indirect measure of plastic strain energy Greek letters a y
Ramberg–Osgood equation parameter half-circumferential crack angle
actual LLD. The testing machine and associated structure, which transfers load to testing component, may undergo significant deformation. Hence, in such testing, it becomes necessary to correct the measured LLD for machine compliance by subtracting the displacements caused by deformation of testing machine members and associated structure. In pipe fracture testing under bending load, the total LLD contains significant contribution of un-cracked component geometry deformation. J-calculation from load-LLD data requires that displacement of un-cracked component geometry be subtracted from the total LLD. Many times displacement due to un-cracked component geometry and machine compliance may not be readily available, which need to be evaluated with additional tests or calculations. In contrast, J-estimation from load and crack mouth opening displacement (CMOD) data is direct and there is no need to subtract any un-cracked or machine compliance CMOD. The CMOD does not include un-cracked deformation contribution and represents the crack-system deformation behaviour only. CMOD is independent from machine compliance since it is directly measured on specimen/component. The CMOD is normally measured using a clip gauge, which gives precise measurement of the deformation behaviour of the component during the experiment. In this study, the J-integral is evaluated using load versus CMOD data. The only difference in the evaluation procedure as compared to the procedure based on the moment-load-point-rotation (M–fpl) curve is the Z and g factors. Here new Zn,pl and gn,pl factors are derived which
687
D total load-line-displacement Del, Dpl elastic and plastic load-line-displacement due to crack-system f total load-point rotation fel, fpl elastic and plastic load-point rotation due the crack n, nel, npl total, elastic and plastic crack mouth opening displacement (CMOD) Zpl a function to multiply the area under the load vs. plastic load-point- deflection curve to get the plastic component of the J-integral gpl a function to correct the plastic J-integral evaluated by Zpl function in crack growth situations Zf,pl Zpl factor based on M–fpl record gf,pl gpl factor based on M–fpl record Zn,pl Zpl factor based on M–npl record gn,pl gpl factor based on M–npl record s stress so reference stress ¼ yield stress e strain corresponding to s eo reference strain (strain corresponding to so) i.e. so/E
are based on the moment-CMOD (M–npl) curve. The derivation of these factors is discussed in Section 3. To validate the proposed Zn,pl and gn,pl factors around 40 elastic and 125 elastic plastic finite element analyses have been performed on circumferential through wall cracked straight pipes subjected to four point bending and discussed in Section 5. Initially the Zf,pl factor has been evaluated from the M, fpl and Jpl curves obtained from finite element analyses. The Zf,pl factor as evaluated using FEM have been validated with equations in the literature. The objective of this exercise was to qualify the analyses and calculation procedures of the Zpl factor from FE data. Then the Zpl based on plastic CMOD (npl) is evaluated. Finally new Zn,pl and gn,pl factors validation with fracture test results on straight pipes have been discussed in Section 6. For these tests the J-integral and modified J integral have been evaluated using conventional as well as the new Z and g factors. 2. Theoretical background for gpl and cpl The plastic part of the J integral is defined as J pl ¼
dU pl , dA
(1)
where Upl is the plastic part of the strain energy and A is crack surface area. For a pipe of mean radius R, thickness t and having a through wall crack subtending an angle 2y at the centre (Fig. 1), the average Jpl given by Eq. (1) can be
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688
∆
P
R t Through wall Crack
2θ
L Z
Fig. 1. Schematic diagram of the circumferential through wall cracked straight pipe subjected to four point bend loading.
expressed as J pl ¼
1 dU pl . 2Rt dy
(2)
The plastic strain energy Upl can be obtained from the area under the moment versus plastic rotation (M–fpl) curve or load versus plastic LLD (P–Dpl) curve. Then, Upl is denoted as Uf,pl and is calculated by Z fpl Z Dpl U f;pl ¼ M dfpl ¼ P dDpl . (3) 0
0
Zahoor and Kanninen [8] have derived an expression for determination of Jpl from M–fpl or P–Dpl curves obtained from pipe fracture tests. It is based on dimensional analysis and on an alternative but equivalent definition of the J integral given by Rice et al. [4]. The resulting expression for evaluating Jpl for a stationary i.e. non-growing crack is Z fpl Z Dpl J pl ¼ Zf;pl M dfpl ¼ Zf;pl P dDpl . (4) 0
0
In the derivation of the Zf,pl, Zahoor and Kanninen [8] assumed that all the plasticity due to the presence of the crack is confined to the crack section. The assumption relies on the hypothesis that the relation between moment (M), crack angle (y) and plastic crack rotation (fpl) can be expressed in variable separable form. This form is given as M ¼ hðyÞgðfpl Þ,
(5)
where h(y) is a function of y alone and does not depend on material properties. The function g(fpl) represents the deformation behaviour and depends on material only. Based on this the resulting expression for Zf,pl as derived by Zahoor and Kanninen [8] is Zf;pl ¼
1 h0 ðyÞ , 2Rt hðyÞ
(6)
where h0 (y) is the derivative of h(y) with respect to y. For the circumferentially through wall cracked pipe geometry, the h(y) function is hðyÞ ¼ cosðy=2Þ 0:5 sinðyÞ.
(7)
The deformation function g(fpl) is material dependent and varies from one material to another. The general equation to evaluate Jpl from experimental data (M–fpl curve) when there is crack growth is as follows [8]: Z fpl Z y Zf;pl M dfpl þ gf;pl J pl dy, (8) J pl ¼ 0
yo
where gn,pl is gf;pl ¼
h00 ðyÞ . h0 ðyÞ
(9)
Here h00 (y) is the second derivative of h(y) with respect to y. The deformations fpl and Dpl are the plastic parts of the crack-system deformations of the pipe under four point bending. The crack-system deformation (i.e. the increased deformation due to the presence of the crack) is evaluated after subtracting the non-crack system deformation from the total deformation. 3. Derivation of gn,pl (CMOD-based) For a particular pipe geometry and material, Kumar et al. [12] have shown that Jpl, fpl, and npl are functions of crack size y and moment M expressed as J pl ¼ J pl ðM; yÞ,
(10)
fpl ¼ fpl ðM; yÞ,
(11)
npl ¼ npl ðM; yÞ.
(12)
From Eqs. (10) and (12), the Jpl can be represented as a function of y and npl J pl ¼ J pl ðnpl ; yÞ.
(13)
Similarly, from Eqs. (11) and (12), the npl can be represented as a function of y and fpl: npl ¼ npl ðfpl ; yÞ.
(14)
For a through wall circumferentially cracked pipe under pure bending, Kumar et al. [12] and Zahoor [13] have given expressions for fpl and npl. For a particular material, the dependence of the plastic rotation fpl and the plastic
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CMOD npl on loading is given as M n fpl / f f ðR; t; yÞ , Mo M n npl / f n ðR; t; yÞ , Mo
(15)
(16)
where ff and fn are functions of pipe size and crack size while Mo is a reference moment, which is usually taken as the limit load for the cracked pipe. Here n is strainhardening parameter of Ramberg–Osgood (R–O) equation representing the material’s stress–strain behaviour. Eqs. (15) and (16) show that the ratio npl/fpl is independent of applied loading and function of only pipe and crack size. Based on this fact following expression is postulated npl ¼ f ðR; t; yÞfpl .
(17)
Then Eq. (4), the expression for Jpl for a non-growing crack, can be rewritten in term of npl, as Z fpl Z npl dfpl M dfpl ¼ Zf;pl M (18) dnpl . J pl ¼ Zf;pl dnpl 0 0 For a stationary crack, dfpl/dnpl, from Eq. (17), is expressed as dfpl 1 . ¼ f ðR; t; yÞ dnpl
(19)
Therefore, Jpl, for a non-growing crack case can be expressed as Z npl M dnpl , (20) J pl ¼ Zn;pl
presence of the crack). A similar hypothesis was used by Smith [14,15] in which fc was related to the crack tip opening displacement CTOD (dpl) about the shifted neutral axis at the cracked section. dpl ¼ fc ðdistance of crack tip from neutral axisÞ, dpl ¼ fc Rðcos y þ cos bÞ.
Eq. (20) gives a linear relation between Jpl and the area under the M–npl curve. The area under the M–npl curve will be denoted by Un,pl. Un,pl does not directly represents the plastic work input but is given by Z npl M dnpl . (22) U n;pl ¼
ð23Þ
In the above equation, the angle b gives the neutral axis position as shown in Fig. 2. For elastic–perfectly plastic material, a free body analysis of the cracked section shows that the point of stress inversion i.e. the angle b is given by py . (24) 2 The logic given above can be extended to CMOD (npl), and Eq. (23), can also be written
b¼
npl ¼ fpl ðdistance of middle of the crack mouth from shifted neutral axisÞ. From Fig. 2, the distance of middle of the crack mouth has been calculated as ‘Ro+R cos b’, then npl ¼ fpl ðRo þ R cos bÞ, y npl ¼ fpl Ro þ R sin . 2
ð25Þ
On comparing Eqs. (17) and (25) the function f(R, t, y) is given as R y sin . (26) f ðR; t; yÞ ¼ Ro 1 þ Ro 2 From Eqs. (6) and (21), the Zn,pl function is given as
0
where Zn,pl is a CMOD based function and is expressed in term of Zf,pl using Eqs. (18)–(20). Zf;pl . (21) Zn;pl ¼ f ðR; t; yÞ
689
Zn;pl ¼
1 h0 ðyÞ 1 . 2Rt hðyÞ f ðR; t; yÞ
(27)
In the above equation the function h(y) is given by Eq. (7) and the function f(R, t, y) is given by Eq. (26). It may be noted that the above hypothesis is perfectly valid only if the material is elastic perfectly plastic and the gross state of stress is uniaxial. If the material is strain Point of CMOD measurement
0
It has been shown that fpl and npl as well as Zf,pl and Zn,pl are linearly related by a function f(R, t, y). This relationship can be appreciated further using the assumption that the deformation discontinuity at the cracked section is entirely in the form of rotation. The rotation of the cracked section fc is a pure manifestation of the applied load point rotation (crack-system) in the pipe subjected to pure bending (four-point bending) which implies that the crack system rotations about the neutral axis will remain constant between the cracked section and the load point. Thus, fc (i.e. the plastic part of the cracked section rotation about the shifted neutral axis) will be equal to fpl (i.e. the increased plastic rotation at the load point due to the
R+0.5t+RCosβ t 2θ R(Cosθ + Cosβ)
R C
C
Pipe Axis β
N
Neutral Axis
N
Fig. 2. Schematic diagram of the cracked section of the circumferential through wall cracked straight pipe.
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690
hardening in nature or if the gross state of stress is highly triaxial then the position of the neutral axis is not exactly the same as calculated above. However, the difference is not likely to be high. The finite element verification of such an assumption will be discussed in Section 6.
For a growing crack the J-integral expression is modified using the g factor. Prior to the initiation of crack growth, the plastic contribution of the J-integral is calculated from Eq. (20). After crack initiation and during propagation, the crack angle y is no longer constant. So it is necessary to know the evolution of the crack length and this also changes the plastic J-integral. The differential increment of Jpl can be derived from Eq. (13) as qJ pl qJ pl dnpl þ dy ¼ dJ pl;1 þ dJ pl;2 . (28) dJ pl ¼ qnpl y qy npl The increment dJpl,1 is further solved along with Eq. (20) and given as qJ pl dnpl dJ pl;1 ¼ qnpl y Z npl q ¼ Z M dnpl dnpl ¼ Zn;pl M dnpl . ð29Þ qnpl n;pl 0 y In the above equation M is expressed as product of h(y) and g(fpl) as mentioned in Eq. (5). The increment dJpl,2 is Z npl qJ pl q Z dJ pl;2 ¼ dy ¼ M dnpl dy. qy n;pl 0 qy npl npl From Eq. (5), the M is replaced in the above equation leading to Z npl q dJ pl;2 ¼ Zn;pl hðyÞ gðfpl Þ dnpl dy. (30) qy 0 npl As discussed earlier (in Section 3) the function g(fpl) does not depend on crack angle y and purely represents the material deformation behaviour. In the above equation Zn,pl and h(y) are functions of y. Then Eq. (31) can be rewritten as Z npl q ðhðyÞZn;pl Þ gðfpl Þ dnpl dy dJ pl;2 ¼ qy 0 leading to Z npl qZn;pl qhðyÞ þ Zn;pl gðfpl Þ dnpl dy dJ pl;2 ¼ hðyÞ qy qy 0
dJ pl;2 ¼
! Z npl 1 qZn;pl 1 qhðyÞ þ M dnpl dy. Zn;pl Zn;pl qy hðyÞ qy 0
The above equation is rewritten dJ pl;2 ¼ gn;pl J pl dy,
gn;pl ¼
1 qZn;pl 1 qhðyÞ . þ Zn;pl qy hðyÞ qy
(32)
From Eqs. (29) and (31), Eq. (28) is solved and leads to dJ pl ¼ dJ pl;1 þ dJ pl;2 ¼ Zn;pl M dnpl þ gn;pl J pl dy.
4. Derivation of cn,pl (CMOD-based)
and
where gn,pl is defined as
(31)
By integrating the above equation, an equation for the plastic J-integral after crack initiation is obtained and is given as Z npl Z y Zn;pl M dnpl þ gn;pl J pl dy. (33) J pl ¼ 0
yo
By differencing Eq. (27) we get 1 qZn;pl q2 hðyÞ=qy2 1 qhðyÞ 1 qf ðyÞ , ¼ Zn;pl qy hðyÞ qy f ðyÞ qy qhðyÞ=qy i.e. 1 qZn;pl h00 h0 f 0 ¼ 0 . Zn;pl qy h h f
(34)
From Eqs. (9), (32) and (34) the simplified equation for gn,pl can be expressed as gn;pl ¼
h00 f 0 f0 ¼ gf;pl , 0 h f f
(35)
where h0 and h00 are the first and second derivatives of h(y) with respect to y and f0 is the first derivative of f(R, t, y) with respect to y. 5. Finite element validations The finite element (FE) validation of the equations derived in Sections 3 and 4 will be discussed here. For FE validation, six pipe cases were considered and the geometry, material and crack sizes details are given in Table 1. The pipe-1, -2 and -3 geometries represent the pipes used in the primary heat transport (PHT) piping system of Indian pressurized heavy water reactors (PHWRs). These pipes are made of C–Mn steel conforming to material grade SA333Gr.6. In addition their geometries (R/t etc.) are similar, although the nominal pipe size and thickness differ. The material and geometry for pipe-1 and -3 are the same as the monotonic fracture experiments carried out by Bhabha Atomic Research Center (BARC), i.e. Chattopadhay et al. [16] while the pipe-2 geometry and material is the same as the cyclic tearing tests carried out by Gupta et al. [17,18]. The pipe-4, -5 and -6 were selected such to cover different pipe geometries (R/t: 5–10) and material stress–strain relation (strain hardening exponent n: 3–7). The materials were assumed to follow widely used and the well-known R–O constitutive law describing material’s
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Table 1 Pipe material and geometry details used for the finite element analysis Parameter
Outer diameter 2Ro (mm) Thickness t (mm) Radius R (mm) R/t Half-crack angle y (deg.) Ramberg–Osgood n Ramberg–Osgood a Yield stress so (MPa) Outer span Z (mm) Inner span L (mm)
Pipe no. Pipe-1
Pipe-2
Pipe-3
Pipe-4
Pipe-5
Pipe-6
219
219
406
219
219
300
15.18 101.91 6.71 30, 32, 34y100
15.5 101.75 6.56 30, 32, 34y100
32.3 186.85 5.78 40, 42, 34y110
10 104.50 10.45 20, 40y100
20 99.5 4.98 20, 40y100
25 137.50 5.50 20, 40y100
s–e curve given in Fig. 3a
s–e curve given in Fig. 3a
s–e in Fig. 3b
288 4000 1480
288 2500 860
312 5820 1480
3 20 250 4000 1480
5 10 250 4000 1480
7 5 250 4000 1480
Fig. 3. Stress–strain curve for (a) pipes 1 and 2, and (b) pipe 3. (c) Finite element mesh used for analysis of the straight pipe with circumferential through wall crack.
post-yield stress–strain (s–e) relation and is given as n s s ¼ , (36) þa o so so
where so is a reference stress, which can be arbitrary, but typically assumed to be the yield stress of the material, o ¼ so =E is the associated reference strain, and a and n are parameters of above power-law model (R–O equation)
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usually chosen from a best fit of the experimental data. Variability in stress–strain curve was considered by considering different a and n values for pipe-4 through pipe-6. From symmetry considerations one-quarter of the through wall circumferentially cracked pipe has been modelled using 20 noded three-dimensional (3D) isoparametric brick and prismatic elements. The thickness was modelled using three elements. The finite element code ADINA [19] has been used for modelling and analysis. The virtual crack extension method was used for J integral calculations. At the cut surfaces, symmetric boundary
conditions were applied. A typical finite element mesh used, is shown in Fig. 3. A spider type mesh pattern was used to mesh the crack tip zone, as shown in Fig. 3. In these analyses, the Load, LLD, J-integral, CMOD, rotations (f) at loading points, etc. have been evaluated. The data have been further processed and plastic parts of various fracture parameters have been calculated. The crack-system plastic parts of LLD, and f have been evaluated as Dpl ¼ D Del;crack Del;non_crack Dpl;non_crack , fpl ¼ f fel;crack fel;non_crack fpl;non_crack .
8000 θ = 30° : Pipe-1 θ = 40° : Pipe-1 θ = 50° : Pipe-1
7000
θ = 70°
θ = 80°
θ = 60°
θ = 60° : Pipe-1 θ = 70° : Pipe-1 θ = 80° : Pipe-1
6000
Jpl (N/mm)
θ = 90°
θ = 100°
θ = 50°
θ = 40°
θ = 90° : Pipe-1
5000
θ = 100° : Pipe-1
4000
θ = 30°
3000 2000 1000 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Uφ = ∫Mdφpl (N.mm)
2 x 107
pl
Fig. 4. Jpl (from FEM) versus Uf,pl (from FEM) for pipe 1.
14000 θ = 40° : PIPE3
12000
θ = 70° : PIPE3 θ = 80° : PIPE3
Jpl (N/mm)
10000
θ = 90° : PIPE3 θ = 100° : PIPE3 θ = 110° : PIPE3
8000
θ = 80°
θ = 90°
θ = 50° : PIPE3 θ = 60° : PIPE3
θ = 70° θ = 60° θ = 50°
θ = 100°
θ = 110° θ = 40°
6000
4000
2000
0
0
5
10 Uφ, pl = ∫Mdφpl (N.mm2)
Fig. 5. Jpl (from FEM) versus Uf,pl (from FEM) for pipe 3.
15 x 107
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The plastic parts of CMOD and J-integral have been evaluated as
It should be noted here that in certain cases the non-crack system rotations constitute a significant part of total rotation in case of shallow cracked geometry where load can go to a high level. This is due to the significant contributions from non-crack plastic LLD and rotation (Dpl,non_crack and fpl,non_crack) for smaller crack angles (generally y/po0.25). However for deeply cracked geometry, the non-crack rotation or LLD constitute an insignificant part of the total LLD and its effect may be small.
5000 4500
J pl ¼ J J el , npl ¼ n nel . In the above calculations, all the elastic crack, elastic non-crack and plastic non-crack terms have been evaluated from the finite element results.
θ = 80°
θ = 100° Pipe-4
693
θ = 60° θ = 100°
θ = 80° Pipe-5
θ = 40°
θ = 60°
4000 3500 Jpl (N/mm)
θ = 40°
3000 2500
θ = 20° Pipe-4
θ = 20° θ = 40° θ = 60° θ = 80°
2000 θ = 20° Pipe-5
1500
: PIPE4 : PIPE4 : PIPE4 : PIPE4
θ = 100° : PIPE4 θ = 20° : PIPE5 θ = 40° : PIPE5 θ = 60° : PIPE5 θ = 80° : PIPE5
1000 500
θ = 100° : PIPE5
0 2
0
4
6
8
10
12
14
Uφ, pl = ∫Mdφpl (N.mm2)
16 x 106
Fig. 6. Jpl (from FEM) versus Uf,pl (from FEM) for pipes 4 and 5.
1
x 10-3 PIPE1 PIPE2 PIPE3 PIPE4
0.8
PIPE5 PIPE6
ηφ, pl
0.6
0.4
0.2
0 20
30
40
50 60 70 80 Half Crack Angle θ (Degree)
90
100
Fig. 7. The value of Zf,pl (FEM) versus half-crack angle y for all pipes.
110
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5.1. Calculation of Zf,pl from FE analyses The value of Uf,pl, i.e. the integral given in Eq. (3), has been calculated using the trapezoidal rule from the finite element results for all six pipe cases. The value of Jpl has been plotted against the calculated Uf,pl for all the pipes. Sample plot for pipe-1, -3, -4 and -5 (which cover all the geometry and crack sizes) are shown in Figs. 4–6. These figures show that the curves are linear except for initial small Uf,pl values. In other words the slopes of these curves are constant and independent of load. This further proves the load separability discussed in Eq. (5).
The calculated value of Zf,pl (from the slope of the Jpl vs. Uf,pl curve) have been plotted in Fig. 7. The Zf,pl is also calculated using Eq. (6) for all cases. Fig. 8 compares RtZf,pl calculated using Eq. (6) (i.e., RtZf;pl ¼ 0:5 h0 ðyÞ= hðyÞ) and using the finite element results, against half-crack angle y. This figure shows that Zf,pl evaluated from the finite element analysis is in good agreement with Zf,pl calculated from the limit load based Eq. (6). After this satisfactory benchmarking of the Zf,pl evaluation procedure, the value of Zn,pl (based on M–npl curve) was calculated using a similar procedure as discussed next.
1.2 FEM : PIPE1 FEM : PIPE2 FEM : PIPE3 FEM : PIPE4 FEM : PIPE5 FEM : PIPE6 EQUATION No. 6
1.1 1
R*t*ηφ, pl
0.9 0.8 0.7 0.6 0.5 0.4 0.3 20
30
40
50 60 70 80 Half Crack Angle 'θ' (Degree)
90
100
110
Fig. 8. The comparison of RtZf,pl versus half-crack angle y calculated from FEM and using the literature Eq. (5) for all pipes.
8000
θ = 30° : Pipe-1 θ = 40° : Pipe-1
7000
θ = 100°
θ = 50° : Pipe-1 θ = 60° : Pipe-1 θ = 70° : Pipe-1 θ = 80° : Pipe-1 θ = 90° : Pipe-1
6000
Jpl (N/mm)
5000
θ = 100° : Pipe-1
4000
θ = 30°
3000 2000 1000 0
0
0.5
1
1.5 Uν = ∫Mdνpl pl
2 (N.mm2)
Fig. 9. Jpl (from FEM) versus Un,pl (from FEM) for pipe 1.
2.5
3 x 109
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6000
695
θ = 100° Pipe-4
θ = 40° Pipe-4
5000
θ = 100° Pipe-5
Jpl (N/mm)
4000
3000 θ = 20° θ = 40° θ = 60° θ = 80°
2000
: PIPE4 : PIPE4 : PIPE4
θ = 100° : PIPE4 θ = 20° : PIPE5 θ = 40° : PIPE5
θ = 20° Pipe-5
1000
0
: PIPE4
θ = 60° : PIPE5 θ = 80° : PIPE5 θ = 100° : PIPE5
0
0.5
1
1.5
2
2.5
Uν, pl = ∫Mdνpl (N.mm3)
x 109
Fig. 10. Jpl (from FEM) versus Un,pl (from FEM) for pipes 4 and 5.
12000 θ = 40° Pipe-6
Jpl (N/mm)
10000
θ = 100° Pipe-6
θ = 110° Pipe-3
8000
θ = 40° Pipe-3
6000
θ = 40° : PIPE3 θ = 50° : PIPE3 θ = 60° : PIPE3 θ = 70° : PIPE3
4000
θ =80° : PIPE3 θ = 90° : PIPE3 θ = 100° : PIPE3 θ = 110° : PIPE3 θ = 20° : PIPE6 θ = 40° : PIPE6 θ = 60° : PIPE6 θ = 80° : PIPE6
2000
0 0
θ = 100° : PIPE6
0.5
1
1.5
2
2.5
Uν, pl = ∫Mdνpl (N.mm3)
3
3.5 x 1010
Fig. 11. Jpl (from FEM) versus Un,pl (from FEM) for pipes 3 and 6.
5.2. Calculation of Zn,pl from FE analyses To verify the relation between Jpl and npl, Un,pl, i.e. Eq. (22), has been evaluated. Figs. 9–11 plot the Jpl versus Un,pl for the pipe-1 to -6 geometries for the crack angles analysed. Figs. 9–11 clearly verify the linear relation between the plastic J-integral and the area under the M–npl curve, i.e. given by Eq. (20). The value of Zn,pl are plotted against the half-crack angle y in Fig. 12 for all six pipes having different crack angles. Eq. (27) has been validated by comparing the term RoRtZn,pl as evaluated directly from
the FE results and by the equation Ro RtZn;pl ¼
1 1 h0 ðyÞ R y 1þ sin . 2 hðyÞ Ro 2
The value of RoRtZn,pl using Eq. (27) as given above, was evaluated for two bounding R/t values, viz, R=t ¼ 5 and 10. The evaluated values are plotted against y in Fig. 13. The figure demonstrates good agreement between RoRtZn,pl evaluated from finite element analyses and Eq. (27). This also indirectly validates the function f.
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6
x 10-6
PIPE1 PIPE2 PIPE3 PIPE4 PIPE5 PIPE6
5
ην, pl
4
3
2
1
0 20
30
40
50 60 70 80 Half Crack Angle θ (Degree)
90
100
110
Fig. 12. The value of Zn,pl (FEM) versus half-crack angle y for all pipes.
0.7 0.65 FEM : PIPE1 FEM : PIPE2 FEM : PIPE3 FEM : PIPE4 FEM : PIPE5 FEM : PIPE6 EQUATION NO. 27 (R/t=10) EQUATION NO. 27 (R/t=5)
0.6
RO*R*t*ην, pl
0.55 0.5 0.45 0.4 0.35 0.3 0.25 20
30
40
50 60 70 80 Half Crack Angle 'θ' (Degree)
90
100
110
Fig. 13. The comparison of RoRtZn,pl versus half-crack angle y calculated from FEM and using Eq. (27) for all pipes.
6. Validations using pipe fracture tests In this section, validation have been made with respect to three-pipe fracture tests, carried out on circumferential through wall cracked pipes of 200 mm nominal diameter and schedule 100, by BARC as a part of a component integrity testing programme [16]. These tests were conducted under four point bending. The test specimens were made of SA333Gr.6 carbon steel that is the same as the primary heat transport (PHT) piping material of the Indian pressurized heavy water reactor (PHWR). The pipe
geometry for all three tests is nearly the same and represented by pipe-1 (Table 1) in finite element analyses. The geometry and material details of these tests are given in Table 2. The load and CMOD for pipe-1 have been calculated, corresponding to the experimentally known data of LLD (corrected for machine/set up compliance) and crack size, from the finite element analyses results. Then the load and CMOD evaluated from FEM and the load and CMOD obtained from the tests are plotted against the LLD in Figs. 14 and 15. These figures show reasonable agreement
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Table 2 Pipe material and geometry details for monotonic fracture experiments Test name
Outer radius Ro (mm)
Thickness t (mm)
Outer span Z (mm)
Inner span L (mm)
Crack angles (2y1) after fatigue precracking
Young’s modulus E (N/mm2)
Material’s s–e curve
Yield stress so
SPBMTWC81 SPBMTWC82 SPBMTWC83
109.5 109.5 109.5
15.15 15.10 15.29
4000 4000 4000
1480 1480 1480
65.6 93.9 126.4
203,000 203,000 203,000
Given in Fig. 3a
288 288 288
2.5
x 105
LOAD 'P' (N)
2
1.5
1 TEST DATA : SPBMTWC81 FEM : Load determined corresponding to SPBMTWC81 test data (∆, θ) TEST DATA : SPBMTWC82 FEM : Load determined corresponding to SPBMTWC82 test data (∆, θ) TEST DATA : SPBMTWC83 FEM : Load determined corresponding to SPBMTWC83 test data (∆, θ)
0.5
0
0
20
40
60
80 LLD (mm)
100
120
140 150
Fig. 14. Comparison of FEM predicted (using stationary crack FEM results) and experimental load versus load line displacement.
40 TEST DATA : SPBMTWC81 FEM : CMOD determined corresponding to SPBMTWC81 test data (∆, θ) TEST DATA : SPBMTWC82 FEM : CMOD determined corresponding to SPBMTWC82 test data (∆, θ) TEST DATA : SPBMTWC83 FEM : CMOD determined corresponding to SPBMTWC83 test data (∆, θ)
35
CMOD 'ν' (mm)
30 25 20 15 10 5 0
0
20
40
60
80 LLD (mm)
100
120
140 150
Fig. 15. Comparison of FEM predicted (using stationary crack FEM results) and experimental CMOD versus load line displacement.
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8000
8000 Test : SPBMTWC81 Test : SPBMTWC82 Test : SPBMTWC83
7000
6000 Jr (νpl) (N/mm)
Jr (φpl) (N/mm)
6000 5000 4000 3000
5000 4000 3000
2000
2000
1000
1000
0 (a)
Test : SPBMTWC81 Test : SPBMTWC82 Test : SPBMTWC83
7000
0
10 20 30 40 Half Crack Extension ∆a (mm)
0
50 (b)
0
10 20 30 40 Half Crack Extension ∆a (mm)
50
Fig. 16. Comparison of J-integral versus crack extension Da, evaluated from M–npl record and from M–fpl record for tests on 219 mm outer diameter pipes. (a) J–R curve evaluated from M–fpl record; (b) J–R curve evaluated from M–npl record.
8000 From M-φ
: SPBMTWC81
From M-φ
: SPBMTWC82
From M-φ
: SPBMTWC83
pl
7000
pl pl
From M-ν pl : SPBMTWC81
6000
From M-ν pl : SPBMTWC82
Jm,pl (N/mm)
From M-ν pl : SPBMTWC83
5000 4000 3000 2000 1000 0
0
5
10
15 20 25 30 Half Crack Extension ∆a (mm)
35
40
45
Fig. 17. Jm,pl-integral versus crack extension Da for tests on 219 mm outer diameter pipes.
between the finite element evaluated and experimental values. This indirectly validates the proposed equations for Zn,pl. Further the J–R curve has been evaluated using the proposed value of Zn,pl, gn,pl and the area under the M–npl curve in Eq. (33) and has been plotted in Fig. 16a. The J–R curve has also been evaluated using the conventional Zf,pl, gf,pl and the area under the M–fpl curve in Eq. (8) and has been plotted in Fig. 16b. It can be observed that for SPBMTWC83 test the LLD based J–R curve and the CMOD based J–R curve are in good agreement while for the other two tests i.e. SPBMTWC81 and SPBMTWC82 the LLD based J–R curve is slightly higher than the CMOD based J–R curve. Moreover, the CMOD based
J–R curve has less scatter. Further Ernst’s [20] modified version of the J-integral has been evaluated by both procedures. The modified plastic J integral is defined as Z y qJ pl J m;pl ¼ J pl dy, yo qy Dpl Z y ) J m;pl ¼ J pl gpl J pl dy. yo
The value of Jm,pl evaluated from the M–npl curve and from the M–fpl curve for all three tests are plotted in Fig. 17. This figure shows that all the points lie in a narrow band and are in agreement.
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7. Conclusion
Acknowledgements
The following conclusions have been drawn from the present work.
The authors gratefully acknowledge Mr. J. Chattopadhaya and Mr. T.V. Pavankumar of RSD, BARC, for providing digital data of the pipe fracture experiments for the validation of the work.
1. It is possible to evaluate the J–R curve from CMOD data obtained in pipe fracture tests. The following equation is solved iteratively along with the proposed Zn,pl and gn,pl factors for evaluation of the J–R curve. Z
Z
npl
J pl ¼
y
Zn;pl M dnpl þ 0
gn;pl J pl dy, yo
where Zn;pl ¼
Zf;pl ; f ðR; t; yÞ
gn;pl ¼ gf;pl
f0 ; f
f ¼ Ro þ R sin
2.
3.
4.
5.
Zf;pl ¼
gf;pl ¼
y ; 2
1 h0 ðyÞ , 2Rt hðyÞ
h00 ðyÞ , h0 ðyÞ
hðyÞ ¼ cosðy=2Þ 0:5 sinðyÞ.
J-calculation from load-LLD or moment-rotation data requires that displacements or rotation of noncrack geometry and machine compliance be subtracted from total LLD or rotation. In contrast, formulation in terms of CMOD helps in the direct evaluation from measured moment-CMOD data with no need to subtract any non-crack or machine compliance deformations. The proposed Zn,pl and gn,pl factors have been rigorously validated by finite element analysis which cover a wide range of parameters such as crack angle y (20–1101), Ramberg–Osgood parameters n (3–7), a (5–20), radius to thickness ratio R/t (4.98–10.45), thickness t (10–32.3 mm), outer diameter (219–406) and different inner and outer spans. The parameters i.e., R/t and n cover the wide range of pipe sizes and materials used in NPP piping. For one case i.e. pipe-1, the load versus LLD and CMOD versus LLD from finite element analyses have been validated with three monotonic fracture experiments on a similar pipe geometry. This is an indirect validation of the Zn,pl and gn,pl factors. The calculation procedure for Z has been validated with established conventional equation, which is based on the M–fpl record. The J–R curve and modified plastic J-integral ‘Jm,pl’ for these tests have been evaluated using the proposed equation i.e., Zn,pl and gn,pl factor with M–npl curve and compared with the J–R curve based on M–fpl curve.
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