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Investigation of Fracture Behavior of Steam Generator Tubes of Indian PHWR using PLT Specimens
Available online at www.sciencedirect.com
Procedia Engineering 55 (2013) 578 – 584
6th International Conference on Cree ep, Fatigue and Creep-Fatigue Interaction [CF-6 6]
Investigation of Fracture Beehavior of Steam Generator Tubess of Indian PHWR R using PLT Specimens G. Sannyal*, M.K. Samal Bhabha Atomic Reseaarch Centre, Trombay, Mumbai-85, India
Abstract Steam generator (SG) tubes made of Ni-Cr-Fe based alloys such as Alloy-800 are used in nuclear power plants likee Indian hangers. Pressurized Heavy Water Reactors (PHWRs) in thee form of thin-walled tubular structural members as heat exch These tubes are exposed to aggressive environments such as high temperature, internal pressure and flow-induced viibration etc. Corrosion aspects like intergranular-stress-corrrosion-cracking (IGSCC) of such alloys received much atten ntion in literature, whereas work on fracture behavior of thesse materials is limited, particularly for thin-walled tubes. For strructural integrity assessment of SG tubes, a valid fracture mecchanics parameter must be obtained. As their thickness is only about 1 mm, it is not feasible to machine standard fracture meechanics specimens from them. Moreover, the plane-strain cond dition of state of stress cannot be achieved at the crack-tip off these tubes which is a requisite of ASTM standard tests for fracture f toughness. Thus, non-standard tests have been carriedd out on axially-cracked tubular specimens, directly machined frrom SG tubes in a test-setup known as Pin-Loading-Tensionn (PLT) setup for derivation of shape function from compliaance for determining stress intensity factor and fracture toughhness. Finite element analysis (FEM) has also been done to dettermine the shape function and the other required geometric ffunctions for obtaining J from experimental data. A load-normaalization technique was used for estimation of crack growth. T The results obtained will be highly useful for residual life assessm ment of SG tubes of Indian PHWRs. © Authors. Published by Elsevier ©2013 2013The The Authors. Published by Ltd. Elsevier Ltd. Selection and/or peer-review Selection peer-review under responsibility Gandhi and Centre for Atomic Research. of the Indira Gandhi Centre for Atomic Research.
under responsibility of the Indira
Keywords: SG tube, stress intensity factor, J-R curve.
1. Introduction bjected Steam generator (SG) tubes used in nuclear rreactors are thin-walled structural members which are sub to aggressive environments and loading condditions such as high temperature, internal pressure and d flowinduced vibration etc. Inter-granular stress-corrrosion-cracking is one of the major causes of failure off these tubes. This has led to the development of diffferent types of Ni-Cr-Fe based alloys by various researchers.
*
Corresponding author. E-mail address: [email protected].
1877-7058 © 2013 The Authors. Published by Elsevier Ltd. Selection and peer-review under responsibility of the Indira Gandhi Centre for Atomic Research. doi:10.1016/j.proeng.2013.03.298
G. Sanyal and M.K. Samal / Procedia Engineering 55 (2013) 578 – 584
While research work into the corrosion aspect of o such alloys has received much attention in literature, work w on fracture behavior of these materials is limited, esspecially the fracture behavior of the thin-walled component. In order to perform structural integrity assesssment of these tubes after periodic in-service inspection ns, one needs to have the fracture toughness as well as the fracture resistance curve. As these tubes have thickn ness of the order of 1 mm, it is not possible to machhine standard fracture mechanics specimens from these tubes. Alteration of geometry of the tube may lead to t incalculable uncertainty in test result. In view of thiss, nonstandard tests have been carried out on axially-ccracked tubular specimens directly machined from these tu ubes in a test-setup known as Pin-Loading-Tension (PL LT) setup [1]. Finite element analysis has been carried out to evaluate the geometric functions as a function of crack-length-to-width ratio. These functions are requ uired to estimate the stress-intensity factors which are used u in the evaluation of fracture toughness of these tub bes [2]. Fracture toughness tests were later conducted on these tubes to evaluate the fracture-resistance (J-R) beh havior. In order to estimate the crack growth during thee tests, a load-normalization technique [3, 4] was used which w is suitable for this loading condition as the unloadding-compliance method cannot be used due to the type of the loading device. Finite element analysis has alsso been used to evaluate the geometric factors [5] whiich are needed for evaluation of J-R curve from thesse non-standard specimens for which these functions are a not available in literature. The results obtained will be highly useful for residual life assessment of steam-gen nerator tubes of Indian PHWRs. 2. Experimental a is SG tube made of Alloy 800 is chosen for frracture toughness evaluation in the current work. This alloy used as material for steam-generator tubing for KWU type of PWRs and new Indian PHWRs. Alloy 800 is an austenitic, precipitation-hardenable type alloy coontaining about 45% Fe, 30% Ni and 20% Cr as major allloying elements. Minor alloying elements like Ti and Al A are added to this alloy to have gamma-prime precipittates of type Ni3(Ti,Al) on ageing for longer duration. The tube has a ID of 14 mm and a wall thickness of 1 mm, resulting in an OD of 16 mm. 2.1. Design of specimen and loading jigs men fabricated from the tube through EDM wire cutting. There Figure 1 (a) describes the design of a specim are two diametrically opposite notches at each end e of the specimen, the notches at both ends lying on thee same diametral plane of the tube. The width of the nottch at one end is 0.25 mm whereas it is 0.5 mm at the other end. The notches at the specimen end (i.e., where thee thinner notches exist) are provided for subjecting the speecimen to mode-I loading. The other end contains two diametrically d opposite rectangular slots of dimension 2 mm×0.5 m mm. These two slots are made for avoiding bucckling of the material (behind the pin) due to compressivee force when the specimen is loaded. A set of seven sppecimens were fabricated with the thinner notch varying from 1 mm to 7 mm for measurement of compliancee through experiment for derivation of shape function and a number of specimens were fabricated with a connstant notch length of 1.5 mm for fracture toughness testin ng.
(a)
(b)
(c)
Fig. 1. (a) Appearance of a specimen with dimensions, (b) ( appearance of specimen fixture assembly with specimen width and crack c length, and (c) schematicc drawing of the fixture elaborating all details.
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The detailed design of loading fixture is deescribed in Fig. 1 (b). The fixture consists of two split--halves which form the cylindrical holder and a rectanggular block. In this figure [Fig. 1(b)], ‘W=19 mm’ denotes the distance between the centre of the axis and thee load line (i.e. line joining the holes of the rectangular block), b whereas ‘b=8 mm’ denotes the distance betweeen the load line and the contact plane of the cylindrical and a the rectangular block. The split halves are loaded through the holes with pins. The split halves which haave the possibility of mutual rotation around an axis, determined by a small pin, are inserted inside the speecimen [Fig. 1 (c)]. The specimen is so aligned that all the four notches are coplanar with the contact plane of th he split halves. The position of the mutual rotation axis is i fixed even after crack initiation and propagation. A clip p at the back end of the fixture is inserted after the sppecimen is mounted on the fixture to avoid axial shift of the specimen along the mandrel away from load linee while loading [Fig. 1 (c)]. 2.2. The loading procedure All the seven specimens with different notchh lengths are attached to the respective fixture one by on ne. The specimens are loaded in displacement control mode with a pull rate of 0.2 mm/min with a servo-hyd draulic actuator controlled by a 24-bit controller havingg load frame of 25 kN capacity [Fig. 2(a)] up to a point where considerable load drop occurs after the peak loaad. The crack opening displacement (COD) is measured using u a clip-on gage and the timed load-COD data are digitally d recorded.
(a)
(b)
(c)
Fig. 2. (a) Loading of a specimen in a servo-hydraulicc load frame with clevis grips, (b) appearance of a failed specimen after heat tinting, and (c) Appeearance of fracture surfaces of a specimen.
Prior to the fracture toughness testing, all thee specimens with same notch length of 1.5 mm (along with w the fixture) were subjected to fatigue loading to gennerate a sharp crack of length ~1.5 mm ahead of the notch h tip in the same table-top servo hydraulic actuator. While W loading cyclically, the crack size was visually mon nitored with a travelling microscope and also indirectlyy, by compliance measurement by insertion of a crack opening displacement (COD) gauge at the knife edge of o the fixture halves. The pre-cracked specimens are theereafter loaded under displacement control upto nine different total applied displacement values so as to generatte nine different extents of crack extensions. Then all the t nine specimens were heated in air at 800oC for one hour h to mark the crack extension and then subjected to fatigue reloading for the final cleave-opening. In Fig. 2 (b), ( the appearance of a heat tinted failed specimen is shown. The crack lengths before and after fracture toughness testing for each specimen were measured undder a stereomicroscope [Fig. 2 (c)] with nine point aveeraging method. 2.3. Analysis for SIF calculation The detailed procedure adopted for calculatiion of shape function from compliance is given in Ref. [2, 5]. The procedures will be briefly discussed heree. From the initial linear elastic straight portion of load d-COD response curve [Fig. 3 (a)], the compliance is calculated for all seven cases and that is plotted against a normalized notch length, a/W [Fig. 3 (b)]. From m the concept of strain energy release rate, G, it can be proved p that [2, 5]
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G. Sanyal and M.K. Samal / Procedia Engineering 55 (2013) 578 – 584
f ( a W ) = [tE
dC 1 2 P ] and K I = f (a W ) d (a W ) 2t W
(1)
where f(a/W) is shape function, specific to the tube, t is tube wall thickness, C is compliance, a is crack length, W is specimen width, P is applied load, and KI is mode I stress intensity factor. Hence, basically, for derivation of f(a/W) from experimental compliance, it is required to differentiate the compliance function of a/W, w.r.t. a/W and proceed through eq. 1. Two independent procedures are adapted for finding 0.0016
Force (N)
2000 1500 1000
0.0014
50
0.0012
40
0.0010
f(a/W)
2500
C (mm/N)
1mm 2mm 3mm 4mm 5mm 6mm 7mm
3000
dC . d (a / W )
30
0.0008
20
0.0006
ASTM Ryder FEM inside FEM outside
10
0.0004
500 0.0002
0 0
1
2
3
COD (mm)
4
0.50
0.55
0.60
0.65
(a/W)
(b)
(a)
0 0.45
5
0.70
0.75
0.5
0.80
0.6
a/W
0.7
0.8
(c)
Fig. 3. (a) Load-COD response for all specimens with different notch lengths, (b) compliance fitted as a function of a/W, and (c) comparison of shape function expressions obtained through experiment and FEM.
In the first approach, the compliance curve [Fig. 3 (b)] is fitted with a polynomial in a/W and the fitted function is differentiated using power law. The resulting f(a/W) expression is a polynomial in a/W. In the second approach, suggested by Ryder et. al. [2, 5, 6, 7], a Z function is incorporated to correlate the discrete compliance values with the discrete a/W values and using the function a pure analytic expression for dC in the form of a/W is derived, involving three constants, specific to tube geometry. Those constants d (a / W )
are evaluated imposing special boundary conditions to find a pure analytic expression for f(a/W). The f(a/W) expressions for both the cases are graphically compared with FEM results [Fig. 3 (c)] in Section 5. 3. FEM analysis Finite element analysis (FEM) has been done for the entire assembly of specimen and fixture to find the shape function and η and γ functions necessary for calculation of Jpl from experimental data. The details will be briefly discussed below. 3.1. Mesh generation Taking into account the symmetry in the specimen geometry, crack and loading configuration, one quarter of the test setup has been modelled as shown in Fig. 4 (a). A prescribed displacement is applied at the end of the mandrel in y-direction and the movement of the loading point is restricted in the x-direction (i.e., direction of length of the tube) as this point is attached to the loading axis of the machine. The un-cracked ligament of the tubular specimen is fixed in y-direction (however, this is free to move in x and z-directions) in order to apply the symmetry condition. The line of the loading pin (which acts like a fulcrum) is also fixed in ydirection only.
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200
P Ni
180
160
140
120 0.00
0.03
0.06
0.09
0.12
Vpli
(a)
(b)
(c)
Fig. 4. (a) The FE model for specimen fixture assembly, (b) ( fitting of normalized load-LLD data, and (c) multiple specimen and single speccimen J-R curve results.
n order The symmetric plane of the tubular specimenn as well as of the mandrel is fixed in z-direction only in to consider the symmetric boundary condition. The T FE discretization consists of 20-noded iso-parametricc brick elements with 3×3×3 Gauss point integration (ffull integration scheme). For the seven specimens with diifferent notch lengths used in the experiment, seven diffe ferent meshes were generated. 3.2. Simulation od, the The analysis has been carried out using an inn-house FE code. Using displacement extrapolation metho SIF for each of the models is calculated using thhe following equation:
KI =
E 4
2π v r
(2)
where r is the distance from the crack tipp and v is the displacement; from these, the shape fu unction expression is derived. Elastic-plastic analysis with von-Mises yield surface and isotropic plastic hardening constitutive equations have been carried out inn this work for evaluation of limit load of the specimen ns as a function of a/W. From limit load (FL) expressioons, η and γ functions are derived using the following equ uations [8]:
§ ©
η = − ¨1 −
a W § ©
· 1 ∂FL ¸ ¹ FL ∂ ( a W )
γ = η − 1 − ¨1 −
a W
· 1 ∂η ¸ ¹η ∂ (a W )
(3)
(4)
4. Load normalization method for estimating J-R curve The method is elaborated in the references [3, 4, 5]. The load line displacement for all specimens is first corrected by subtracting the contribution from thhe testing machine and loading arrangements. Then all lo oad and displacements up to Pmax for each specimen aree normalized using the following equations based on thee crack length prior to fracture toughness testing and connsidering a blunting corrected crack growth as per eq. (7) where σY is the flow stress:
PNi =
Pi η
2tW ª¬(W − abi' ) W º¼
(5)
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G. Sanyal and M.K. Samal / Procedia Engineering 55 (2013) 578 – 584
v 'pli = ( vi − PC i i) W
(6)
abi' = a0' + J i 2σ Y
(7)
The final data point is also normalized in the same way but in this case the crack length after fracture toughness testing [Fig. 2 (c)] is considered. The resulting normalized data set is fitted [Fig. 4 (b)] using the following equation:
a + bv′pl + cv′pl2 PN = d + v′pl
(8)
′ values are forced to change allover the entire data set to lie in eq. (8). Thus estimation of Thereafter all abi actual crack growth is achieved and then Jpl for all data points are derived using the following equation:
J pl (i )
ª § η(i −1) · Apl (i ) − Apl (i −1) º ª a(' i ) − a(' i −1) º = « J pl (i −1) + ¨ » «1 − γ (i −1) » ¨ ¸¸ b(i −1) »¼ 2t «¬ »¼ «¬ © b(i −1) ¹
(9)
Jel is derived by
J el = K I2 E , where K I =
P f (a W ) 2t W
(10)
Since all the required geometric functions are now known, J can be derived for all data points and can be plotted against Δa to find J-R curve [Fig. 4 (c)]. 5. Results and discussion The shape function results are graphically compared in Fig. 3 (c). Between 0.6 and 0.75 values of a/W, the experimental curves are found to match well with each other. Outside this band, for lower a/W, approach 1 is found to predict a higher f(a/W) value whereas approach 2 does the same for higher a/W. However, the curve for approach 2 is found to match well with the FEM findings for the highest a/W considered here unlike the curve for approach 1. Overall, also, FEM results show closer resemblance with the curve for approach 2 but the FEM result is always a little lower than the experimental findings. In fact, there is variance in FEM result if one considers the variation of SIF across the tube wall thickness and from Fig. 3 (c), it is clear that the FEM result for inside surface of the tube is matching more closely with experimental findings. Considering Fig. 4 (c), it can be said that ample reproducibility is achieved when all nine single specimen results are considered together, at least up to a crack growth of ~1mm. also, from J values for each loaddisplacement curve, a multiple specimen J-R curve is constructed as shown in Fig. 4(c). As the multiple specimen result is not crack growth corrected, it always underestimates J a little bit for the same Δa. 6. Conclusions A new method is presented for finding the crack resistance curve of thin-walled steam generator tubes. Specimens and loading fixtures are fabricated after suitable customization. A number of geometric functions, necessary for calculation of SIF and J have been derived through experiment and FEM and using those functions J-R curve for SG tube used in Indian PHWR has been obtained.
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References [1] V.Grigoriev, B.Josefsson, A.Lind, B.Rosborg, A pin-loading tension test for evaluation of thin-walled tubular materials. Scr. Met. Mater. 33(1)(1995)109-114. [2] M.K.Samal, G.Sanyal, J.K.Chakravartty, An experimental and numerical study of the fracture behaviour of tubular specimens in a pinloading-tension set-up. J Mech. Engg. Sci. 224(1)(2010)1-12. [3] G.Sanyal, M.K.Samal, J.K.Chakravartty, K.K.Ray, A.K.Suri, S.Banerjee, Prediction of J-R curves of thin-walled fuel pin specimens in a PLT setup. Engg Frac Mech 78(6)(2011)1029-1043. [4] M.K.Samal, G.Sanyal, J.K.Chakravartty, Investigation of failure behavior of two different types of Zircaloy clad tubes used as nuclear reactor fuel pins. Engg. Fail. Anal. 18(8)(2011)2042-2053. [5] M.K.Samal, G.Sanyal, J.K.Chakravartty, Estimation of fracture behaviour of thin walled nuclear reactor fuel pins using pin-loadingtension (PLT) test. Nucl Engg. & Design, 240(12)(2010)4043-4050. [6] J.T.Ryder, G.E.Browie, D.E.Pettit, Recent considerations in experimental compliance calibration of the WOL specimen. Eng Frac Mech 9(4)(1977)901-23. [7] T.V.Duggan, M.W.Proctor, L.J.Spence, Stress intensity calibrations and compliance functions for fracture toughness and crack propagation test specimens. Int. J. Fatigue 1(1)(1979)37-47. [8] N.Miura, G.M.Wilkowski, J–R curves from circumferentially through-wall cracked pipe tests subjected to combined bending and tension-part i: theory and numerical simulation. J Press Vessel Technol, Trans, ASME 120(1998)406-411.
Procedia Engineering 55 (2013) 578 – 584
6th International Conference on Cree ep, Fatigue and Creep-Fatigue Interaction [CF-6 6]
Investigation of Fracture Beehavior of Steam Generator Tubess of Indian PHWR R using PLT Specimens G. Sannyal*, M.K. Samal Bhabha Atomic Reseaarch Centre, Trombay, Mumbai-85, India
Abstract Steam generator (SG) tubes made of Ni-Cr-Fe based alloys such as Alloy-800 are used in nuclear power plants likee Indian hangers. Pressurized Heavy Water Reactors (PHWRs) in thee form of thin-walled tubular structural members as heat exch These tubes are exposed to aggressive environments such as high temperature, internal pressure and flow-induced viibration etc. Corrosion aspects like intergranular-stress-corrrosion-cracking (IGSCC) of such alloys received much atten ntion in literature, whereas work on fracture behavior of thesse materials is limited, particularly for thin-walled tubes. For strructural integrity assessment of SG tubes, a valid fracture mecchanics parameter must be obtained. As their thickness is only about 1 mm, it is not feasible to machine standard fracture meechanics specimens from them. Moreover, the plane-strain cond dition of state of stress cannot be achieved at the crack-tip off these tubes which is a requisite of ASTM standard tests for fracture f toughness. Thus, non-standard tests have been carriedd out on axially-cracked tubular specimens, directly machined frrom SG tubes in a test-setup known as Pin-Loading-Tensionn (PLT) setup for derivation of shape function from compliaance for determining stress intensity factor and fracture toughhness. Finite element analysis (FEM) has also been done to dettermine the shape function and the other required geometric ffunctions for obtaining J from experimental data. A load-normaalization technique was used for estimation of crack growth. T The results obtained will be highly useful for residual life assessm ment of SG tubes of Indian PHWRs. © Authors. Published by Elsevier ©2013 2013The The Authors. Published by Ltd. Elsevier Ltd. Selection and/or peer-review Selection peer-review under responsibility Gandhi and Centre for Atomic Research. of the Indira Gandhi Centre for Atomic Research.
under responsibility of the Indira
Keywords: SG tube, stress intensity factor, J-R curve.
1. Introduction bjected Steam generator (SG) tubes used in nuclear rreactors are thin-walled structural members which are sub to aggressive environments and loading condditions such as high temperature, internal pressure and d flowinduced vibration etc. Inter-granular stress-corrrosion-cracking is one of the major causes of failure off these tubes. This has led to the development of diffferent types of Ni-Cr-Fe based alloys by various researchers.
*
Corresponding author. E-mail address: [email protected].
1877-7058 © 2013 The Authors. Published by Elsevier Ltd. Selection and peer-review under responsibility of the Indira Gandhi Centre for Atomic Research. doi:10.1016/j.proeng.2013.03.298
G. Sanyal and M.K. Samal / Procedia Engineering 55 (2013) 578 – 584
While research work into the corrosion aspect of o such alloys has received much attention in literature, work w on fracture behavior of these materials is limited, esspecially the fracture behavior of the thin-walled component. In order to perform structural integrity assesssment of these tubes after periodic in-service inspection ns, one needs to have the fracture toughness as well as the fracture resistance curve. As these tubes have thickn ness of the order of 1 mm, it is not possible to machhine standard fracture mechanics specimens from these tubes. Alteration of geometry of the tube may lead to t incalculable uncertainty in test result. In view of thiss, nonstandard tests have been carried out on axially-ccracked tubular specimens directly machined from these tu ubes in a test-setup known as Pin-Loading-Tension (PL LT) setup [1]. Finite element analysis has been carried out to evaluate the geometric functions as a function of crack-length-to-width ratio. These functions are requ uired to estimate the stress-intensity factors which are used u in the evaluation of fracture toughness of these tub bes [2]. Fracture toughness tests were later conducted on these tubes to evaluate the fracture-resistance (J-R) beh havior. In order to estimate the crack growth during thee tests, a load-normalization technique [3, 4] was used which w is suitable for this loading condition as the unloadding-compliance method cannot be used due to the type of the loading device. Finite element analysis has alsso been used to evaluate the geometric factors [5] whiich are needed for evaluation of J-R curve from thesse non-standard specimens for which these functions are a not available in literature. The results obtained will be highly useful for residual life assessment of steam-gen nerator tubes of Indian PHWRs. 2. Experimental a is SG tube made of Alloy 800 is chosen for frracture toughness evaluation in the current work. This alloy used as material for steam-generator tubing for KWU type of PWRs and new Indian PHWRs. Alloy 800 is an austenitic, precipitation-hardenable type alloy coontaining about 45% Fe, 30% Ni and 20% Cr as major allloying elements. Minor alloying elements like Ti and Al A are added to this alloy to have gamma-prime precipittates of type Ni3(Ti,Al) on ageing for longer duration. The tube has a ID of 14 mm and a wall thickness of 1 mm, resulting in an OD of 16 mm. 2.1. Design of specimen and loading jigs men fabricated from the tube through EDM wire cutting. There Figure 1 (a) describes the design of a specim are two diametrically opposite notches at each end e of the specimen, the notches at both ends lying on thee same diametral plane of the tube. The width of the nottch at one end is 0.25 mm whereas it is 0.5 mm at the other end. The notches at the specimen end (i.e., where thee thinner notches exist) are provided for subjecting the speecimen to mode-I loading. The other end contains two diametrically d opposite rectangular slots of dimension 2 mm×0.5 m mm. These two slots are made for avoiding bucckling of the material (behind the pin) due to compressivee force when the specimen is loaded. A set of seven sppecimens were fabricated with the thinner notch varying from 1 mm to 7 mm for measurement of compliancee through experiment for derivation of shape function and a number of specimens were fabricated with a connstant notch length of 1.5 mm for fracture toughness testin ng.
(a)
(b)
(c)
Fig. 1. (a) Appearance of a specimen with dimensions, (b) ( appearance of specimen fixture assembly with specimen width and crack c length, and (c) schematicc drawing of the fixture elaborating all details.
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G. Sanyal and M.K. Samal / Procedia Engineering 55 (2013) 578 – 584
The detailed design of loading fixture is deescribed in Fig. 1 (b). The fixture consists of two split--halves which form the cylindrical holder and a rectanggular block. In this figure [Fig. 1(b)], ‘W=19 mm’ denotes the distance between the centre of the axis and thee load line (i.e. line joining the holes of the rectangular block), b whereas ‘b=8 mm’ denotes the distance betweeen the load line and the contact plane of the cylindrical and a the rectangular block. The split halves are loaded through the holes with pins. The split halves which haave the possibility of mutual rotation around an axis, determined by a small pin, are inserted inside the speecimen [Fig. 1 (c)]. The specimen is so aligned that all the four notches are coplanar with the contact plane of th he split halves. The position of the mutual rotation axis is i fixed even after crack initiation and propagation. A clip p at the back end of the fixture is inserted after the sppecimen is mounted on the fixture to avoid axial shift of the specimen along the mandrel away from load linee while loading [Fig. 1 (c)]. 2.2. The loading procedure All the seven specimens with different notchh lengths are attached to the respective fixture one by on ne. The specimens are loaded in displacement control mode with a pull rate of 0.2 mm/min with a servo-hyd draulic actuator controlled by a 24-bit controller havingg load frame of 25 kN capacity [Fig. 2(a)] up to a point where considerable load drop occurs after the peak loaad. The crack opening displacement (COD) is measured using u a clip-on gage and the timed load-COD data are digitally d recorded.
(a)
(b)
(c)
Fig. 2. (a) Loading of a specimen in a servo-hydraulicc load frame with clevis grips, (b) appearance of a failed specimen after heat tinting, and (c) Appeearance of fracture surfaces of a specimen.
Prior to the fracture toughness testing, all thee specimens with same notch length of 1.5 mm (along with w the fixture) were subjected to fatigue loading to gennerate a sharp crack of length ~1.5 mm ahead of the notch h tip in the same table-top servo hydraulic actuator. While W loading cyclically, the crack size was visually mon nitored with a travelling microscope and also indirectlyy, by compliance measurement by insertion of a crack opening displacement (COD) gauge at the knife edge of o the fixture halves. The pre-cracked specimens are theereafter loaded under displacement control upto nine different total applied displacement values so as to generatte nine different extents of crack extensions. Then all the t nine specimens were heated in air at 800oC for one hour h to mark the crack extension and then subjected to fatigue reloading for the final cleave-opening. In Fig. 2 (b), ( the appearance of a heat tinted failed specimen is shown. The crack lengths before and after fracture toughness testing for each specimen were measured undder a stereomicroscope [Fig. 2 (c)] with nine point aveeraging method. 2.3. Analysis for SIF calculation The detailed procedure adopted for calculatiion of shape function from compliance is given in Ref. [2, 5]. The procedures will be briefly discussed heree. From the initial linear elastic straight portion of load d-COD response curve [Fig. 3 (a)], the compliance is calculated for all seven cases and that is plotted against a normalized notch length, a/W [Fig. 3 (b)]. From m the concept of strain energy release rate, G, it can be proved p that [2, 5]
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G. Sanyal and M.K. Samal / Procedia Engineering 55 (2013) 578 – 584
f ( a W ) = [tE
dC 1 2 P ] and K I = f (a W ) d (a W ) 2t W
(1)
where f(a/W) is shape function, specific to the tube, t is tube wall thickness, C is compliance, a is crack length, W is specimen width, P is applied load, and KI is mode I stress intensity factor. Hence, basically, for derivation of f(a/W) from experimental compliance, it is required to differentiate the compliance function of a/W, w.r.t. a/W and proceed through eq. 1. Two independent procedures are adapted for finding 0.0016
Force (N)
2000 1500 1000
0.0014
50
0.0012
40
0.0010
f(a/W)
2500
C (mm/N)
1mm 2mm 3mm 4mm 5mm 6mm 7mm
3000
dC . d (a / W )
30
0.0008
20
0.0006
ASTM Ryder FEM inside FEM outside
10
0.0004
500 0.0002
0 0
1
2
3
COD (mm)
4
0.50
0.55
0.60
0.65
(a/W)
(b)
(a)
0 0.45
5
0.70
0.75
0.5
0.80
0.6
a/W
0.7
0.8
(c)
Fig. 3. (a) Load-COD response for all specimens with different notch lengths, (b) compliance fitted as a function of a/W, and (c) comparison of shape function expressions obtained through experiment and FEM.
In the first approach, the compliance curve [Fig. 3 (b)] is fitted with a polynomial in a/W and the fitted function is differentiated using power law. The resulting f(a/W) expression is a polynomial in a/W. In the second approach, suggested by Ryder et. al. [2, 5, 6, 7], a Z function is incorporated to correlate the discrete compliance values with the discrete a/W values and using the function a pure analytic expression for dC in the form of a/W is derived, involving three constants, specific to tube geometry. Those constants d (a / W )
are evaluated imposing special boundary conditions to find a pure analytic expression for f(a/W). The f(a/W) expressions for both the cases are graphically compared with FEM results [Fig. 3 (c)] in Section 5. 3. FEM analysis Finite element analysis (FEM) has been done for the entire assembly of specimen and fixture to find the shape function and η and γ functions necessary for calculation of Jpl from experimental data. The details will be briefly discussed below. 3.1. Mesh generation Taking into account the symmetry in the specimen geometry, crack and loading configuration, one quarter of the test setup has been modelled as shown in Fig. 4 (a). A prescribed displacement is applied at the end of the mandrel in y-direction and the movement of the loading point is restricted in the x-direction (i.e., direction of length of the tube) as this point is attached to the loading axis of the machine. The un-cracked ligament of the tubular specimen is fixed in y-direction (however, this is free to move in x and z-directions) in order to apply the symmetry condition. The line of the loading pin (which acts like a fulcrum) is also fixed in ydirection only.
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200
P Ni
180
160
140
120 0.00
0.03
0.06
0.09
0.12
Vpli
(a)
(b)
(c)
Fig. 4. (a) The FE model for specimen fixture assembly, (b) ( fitting of normalized load-LLD data, and (c) multiple specimen and single speccimen J-R curve results.
n order The symmetric plane of the tubular specimenn as well as of the mandrel is fixed in z-direction only in to consider the symmetric boundary condition. The T FE discretization consists of 20-noded iso-parametricc brick elements with 3×3×3 Gauss point integration (ffull integration scheme). For the seven specimens with diifferent notch lengths used in the experiment, seven diffe ferent meshes were generated. 3.2. Simulation od, the The analysis has been carried out using an inn-house FE code. Using displacement extrapolation metho SIF for each of the models is calculated using thhe following equation:
KI =
E 4
2π v r
(2)
where r is the distance from the crack tipp and v is the displacement; from these, the shape fu unction expression is derived. Elastic-plastic analysis with von-Mises yield surface and isotropic plastic hardening constitutive equations have been carried out inn this work for evaluation of limit load of the specimen ns as a function of a/W. From limit load (FL) expressioons, η and γ functions are derived using the following equ uations [8]:
§ ©
η = − ¨1 −
a W § ©
· 1 ∂FL ¸ ¹ FL ∂ ( a W )
γ = η − 1 − ¨1 −
a W
· 1 ∂η ¸ ¹η ∂ (a W )
(3)
(4)
4. Load normalization method for estimating J-R curve The method is elaborated in the references [3, 4, 5]. The load line displacement for all specimens is first corrected by subtracting the contribution from thhe testing machine and loading arrangements. Then all lo oad and displacements up to Pmax for each specimen aree normalized using the following equations based on thee crack length prior to fracture toughness testing and connsidering a blunting corrected crack growth as per eq. (7) where σY is the flow stress:
PNi =
Pi η
2tW ª¬(W − abi' ) W º¼
(5)
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v 'pli = ( vi − PC i i) W
(6)
abi' = a0' + J i 2σ Y
(7)
The final data point is also normalized in the same way but in this case the crack length after fracture toughness testing [Fig. 2 (c)] is considered. The resulting normalized data set is fitted [Fig. 4 (b)] using the following equation:
a + bv′pl + cv′pl2 PN = d + v′pl
(8)
′ values are forced to change allover the entire data set to lie in eq. (8). Thus estimation of Thereafter all abi actual crack growth is achieved and then Jpl for all data points are derived using the following equation:
J pl (i )
ª § η(i −1) · Apl (i ) − Apl (i −1) º ª a(' i ) − a(' i −1) º = « J pl (i −1) + ¨ » «1 − γ (i −1) » ¨ ¸¸ b(i −1) »¼ 2t «¬ »¼ «¬ © b(i −1) ¹
(9)
Jel is derived by
J el = K I2 E , where K I =
P f (a W ) 2t W
(10)
Since all the required geometric functions are now known, J can be derived for all data points and can be plotted against Δa to find J-R curve [Fig. 4 (c)]. 5. Results and discussion The shape function results are graphically compared in Fig. 3 (c). Between 0.6 and 0.75 values of a/W, the experimental curves are found to match well with each other. Outside this band, for lower a/W, approach 1 is found to predict a higher f(a/W) value whereas approach 2 does the same for higher a/W. However, the curve for approach 2 is found to match well with the FEM findings for the highest a/W considered here unlike the curve for approach 1. Overall, also, FEM results show closer resemblance with the curve for approach 2 but the FEM result is always a little lower than the experimental findings. In fact, there is variance in FEM result if one considers the variation of SIF across the tube wall thickness and from Fig. 3 (c), it is clear that the FEM result for inside surface of the tube is matching more closely with experimental findings. Considering Fig. 4 (c), it can be said that ample reproducibility is achieved when all nine single specimen results are considered together, at least up to a crack growth of ~1mm. also, from J values for each loaddisplacement curve, a multiple specimen J-R curve is constructed as shown in Fig. 4(c). As the multiple specimen result is not crack growth corrected, it always underestimates J a little bit for the same Δa. 6. Conclusions A new method is presented for finding the crack resistance curve of thin-walled steam generator tubes. Specimens and loading fixtures are fabricated after suitable customization. A number of geometric functions, necessary for calculation of SIF and J have been derived through experiment and FEM and using those functions J-R curve for SG tube used in Indian PHWR has been obtained.
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References [1] V.Grigoriev, B.Josefsson, A.Lind, B.Rosborg, A pin-loading tension test for evaluation of thin-walled tubular materials. Scr. Met. Mater. 33(1)(1995)109-114. [2] M.K.Samal, G.Sanyal, J.K.Chakravartty, An experimental and numerical study of the fracture behaviour of tubular specimens in a pinloading-tension set-up. J Mech. Engg. Sci. 224(1)(2010)1-12. [3] G.Sanyal, M.K.Samal, J.K.Chakravartty, K.K.Ray, A.K.Suri, S.Banerjee, Prediction of J-R curves of thin-walled fuel pin specimens in a PLT setup. Engg Frac Mech 78(6)(2011)1029-1043. [4] M.K.Samal, G.Sanyal, J.K.Chakravartty, Investigation of failure behavior of two different types of Zircaloy clad tubes used as nuclear reactor fuel pins. Engg. Fail. Anal. 18(8)(2011)2042-2053. [5] M.K.Samal, G.Sanyal, J.K.Chakravartty, Estimation of fracture behaviour of thin walled nuclear reactor fuel pins using pin-loadingtension (PLT) test. Nucl Engg. & Design, 240(12)(2010)4043-4050. [6] J.T.Ryder, G.E.Browie, D.E.Pettit, Recent considerations in experimental compliance calibration of the WOL specimen. Eng Frac Mech 9(4)(1977)901-23. [7] T.V.Duggan, M.W.Proctor, L.J.Spence, Stress intensity calibrations and compliance functions for fracture toughness and crack propagation test specimens. Int. J. Fatigue 1(1)(1979)37-47. [8] N.Miura, G.M.Wilkowski, J–R curves from circumferentially through-wall cracked pipe tests subjected to combined bending and tension-part i: theory and numerical simulation. J Press Vessel Technol, Trans, ASME 120(1998)406-411.