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and fracture resistance behaviour of thin-walled axially cracked tubular ... Abstract: Integrity assessment of thin-walled tubular components such as nuclear reactor fuel ... In this article, the crack growth data during loading of the axially cracked.
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A load-separation technique to evaluate crack growth and fracture resistance behaviour of thin-walled axially cracked tubular specimens M K Samal1* and G Sanyal2 1 Reactor Safety Division, Bhabha Atomic Research Centre, Trombay, Mumbai, India 2 Mechanical Metallurgy Division, Bhabha Atomic Research Centre, Trombay, Mumbai, India The manuscript was received on 8 April 2011 and was accepted after revision for publication on 8 September 2011. DOI: 10.1177/0954406211424978

Abstract: Integrity assessment of thin-walled tubular components such as nuclear reactor fuel pins during postulated and beyond-design-basis accident scenarios is an important issue for limiting the release of radioactivity into the surrounding fluid. The fracture resistance behaviour of these tubes cannot be evaluated using standard ASTM techniques. It is because of the inability of these axially cracked specimens to meet the stringent plane strain requirement due to their geometry and the high ductility of the zirconium alloy used for their fabrication. Moreover, the measurement of crack growth during the testing by conventional methods is a cumbersome process and sometimes it is not possible to use them due to the small size of these specimens. Alternative methods such as load-normalization and load-separation techniques are suitable for these types of situations. In this article, the crack growth data during loading of the axially cracked tubular specimens have been evaluated using a load-separation method. Several specimens have been tested in order to evaluate their fracture resistance behaviour. These specimens have different values of initial crack lengths. A modified procedure for the estimation of the material constant ‘m’ has been developed in this study so that data from multiple specimens can be used. This will lead to a more reliable estimation of crack growth during the loading process. The J–R curves, obtained using the current method, have also been compared with those obtained using load-normalization technique. It was observed that this modified method of load separation is a convenient and reliable technique for the evaluation of fracture resistance behaviour of nonstandard specimens. Keywords: zircaloy fuel clad, fracture resistance, J–R curve, load-separation method pin-loading-tension setup

1

INTRODUCTION

Thin-walled small-diameter tubular structures are commonly used in many modern engineering systems. These tubes are usually subjected to highpressure and high-temperature environments such as those in super-heater tubes, re-heater tubes, and steam-generator tubes of thermal power plants, heat exchanger tubes in chemical and process industries, *Corresponding author: Reactor Safety Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400085, India. email: [email protected], [email protected]

and fuel clad tubes of nuclear reactors. These clad tubes in light water and pressurized heavy water reactors are made of zirconium alloys which have a very low neutron absorption cross-section and hence, are suitable to sustain a fission chain reaction. The development and applications of different types of zirconium alloys for nuclear applications are discussed by Northwood in Ref. [1]. The tubes also act as a barrier to release radioactivity to the surrounding coolant and a conduit for transfer of heat generated due to nuclear fission from nuclear fuel pellets (i.e. uranium oxide or mixed oxide fuels) to the coolant (i.e. water in light water Proc. IMechE Vol. 226 Part C: J. Mechanical Engineering Science

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reactors and heavy water in pressurized heavy water reactors). In order to enhance the heat transfer, the tubes are manufactured as thin-walled tubes (thickness ranging from 0.38 to 0.9 mm approximately). The inner diameter varies from 10 to 15 mm approximately. It is important to ensure the structural integrity of these fuel clad tubes under various types of operational transients as well as accident scenarios. One of the postulated design-basis accidents for the fuel clad tubes is the reactivity-initiated accident (RIA). In such a situation, the control rod drops or ejects very rapidly from the reactor. This in turn deposits a large amount of energy in the fuel pellets and leads to adiabatic heating and large fission gas release in the fuel pins [2]. The fuel pellets expand thermally and may cause fast straining of the surrounding zircaloy clad tube through pellet-clad mechanical interaction. At the early heat-up stage of the RIA, the clad tube material may still be experiencing a fairly low temperature. The fast loading imposed by the mechanical expansion of fuel pellet may cause rapid propagation of preexisting cracks. These pre-existing defects may have been initiated during the service life due to mechanical and chemical interactions of fuel pellets with the clad. In the regime of temperatures experienced in pressurized water reactors, the oxide growth occurs initially by the formation of a dense, adherent oxide layer. Above a certain critical thickness, the growth kinetics transforms to a more accelerated linear growth rate, which is associated with fracture or buckling of the oxide which leads to the formation of small cracks [3]. In the oxygenated boiling water reactors (BWRs), the zirconium alloys exhibit a peculiar form of corrosion called nodular corrosion that occurs in the form of discrete disc-shaped white nodules on the surface, which eventually coalesce to form a continuous layer of defects [3]. Another mechanism for the formation of defects is the phenomenon of delayed hydride cracking due to the ingress of hydrogen/deuterium into the zircaloy tubes [4]. Again, at high burn-up condition, there is significant degradation of mechanical and fracture properties of clad tubes due to the presence of various types of micro-structural transformations (thermomechanical as well as irradiation-assisted processes) which can lead to aging and precipitation in these alloys. This can alter the crack propagation rate in the event of occurrence of a RIA [5–7]. In order to ensure that the clad remains intact for a longer period in the reactor, fracture mechanics principles can be applied for the fitness-for-purpose of service assessment of these fuel pins. As these fuel pins are of very small thickness (less than 1 mm) Proc. IMechE Vol. 226 Part C: J. Mechanical Engineering Science

and the material has good ductility, the plane strain condition of state of stress cannot be satisfied at the crack-tip. ASTM standard [8] enforces a stringent condition on the dimensions of the specimens (based on crack initiation toughness and yield stress of the material). Hence, the fracture resistance curve cannot be evaluated following this standard method. In order to evaluate the fracture resistance behaviour for carrying out structural integrity analysis of these fuel pins, the researchers worldwide have resorted to the use of non-standard specimens [9–17]. In this study, the pin-loading tension (PLT) test, which was earlier proposed by Grigoriev et al. [13, 14], has been used to evaluate the fracture toughness of zircaloy-2 fuel claddings which are used in Indian BWRs. The fuel clad is of 0.9 mm thickness and 12.4 mm internal diameter. To estimate the instantaneous crack growth during loading of the axially cracked thin-walled fuel clad specimens, an indirect method based on the so-called load-separability parameter ‘Spb’ has been adopted. A modified procedure for estimation of the material constant ‘m’ has been developed in this study so that the data from many specimens can be used. This will lead to a more reliable estimation of crack growth during loading. Such a study for the estimation of fracture resistance behaviour of PLT specimens made of zircaloy fuel pins has not been previously reported in literature. In this study, several identical fuel pin specimens (with different initial crack lengths) machined from fuel clad tubes of Indian BWRs have been tested in the PLT setup. The load–displacement responses of the cracked specimens have been evaluated by finite element (FE) analysis. The elastic–plastic constitutive model (i.e. J2 plasticity) has been used for the material response. The load-separability parameter Spb has been evaluated as a function of the final crack length ratios of these specimens. The final crack lengths have been estimated from the specimen fracture surface using heat-tinting method. From the variation of load-separability parameter Spb with crack length ratio, the material parameter m has been derived. The method has been used to evaluate the J–R curves of the fuel pin specimens with different crack length to width ratios with the help of earlier derived  and  functions for these specimens. This article is divided into seven sections. A brief overview of the current state of research in literature regarding fracture toughness evaluations of thinwalled small-diameter tubular fuel clad specimens is presented in Section 2. The technique of evaluation of crack growth during loading of PLT specimens using the load-separability parameter Spb has been discussed in Section 3. Section 4 discusses the

A load-separation technique to evaluate crack growth and fracture resistance behaviour

experimental procedure followed in this study for fracture toughness test of PLT specimens. The details of elastic–plastic FE analysis of the axially cracked specimens in the PLT setup are presented in Section 5. The methods of determination of load-separability parameter, calibration of material constant, and use of the same to evaluate J–R curves of PLT specimens with different initial crack length to width ratios are presented in Section 6‘Results and discussion’. The concluding remarks from this study along with the scope for research are presented in Section 7. 2

BRIEF OVERVIEW OF RESEARCH LITERATURE REGARDING FRACTURE TOUGHNESS EVALUATION OF THINWALLED FUEL CLAD TUBES

Evaluation of fracture resistance curves of thinwalled tubular specimens has been a long-standing concern in nuclear industry. As standard ASTM fracture mechanics specimens cannot be machined out of these tubes, one has to resort to the testing of non-standard specimens. In this section, the state of current research in literature regarding fracture toughness evaluation of thin-walled fuel clad tubes has been presented. An X-shaped specimen made out of the thin-walled clad tubes was designed by Hsu et al. [10] to measure their fracture toughness. The ring tension test designed for measuring the transverse direction properties of structural tubes has been discussed in Arsene and Bai [11]. An internal conical mandrel technique has been used in Sainte Catherine et al. [12] to evaluate the fracture resistance behaviour of fuel clad tubes. In this method, an internal mandrel is forced through the cladding tubes leading to initiation and propagation of cracks in the axial direction. A PLT test was proposed by Grigoriev et al. [13, 14] to evaluate the fracture toughness of zircaloy-2 fuel claddings. In this method, two split cylindrical mandrels are placed inside the axially pre-cracked fuel pin specimen machined from the actual nuclear fuel clad tubes. The plane of split is aligned with the two diametrically opposite axial cracks in the fuel pin specimens. By restricting one end of the fuel pin from moving out of the mandrels, the tubular cracked geometry is loaded at the other end through a loading fixture which is composite to the mandrel. Later, Josefsson and Grigoriev [15] also developed a modified ring tension specimen to evaluate fracture toughness of irradiated claddings. They machined a reduced section in the ring specimen so as to localize the crack initiation and propagation in a particular region.

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Bertsch and Hoffelner [16] used a pair of split mandrels to load a specimen machined out of fuel pin with co-axial longitudinal cracks on both sides of the pin. They used specimens made of aluminium as well as zircaloy cladding tubes in order to compare the crack initiation toughness values of the two materials obtained from these types of thin-walled tubular specimens. However, they evaluated the fracture resistance behaviour (J–R curve) using  and  functions of standard fracture mechanics specimens. A curved compact tension (CT) specimen has also been proposed by Yagnik et al. [17] to determine the fracture toughness of pressure tubes. However, these specimens cannot be directly applied to measure fracture toughness of the fuel cladding due to the geometry. The cladding tube, much thinner than the pressure tube, does not meet the requirements of plane strain condition. During testing, the bending effects resulting from the cladding specimen curvature could also complicate the stress field around the crack tip. The PLT test setup has also been used by Grigoriev and Jakobsson [18, 19] to measure fracture toughness of hydrogen-embrittled zircaloy tubes. Due to the phenomenon of hydrogen embrittlement, the velocity of crack propagation changes in these materials. In Bertolino et al. [20–22], CT and single edge bending specimens have been machined from rolled zircaloy plates (with and without hydrogen charging) and these have been tested to evaluate the fracture resistance curve. Fukuda et al. [23] and Edsinger et al. [24] have also extensively studied the effect of hydrogen on degradation in fracture properties of zircaloy cladding tubes. The interaction of fuel pellet and zircaloy clad by mechanical and chemical processes (due to fission product environment such as iodine) is an important phenomenon which leads to initiation and propagation of stress corrosion cracks and has been considered in the works of Videm and Lunde [25]. The presence of neutron irradiation further degrades the mechanical and fracture properties of zircaloy fuel clad and the deterioration progresses with increasing burn-up seen by the fuel pins in the nuclear reactor. The fatigue and fracture behaviours of irradiated fuel clads are very complex and these have been studied by Nakatsuka et al. [26] and Grigoriev et al. [27] which are of extreme importance for the long-term integrity analysis of these tubes in the actual reactor environment. Recently, several authors [28–31] have attempted to evaluate the fracture toughness of the thin-walled metallic as well as thin film coating structures using cohesive zone elements in FE analysis. The interaction between buckling and crack extension behaviour Proc. IMechE Vol. 226 Part C: J. Mechanical Engineering Science

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of thin-walled aluminium structures has been studied by Li [28] using the above approach. It was concluded by Li [28] that the cohesive zone model approach is an attractive option in investigation of structural integrity of thin-walled structures, and provides a solution to the problem of geometry and size dependence of conventional crack growth parameters. As mentioned in the previous paragraphs, there have been significant efforts worldwide to develop an easy and reliable method for fracture property determination of these thin-walled fuel clad tubes. However, the geometric functions for the evaluation of fracture properties such as linear elastic stress intensity factor, and  and  functions for the evaluation of plastic part of J-integral from experimental load–displacement data for these particular nonstandard fracture mechanics specimens are not well established in literature. Recently, the authors have modelled the PLT test setup using 3D elastic–plastic FE analysis. Both the tube as well as the loading mandrel have been incorporated in the FE model. A particular fuel pin geometry (used in Indian BWRs as fuel clad) has been considered in the analysis with various values of initial crack lengths. The geometric function for evaluation of stress intensity factor for this particular PLT test setup has been derived using both analytical and numerical techniques in Samal et al. [32]. The geometric functions such as  and  functions required for the evaluation of plastic part of J-integral have been derived from the limit load solutions of these tubes obtained from elastic–plastic FE analysis. Later, a load-normalization technique was used in Samal et al. [33] and Sanyal et al. [34] to estimate the crack growth during the loading of the PLT specimen for a particular specimen geometry with a given initial crack length. This information along with the  and  functions have been used to obtain a first-hand result regarding the fracture toughness of these particular fuel clad tubes used in Indian BWRs. A comparison of fracture behaviour of two different types of fuel pins (with different dimensions, microstructures, and material properties) along with a detailed investigation for their micro-structural origin has been presented in Samal et al. [35]. A FE analysis of deformation behaviour of tubular structures was also presented in Dhia et al. [36]. Direct evaluation of crack growth during loading of small and thin tubular specimens is often cumbersome. Especially, in high rate loading conditions, aggressive environments, such as high temperature, hydrogen atmosphere, and irradiation, may not be possible to directly measure the crack growth using conventional techniques such as alternating current potential difference, direct current potential Proc. IMechE Vol. 226 Part C: J. Mechanical Engineering Science

difference, acoustic emission and ultrasonic techniques. Use of multi-specimen test techniques may be expensive and time consuming. Moreover, there can be variability in the toughness properties among the specimens. The unloading compliance method may also incur significant error in crack growth estimation if there is significant nonlinearity in the unloading load–displacement curve due to various conditions such as change in contact conditions, nonlinear material relaxations, and precision of instrument for compliance measurement. In order to circumvent the above problem, Ernst et al. [37] have proposed the socalled load-separation theory in 1979. Using this method, J-resistance curves have been directly estimated from load–displacement records in Ernst et al. [37]. Based on this load-separation theory, a load-normalization method was at first proposed by Herrera and Landes [38]. In this method, the load is normalized by a geometry function and then a deformation function can be expressed as the relationship between the normalized load and the normalized plastic displacement. To determine the instantaneous crack growth during loading, the deformation function can be described by a power function [39] with two calibration points or by LMN function [40] with three calibration points. The calibration points are the data points at which the load, displacement, and crack length must be known simultaneously. These data points are usually determined from the fracture surfaces of tested specimens. Sharobeam and Landes [41, 42] have successfully used this method to derive the pl function of several fracture mechanics specimens, which is required for the evaluation of plastic part of J-integral from the load–displacement records obtained in the specimen tests. This load-normalization method was applied to evaluate the fracture toughness of 6061-T6 grade aluminium alloy in Cassanelli et al. [43], various kinds of steels in Dzugan and Viehrig [44], Zhu and Joyce [45], and Zhu et al. [46] and various types of polymers in Baldi and Ricco` [47], Morhain and Velasco [48], Salazar and Rodrı´guez [49], and Varadarajan et al. [50] However, the load–displacement data of each specimen must be first normalized and a suitable set of initial data points and the last data point (corresponding to final crack growth) must be properly selected to obtain the constants of the fitted function. These constants can only be used for the estimation of crack growth in the specimen from which these were obtained by best fit of normalized load– displacement data. These constants are not transferable to other specimens with different initial crack lengths.

A load-separation technique to evaluate crack growth and fracture resistance behaviour

An alternative technique based on the load-separability parameter Spb has recently emerged in literature [51]. It has been successfully used to estimate the fracture resistance behaviour of a polymer specimen and a steel specimen in Wainstein et al. [52] and Chen Bao and Cai [53], respectively. This method relies on the ratio of load-carrying capability of a specimen with growing crack and other with stationary crack for a given value of plastic displacement. This ratio is a function of the ratio of crack lengths (instantaneous crack length of the specimen with growing crack to the initial crack length of the specimen with stationary crack). A brief overview of this method is presented in the following section. 3

EVALUATION OF LOAD-SEPARABILITY PARAMETER AND ITS USE IN ESTIMATION OF INSTANTANEOUS CRACK GROWTH

According to the load-separation theory [41–42], the load P at any instant of loading in a specimen, with crack length to width ratio, a=W , can be expressed as the product of a geometry G ða=W Þ, and a  function  deformation function H Vp =W in the following way    a  V   Vp a m pl P¼G H H ¼ Wt 1  ð1Þ W W W W where t is the thickness of the specimen, Vp the plastic part of displacement at load P, and m a material constant. Sharobeam and Landes [41] introduced a separable parameter Sij , which is defined as the ratio of load of two blunt-cracked specimens (with different crack lengths ai and aj ) at the same value of plastic displacement Vp , i.e.    G ðai =W ÞH Vp =W  Pi ðai Þ  G ða =W Þ    ¼  i  Sij ¼   ¼  G aj =W H Vp =W  Pi a j  G aj =W Vp

Vp

ð2Þ The parameter Sij is independent of the deforma tion function H Vp =W and depends only on the geometry function G ða=W Þ. For a stationary crack, the geometry function remains constant and hence the parameter Sij remains constant over the whole domain of the plastic displacement. In order to study the load-separation property for the specimens with crack growth during loading, the parameter was redefined in Sharobeam and Landes [42] as       m  G ap =W H Vp =W  Pp ðai Þ  ap    ¼ Spb ¼   ¼   ab Vp G ðab =W ÞH Vp =W Pb a j Vp

Vp

ð3Þ where the subscripts ‘p’ and ‘b’ represent the sharpcracked and the blunt-cracked specimens,

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respectively. The parameter Spb remains constant while there is no crack growth in the sharp cracked specimen. With crack growth, the parameter Spb changes according to equation (3). The material constant m need to be determined from testing a set of specimens with and without crack growth during loading, obtaining a database for Spb versus rpb (i.e. ap =ab ) and evaluation of m from the best fit of equation (3) with experimental data points. Usually, the initial and final crack lengths are measured on the fracture surface of the broken cracked specimen. The specimens were broken after heat tinting of the crack surfaces. These surfaces were observed under the stereo-microscope and the crack lengths measured at different points along the thickness of the specimens. The crack growth measurements were carried out at nine points and the average value was computed using the standard procedure. In addition to the initial and final crack length values, a third calibration point is required which corresponds to the unit value of Spb (i.e. Spb ¼ 1 when rpb ¼ 1). With these three points, a power law according to equation (3) is fitted. In some situations, these three points may not be enough for a reliable estimation of the material constant m. For this purpose, a modified procedure for estimation of the material constant m has been developed in this study so that data from many specimens can be used. This will lead to a more reliable estimation of crack growth during loading. The modified procedure is described below. Different fatigue pre-cracked specimens with known initial crack lengths are tested and their load versus displacement curves are recorded. Let the data for these specimens with growing crack be denoted by the subscript ‘g’. The load versus displacement responses of the specimens are to be determined with same values of initial crack length, however, with no crack growth or stationary crack during loading. Let the data for these specimens with growing crack be denoted by subscript ‘s’. It may not be possible to obtain such a load versus displacement data with a specimen with non-growing crack, especially obtaining a load point corresponding to the same value of plastic displacement Vp as recorded at the final load point of the specimen with propagating crack (the crack growth in the growing crack specimen at final point being equal to the final crack length as recorded in the specimen fracture surface). This is because the crack may not remain stationary even if an initial highly blunted crack is chosen in such a test setup (i.e. the PLT test setup) due to the highly constrained nature of loading. However, the authors chose to perform a 3D elastic–plastic FE analysis in order to calculate the load versus displacement response of the specimen with different initial crack Proc. IMechE Vol. 226 Part C: J. Mechanical Engineering Science

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lengths. The initial cracks remain stationary during loading due to elastic–plastic material constitutive equations which incorporate the only phenomenon of plastic hardening and there is no material softening. However, there can be global geometric softening due to geometry nonlinearity in the analysis which is synonymous with plastic collapse and necking of the remaining ligament in the specimen. The plastic part of displacement Vp is evaluated from the total displacement once the compliance of the specimen in the PLT setup (as a function of crack length to width ratio) is known. The compliance of the PLT setup for the specimen under consideration in this study has been estimated by both experiment and FE analysis in a previous work by Samal et al. [32] and the same has been used here. For the specimen with stationary crack, the compliance is constant throughout the loading and hence the computation of plastic displacement Vp is straightforward. Nevertheless, the calculation process of plastic displacement Vp for the specimen with growing crack needs an iterative procedure as the compliance of the system has to be updated at every load point with a new value of crack length to width ratio. The new load ratio Sgs is defined as the ratio of load for a growing crack to the load for a stationary crack with exactly the same initial geometry and crack lengths at a particular value of plastic displacement. The crack length ratio rgs is defined as the current crack length of the specimen with growing crack to the initial crack length of the specimen with the stationary crack (which is constant during loading). The current crack length ag in the specimen with growing crack can be estimated from the following equation, i.e. h  im1 ag ¼ as Sgs Vp ð4Þ For a given plastic displacement Vp of the stationary cracked specimen, the load of the specimen with growing crack can be estimated with the known value of crack length (i.e. the initial crack length or crack length computed for a previous point with a lesser value of plastic displacement). However, this is a first estimation of the load Pg for a given value of Vp . Once Sgs is evaluated, the new crack length ag can be computed. The above procedure can be repeated till a converged value of crack length ag is obtained. However, one must know the material constant m in order to use equation (4) in the estimation of crack growth in the PLT specimen. The evaluation procedure of the material constant m from several specimen test data will be discussed in Section 6 of ‘Results and discussion’. With this discussion on the Proc. IMechE Vol. 226 Part C: J. Mechanical Engineering Science

procedure for estimation of crack growth in PLT specimen, the authors proceed to describe the experimental details of the fracture toughness test of thinwalled zircaloy tubes in the PLT test setup. 4 FRACTURE EXPERIMENTS ON FUEL CLAD TUBES IN THE PLT SETUP The axially cracked specimens used in this study have been machined from the cladding tubes used in the Indian BWRs. These tubes have inner diameter and wall thickness 12.4 and 0.9 mm, respectively. The sketch of the tubular specimens with axial cracks is shown in Fig. 1. The as-received tubes used in these experiments are in the re-crystallization annealed (RXA) condition. The material is an alloy of zirconium with 1.5 wt% Sn, 0.09 wt% Fe, 0.1 wt% Cr, 0.05 wt% Ni, and 0.12 wt% oxygen and it is designated as zircaloy2. The microstructure contains equiaxed grains with an average grain diameter of approximately 5 mm. The matrix is more or less uniformly dispersed with intermetallic precipitates [32]. As seen from Fig. 1, there are two diametrically apart axial cracks at each end of the specimen. The loading fixture consists of a pair of split semi-cylindrical halves which form the cylindrical holder when put together. The fixture halves are loaded through the pins after being inserted into the tubular cracked specimen. The loading arrangement is shown in Fig. 2. The specimen is aligned such that the plane of axial cracks is co-planar with the openings of the split-halves of the mandrel. A pin at the back end of the fixture is inserted into a grove machined in the mandrels. It serves as an anchor point about which the semi-cylindrical fixtures rotate during the loading process.

Fig. 1

Geometric dimensions of the tubular specimen with longitudinal through-wall cracks

A load-separation technique to evaluate crack growth and fracture resistance behaviour

Fig. 3 Fig. 2

Specimen along with the loading fixture in the PLT setup

The tubular specimen is prevented from slipping from the loading mandrels by inserting a clip at the rear end of the mandrels (opposite to loading groves). The loading pins at the right end of the fixtures in Fig. 2 are connected to the machine frames through the loading attachments. A notch is provided at the left end of the specimen in order to prevent any possible buckling during the loading. The initial crack length a0 and width W (19 mm in this study) of the PLT setup are labelled in Fig. 2. In order to determine the material constant m for the evaluation of crack growth using load-separability parameter Sgs , four tubular specimens (labelled as specimens 1–4) machined from the above-mentioned zircaloy tubes have been tested initially. The average initial crack lengths (i.e. average values of initial crack lengths on two diametrically opposite sides) after fatigue pre-cracking have been measured as 11.2, 11.3, 11.35 and 11.6 mm, respectively, for the specimens 1–4. The crack lengths on both sides were originally measured as (11.4, 11.0), (10.7, 11.9), (11.7, 11.0), and (12.1, 11.1) mm, respectively, for the specimens 1–4. These data were obtained through measurement of the crack lengths in different regions on the heat-tinted surface of the specimens by stereomicroscope. It may be observed that the initial crack lengths of the four specimens are almost the same except that of specimen 4. The specimens with different a0/W ratios were tested in the PLT setup and the load versus load line displacement data obtained from the tests have been plotted in Fig. 3. The load line displacement is measured through a clip-gauge attached to the grooved end on the right-hand side of the test setup, as shown in Fig. 2, (i.e. between the loading mandrels along the

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Experimental load–load line displacement curves of four different specimens with different initial crack sizes

line of application of load). The specimens have been loaded to different extents of displacements (Fig. 3) in order to obtain different values of final crack lengths for the specimens after the loading is stopped. After the final load point, the specimens were broke open by fatigue loading so that the different regions of crack propagation can be identified in the fracture surface in order to accurately measure the initial and final crack lengths. After the load versus displacement data are obtained from the specimens with growing cracks from PLT tests, the corresponding load versus displacement data need to be evaluated for the specimens with stationary cracks. This is achieved by the use of elastic–plastic FE analysis of the PLT setup as described in the next section. 5 FE SIMULATION OF THE THIN-WALLED TUBULAR SPECIMENS WITH STATIONARY CRACK IN THE PLT SETUP The FE mesh used in this analysis is shown in Fig. 4. Only one-fourth of the loading setup has been modelled considering symmetry. The material true stress versus true plastic strain curve is required for the elastic–plastic FE analysis and the data at room temperature is presented in Fig. 5. The sharp crack is modelled in the FE mesh through the use of symmetric boundary conditions and hence the details of notch acuity are not considered in this analysis. As the authors are interested in calculating the load–displacement response, the convergence criteria used here is the ratio of the norm of the unbalanced load vector to the norm of the applied load vector. A value of 1 e6 is used for this ratio. The stress–strain curve is obtained from the ring tensile tests. Ring type of specimens are machined from Proc. IMechE Vol. 226 Part C: J. Mechanical Engineering Science

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Fig. 4

FE model of the test setup along with the boundary and loading conditions

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Fig. 5

a 0/W=0.566

Material stress–strain curve for the RXA zircaloy-2 fuel pin specimen

these tubes and loaded through split semi-cylindrical fixtures in order to obtain the load–displacement data which are converted to true stress–strain curves. Multiple data points along the stress–strain curves are used in the FE analysis as material property set. All the four specimens with different initial crack lengths have been analysed in order to obtain the load–displacement responses which correspond to the responses of the specimens with stationary cracks. The load–displacement response for a specimen with initial crack length to width ratio ða0 =W Þ of 0.566 as obtained from FE analysis is shown in Fig. 6. This corresponds to the load–displacement response of the specimen with stationary crack. The experimental data for the specimen with growing crack are also shown in Fig. 6. Both the Proc. IMechE Vol. 226 Part C: J. Mechanical Engineering Science

Fig. 6

0.5

1.0 1.5 2.0 load line displacement (mm)

2.5

Load versus load line displacement for the PLT specimen obtained from elastic–plastic FE analysis (with stationary crack) and comparison with experimental data (growing crack)

specimens have same initial crack lengths. The load–displacement curves of the identical specimens with stationary and growing cracks start deviating after a displacement of about 1 mm. This is due to crack propagation in the actual PLT experiment compared to the FE analysis where the crack is stationary. This information is used to evaluate the load-separation parameter Sgs for a given value of plastic displacement Vp as discussed in Section 3. The procedure for the evaluation of the material constant m from the above data of FE analysis and PLT experiment is discussed in the next section.

A load-separation technique to evaluate crack growth and fracture resistance behaviour

load ratio (Sgs)

1.00

Fig. 7

Measurement of initial and final crack lengths from the specimen fracture surface

data best fit

0.96

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0.88 1.00

6

RESULTS AND DISCUSSION

Once the load–displacement results have been evaluated for the specimens with growing and stationary cracks with different initial crack length to width ratios, the final crack lengths at the end point of the load–displacement curves need to be evaluated, as shown in Fig. 3 for each specimen. The final crack lengths have been measured on the fracture surface after heat-tinting operation in order to distinguish different regions of crack propagation. One such heat-tinted fracture surface for specimen 1 is shown in Fig. 7. It may be noted that four different regions are clearly visible on the fracture surface (i.e. notch region, fatigue pre-crack region, and actual zone of crack growth during loading and fracture surface obtained from final split-opening of the specimens by fatigue loading). The initial as well as the final crack fronts are not straight which is due to difference in the state of stress at the inner and outer radii of the tube. The crack tends to grow more at the inner radius of the tube compared to the outer surface. The crack lengths have been measured by taking measurements at nine different points along the thickness of the tubular specimens and the average crack lengths on both sides have been calculated. Again, it may be noted that the average crack lengths on both sides (i.e. diametrically opposite sides) of the specimens are not exactly the same due to slight difference in loading conditions for the two crack fronts. Moreover, the crack front with larger initial crack will tend to have larger crack growth compared to the opposite crack front. The crack length ratio rgs for each specimen at the final loading point has been calculated as the ratio of final crack length of the specimen with growing crack (as obtained from the final fracture surface) to the initial crack length which is same as initial crack length of the stationary specimen. For calculation of the load ratio Sgs , the plastic displacement component Vp needs to be computed. As has been discussed

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Fig. 8

1.02 1.04 crack length ratio (rgs)

1.06

Evaluation of parameter ‘m’ from experimental data of load and crack length ratios

earlier, this parameter can be computed from the displacement at the final point of load–displacement diagram once the compliance function for the PLT specimen with the initial a0 =W is known. In a previous work by Samal et al. [32], the compliance functions for this particular BWR fuel pin have been evaluated using both analytical and numerical schemes. The details can be found in Samal et al. [32]. The compliance function C ða=W Þ used in this study can be written as (in mm/N) a a  a 2 C ¼ 0:01101 þ 0:06113  0:11156 W W W  a 3 þ 0:0691 W ð5Þ Using the above equation (5), the plastic displacement Vp for the specimen with stationary crack can be evaluated in a straightforward manner. For the specimen with growing crack, an iterative procedure is followed to evaluate the plastic displacement Vp as discussed earlier in Section 3. For the same value of the plastic displacement Vp (at the final point of load– displacement data), the load ratio parameter Sgs is evaluated as the ratio of the load of the specimen with growing crack to the load of the specimen with stationary crack. Similar exercises were conducted for all the specimens and the data of Sgs is plotted as a function of rgs in Fig. 8. Equation (4) has been used to obtain a best fit to the experimental data points. The material constant parameter m has thus been evaluated as 2.043. This value has been found to be of same order as presented in Ref. [51]. Once the material constant m is obtained, it has been used to evaluate crack growth in other PLT specimen with different initial crack lengths and loaded to different amount of Proc. IMechE Vol. 226 Part C: J. Mechanical Engineering Science

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3000

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load (N)

load (N)

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a0/W=0.526 growing crack (exp.) stationary crack (FE) 1000

growing crack (exp.) stationary crack (FE)

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1 2 3 load line displacement (mm)

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Comparison of load versus load line displacement response for stationary crack and growing crack scenarios for the PLT specimen with a0/W ratio 0.526

2000

Fig. 11

a0/W=0.578 growing crack (exp.) stationary crack (FE)

1000

2 3 load line displacement (mm)

4

5

Comparison of load versus load line displacement responses for stationary and growing crack scenarios for the PLT specimen with a0/W ratio 0.631

a0 /W=0.685 growing crack (exp.) stationary crack (FE)

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0

0 0

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0

5

Comparison of load versus load line displacement responses for stationary and growing crack scenarios for the PLT specimen with a0/W ratio 0.578

displacements. Four such specimens have been machined from the same zircaloy fuel pins of Indian BWR. The initial a0 =W of these specimens have been measured as 0.526, 0.578, 0.631, and 0.685, respectively. The aim is to evaluate the fracture resistance (J–R) curves from the experimental load–displacement data. The necessary geometric functions, i.e.  and  functions have been evaluated for these tubes as a function of a=W ratio in a previous work by Samal et al. [33]. Elastic–plastic FE analyses for these specimens have been carried out in order to evaluate the load–displacement response for the corresponding specimens with stationary cracks. The load–displacement results obtained from the FE analysis of these Proc. IMechE Vol. 226 Part C: J. Mechanical Engineering Science

1

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load line displacement (mm)

load line displacement (mm)

Fig. 10

1

1000

load (N)

load (N)

a0/W=0.631

500

0

Fig. 9

1000

Fig. 12

Comparison of load versus load line displacement responses for stationary and growing crack scenarios for the PLT specimen with a0/W ratio 0.685

specimens with stationary cracks have been plotted along with the load–displacement data obtained from PLT tests for specimens with growing cracks (Figs 9–12). As discussed earlier, the difference in the response between the two results is due to the growth of crack in the PLT experiment, whereas the crack is stationary in FE analysis. The load ratio parameter Sgs has been evaluated as a function of plastic displacement for the four different specimens and is plotted in Fig. 13. The parameter Sgs decreases as the plastic displacement increases due to more and more crack growth due to increased loading. The behaviour of the four specimens as regards to variation of the parameter Sgs as a function of plastic displacement Vp is nearly the same except

A load-separation technique to evaluate crack growth and fracture resistance behaviour

a0 /W=0.526

1.2

1.4

a0 /W=0.578

1.2 crack growth (mm)

a0 /W=0.684

Sgs

0.8 0.6 0.4 0.2

a0/W=0.631 a0/W=0.684

1.0 0.8 0.6 0.4 0.2

0.0 0

Fig. 13

a0/W=0.526 a0/W=0.578

a0 /W=0.631

1.0

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2 3 plastic displacement (mm)

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0.0

5

Variation of load-separation parameter with plastic displacement for four PLT specimens with different initial crack lengths

for the region beyond the plastic displacement of 3 mm approximately. In this region, the specimen with a higher value of a0 =W ratio exhibits a higher drop in the load ratio Sgs , which corresponds to the higher crack growth for this specimen. At small plastic displacements, the crack growth is very small and hence the load-separation parameter Sgs may not be very sensitive in this region. However, when there is appreciable stable crack growth, the variation of load-separation parameter (with applied plastic displacement) becomes almost independent of the initial a0/W ratios of the different specimens as it is a normalized parameter. At larger plastic displacements, the remaining ligament is small and different for different specimens. The crack growth in the remaining ligaments becomes unstable at larger values of plastic displacements. The FE analysis represents the response of a cracked specimen (with stationary crack) as a function of applied plastic displacement. Hence, the loadseparation parameter (which is the ratio of the load carried by the specimen with stationary crack to the load carried by the specimen with growing crack, for a given value of plastic displacement) becomes different for different specimens at larger values of applied plastic displacement. The crack growth information a as a function of plastic displacement can be evaluated using an iterative procedure as discussed earlier along with the use of equation (4). These crack growth data are required along with the experimental load– displacement curve in order to evaluate the necessary fracture resistance (J–R) curve of these specimens which have different initial a0 =W ratio. The variation of crack growth with plastic displacement is shown in Fig. 14 for the four different specimens. The specimen

0

1

2

3

4

5

plastic displacement (mm)

Fig. 14

Variation of crack growth with plastic displacement for four PLT specimens with different initial crack lengths

with higher initial a0 =W ratio exhibit higher crack growth in the regions of larger plastic displacement. The value of J-integral in the J–R curve (at any loading point i) is evaluated using elastic and plastic parts of J as   2 KðiÞ 1  2 þ JplðiÞ ð6Þ JðiÞ ¼ E where the first term of the right-hand side of equation (6) is the elastic part of the J-integral, and the plastic part of the J-integral, Jpl is evaluated as  

ði1Þ AplðiÞ  Aplði1Þ JplðiÞ ¼ Jplði1Þ þ bði1Þ 2t " # 0 0 ð7Þ aðiÞ  aði1Þ 1  ði1Þ bði1Þ The method of evaluation of stress intensity factor KI needed for calculation of elastic part of J-integral for these BWR fuel clad tubes (in PLT setup) has been discussed by Li [28]. The expression for KI can be written as load P, thickness t , and geometric function f ða=W Þas KI ¼

P pffiffiffiffiffiffi f ða=W Þ 2t W

ð8Þ

These geometric functions f ða=W Þ have been evaluated for these particular PLT specimens by both analytical and numerical techniques in Li [28]. The expression for f ða=W Þ used in this study is given here as f

a

a  a 2 ¼ 416:9  1903:9 þ 2847:7 W W W  a 3  1330:3 W

ð9Þ

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a0 /W=0.526 a0 /W=0.578

J-integral (N/mm)

600

400

200

Fig. 17

Fig. 16

Variation of  factor with a=W for the PLT specimen under consideration

Variation of  factor with a=W for the PLT specimen under consideration

For evaluation of the plastic part of the J-integral using equation (7), the geometric functions, i.e.  and  functions are required to be evaluated as function of a=W ratio. These functions have been evaluated from the limit load expressions of these PLT specimens in Zavattieri [29]. The limit load values were previously evaluated from a detailed elastic–plastic FE analysis of this PLT setup. The variations of  and  functions required for evaluation of the plastic J-integral as a function of a=W ratio for this BWR fuel clad PLT specimen are shown in Figs 15 and 16, respectively. The  and  factors depend upon the loading conditions prevalent in the cracked geometries. The value of  is ‘1’ for pure tensile loading and ‘2’ for pure bending case of loading. However, for other loading conditions, it depends upon the a/W ratio of the specimen. Proc. IMechE Vol. 226 Part C: J. Mechanical Engineering Science

a0 /W=0.684 load-normalization method

0 0.0

Fig. 15

a0 /W=0.631

0.2

0.4 0.6 0.8 crack growth (mm)

1.0

1.2

Estimation of fracture resistance behaviour (J–R curve) for four PLT specimens with different initial a0 =W ratios by the method of load separation and comparison of results with those obtained by load-normalization technique

As the loading condition in our specimen geometry is complex, the values of  factor range from 1.4 to 3 for a/W ratios varying from 0.46 to 0.79. Similar values for  factor are also reported for the double edged notch tensile specimen geometry in the EPRI report (Kumar et al. [54]). For this specimen, the  factor varies from 1.6 to 0.9 for a/W ratios of 0.125 to 0.375 and 0.9 to 2.9 for a/W ratios of 0.375 to 0.8. This aspect of the nature of variation of  factor has also been discussed by Wilson and Mani [55]. Similarly, the gamma factor depends upon the loading condition and for a complex loading situation (as encountered in this work), such a variation can be observed. Using the area under the experimental load–displacement record from Figs 9–12, crack growth data from Fig. 14 and  and  functions from Figs 15 and 16, the J–R curves have been evaluated for the four different PLT specimens with different values of initial a0 =W ratios and the results are presented in Fig. 17. It can be observed that the fracture resistance behaviours of the four specimens are nearly similar as the initial a0 =W ratios are very close to each other. However, the J–R curves are slightly lower for specimens with higher values of a0 =W ratios beyond 0.8 mm of crack growth (a) approximately. The J–R curves obtained by this load-separation technique have also been compared in the above figure with that obtained from the load-normalization technique for a similar PLT specimen (made from same material and same fuel clad tube). The details of the loadnormalization technique as applied to a BWR fuel clad PLT specimen and the results of fracture resistance behaviour has been discussed by Samal et al. [33]. It can be observed from Fig. 17 that the results

A load-separation technique to evaluate crack growth and fracture resistance behaviour

obtained by both the techniques are very close to each other. In the load-normalization method, the crack growth is calculated from the normalized load–displacement curve of each specimen using a specific type of function. The values of the constants in this function are different for different specimens. However, in our current method, data from many specimens have been used to obtain the exponent ‘m’ which is valid for all these specimens. As the methods of estimation of crack growth are different in these two methods (i.e. modified load-separation versus load-normalization techniques), the crack lengths computed during loading are slightly different. This is the reason why the slope of the J–R curve as obtained from the use of load-normalization technique is a little bit different compared to that obtained from the modified loadseparation technique. Nevertheless, the J–R curves are comparable in both the cases of analysis. The new load-separation method presented in this study can be used easily for other specimens with different initial crack length values as the material constant m is same for all the values of a0 =W . Elastic–plastic FE analysis can replace the need for testing of specimen with stationary crack. Once the experimental data of the PLT specimen is obtained from the test, this can be directly used to evaluate the J–R curve without the need to measure the crack growth information by other conventional methods which may be cumbersome, inaccurate, and time consuming. Hence, this method is very suitable for evaluation of fracture resistance behaviour of thinwalled tubular specimens and can be extended to other loading environments such as high temperature, hydrogen, and irradiations environments as encountered in nuclear fuel pin applications. 7

CONCLUSIONS

Evaluation of fracture resistance behaviour of thinwalled small-diameter tubes are of utmost importance in thermal power, chemical, process, and nuclear power plants. Especially, it is important for designers and safety analysts to ensure that the zirconium alloy fuel clad tubes do not burst during the reactor operation either due to abnormal plant transients or RIA scenarios. The bursting of these tubes can release fission products into the coolant and in turn increase the activity in the coolant. Estimation of fracture behaviour of these thin-walled tubes cannot be evaluated using standard fracture mechanics specimens. The PLT test technique as discussed in this study for evaluation of fracture resistance behaviour zircaloy-2 fuel clad tubes of Indian BWRs seems to be a promising and reliable method. All the necessary

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geometric functions for this tube geometry have been evaluated by the authors through a detailed FE analysis of the PLT test setup. Both load-normalization and load-separation parameter techniques have been explored in order to evaluate the crack growth information directly from the experimental load–displacement data of a PLT specimen. The modified procedure as developed in this study can make use of data from many specimens in order to reliably evaluate the material constant m. With combined FE analysis and experimental data, the J–R curves can be evaluated in a straightforward manner though an iterative procedure is needed for evaluation of the crack growth information at different load steps. The loadnormalization technique requires parameters of the normalization functions to be fitted for each specimen, whereas the same parameter m can be used here for each specimen with different a=W ratios in case of this method which uses the load-separation parameter Sgs . The applications of this method to other types of fuel clad specimens and other loading environment including the high-temperature environment will be a scope of further research. FUNDING This research received no specific grant from any funding agency in the public, commercial, or notfor-profit sectors. ACKNOWLEDGEMENT This work has been carried out under the XI-plan project of the Department of Atomic Energy, Government of India with Sub-project No. 06_R&D_N_06.17/ Development of techniques for assessing structural integrity of tubular nuclear structural components: fuel cladding and pressure tubes. The authors acknowledge the kind support of Dr. J.K. Chakravartty, Mr. V. Bhasin and Mr. K.K. Vaze for this work.

ß Authors 2011 REFERENCES 1 Northwood, D. O. The development and applications of zirconium alloys. Mater. Des., 1985, 6(2), 58–70. 2 Meyer, R. O., McCardell, R. K., Chung, H. M., Diamond, D. J., and Scott, H. H. A regulatory assessment of test data for reactivity initiated accidents. Nucl. Saf., 1996, 37(4), 271–288. 3 Sridharan, K., Harrington, S. P., Johnson, A. K., Licht, J. R., Anderson, M. H., and Allen, T. R. Oxidation of plasma surface modified zirconium Proc. IMechE Vol. 226 Part C: J. Mechanical Engineering Science

1460

4

5

6

7

8

9

10

11

12

13

14

15

16

17

M K Samal and G Sanyal

alloy in pressurized high temperature water. Mater. Des., 2007, 28(4), 1177–1185. Gou, Y., Li, Y., Liu, Y., Chen, H., and Ying, S. Evaluation of a delayed hydride cracking in Zr– 2.5Nb CANDU and RBMK pressure tubes. Mater. Des., 2009, 30(4), 1231–1235. Fuketa, T., Sasajima, H., and Sugiyama, T. Behavior of high-burnup PWR fuels with low-tin zircaloy-4 cladding under reactivity initiated accident conditions. Nucl. Technol., 2001, 133, 50–62. Chung, H. M. and Kassner, T. F. Cladding metallurgy and fracture behaviour during reactivityinitiated accidents at high burnup. Nucl. Eng. Des., 1998, 186, 411–427. Link, T. M., Koss, D. A., and Motta, A. T. Failure of zircaloy cladding under transverse plane-strain deformation. Nucl. Eng. Des., 1998, 186, 379–394. ASTM Standard E 1820-01. Standard test method for measurement of fracture toughness, Vol. 03.01, 2000, pp. 1–21 (ASTM, Philadelphia, Pennsylvania). Rashid, Y. R., Lemaignan, C., and Strasser, A. Evaluation of fracture toughness and extension in fuel cladding, American Nuclear Society, Park City, UT, April 2000. Hsu, H. H., Chien, K. F., Chu, H. C., and Liaw, P. K. An X-specimen test for determination of thin walled tube fracture toughness. In Fatigue and fracture mechanics (Ed. R. Chona), 2001, vol. 32, ASTM STP 1406, pp. 214–226 (American Society for Testing and Materials, West Conshohocken, Pennsylvania). Arsene, S. and Bai, J. New approach to measuring transverse properties of structural tubing by a ring test. ASTM J. Test. Eval., 1996, 24(6), 386–391. Sainte Catherine, C., Le Boulch, D., Carassou, S., Ramasubramanian, N., and Lemaignan, C. An internal conical mandrel technique for fracture toughness measurements on nuclear fuel cladding. ASTM J. Test. Eval., 2006, 34(5), 373–382. Grigoriev, V., Josefsson, B., Lind, A., and Rosborg, B. A pin-loading tension test for evaluation of thin-walled tubular materials. Scr. Metall. Mater., 1995, 33(1), 109–114. Grigoriev, V., Josefsson, B., and Rosborg, B. Fracture toughness of zircaloy cladding tubes. In Zirconium in nuclear industry. Proceedings of the 11th International Symposium (Eds. E. R. Bradely and G. P. Sabol), 1996, ASTM STP 1295, pp. 431– 447 (American Society for Testing and Materials, West Conshohocken, Pennsylvania). Josefsson, B. and Grigoriev, V. Modified ring tensile testing and a new method for fracture toughness testing of irradiated cladding. In Proceedings of the European Working Group Hot Laboratories and Remote Handling Meeting, Petten, Netherlands, 14–16 May 1996, pp. 1–14. Bertsch, J. and Hoffelner, W. Crack resistance curve determination of tube cladding material. J. Nucl. Mater., 2006, 352, 116–125. Yagnik, S. K., Ramasubramanian, N., Grigoriev, V., Sainte-Catherine, C., Bertsch, J., Adamson, R., Kuo, R.-C., Mahmood, S. T., Fukuda, T., Efsing, P., and

Proc. IMechE Vol. 226 Part C: J. Mechanical Engineering Science

18

19

20

21

22

23

24

25

26

27

28

Oberla¨nder, B. C. Round-robin testing of fracture toughness characteristics of thin-walled tubing. J. ASTM Int., 2008, 5(2), 1–21. Grigoriev, V. and Jakobsson, R. Application of the pin loading tension test to measurements of delayed hydride cracking velocity in zircaloy cladding, SKI Rapport 00:57, Studsvik Nuclear AB, SE-611 82 Nyko¨ping, Sweden, November 2000. Grigoriev, V. and Jakobsson, R. Delayed hydrogen cracking velocity and J-integral measurements on irradiated BWR cladding. J. ASTM Int., 2005, 2(8), 1–16. ˜ a, J. P. Bertolino, G., Meyer, G., and Ipin Degradation of the mechanical properties of zircaloy-4 due to hydrogen embrittlement. J. Alloys Compd. 2002, 330–332, 408–413. Bertolino, G., Meyer, G., and Perez, I. J. Effects of hydrogen content and temperature on fracture toughness of zircaloy-4. J. Nucl. Mater., 2003, 320, 272–279. Bertolino, G., Meyer, G., and Perez, I. J. In-situ crack growth observation and fracture toughness measurement of hydrogen charged zircaloy-4. J. Nucl. Mater., 2003, 322, 57–65. Fukuda, T., Itagaki, N., Kamimura, J., Mozumi, Y., and Furuya, T. Fracture toughness of hydrogenabsorbed zircaloy-2 cladding tube. Paper No. 21. In Proceedings of the International Conference on Nuclear Fuel ENS Top Fuel 2003/ ANS LWR Fuel Performance Meeting, Wurzburg, Germany, 16–19 March 2003. Edsinger, K., Davies, J. H., and Adamson, R. B. Degraded fuel cladding fractography and fracture behaviour. In Zirconium in nuclear industry: 12th International symposium (Eds. G. P. Sabol and G. D. Moan), 2000, ASTM STP 1354, pp. 316– 339(American Society for Testing and Materials, West Conshohocken, Pennsylvania). Videm, K. and Lunde, L. Stress corrosion cracking and growth formation of pellet-clad interaction defects. In Zirconium in nuclear industry: 4th International symposium (Eds. J. H. Schemel and T. P. Papazoglou), 1979, ASTM STP 681, pp. 230–237(American Society for Testing and Materials, West Conshohocken, Pennsylvania). Nakatsuka, M., Kubo, T., and Hayashi, Y. Fatigue behavior of neutron irradiated zircaloy-2 fuel cladding tubes. In Proceedings of the Ninth International Symposium on Zirconium in the nuclear industry, (Eds. C. M. Eucken and A. M. Garde), 1991, ASTM STP 1132, pp. 230–245 (American Society for Testing and Materials, West Conshohocken, Pennsylvania). Grigoriev, V., Josefsson, B., Rosborg, B., and Bai, J. A novel fracture toughness testing method for irradiated tubing-experimental results and 3D numerical evaluation. In Transactions of the 14th International Conference on Structural mechanics in reactor technology, Lyon, France, 17–22 August 1997, 57–64. Li, W. Studies of fracture in thin-wall and thin-film structures by use of cohesive zone model approach,

A load-separation technique to evaluate crack growth and fracture resistance behaviour

29

30 31

32

33

34

35

36

37

38 39 40

41 42

2003 (Doctoral Dissertation, Purdue University, West Lafayette, Indiana). Zavattieri, P. D. Modeling of crack propagation in thin-walled structures using a cohesive model for shell elements. Trans. ASME J. Appl. Mech., 2006, 73, 948–958. Li, W. and Siegmund, T. An analysis of crack growth in thin-sheet metal via a cohesive zone model. Eng. Fract. Mech., 2002, 69(18), 2073–2093. Cirak, F., Ortiz, M., and Pandolfi, A. A cohesive approach to thin-shell fracture and fragmentation. Comput. Meth. Appl. Mech. Eng., 2005, 194(21–24), 2604–2618. Samal, M. K., Sanyal, G., and Chakravartty, J. K. An experimental and numerical study of the fracture behaviour of tubular specimens in a pin-loading-tension set-up. J. Mech. Eng. Sci., 2010, 224(1), 1–12. Samal, M. K., Sanyal, G., and Chakravartty, J. K. Estimation of fracture behavior of thin walled nuclear reactor fuel pins using pin-loading-tension (PLT) test. Nucl. Eng. Des., 2010, 240, 4043–4050. Sanyal, G., Samal, M. K., Ray, K. K., Chakravartty, J. K., Suri, A. K., and Banerjee, S. Prediction of J–R curves of thin-walled fuel pin specimens in a PLT setup. Eng. Fract. Mech., 2011, 78, 1029–1043. Samal, M. K., Sanyal, G., and Chakravartty, J. K. Investigation of failure behavior of two different types of zircaloy clad tubes used as nuclear reactor fuel pins. Eng. Fail. Anal 2011, in press. doi:10.1016/ j.engfailanal.2011.06.009. Dhia, A. B., Bai, J. B., and Francois, D. 3D finite element analyses of a new fracture toughness testing method for tubular structures. Int. J. Press. Vessels Pip., 1997, 71, 189–195. Ernst, H. A., Paris, P. C., Rossow, M., and Hutchinson, J. W. Analysis of load-displacement relationships to determine J-R curve and tearing instability material properties. In Proceedings of the Fracture mechanics: Eleventh Conference (Ed. C. W. Smith), 1979, ASTM STP 677, pp. 581–599 (American Society for Testing and Materials, West Conshohocken, Pennsylvania). Herrera, R. and Landes, J. D. A direct J-R curve analysis of fracture toughness tests. J. Test. Eval., 1988, 16(5), 427–449. Landes, J. D. and Herrera, R. A new look at J-R curve analysis. Int. J. Fract., 1988, 36, 9–14. Landes, J. D., Zhou, Z., Lee, K., and Herrera, R. Normalization method for developing J-R curves with the LMN function. J. Test. Eval., 1991, 19(4), 305–311. Sharobeam, M. H. and Landes, J. D. The load separation criterion and methodology in ductile fracture mechanics. Int. J. Fract., 1991, 47(2), 81–104. Sharobeam, M. H. and Landes, J. D. The load separation and Zpl development in precracked

43

44

45

46

47

48

49

50

51

52

53

54

55

1461

specimen test records. Int. J. Fract., 1993, 59(3), 213–226. Cassanelli, A. N., Ortiz, H., Wainstein, J. E., and de Vedia, L. A. Separability property and load normalization in AA 6061-T6 aluminum alloy. In Fatigue and fracture mechanics, vol. 32, 2001, ASTM STP 1406, pp. 49–72 (American Society for Testing and Materials, West Conshohocken, Pennsylvania). Dzugan, J. and Viehrig, H. W. Application of the normalization method for the determination of J-R curves. Mater. Sci. Eng., 2004, A387–389, 307–311. Zhu, X. K. and Joyce, J. A. J-resistance curve testing of HY80 steel using SE(B) specimens and normalization method. Eng. Fract. Mech., 2007, 74(14), 2263–2281. Zhu, X. K., Lam, P. S., and Chao, Y. J. Application of normalization method to fracture resistance testing for storage tank A285 carbon steel. Int. J. Press. Vessels Pip., 2009, 86(10), 669–676. Baldi, F. and Ricco`, T. High-rate J-testing of toughened polyamide 6/6: applicability of the load separation criterion and the normalization method. Eng. Fract. Mech., 2005, 72(14), 2218–2231. Morhain, C. and Velasco, J. I. Determination of J-R curve of polypropylene copolymers using the normalization method. J. Mater. Sci., 2001, 36(6), 1487–1499. Salazar, A. and Rodrı´guez, J. The use of the load separation parameter Spb method to determine the J-R curves of polypropylenes. Polym. Test., 2008, 27(8), 977–984. Varadarajan, R., Dapp, E. K., and Rimnac, C. M. Static fracture resistance of ultra high molecular weight polyethylene using the single specimen normalization method. Polym. Test., 2008, 27(2), 260–268. Wainstein, J., Vedia, L. A., and Cassanelli, A. N. A study to estimate crack length using the separability parameter Spb in steels. Eng. Fract. Mech., 2003, 70(17), 2489–2496. Wainstein, J., Frontini, P. M., and Cassanelli, A. N. J-R curve determination using the load separation parameter Spb method for ductile polymers. Polym. Test., 2004, 23, 591–598. Chen Bao, C. and Cai, L. Estimation of the J-resistance curve for Cr2Ni2MoV steel using the modified load separation parameter Spb method. J Zhejiang Univ. Sci. A (Appl. Phys. Eng.), 2010, 11(10), 782–788. Kumar, V., German, M. D., and Shih, C. F. An engineering approach for elastic–plastic fracture analysis, EPRI report no. NP-1931, Electric Power Research Institute, Palo Alto, California, 1981. Wilson, C. D. and Mani, P. Plastic J-integral calculations using the load separation method for the double edged notched tensile specimen. Eng. Fract. Mech., 2008, 75, 5177–5186.

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