PLT paper Mech Eng Sc

1

An experimental and numerical study ofthe fracture behaviour of tubular specimensin a pin-loading-tension set-upM K Samal1∗, G Sanyal2, and J K Chakravartty2

1Reactor Safety Division, Bhabha Atomic Research Centre, Mumbai, India2Mechanical Metallurgy Section, Bhabha Atomic Research Centre, Mumbai, India

The manuscript was received on 12 June 2009 and was accepted after revision for publication on 13 July 2009.

DOI: 10.1243/09544062JMES1781

Abstract: Evaluation of a valid crack tip loading parameter using standard ASTM practice maynot be feasible in many situations, e.g. in thin tubular specimens with diametrically opposite axialthrough-wall cracks (loaded in opening mode) that are obtained from zircaloy fuel pins of nuclearreactors. Recently, a pin-loading-tension (PLT) experimental set-up has evolved for this purpose.However, the required geometry functions (which should be obtained from a three-dimensional(3D) finite-element (FE) analysis procedure) for evaluation of the stress intensity factor (SIF) forsuch a set-up is not available in the literature. In this work, a detailed 3D FE analysis of the PLTset-up has been carried out where both the pin and the mandrel have been modelled. Analysishas been carried out for the full loading path (i.e. for applied displacements beyond the limitload of the specimen) in order to demonstrate the effectiveness of the model and the results havebeen compared with those of the experiment. In addition, the SIFs (and geometry functions)of the PLT set-up (with different initial crack lengths) have been evaluated from the FE resultsusing extrapolation techniques and were compared with the results from two different analyticaltechniques.

Keywords: zircaloy fuel pin, fuel clad material, pin-loading-tension set-up, stress intensity factor,geometry functions

1 INTRODUCTION

The failure of thin-walled tubes like zircomium-2.5per cent niobium pressure tubes and zircaloy fuelcladdings due to the presence of axial cracks andinternal pressure has been a long standing concern ofthe nuclear industry and has received much attentionrecently in the research literature [1–11]. These tubesmay have defects in the form of small cracks during thefabrication process or the small cracks may developduring service. A number of test techniques have beenproposed [2–4, 7, 8, 11–17] that can approximate thestress state experienced by the cladding for crack initi-ation and propagation in the axial direction, and hence

∗Corresponding author: Reactor Safety Division, Bhabha Atomic

Research Centre, Hall-7, Trombay, Mumbai, Maharashtra 400085,

India.

email: [email protected]

these may provide a measure of the crack propagationresistance. However, the classical fracture mechan-ics procedure (e.g. ASTM standard test techniques[18]) cannot be applied directly to the thin-walledcladding geometry. This is because of the presence oflarge plastic zone sizes (compared to the thickness)at the crack tips of the cladding and hence it violatesthe assumptions of linear elastic fracture mechan-ics. Thus, the attempts to measure fracture toughnesshave been influenced not only by material character-istics but also by the technique used to measure it. Alarge scatter also exists in the reported data from lit-erature [2]. In addition, crack propagation resistancein the radial direction of the thin cladding tubes iseven harder to quantify due to the small wall thick-ness. Because of the above limitations, investigationrelated to fracture toughness assessment of claddingtubes for nuclear fuel is limited, although severalother components of the nuclear reactors are currentlybeing designed and fabricated based on the structuralintegrity assessment concepts [19–21].

JMES1781 Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science

2 M K Samal, G Sanyal, and J K Chakravartty

Any new attempt to assess the fracture resistanceof the currently used or the emerging cladding tubematerials requires design and fabrication of the exper-imental set-ups as well as theoretical analyses toestimate the critical stress intensity factor (SIF) orany such parameter based on J -integral or crack tipopening displacement, etc. Till now, very few num-bers of attempts were made by researchers for eval-uation of fracture behaviour of thin walled tubes[3–5, 7, 10–12, 19]. A curved compact tension (CT)specimen has been proposed in reference [2] to deter-mine the fracture toughness of pressure tubes. How-ever, these specimens cannot be directly applied tomeasure fracture toughness of the fuel cladding dueto the geometry. The cladding wall, much thinner thanthe pressure tube, does not meet the requirements ofplane strain condition. During testing, the bendingeffects resulting from the cladding specimen curva-ture could also complicate the stress field around thecrack tip. A pin-loading tension (PLT) test was pro-posed by Grigoriev et al. [12] to evaluate the fracturetoughness of zicaloy-2 fuel claddings. Axially notchedring specimens cut from the fuel cladding sectionhave been tested in a way similar to that used totest compact tension specimens. Hsu et al. [3] havepresented an X-specimen test technique for determin-ing the fracture toughness of the thin-walled tubularmaterials. In their method, two curved CT specimenswere glued together by epoxy adhesive to make anX-shaped specimen so as to minimize the bendingeffects of the loading. A modified ring tension testwas carried out by Josefsson and Grigoriev [19] forfracture toughness testing of cladding where flatten-ing of the ring is prevented by means of a centralpiece that is inserted into the ring and has almost thesame dimension as the ring’s internal diameter. A highstrain rate mechanical testing technique called theexpansion-due-to-compression test has been devel-oped at Studsvik, Sweden, to simulate the pellet–cladmechanical interaction loading of the cladding duringa reactivity initiated accident.

In the present work, a detailed experimental inves-tigation has been carried out on thin walled claddingtubes used in Indian boiling water reactors (BWR).Specimens with different initial notch lengths havebeen tested in order to establish the geometry-dependent functions required for the evaluation ofSIFs. Finite-element (FE) simulations have been car-ried out for computing the stress and strain fieldnear the crack tip of the specimens. The actual three-dimensional (3D) geometry of the tube along with theloading mandrels has been modelled in this work.The load–displacement behaviour of the specimenobtained from FE analysis has been compared withthose of the experiment for applied displacementranges beyond the limit load of the specimen in orderto demonstrate the effectiveness of the model. Inaddition, extrapolation techniques [21–25] have been

used to determine the SIF solutions for the crackedtubes as a function of crack length. Based on exper-imental data of compliance of the specimens, thegeometric function has been evaluated using two dif-ferent analytical techniques [26–29]. The results ofgeometry functions obtained from FE analysis havebeen compared with those from analytical methods.The results have also been compared with those fromliterature, which are based on specimens of differentgeometrical configurations.

This article is organized into seven sections.Section 2 describes the details of the material andits properties used in this work. Section 3 discussesthe details of the experimental procedure followed inthis work, including the description of the test set-up. Section 4 gives an overview of the two analyticalmethods followed to derive the geometric functionsfor the SIF solution from the experimental data basedon the compliance of the specimens. Section 5 dis-cusses the procedure followed for developing FE mod-els of the specimens. The results obtained from FE andanalytical methods are discussed in section 6 followedby concluding remarks in section 7.

2 MATERIAL

Zirconium-based alloys like zircaloy-2, zircaloy-4,zirlo, etc. are some of the candidate materials thatare being currently used as fuel claddings in thecore of thermal nuclear reactors. Typically, the innerdiameter and the wall thickness of these tubes are14.2 and 0.4 mm, respectively, for pressurized heavywater reactors, and 12.4 and 0.9 mm, respectively, forBWRs used in India. The as-received tubes used inthis experiment are in the re-crystallization annealed(RXA) condition. The chemical composition of thematerial (zircaloy-2) is given in Table 1. This mate-rial exhibits anisotropy in the mechanical propertiesand these are studied in references [30] to [33]. Themicrostructure of this alloy is shown in Fig. 1. As canbe seen from this figure, the microstructure containsequi-axed grains with an average grain diameter ofapproximately 5 μm. The matrix is more or less uni-formly dispersed with inter-metallic precipitates. Thefracture toughness of this material depends on themetallurgical characteristics controlling its properties,which include chemical composition, microstructure,crystallinity, and dislocation density as well as onthe extent of hydrogen degradation (due to differentmechanisms).

Table 1 Composition of the fuel pin material

Element Sn Fe Cr Ni O2 Zr

Wt% 1.5 0.09 0.1 0.05 0.12 Rest

Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science JMES1781

Fracture behaviour of tubular specimens in a pin-loading-tension set-up 3

3 EXPERIMENTAL WORK

3.1 Fabrication of specimen

A schematic diagram of the specimen (along withthe dimensions) is shown in Fig. 2. There are two

Fig. 1 Microstructure of RXA zicalloy-2 (material ofIndian BWR fuel pin)

Fig. 2 Geometric dimensions of the tubular specimenwith longitudinal through-wall cracks

diametrically apart notches at each end of the speci-men. The width of the notch at one end is 0.25 mm,whereas it is 0.5 mm at the other end. The notchesat the specimen end (i.e. where the thinner notchesexist) are provided for subjecting the specimen tomode-I loading. The other end contains two dia-metrically opposite rectangular slots of dimension2 mm × 0.5 mm, respectively. These two slots are madefor avoiding buckling of the material (behind thepin) when the specimen is loaded. Seven specimenswith different initial notch lengths ranging from 1 to7 mm (with an interval of 1 mm) are fabricated byusing the electro-discharge machining (i.e. wire cut-ting) technique. The actual photographs of all theseven specimens used in this study are shown inFig. 3.

3.2 Fabrication of fixture

The fixture consists of two split-halves that form thecylindrical holder. Figure 4(a) shows the schematicdrawing of the fixture half (along with the dimensions)used for loading the specimens. The fixture halves areloaded through the pins that have the possibility ofmutual rotation around an axis as shown in the 3Dview of Fig. 4(b). These split-halves are inserted insidethe specimen and the specimen is aligned such thatthe axial cracks are in line with the openings of thesplit-halves of the mandrel upon loading. A pin at theback end of the fixture is inserted into the positionas shown in Fig. 4(b) after the specimen is mountedon the fixture. The dimension of the pin is such thatit keeps the specimen in tight contact with the split-halves of the mandrel. There is no slippage betweenthe mandrel and the specimen because of this config-uration and hence a bi-axial loading state is achievedfor the cladding tube (similar to that occurs in actualloading conditions in service). The position of the axisis fixed even after crack initiation and propagation asopposed to a usual fracture testing. The fixture half isa combination of a semi-cylindrical block and a rect-angular block. In Fig. 4(a), ‘W ’ denotes the distancebetween the centre of the axis and the load line (i.e. linejoining the holes of the rectangular block), whereas ‘b’denotes the distance between the load line and the

Fig. 3 Photograph of actual specimens (with different initial notch lengths) used in the experiment



Fig. 4 (a) Geometric dimensions of the loading fixture used in the test set-up; (b) 3D view of theloading fixture along with the loading arrows; and (c) actual photograph of the specimen asloaded in the loading fixture

contact plane of the cylindrical and the rectangularblock.

3.3 Loading procedure

All the seven specimens are attached to the fixture oneby one. While making the specimen-fixture assembly,care is taken to ensure that all the four notches lie onthe contact plane of the fixture halves. The actual pho-tograph of the specimen loaded in the fixture is shownin Fig. 4(c). The specimens are loaded in displace-ment control mode with a pull rate of 0.2 mm/minwith a servo-hydraulic actuator controlled by a 24-bit controller having a load frame of 25 kN capacityupto a point where considerable load drop occursafter the peak load. The crack opening displacement(COD) is measured using a clip gauge and the timed

load–COD data are digitally recorded and presentedin Fig. 5.

4 DETERMINATION OF THE GEOMETRICFUNCTION FROM EXPERIMENTAL DATA

From the load–COD response (Fig. 5), the best straightline part is fitted by linear regression analysis to findthe compliance for each specimen. Also, the notchlengths of the specimens are normalized in the formof a/W . The resultant data are analysed by meth-ods proposed in references [26] to [29]. The detailsof the analytical approaches for calculation of thegeometric function for evaluation of SIF in the open-ing mode as described in the test set-up are givenbelow.



0 5 10 150

500

1000

1500

2000

2500

3000

crack lengths

1mm

2mm

3mm

4mm

5mm

6mm

7mm

COD (mm)

Load

(N

)

Fig. 5 Experimental load–COD data of the tubular spec-imens with various initial notch lengths rangingfrom 1 to 7 mm

4.1 Approach 1

According to the strain energy approach [13], strainenergy release rate G is a measure of energy availablefor an increment of area of crack extension, dA, andthe specimen with edge crack a can be expressed as

G = P2

2dCdA

= P2

2BdCda

(1)

where P is the load applied to the specimen, B is thespecimen thickness, C is the specimen compliance,and A is the crack area. The strain energy release ratecan also be expressed as

G = K 2I

E=

(PB

)2 1WE ′ f 2

( aW

)(2)

where KI is the SIF, W is the distance between the load-line and the crack tip, and E is the Young’s modulus(E ′ = E for plane stress and E ′ = E/(1 − ν2) for planestrain). Combining equations (1) and (2) yields

f( a

W

)=

[12

BEdC

d(a/W )

]1/2

(3)

The PLT specimen fabricated from the thin walledtubing contains two edge-notches at the loaded edgeof the specimen. E ′ = E due to the small thickness ofthe specimen. In the specimen, there are two crackedges and hence the total thickness B is equal to 2t inequation (3), where t is the tube wall thickness. Thisgives

f( a

W

)=

[tE

dCd(a/W )

]1/2

and KI = P

2t√

Wf

( aW

)

(4)

Differentiation of the fitted polynomial functionderived from the compliance versus a/W curve (Fig. 6)

Fig. 6 Compliance as a function of crack length to widthratio obtained from the experiment

is carried out by power law to obtain dC/d(a/W ) as afunction of a/W . The dC/d(a/W ) versus a/W curve isagain fitted by non-linear regression analysis to finda polynomial function of a/W . Some of the values ofdC/d(a/W ) corresponding to the a/W values rangingform 0.45 to 0.8 are evaluated using the polynomialfunction and those values are multiplied with elasticmodulus (E = 92 GPa for zirconium) and the specimenthickness (t = 0.9 mm). The reason behind choosingthe range of a/W from 0.45 to 0.8 is to cover the wholerange in which the corresponding notch lengths ofall the seven specimens lie. Square root of the prod-ucts of dC/d(a/W ), t and E are evaluated (accordingto equation (4)) and those values are further replot-ted to obtain f (a/W ) as a function of a/W (Fig. 7for approach 1). The least-square fitting of the curvesis done for finding the K -calibration function in the

Fig. 7 Comparison of the geometric functions obtainedfrom experimental data using both approaches



form of polynomial and the derived expression for thegeometric factor is obtained as

f( a

W

)= 416.9 − 1903.9

( aW

)+ 2847.7

( aW

)2

− 1330.3( a

W

)3

(5)

The maximum difference between the fitted polyno-mial values and the actual point values of the functionf (a/W ) at all the points of a/W lies within 2 per cent.

4.2 Approach 2

Duggan et al. [27] presented a unique method of find-ing the stress intensity function proposed by Ryderet al. [26]. This method has also been used in ref-erences [28] and [29] and it provides fairly accurateresults for the functions of SIF. We describe here a briefdescription of the method. There are seven discretedata points, i.e. seven compliance values against sevennormalized crack lengths, one for each specimen. Thecompliance values are multiplied with elastic mod-ulus E and respective wall thicknesses t to obtaindimensionless normalized CEt values. The ratio of un-cracked ligament length to the width is usually writtenas an exponential function of a new variable Z as

1 −( a

W

)= exp(−Z k) (6)

where

Z = ln[ln(CEt + 2.718 28)] − ev − e

(7)

and k, e, and v are numerical constants. Combningequations (6) and (7), we obtain

CEt = exp{

exp[

g( a

W

)]}− 2.718 28 (8)

where

g( a

W

)= e + (v − e)

[− ln

(1 − a

W

)]1/k

(9)

Now, the normalized compliance (CEt) can be dif-ferentiated mathematically, that is

d(CEt)d(a/W )

= exp{

exp[

g( a

W

)]}exp

[g

( aW

)]g ′

( aW

)

(10)

where

g ′( a

W

)= v − e

k

[− ln

(1 − a

W

)](1/k)−1(

11 − a/W

)

(11)

As a/W approaches unity, CEt continuouslyincreases, consistent with the physical reasoning: i.e.the more is the crack length, the more is the com-pliance. When a/W approaches zero, e approachesln[ln(CEt + 2.718 28)]; v and k are fixed by CEt ver-sus a/W data. Hence, once e, v, and k are foundout, dCEt/d(a/W ) can be obtained as a function ofa/W by deploying equations (9) to (11). Square rootof dCEt/d(a/W ), if expressed as a function of a/W ,will yield the calibration function as per equation (4).The total expression for the K -calibration function canthen be written as

f( a

W

)

=

√√√√√√√√√√√√

exp(

exp{

e + (v − e)[− ln

(1 − a

W

)]1/k})

× exp{

e + (v − e)[− ln

(1 − a

W

)]1/k}

v − ek

×[− ln

(1 − a

W

)](1/k)−1(

11 − a/W

)

(12)

As a/W approaches zero, CEt also approaches zero.Hence the value of e is set as zero. Then, eliminating Zafter suitable shuffling with equations (6) and (7), thefollowing relation is deduced

ln{ln[ln(CEt +2.718 28)]}− ln v = ln[−ln(1−a/W )]k

(13)

Since there are seven discrete data points (i.e. sevenvalues of CEt against seven known values of a/W ),seven sets of simultaneous linear equations are madetaking ln v and 1/k as the unknowns. Being constantfor all the seven equations, v and k values should besame. All the seven values of ln(ln(ln(CEt + 2.718 28)))

are plotted against the values of ln(− ln(1 − a/W )).The data are fitted with a straight line. From theslope (i.e. 1/k) and the y-axis intercept (i.e. ln v), theconstants k and v are found to be 2.73 and 1.35,respectively. Using these values, the derived expres-sion for the geometric factor is obtained as (as perequation (12))

f( a

W

)=

[0.49 exp

(exp

{1.35

[− ln

(1 − a

W

)]0.36})

× exp{

1.35[− ln

(1 − a

W

)]0.36}

×[− ln

(1 − a

W

)]−0.63[

1(1 − a/W )

]]1/2

(14)

The maximum difference between the fitted polyno-mial values and the actual point values of the functionf (a/W ) at all the points of a/W lies within 2.5 per cent.



5 FE ANALYSIS OF THE PLT SET-UP

In this work, the whole PLT experimental set-up (i.e.the tubular specimen as well as the mandrel) hasbeen discretized in 3D domain for FE analysis. Thisis very essential as the load is applied at the end ofthe mandrel and this in turn loads the cracked tubularspecimen in the opening mode. Taking into accountthe symmetry in specimen geometry, crack and load-ing configuration, one-fourth of the test set-up hasbeen modelled as shown in Fig. 8. A prescribed dis-placement is applied at the end of the mandrel in they-direction and the movement of the loading point isrestricted in the x-direction (i.e. direction of the lengthof the tube) as this point is attached to the loading axisof the machine.

The uncracked ligament of the tubular specimenis fixed in the y-direction (however, this is free tomove in the x- and z-directions) in order to take intoaccount the symmetry condition. The line of the load-ing pin (which acts like a fulcrum) is also fixed in they-direction only. The symmetric plane of the tubu-lar specimen as well as of the mandrel is fixed inthe z-direction only in order to consider the sym-metric boundary condition. The material propertiesused in the analysis are given in Table 2. The tubularspecimens are loaded beyond the limit loads of thespecimens and hence elasto-plastic FE analysis hasbeen carried out in this work. The aim is to demon-strate the effectiveness of the model to simulate theload–displacement behaviour of the whole set-up andhence the analysis has been carried out for applieddisplacements which are well beyond the limit load ofthe specimen. The results should be able to demon-strate not only the correct elastic behaviour but also

Fig. 8 FE model of the test set-up along with the bound-ary and loading conditions

the whole load–response of the experimental set-up atlarge applied displacements. In addition, the SIF havebeen obtained from the FE solution in the linear elasticportion of the response.

The stress–strain curve (i.e. true stress versus trueplastic strain) of the material of the tubular specimenused for FE analysis is shown in Fig. 9. In the FE dis-cretization, very fine elements are used in the crackfront in order to simulate the large stress gradient.The sizes of the 3D elements are 0.2 × 0.2 × 0.2 mm3,respectively. The FE discretization consists of 20-noded iso-parametric brick elements with 3 × 3 × 3Gauss point integration. These elements show a verygood performance in evaluation of stress and strainfields in 3D solid FE models and hence have been usedin this analysis. The 3 × 3 × 3 full Gaussian integra-tion helps to avoid spurious locking of the FE meshas deformation proceeds. The details of the elementcan be found in reference [34]. Other elements in3D that can be used is the 27-noded iso-parametricbrick element with full Gaussian integration. However,the 20-noded elements have been shown to have bet-ter performance compared to the 27-noded element[34]. The degrees of freedom used at each FE nodeare the displacements in the x, y, and z-directions,respectively.

The material of the mandrel (i.e. steel with Young’smodulus of 210 GPa and Poisson’s ratio of 0.3) is dif-ferent from that of the tubular specimen, and hencethe corresponding material properties have been usedin the FE analysis for the elements that fall in the

Fig. 9 Material stress–strain curve of zircalloy-2 fuel pinused in the FE analysis

Table 2 Mechanical properties of the fuel pin material

Young’s Poisson’s Yield StrainProperty modulus (GPa) ratio stress (MPa) UTS (MPa) hardening exponent

Value 92 0.35 452 619 0.113



domain of the mandrel. It may be noted that we haveused seven different tubular specimens with differentinitial crack lengths that vary from 1 to 7 mm. Apartfrom the initial crack lengths, other dimensions aresame for all the specimens. In order to make use ofoptimum number of 3D elements in FE mesh, finitediscretization has only been used near the crack tip forall the specimens. These fine elements have been ableto reproduce the stress gradient near the crack tip, andhence the usual quarter point elements have not beenused in this analysis. The front views of the FE mesh ofall the specimens are shown in Figs 10(a) to (g).

In order to evaluate the linear elastic SIF (i.e. K ) forthis experimental set-up, the data from elastic analysisare required. However, another purpose of this workis to demonstrate the effectiveness of this model incorrectly modelling the response of the experimentalset-up, not only in the elastic range, but also in the

whole elasto-plastic range at large applied displace-ments. Hence, a complete elasto-plastic analysis hasbeen carried for each specimen for applied displace-ments of the order of 7 mm, which are well beyond thelimit loads of the specimens.

For evaluating the SIF, the analysis is carried out forthe applied displacement of 0.01 mm in each spec-imen and the corresponding compliance values arecalculated for each value of a/W . There are two dif-ferent approaches that can be followed in employingFE procedures to arrive at the required SIF. One isthe direct method that follows from the stress fieldor from the displacement field. However, several otherapproaches exist in the literature [21–25] to determinethe SIF values (e.g. J -integral-based methods). Theextrapolation techniques are based on asymptotic dis-placement and stress fields near the crack front [21]:by the displacement field approach [22] and by nodal

Fig. 10 (a) to (g) FE meshes of the test set-up with crack lengths of 1–7 mm in the fuel tube



force approach [23]. The second is an indirect one inwhich K is determined via its relation with other quan-tities such as the compliance, the elastic energy, or theJ integral [24, 25]. The direct method utilizes the resultof the general analytic solutions of crack problems.

For indirect methods, it is possible to calculatethe compliance for a range of crack sizes; these datahave to be numerically differentiated with respect tocrack size to evaluate SIF. Alternatively, the total elas-tic energy contained in the plate can be calculatedfor a number of crack sizes. Numerical differentiationyields the elastic energy release rate, from which Kcan be calculated as K 2 = EG. Finally, for the elasticcase, since G = J , SIF can also be obtained from J inte-gral, numerically evaluated from FEM solution. In thiswork, we have followed the displacement extrapola-tion method to calculate the SIF, where the SIF can bewritten as (neglecting higher order terms)

KI = E ′

4

√2π

rv (15)

where KI is SIF in the opening mode of the crack, v isthe opening displacement ahead of crack tip, r is thedistance from the crack tip, E ′ = E/(1 − ν2) for planestrain loading condition and E ′ = E for plane stressloading condition, E and ν being the Young’s modulusand Poisson’s ratio for the material, respectively.

6 RESULTS AND DISCUSSION

In section 4, the method for obtaining the geometricfunction f (a/W ) as a function of (a/W ) was discussedand the corresponding results for the two analyticalapproaches are shown in Fig. 7. It can be observedthat both the methods give similar values for the abovefunction. The difference in the two approaches lies inthe method of obtaining the function of the derivativeof compliance with respect to a/W . In approach 1, afunction is fitted for the variation of the derivative ofcompliance with respect to a/W and used in the calc-ulation of f (a/W ), whereas in approach 2, a functionis fitted for the compliance with respect to a/W andused in the calculation of f (a/W ).

The compliance results of the specimens with dif-ferent a/W ratios as obtained from FE analysis areshown in Fig. 11 and these are very close to the exper-imental results of Fig. 6. This is very important as theFE simulation requires prescription of all the bound-ary conditions correctly, which is many times difficultto interpret from the complex experimental set-upslike the one used here. Moreover, an accurate 3D dis-cretization has been used and the loading mandrel hasbeen modelled explicitly, which is usually neglectedmany times for simplicity. In order to obtain the SIFusing the displacement extrapolation technique, theCOD is plotted against the distance from the crack

tip for each specimen. One such result for the tubu-lar specimen with 3 mm initial crack length is shownin Fig. 12.

In order to compare the experimental load–CODdata with that obtained from analysis, we carry outthe elasto-plastic analysis beyond the maximum load-carrying capacity of the specimen and the results areshown in Fig. 13. It can be seen that the present FEmodel could very well reproduce the experimentalresults, not only in the elastic regime, but also in thewhole range of the experimental load–COD curve. Inorder to obtain the SIF, the data of COD versus squareroot of distance from the crack tip are plotted forall the seven specimens and the resulting data arefitted through straight lines to obtain the constantsaccording to equation (15).

Fig. 11 Compliance as a function of a/W as obtainedfrom FE analysis

Fig. 12 Variation of COD with distance from the cracktip for the specimen with 3 mm crack length inthe fuel tube



Fig. 13 Load versus COD of the test set-up (experimentversus FE analysis)

The SIF obtained at an applied load line displace-ment of 0.01 mm for all the specimens as a function ofa/W is shown in Fig. 14. The SIF values as a functionof a/W for the inside, middle, and outside surfaces foran applied displacement of 0.01 mm can be fitted withsuitable polynomials and are represented

Kinside surface

( aW

)= −256.81 + 1303.78

( aW

)

−1887.84( a

W

)2

+ 927.12( a

W

)3

Kinside surface

( aW

)= −226.35 + 992.79

( aW

)

−1180.82( a

W

)2

+ 469.88( a

W

)3

Fig. 14 Variation of SIF in the opening mode of the tubu-lar specimen with a/W at different points of thecrack front

Kinside surface

( aW

)= −376.83 + 1634.19

( aW

)

−2108.87( a

W

)2

+ 926.14( a

W

)3

(16)

It may be noted that the SIF solution will be dif-ferent at different locations along the crack front (i.e.at the inside, middle, and outside surfaces of the fueltube). The polynomials in equation (16) have been fit-ted with a maximum error of 2 per cent. As the insidesurface of the crack front is closer to the centre of load-ing compared to the outside surface, the SIF solutionis higher for smaller a/W values and the differencesamong the results at the three locations along the crackfront vanish as the a/W is increased. However, the SIFsolutions obtained from the analytical methods pro-vide a single value that can be deemed as the averagevalue across the thickness of the specimen. In orderto obtain the geometric function f (a/W ) as a func-tion of a/W (Fig. 15), we use equation (4) as the SIFsolutions are known as functions of a/W . The resultsof the geometric functions obtained from FE analy-sis are compared with those of analytical approachesand with a similar result from reference [13] (for speci-mens that are of different dimensions) in Fig. 15. It canbe observed that the results from FE analysis com-pare very well with those of the analytical solutions.The results of the geometric functions f (a/W ) for theinside, middle, and outside surfaces as a function ofa/W can be fitted with suitable polynomials and arerepresented as

f( a

W

)∣∣∣inside surface

= −416.48 + 2220.26( a

W

)

−3899.27( a

W

)2

+2330.56( a

W

)3

Fig. 15 Variation of geometric function for evaluation ofSIF with a/W



f( a

W


= −346.58 + 1835.49( a

W

)

−3228.79( a

W

)2

+1957.98( a

W

)3

f( a

W


= −424.62 + 2213.44( a

W

)

−3846.07( a

W

)2

+2296.22( a

W

)3

(17)

The polynomials in equation (17) have been fittedwith a maximum error of 2 per cent. Hence, usingthese geometric functions, the SIF values in the open-ing mode of the tubular specimens with any value ofa/W and loaded in the similar condition as prevails inthis set-up can be determined. These solutions can befurther used in structural integrity assessment meth-ods of these types of thin tubular specimens with axialthrough-wall cracks.

7 CONCLUSIONS

In this work, a new test technique is presented to assessthe fracture behaviour of thin-walled fuel-pins usedin Indian BWRs. Two analytical approaches have beenused to obtain the geometric function needed for eval-uation of SIF from experimental data. FE analysis hasbeen performed using a 3D model of the whole exper-imental set-up where the cracked tubular specimenhas been modelled along with the mandrel, which isvery essential for accurate simulation of the test set-upas well as the evaluation of SIFs of the specimens. A fullelasto-plastic analysis has been performed in order topredict the load–COD curve of the test set-up beyondthe elastic limit and the results of the FE analysis showexcellent agreement with those of the experiment. Theresults of SIF solutions obtained from FE analysis havebeen compared with those obtained from the two dif-ferent analytical techniques and it was seen that theresults of FE analysis show a very good agreementwith those obtained with analytical approaches. Usingthese geometric functions, the SIF values in the open-ing mode of the tubular specimens with any value ofa/W and loaded in the similar condition as prevails inthis set-up can be determined.

© Authors 2010

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