BE9800004 SCK« CEN STUIMKChNTRUM VOOR Kl-.RNfcNLRGlt CTNTRF D'FTUDl 1)1 I'fcNl-RGIl- MICI.FAIRF Precracking of round notched bars
Progress Report
M. Scibetta SCK-CEN Mol* Belgium Precracking of round notched bars
Progress Report
M. Scibetta SCK»CEN Mol, Belgium
February, 1996
BLG-708 Precracking of round notched bars
1 Abstract 3
2 Introduction 3
3 Overview of precracking techniques 4 3.1 Introduction 4 3.2 Equipment 4 3.3 Eccentricity 5 3.4 Time required to precrack 5 3.5 Crack depth determination 5 3.5.1 The compliance method 6 3.5.1.1 Measuring force 6 3.5.1.2 Measuring displacement 6 3.5.1.3 Measuring an angle with a laser 6 3.5.1.4 Measuring force and displacement 7 3.5.2 Optical methods 7 3.5.3 The potential drop method 7 3.5.4 The resonance frequency method 8 3.6 Summary 8
4 Stress intensity factor and compliance of a bar under bending 9 4.1 Analytical expression 9 4.2 Stress intensity factor along the crack front 10 4.4 Combined effect of contact and eccentricity 12 4.5 Effect of mode II 12 4.6 Compliance 13
5 Stress intensity factor and compliance of a bar under bending using the finite element method (FEM) 15 5.1 Stress intensity factors 15 5.2 Compliance 17
6 Precracking system 18 6.2 Schematic representation of the experimental system 19 6.3 Initial stress intensity factor 19 6.4 Initial force 20 6.5 Initial contact pressure 20 6.6 Final force 22 6.7 Final stress intensity factor 22
7 Stress intensity factor of a precracked bar under tension 23 7.1 Analytical formulations 23 7.2 Finite element method 24 8 Compliance of a precracked bar under tension 26 8.1 Analytical formulations 26 8.2 Finite element method 27
9 Conclusions 28
10 Future investigations 29
11 References 30
Annex 1 Stress intensity factor and compliance function of a precracked bar under bending 33
Annex 2 Neutral axis position and stress intensity factor correction due to contact .34
Annex 3 Stress intensity factor and compliance function of a precracked bar under tension 35 1 Abstract
Precracking round notched bars is the first step before fracture mechanics tests. This report gives an overview of the different techniques described in the literature. Difficulties generally encountered are linked to the crack length determination and the creation of eccentric cracks. As the compliance technique is often used, a detailed study of the stress intensity factor and the compliance of the precracked bar under bending and tension is presented. Comparison with finite element calculations is made to validate the proposed analytical formulation. Finally a practical way for precracking is described. 2 Introduction
Mechanical properties of reactor pressure vessel steels are subject to degradation caused by neutron irradiation and thermal ageing. Safety and reliability of nuclear power plants depend largely on the monitoring of material properties through the evaluation of tests on surveillance specimens. The transition temperature shift obtained by the Charpy impact test is used to evaluate the fracture behaviour through semi-empirical correlations, that are mainly based on a large experimental database. Sometimes, this results in large conservatism penalizing the operation of some reactors. Therefore, direct fracture toughness measurements are desirable. As a consequence, the three-point bending technique applied to Charpy-size samples has been thoroughly investigated. However, this technique suffers from the complexity of the method together with a large scatter due to specimen size and loss of constraint. Another important aspect, is the use of miniaturized samples to investigate the radiation damage of materials, because the amount of irradiated material available is very limited. Therefore, the precracked tensile bar seems very promising to address both aspects, i. e. fracture toughness measurements to monitor the transition temperature shift as well as the implications of fracture toughness parameters derived from miniaturised samples on the assessment of radiation damage.
However, before developing the testing of the precracked tensile bar into a standard testing procedure, it is required to determine the fracture toughness function for this geometry and to get a good precracking procedure. 3 Overview of precracking techniques
3.1 Introduction
From a machined notched round bar, a circumferentially crack of a specified length needs to be generated by fatigue. The main specifications of a good precracking method are: - generation of a circular crack without eccentricity, - a good accuracy in the crack depth determination. - a simple method with few manipulations, - application to a wide range of bar diameters, - a short time to precrack the bar, - not too expensive.
There are different techniques to precrack cylindrical bars. But the most popular method to generate a circular crack with minor eccentricity is rotating bending fatigue. A bending moment can be applied with two different techniques; namely, cantilever beam and four-points bending (see Figure 1). In either technique, the displacement or the bending moment can be chosen to remain constant. v^a A
a) b)
Figure 1 : a) Cantilever beam and fixation b) Four points bending configuration
It is interesting to notice that cantilever beams are more sensitive to vibrations due to their higher flexibility. To reduce these vibrations some systems are equipped with a damper.
3.2 Equipment
A commercially available rotating bending fatigue system, the design of a new machine or a modified drilling tour are different possibilities that allow to generate precracked tensile bars. Generally, a drilling tour is chosen because it can be easily instrumented, has a good rigidity and stability, allows easy centring of the bar and last but not least the cost is generally the lowest. 3.3 Eccentricity
The first way to deal with possible eccentricity of the crack is to try to reduce it to a negligible value. The main parameters inducing eccentricity are in the order of importance:
- the accuracy and concentricity in mounting the specimen on the fatigue machine [LUC"90>SHE-8^ LUC 90 - the material intrinsic variability [ - '; - the crack depth ISTA-86' (see also paragraph 5.4), - a high rotation speed P-uc-9°].
The second way to take eccentricity into account is in the analysis of the results. For example, a thorough 3D study on eccentricity using elastic FEM has been performed by Ibraham and al 010 93 [IBR-90] Howeverj simpier models could also be used I ' '.
3.4 Time required to precrack
The time to precrack a bar is strongly dependent on three parameters: - the rotation speed of the system, - the notch radius, - the applied stress intensity factor.
A high rotation speed increases the number of fatigue cycles and thus determines the crack propagation time. However, according to Lucon tLuc"9°l a too high rotation speed generates eccentric cracks and vibrations. Nevertheless rotation speeds used [EIS"85' GIO'93> LUC'93'STA"861 are in the range of 200 to 10000 rpm.
At the beginning of the process, the bar has no crack. Then the time to initiate a crack is linked to the sharpness of the notch. Notches with a radius lower than 0.01 mm cannot be manufactured. Radii generally used ^Els-si] are in the range of 0.2 to 0.03 mm. Initiating the crack using a sharp knife is common tGUp-84'SHE-82) The time to initiate the crack can take one third of the whole precracking process.
The crack growth rate is strongly dependent on the stress intensity factor. This relation is given by the Paris law [AND"95). However, in order to avoid creation of a large plastic zone at the crack tip, the stress intensity factor should not be too high. Generally |GI°"83' LUC'90'SHE"821 the ratio between the stress intensity factor during fatigue and the fracture toughness of the material at the tested temperature should be in the range of 0.2 to 0.6.
The total time to precrack is generally in the range of 10 minutes to 4 hours. 3.5 Crack depth determination
The crack depth must be evaluated to stop the precracking process at the right moment. The main techniques for crack depth evaluation are: - the compliance method, - optical methods, - the AC or DC potential drop method, - the measurement of the resonant frequency.
The main advantages of a specific method are generally linked to the accuracy, the number of manipulations and the complexity.
3.5.1 The compliance method
The compliance of a bar under bending is defined as the ratio between the displacement and the applied force. So the compliance of a bar increases as the crack grows. The relation between the compliance and the crack length can be established by an experimental calibration, an analytical method or a numerical method.
To measure the compliance, four possibilities exist: - by measuring the force, - by measuring the displacement, - by measuring the angle, - by measuring the force and the displacement.
The compliance method is very popular as it is a simple method with usual instrumentation.
3.5.1.1 Measuring force
This is the most popular method that imposes a constant displacement during the rotary bending process. The force under the load decreases as the crack grows. To evaluate the crack length, the force is measured using a load cell IQI°-931 or strain gauges [BOL-85'HAW"85'HOU-79-SHE"821.
3.5.1.2 Measuring displacement
The bending moment load is imposed to be constant during the rotary bending process. The displacement increases as the crack grows and is generally measured using a LVDT ^LUC-93' STA-86i
3.5.1.3 Measuring an angle with a laser
This technique has been used in addition to one of the two former techniques, namely, constant displacement f801-'851 or constant bending moment[EIS-851. The technique allows to get a better accuracy in the crack depth determination. It is based on the compliance defined as the ratio between the angle variation and the bending moment near the circumferentially crack. To get an accurate measure of angles, two mirrors are sticked to the bar (see Figure 2). A laser beam is projected on the two mirrors and reflected on a screen. The variation of the distance between the two reflected spots is related to the crack length through calibration. Laser / / Screen
Figure 2 : Angle deflection
3.5.1.4 Measuring force and displacement
The force and the displacement are measured during an unloading. The slope of this ratio gives the actual compliance. This technique is more complicated but has the advantage that the load or displacement can be changed during the rotating process.
3.5.2 Optical methods
Optical methods are based on direct observation of the crack length. These methods are more easy to set-up if the specimen is large and if the crack can be seen at the free surface. It is commonly applied for compact tension and three points bending specimens.
3.5.3 The potential drop method
An AC or DC constant current passes through the specimen. The potential difference across the crack mouth, the so-called potential drop, is measured. This potential drop increases as the crack grows. Calibration relates the potential drop to the crack length. The technique can be used continuously for precracking compact tension and three points bending, but for the axisymmetric specimen the rotation must be interrupted to allow the measurement[GAG-96'AMA"86' HOU-79> DEV"851.
In order to measure a significant potential drop a high DC current must pass through the specimen (5 to 10 Ampere). The current is less for the AC potential drop method due to the skin effect (1 to 2 Ampere).
For the AC potential drop method, the optimum frequency is in the range of 1 to 10 kHz and is limited due to induced electromotive forces. Another important feature is the ability to get a linear response depth versus volts due to the skin effect.
The DC potential drop method is more simple to install, but thermally induced DC electromotive forces can appear due to thermocouple effects at junctions of dissimilar metals. This problem can be overcome by repeatedly reversing the current flow. High current can lead to specimen heating and requires heavy equipment. The resolution of both techniques is in the range of 0.05 to 0.2 mm. But the AC potential drop seems to be more accurate. Some details and comparisons can be found in (GIB-87>HWA-92. NAK-«9, WOJ-95]
3.5.4 The resonance frequency method
The resonance frequency of a specimen varies with crack growth. As the crack grows, the resonance frequency decreases because the compliance increases. The resonance frequency can be measured by analysing the signal of a hammer impact with the Fast Fourier Transform algorithm.
The technique requires to fix an accelerometer on the specimen and to stop the rotation. So it seems to require many manipulations.
The resonance frequency is linked to the square root of the compliance. Consequently, a variation of crack length has a smaller effect (factor 2) on the resonance frequency than on the compliance [GOL-95] Y^ method can be successful if the fixations induce no perturbations and no damping. These conditions are fulfilled if the specimen is simply supported. Up to now, to the author knowledge, this technique has not been used in any precracking technique.
3.6 Summary
The crack length determination and the generation of crack without eccentricity are the most important requirements of a good precracking technique.
The compliance and the potential drop methods are generally used to monitor the crack growth. According to literature, each technique gives good results. However the potential drop technique and the measure of the angle with a laser use a more complicated instrumentation increasing the accuracy of the crack length determination. Up to now, the resonance frequency method has not been used but could also give interesting results. 4 Stress intensity factor and compliance of a bar under bending
To have a good understanding of the rotating bending process, it is important to get good idea of stress intensity factors and compliance functions. Because the stress intensity factor will determine the crack growth rate and the compliance will determine the force drop. 4.1 Analytical expression
The maximal stress intensity factor of a bar under a uniform bending moment has been established using an approximate analytical and numerical method. The formulation of Benthem and Koiter has the following expression PEN'73l using notation of Figure 3:
aM 4M K • G(-) a with ° • — (1) b I
2 3 4 and o(i) - - [l • -(-) • -(-) • —(-)( • —(-)( • 0.531 (i-) (2) b 8 2 VV 8 * 16 b 1288 b b where K is the stress intensity factor, 2a is the ligament diameter, 2b is the bar diameter, M is the bending moment, I is the moment of inertia of the smallest section and a is the maximal stress for a bar of diameter 2a under bending.
Figure 3 : Geometry of the specimen
To simplify the expression, a function F, is defined as: £ (3) The functions F, and G are tabulated in annex 1.
Harris [HAR"67i proposed the relation:
na(b - a) K • a (4) 0.8a +7.12(6-o) The two functions (Equation 1 and 4) are compared in Figure 4. The maximum difference is 11.7% for a/b = 0.82.
1
0.8 o °-6 -
0.4 -I
0.2 [BEN-73]
0 -• 1 1 1 1 0 0.2 0.4 0.6 0.8 a/b Figure 4 : Stress intensity factor comparison
Finite element calculations [KIEG"90i have shown very good agreement between these two formulations for a/b < 0.6. Moreover finite element studies (MEG'90) have shown that the notch-tip radius r and the angle of the notch (J> have a very small influence on the stress intensity factor. 4.2 Stress intensity factor along the crack front
The crack front is not uniformly loaded when loaded in bending. This study proposes an expression for the stress intensity factor along the crack front. The stress intensity factor is zero on the neutral axis and maximum at the two opposites of the neutral axis. The relation between the stress and the angular position (see Figure 5) for a uniform bending bar is supposed to be the same for the stress intensity factor.
max
(T= CT sin® max
neutral axis
Figure 5 : Stress along the circumference
Therefore the proposed relation is:
K(0) = Kmaxsin(e) (5)
This formula is verified using finite element calculations (see paragraph 5.1).
10 4.3 Contact
During the rotary bending fatigue process, the crack lips are in contact at the compressive side of the bar. The contact increases the rigidity of the structure and decreases the stress intensity factor. The notch depth is therefore an important parameter to consider as it changes the contact area, the position of the neutral axis and the moment of inertia.
Figure 6 . New location of the neutral axis
A model proposed by Sawaki ISAW"82) based on the beam theory, is described below. This model has also been compared using the FEM by Meguid 0
The position of the neutral axis is found by solving the equation:
1 - ».
l- Ov yf The modified moment of inertia is: 4 2 /„- 4 c J y Jl- {nc,yfdy (7) The modified maximal stress for the bar with this moment of inertia is: o_ - M (8) The dimensionless function S is defined as the ratio between the maximal stress with contact and the maximal stress without contact: In a (9) 16 f y2 ^1- Ov yf dy The functions S and nc are tabulated in Annex 2 as a function of ac. Contact effect tends to decrease the stress intensity factor. Consequently the S function is always less or equal to 1. The modified maximal stress intensity factor is found using the modified stress om in the formulation without contact. 11 - alb) (10) *m« " °(-> "m The stress intensity factor along the crack tip is null under the neutral axis. For the same reasons as in paragraph 4.2 (Equation 5), the stress intensity factor for points above the neutral axis is given by: n * a sin (9) (11) n * a When contact effect are taken into account, Equation 1 and 5 have to be corrected using Equation 9 and 11. This correction is large if the ligament diameter is small compared to the notch diameter (a/c < 0.8). 4.4 Combined effect of contact and eccentricity As the crack grows, an instability can occur that leads to an eccentric crack (STA-86i This observation can be explained by considering contact effects. An eccentric crack is submitted to a positive and a negative bending moment (see Figure 7). The moment of inertia of the section a (Ia see hatched part of Figure 7a) is smaller than the moment of inertia Ib, so the stress intensity factor Ka is larger than Kb. This difference tends to increase the eccentricity because the crack growth rate is larger in the configuration a. section b Action a b) Figure 7 : a) Eccentric crack with a positive bending moment b) with a negative bending moment 4.5 Effect of mode II Mode II is induced by shear loading. This mode induces an angle deflection. Therefore, the crack lips are not in one plane. This effect can be observed with the cantilever configuration because then the bending moment is non-uniform and a shear loading exists. 12 Figure 8 : Effect ofKu on the fracture surface This effect can be observed in all broken specimens, but is generally negligible. 4.6 Compliance An analytical expression for the compliance in the cantilever beam configuration is established. This expression is verified using the finite element method in paragraph 5.2. Here contact effects are not taken into account. 2a { L P Figure 9 : Geometry of cantilever beam configuration The relation between the energy release rate and the variation of compliance due to crack growth is given by: G - L (12) 2 d(b)(b-a) 2 da J E where G is the energy release rate, C is the compliance and K is the stress intensity factor along the crack front. The substitution of Equation 5 gives: n/2 (13) 2 da E -n/2 Then its integration gives: 13 C(a) . Ca* f 2^-—n a'da' (14) J E p2 a The compliance of a non-cracked bar Co is the sum of the system attachment compliance and the bar compliance under bending and shear loading: L r 4 L* L 1175 0 * 3EI EA> "*• lEnb* Enb2 where A' is called the reduced section and Catt is the system attachment compliance. The contribution of the crack is: 2 } ' da'. 32^ ^ // E P* J E The function H, tabulated in annex 1, is defined as: "(?)•/?•*' b The exact integration of Equation 17, based on Benthem's equation using polynomial manipulation gives: H^) . 0046g75 - 0.06303639 - 0.04006494 i\2 • 0.034S0366 i)J * 0.009301849 t/ /jgx * 0.004570464 i\s * 0.002724667 x\6 * 0.0001693451 t^7 * 0.004956346 x\* The compliance of the total system is: C(.) . r . -^-[ 1 . 3S2Sfe2 • 24 ,i^! ft /2 iyj (19) 3£fr4 4L2 L3 This compliance function can be used to determine the crack length, an example of application is done in Paragraph 6.6. 14 5 Stress intensity factor and compliance of a bar under bending using the finite element method (FEM) Three dimensional finite element analyses are performed with the SYSTUS code to derive the stress intensity factor and the compliance. Finite element results are compared to the analytical formulation developed in the former paragraph. In the present analysis, contact is not taken into account. The geometry used is described by the following parameters (see Figure 3 and Figure 9): L = 40 mm, 1 = 30 mm, r = 0,<|) = 60, b = 6 mm, c = 4.2 mm. The material has the properties: Young's modulus: E = 205000 MPa Poisson's ratio: v = 0.3 The applied load P correspond to a punctual force of 1 kN. Due to symmetric conditions only half the structure needs to be modelled using quadratic finite elements. Figure 10 . Deformed mesh 5.1 Stress intensity factors Several analyses (see Table 1) for different a/b ratios are performed and the derived maximal stress intensity factors (for mode I) are compared using Benthem and Harris equations (Equations 1 and 4). For this configuration, the shear loading induces a mode II stress intensity factor. This mode is limited to about 5% of mode I. 15 a/b K FEM (MPa mm) T£ [HAR-67] j£ [BEN-73] Max. percent diff. (%) 0.4 84.90 86.73 90.08 5.9 0.5 48.22 48.80 51.59 6.7 0.6 29.88 30.19 32.62 8.7 Table 1 : Stress intensity factors The small difference between the results can be explained by the following reasons: - The finite element method gives an approximate solution, - The theoretical solution was found using some approximations. Moreover, it was derived for an infinite bar under a pure bending moment neglecting shear loading. For a/b = 0.5, the stress intensity factor along the crack front is compared (see Table 2) to the proposed formulation of paragraph 4.2. e K(9) / K^ FEM sin(9) (Equation 5) Percent difference (%) 90 1 1 0 54 0.8090 0.809 0.02 18 0.3090 0.309 0.02 -18 -0.3090 -0.309 0.02 -54 -0.8090 -0.809 0.02 -90 -1 -1 0 Table 2 : Stress intensity factors along the crack front The proposed formula is verified quasi exactly using finite element calculations. 16 5.2 Compliance The former finite element analysis allows to calculate the compliance of the specimen. This compliance is compared in Table 3 using Equation 19. The accuracy of analytical formulations is very good. a/b C FEM mm kN'1 C theoretical mm kN"1 Percent difference % 0.4 0.4905 0.4983 1.5 0.5 0.2815 0.2861 1.6 0.6 0.190 0.1924 1.3 0.7 0.1405 0.1454 3.4 Table 3 : Compliance under bending 17 6 Precracking system A first experimental system based on the cantilever beam and on the compliance method is described. The main components needed to set up the test bench are a drilling tour, a load cell and a LVDT to control displacements. The goal is to precrack 18MND5 steel of a given geometry within an hour. To design the system, calculations are made for the configuration described below. 6.1 Material The material used to implement the experiment is a French forged steel used in stream generator fabrication (18MND5). The chemical composition is : 0.18 C, 0.26 Si, 1.55 Mn, 0.007 P, 0.002 S, 0.14 Cu, 0.18 Cr, 0.50 Mo, 0.65 Ni. The mechanical properties are given (Table 4) at room temperature in the orientation S, T and L [CHA-96] wnere OY is ^g yieid stress, ou the ultimate strength, 8 the total elongation and RA the reduction of area. Material Orientation aY (MPa) cu (MPa) Bu (%) 6 (%) RA (%) S 505 645 12 24 68 18MND5 T 515 662 12 24 72 L 518 661 12 24 76 Table 4 : Mechanical properties of 18MND5 at room temperature Young's modulus: E = 205000 MPa Poisson's ratio: v = 0.3 18 6.2 Schematic representation of the experimental system The fully equipped experimental system with a load cell and a LVDT is illustrated in Figure 11. t specimen load cell U Figure 11 : Experimental system The rotation speed of the drilling tour (co) is about 2000 rpm. Y 2b 2a A i 1 L P Figure 12 : Geometry of the specimen For the example investigated here, the dimensions (see Figure 12) are : 2a = 8.4 mm 2b = 12 mm 1 = 30 mm L = 40 mm ((> = 60° 6.3 Initial stress intensity factor An initial stress intensity factor of 20 MPa m1/2 is chosen. This stress intensity factor is sufficiently small to obtain a very small plastic zone size at the crack tip. Using the Irvvin approach for plane strain conditions, the plastic zone size (r) is: 19 s r - 8.5 10 m - 85 |im (20) 6*1 aT) 6n{ 500 The plastic zone size is very small compared to ligament size (4200 |im » 50 r). 6.4 Initial force The initial force applied to get a stress intensity factor of 20 MPa m1/2 is found using the stress intensity factor formulation. a" where M is the bending moment and F, is a function tabulated in Annex 1. In this case F(±\ . F/MI. F,(0.7) - 0.369 , and the applied force is: 20 °-0042" - 9.128 \0^ MN.0.912kN (22) 4F, f I 4 0.369 0.03 6.5 Initial contact pressure The maximal contact pressure applied to the specimen can be calculated using the Hertz formulation [WAR-89). P 1 11 P max ** 1-v2 l.v? r, r, (23) \ E2 Where Eb E^, Vj and v2 are the Young's moduli and the Poisson coefficient of the two materials in contact; r, and r2 the radius of the two surfaces in contact and B the length of the contact zone. In our case the two materials in contact are steel, so the equation is reduced to: _P *- --1 (24) 2 n fil-v^r, r2) The relation between the radial force on the specimen and the measured force is linked to the geometry. The first system was based on two ball bearings linked to the load cell. 20 specimen section \2 load cell F Figure 13 : System with two ball bearings The force applied to one ball bearing is: P . 0.5996 (25) 2 cos(a) where a is the contact angle (see Figure 13). The maximal pressure is thus: 0.5996 e-3 E 1 1 • 1061 MPa (26) \ 2 * 0.005 1-vH 0.006 0.0105 This pressure is too high as indicated by the plastic prints that remains on the specimen. Therefore a new system is chosen to reduce the contact pressure. A self-align ball bearing is passed around a specimen having a slightly smaller diameter. Now the contact pressure is lower as the curvatures are in the same direction. specimen section F Figure 14 : System with one ball bearing If the ball bearing has the same width, and an internal diameter of 12.2 mm, the maximal contact pressure is given by: 0.9128 e-3 E 1 1 133 MPa (27) 2 n 0.005 1-vH 0006 00061 21 6.6 Final force The displacement (A) under the load is constant in the fatigue process so: Initially: A = C, Fj and finally: A = C2 F2 with C the compliance of the cracked bar. Combination of the two last equations gives: C2 F2 = C, F, The compliance of the attachment system must be as rigid as possible to get a better sensitivity. In this configuration, with a common drilling tour, a compliance of 0.5 mm kN'1 is found. C(.>.C-. -^[1.^.24,^*1'*, (28) 4 412 L3 Using the table in Annex 1, we get for a/b = 0.7, H=0.06958 Substitution in Equation 28 gives the initial compliance to be C, = 0.645 mm /kN. The parameters used to calculate the final compliance are : a/b = 0.5 and H=0.3073, so C2 = 0.786 mm /kN. The final load is thus: PA,-— P^H , -^^- 0.912 -0.748kN (29) final n mttel „ _„, If contact effects are considered, a higher final force will be found because contact increases the rigidity of the structure. The load has decreases by 20%. This is not a very large variation as there are other causes of force variations : - the thermal expansion of the system can induce thermal forces, - the difficulty to impose a constant displacement due to vibration . Nevertheless, a good designed system should be able to reduce these unwilling effects. 6.7 Final stress intensity factor After precracking the ligament diameter is smaller but the applied force has also decreased. '-"5 (30) _L. 4 0.3756 a" ft 0.003 The final stress intensity factor is high and in order to get a valid Klc measure, 0.6 Klc at the desired test temperature should be higher than the final stress intensity factor [LUC-9°i. if this condition is not fulfilled another lower initial force should be used. 22 7 Stress intensity factor of a precracked bar under tension [BEN 7i The literature ' - HAR-67, BUE-65] prOpOses various stress intensity factor functions. The functions were always established using analytical or numerical approximations. A review and a comparison between the different formulations is made. In the following paragraph, finite element analyses are performed to verify the analytical formulations. 7.1 Analytical formulations Three functions are found in the literature. The parameters used to describe the specimen are given in Figure 15. 2b Figure 15 : Geometry of a precracked tensile bar 1) Benthem and Koiter tBEN-73' proposed the following relation based on asymptotic approximations: K • G(-() ) a sl*a{\l{ - alb)) with o - —^— (32) b - a- and G(-) . I [l • !(-) • h-)2 - 0.363 (-)* • o.73l c^-)4] (33) b 1 2 b 8 b b b where P the tensile load. In order to simplify the expression, a function Fj is defined as: F,(i-) - G(^) vftl - alb) (34) b b The functions F, and G are tabulated in Annex 3 2) The function of Harris [HAR-67i for an external crack in a tube for the application on a round bar is reduced to: (35) K. a 0.8a * 4{b-a) (inner diameter of the tube set to be zero). This function was confirmed by Rooke and Cartwright [ROO-76] usjng Neuber's approximate stress concentration factors for notches. 23 3) The function of Bueckner [BUE-65l derived from a singular integral equation is: it a K - c n a (1 - -) - (1.72-1.27—) 2 b (36) 6 2 6 this function is valid in the range a/b of 0.4 to 0.8. Figure 16 shows good agreement between Benthem and Harris function ( less than 5% difference) and demonstrates the inadequacy of the Bueckner function out of the range of0.4^a/b^0.5. / •'/ / •'' -- 0.8 0.6 [BEN-73] [HAR-67] / rRTIF-fiSl • FEM 0.2 / 1 1 0.11 0.12 0.13 0.14 0.15 0.16 0.1—7 0.8 0.9 Figure 16 : Stress intensity factor comparison 7.2 Finite element method Nine finite element analyses have been performed using different crack lengths ( a/b = 0.1 to 0.9). The ratio between the bar length L and the bar radius b is 11. Figure 17 shows a typical deformed mesh. 24 Figure 17 : Deformed mesh for a/b=0.5 The finite element analyses show a very good agreement with the Benthem function (see Figure 17). This confirms the results of Neale P^^96!. So the Benthem function is strongly advised for the stress intensity factor calculation. 25 8 Compliance of a precracked bar under tension 8.1 Analytical formulations The load line compliance is the displacement under the load point for a unit load. The use of the well-known relation between the energy released rate and the compliance is an elegant way to find out this function. The relation between the energy release rate and the variation of compliance due to crack growth is given by: G . LP*J£- .. LpiJL.^. 1^1K2 (37) 2 dAir 2 2na da E Integration gives: C(a).C0. fllllZllva'da' (38) J E p2 The compliance of an uncracked bar under tension Co is: -L- . L (39) EA The contribution of the crack is: 6 •> •> 6 7 E"2 , 1 J , , \-v*K\ ,, , , , l-vz ^l , , l-v t\P I 2 2na da - f 2 2%a da - 4A /r dr\ 3 2 J E p* J E -fl' E b 1 n I A new function H is defined as follows: i ^2 7^ (41) This function is tabulated in Annex 3. The ratio between the compliance of a cracked and an uncracked bar is: i. * *±llbH (42) Haigh and Richards !HAKM1 have proposed a polynomial approximation of the H function. H{r\) - ^?- - 0.35125 • 0.16875 T^2 - 0.1325 r\3 * 0.015 ^4 • 0.05 T^S (43) 26 Neale i^*-96! has used all the terms of the H function. His function (Equation 44) can be considered as the exact integration of Equation 41. 0.25 2 - 0.35318 • 0.16888 i\ - 0.13255 0.04881 (44) - 0.01106 r\5 • 0.05045 i\6 - 0.03804 T|7 • 0.0167 i\* In this work a verification of Neale's equation is done. The function found is: - 0.3531717 • 0.168875 ri2 - 0.1325521 r\* • 0.04880469 i\A (45) - 0.01106345 • 0.05044688 r\6 - 0.03803811 r\7 • 0.01669878 i\* For high value of r\, H is very small and the crack length has a very week influence on the compliance (see Figure 18). If the function 1+H is considered, than the accuracy of Haigh's equation is better than 0.1 % in all the range. 8.2 Finite element method Using the same analyses described in paragraph 7.2, the computed finite element compliance is compared to analytical expressions. All results are expressed in the form of the dimensionless function (see Equation 42). The finite element results are in very good agreement with the Haigh [HAI-741 and Neale r^"961 function (less than 1% difference). 3.50 3.00 • 2.00 1.50 1.00 0.00 0.10 0JO 0.30 0.40 0.50 0.60 0.70 0.80 0.90 Figure 18 : Dimensionless compliance 27 9 Conclusions Different systems to precrack notched bars based on rotating bending fatigue are described in the literature. A good attention should be paid to the generation of a crack with minor eccentricity and the measurement of the crack length. The analysis of the stress intensity factor under bending shows that: - the stress intensity factor is not uniform along the crack front, - a correcting factor of the stress intensity factor for deep crack must be used to account for contact effects, - contact effects on deep crack accelerate the formation of an eccentric crack, - shear loading (mode II) has a small effect on the crack growth propagation. A theoretical formulation of the compliance of a cracked bar under bending has been developed and validated. The stress intensity factor and the compliance under bending and tension has been computed using finite element calculations. Results show that most of analytical formulations are very accurate. Analytical considerations has led to design a new configuration to reduce the contact pressure. The investigation of the cantilever beam under constant displacement for a specific configuration has shown that the stress intensity factor increases by a factor 2 and the force decreases by 20%, due to the crack growth. This load drop will be used to monitor the crack extension. 28 10 Future investigations From an experimental point of view, the three different techniques: - the compliance, - the AC potential drop, - the vibration technique, will be used on five specimens of different crack length. Then the five specimens will be broken at a very low temperature to measure the crack length. So experimental calibration curves will be obtained. Then a complete experimental procedure to precrack round notched bars will be established and the most suitable method will be chosed. Finite element calculation will be made to obtain a calculated function of the crack length versus natural bending frequency. This function will be very useful to support experimental measurement. From a theoretical point of view, the possibility to measure fracture toughness on the precrack round bar will be investigated in elastic and elasto-plastic conditions. The study will be focused on the J-integral for this geometry. Finite element calculation will be made to validate the different J-integral formulations proposed in the literature and to compare axisymmetric specimens with three point bending specimens. 29 11 References [AMA-86] Amar E. "Application de l'approche locale de la rupture a letude de la transition ductile-fragile dans l'acier 16MND5", Doctor thesis, Ecole nationale superieur des mines de Paris, Paris 1986. [AND-95] Anderson T.L. "Fracture mechanics fundamentals and applications", CRC Press second edition, London, pp. 1-680, 1995 [BEN-73] Benthem J.P. and Koiter W.T. "Asymptotic approximations to crack problems", in Methods of Analysis and Solutions of Crack Problems (Edited by G.C. Shi), chap. 3, pp. 131-178, Noordhoft International Publishing, Groningen, 1973 [BOL-85] Boliang Li, Hui Liang and Hao Li "Controling crack by laser beams", Engineering Fracture Mechanics Vol. 21, No. 6, pp. 1071-1081, 1985 [BUE-65] Bueckner H.C. "Discussion on stress analysis of cracks", by Paris PC. and Sih G.C., ASTM STP 381, pp. 82-83, 1965 [CHA-96] Chaouadi R. "Micromechanically-based damage modeling of crack initiation in reactor materials", doctor thesis, Katholieke Universiteit te Leuven Belgium, February 1996 [DEV-85] Devaux J.C. "An experimental program for the validation of local ductile fracture criteria using axisymmetrically cracked bars and compact tension specimens", Engineering Fracture Mechanics Vol. 21, No. 2, pp. 273-283, 1985 [EIS-85] Eisele U., Roos E., Silcher H. and Zimmermann C. "A method for the defined fatigue precracking of round notched bars", Journ. of Testing and Evaluation, JTEVA, Vol. 13, No. 1, pp. 46-48, 1985 [GAG-96] Gage G. "Measurement of transition fracture behaviour using miniature-scale specimens", TAGSI Symposium, TWI Cambridge, 9-10 January 1996 [GIB-87] Gibson G.P. "The use of alterating current potential drop for determining J-crack resistance curves", Engineering Fracture Mechanics, Vol. 26, No. 2, pp. 213-222, 1987 [GIO-93] Giovanola J.H., Homma H., Lichtenberger M., Crocker J.E. and Klopp R.W. "Fracture toughness measurements using small cracked round bars", ASTM 2nd Symposium on Constraint Effects in Fracture, 17-18/11/93, Fort Worth,1993 [GOL-95] Golinval J.C., Professor at the University of Liege, Private communication, Belgium, decembre 1985 30 [GUP-84] Gupte K. A. and Banerjee S. "Fracture of round bars loaded in mode II and a procedure for KIIIc determination", Engineering Fracture Mechanics, Vol. 19, No. 5, pp. 919-946, 1984 [HAR-67] Haris D.O."Stress intensity factors for hollow circumferentially notched round bars" J. Basic Engng. 89, pp 49-54, 1967 [HAW-85] Hawley R.H., Duffy J. and Shih C.F."Dynamic notched round bar testing", ASM Metal Handbook, Mech. Testing, American Society for Metals, 9th edition, Vol. 8, pp. 275-282, 1985 [HOU-79] Hourlier F. and Pineau A. "Fissuration par fatigue sous solicitation polymodale (mode I ondule + mode III permanent) d'un acier rotor 26NCDV14" Memoires Scientifiques, Revue Metallurgie, pp. 175-185, March 1979 [HWA-92] Hwang IS. and Ballinger R.G. "A multi-freqency AC potential drop technique for the detection of small cracks", Meas. Sci. Technol., Vol. 3, pp 62-74, 1992 [LUC-93] LuconE., Bicego V., D'Angelo D. and Fossati C. "Evaluating a service-exposed component's mechanical properties by means of subsized and miniature specimens", Small Specimen Test Techniques Applied to Nuclear Reactor Vessel Thermal Annealing and Plant Life Extension, ASTM STP 1204, pp. 311-323, 1993 [LUC-90] Lucon E. "Fracture toughness testing using small cylindrical specimens with ring- shaped cracks", ECF 8 Fracture Behaviour and design of Materials and Structures", Fracture Behavior and Design of Material and Structures, Proceeding of ECF8, 8th Biennial European Conference on fracture, D. Firrao, Ed., Vol. II, EMAS, London, pp. 1077-1082, 1990 [MEG-90] Meguid S.A. "Three-dimensional finite element analysis of circumferentialy- cracked notched and un-notched cylindrical components", Engineering Fracture Mechanics, Vol. 37, No. 2, pp. 361-371, 1990 [NAK-89] Nakai Y. and Wei R.P. "Measurement of short crack lengths by an AC potential method", Engineering Fracture Mechanics Vol. 32, No. 4, pp. 581-589, 1989 [NEA-96] Neale B. K. "The fracture behaviour of a circumferentially cracked bar in tension", TAGSI Symposium, TWI Cambridge, 9-10 January 1996 [ROO-76] Rooke DP. and Cartwright D.J. "Stress intensity factors", HMSO, pp. 237-239, 1976 [SAW-82] Sawaki Y., Aoyama N., Abe Y., Ohshima T. and Kawasaki T. "Determination of fatigue fracture toughness by conventional rotary bending specimen", Joint Conference on Experimental Mechanic,pp. 555-560, Oahu-Maaui, Hawaii, 22-23 May 1982 31 [SHE-82] Shen Wei, Zhao Tingshi, Gao Daxing, Li Dunkang, Li Poliang and Qui Xiaoyun "Fracture toughness measurement by cylindrical specimen whith ring-shaped crack", Engineering Fracture Mechanics Vol. 16, No. 1, pp. 69-82, 1982 [STA-86] Stark H.L. and Ibrahim R.N. "Estimating fracture tougness from small specimens", Engineering Fracture Mechanics Vol. 25, No. 4, pp. 395-401, 1986 [WAR-89] Warren C.Y. "Roak's formulas for stress and strain", McGraw-Hill Book Company, sixth edition, pp. 1-763 New York, 1989 [WOJ-95] Wojcik A.G. "Potential drop techniques for crack characterisation", Materials World, Vol. 3., No. 8, pp 379-381, 1995 32 Annex 1 Stress intensity factor and compliance function of a precracked bar under bending (T)4 * 0.531 (°)5] (46) D K Z o BO lt> 128 - a/fc) (47) a/b G Fl H a/b G Fl H 0.00 0.37500 0.37500 +INF 0.51 0.53655 0.37559 2.8535e-01 0.01 0.37689 0.37500 4.6875e+04 0.52 0.54207 0.37556 2.6529e-01 0.02 0.37881 0.37500 5.8593e+03 0.53 0.54774 0.37551 2.4672e-01 0.03 0.38075 0.37500 1.7360e+03 0.54 0.55357 0.37545 2.2951e-01 0.04 0.38273 0.37500 7.3236e+02 0.55 0.55957 0.37537 'M353e-01 0.05 0.38474 0.37500 3.7494e+02 0.56 0.56573 0.37526 ,9868e-01 0.06 0.38678 0.37500 2.1695e+02 0.57 0.57207 0.37513 .8486e-01 0.07 0.38886 0.37500 1.3660e+02 0.58 0.57859 0.37497 .7199e-01 0.08 0.39096 0.37500 9.1489e+01 0.59 0.58531 0.37478 .5999e-01 0.09 0.39311 0.37500 6.4237e+01 0.60 0.59221 0.37455 .4878e-01 0.10 0.39529 0.37500 4.6812e+01 0.61 0.59932 0.37427 .3832e-01 0.11 0.39750 0.37500 3.5154e+01 0.62 0.60663 0.37395 .2853e-01 0.12 0.39975 0.37500 2.7063e+01 0.63 0.61416 0.37358 .1937e-01 0.13 0.40205 0.37500 2.1272e+01 0.64 0.62190 0.37314 .1080e-01 0.14 0.40438 0.37500 1.7019e+01 0.65 0.62988 0.37264 .0276e-01 0.15 0.40675 0.37501 1.3825e+01 0.66 0.63809 0.37207 9.5227e-02 0.16 0.40917 0.37501 1.1380e+01 0.67 0.64654 0.37141 !?.8159e-02 0.17 0.41163 0.37501 9.4770e+00 0.68 0.65525 0.37066 8.1526e-02 0.18 0.41413 0.37502 7.9734e+00 0.69 0.66421 0.36982 7.5298e-02 0.19 0.41669 0.37502 6.7699e+00 0.70 0.67344 0.36886 6.9450e-02 0.20 0.41929 0.37502 5.7950e+00 0.71 0.68294 0.36777 6.3957e-02 0.21 0.42194 0.37503 4.9971e+00 0.72 0.69273 0.36656 .>.8798e-02 0.22 0.42465 0.37504 4.3377e+00 0.73 0.70280 0.36519 i).3951e-02 0.23 0.42740 0.37505 3.7879e+00 0.74 0.71318 0.36365 4.9400e-02 0.24 0.43022 0.37506 3.3260e+00 0.75 0.72386 0.36193 '*.5126e-02 0.25 0.43309 0.37507 2.9350e+00 0.76 0.73487 0.36001 't.H15e-02 0.26 0.43602 0.37508 2.6019e+00 0.77 0.74620 0.35786 :!.7352e-02 0.27 0.43901 0.37509 2.3163e+00 0.78 0.75786 0.35547 :!.3824e-02 0.28 0.44207 0.37511 2.0700e+00 0.79 0.76988 0.35280 :!.0521e-02 0.29 0.44519 0.37513 1.8565e+00 0.80 0.78225 0.34983 :2.743Oe-O2 0.30 0.44838 0.37515 1.6705e+00 0.81 0.79499 0.34653 :!.4542e-02 0.31 0.45165 0.37517 1.5077e+00 0.82 0.80810 0.34285 ;..1848e-02 0.32 0.45498 0.37519 1.3646e+00 0.83 0.82161 0.33876 .9339e-02 0.33 0.45840 0.37521 1.2383e+00 0.84 0.83551 0.33420 .7009e-02 0.34 0.46189 0.37524 1.1265e+00 0.85 0.84982 0.32914 ,4850e-02 0.35 0.46546 0.37527 1.0270e+00 0.86 0.86456 0.32349 .2857e-02 0.36 0.46912 0.37530 9.3827e-01 0.87 0.87972 0.31719 .1023e-02 0.37 0.47287 0.37532 8.5886e-01 0.88 0.89534 0.31015 <>.3428e-03 0.38 0.47670 0.37536 7.8757e-01 0.89 0.91141 0.30228 7.8130e-03 0.39 0.48063 0.37539 7.2340e-01 0.90 0.92794 0.29344 6.4289e-03 0.40 0.48466 0.37542 6.6548e-01 0.91 0.94496 0.28349 '.. 1870e-03 0.41 0.48880 0.37545 6.1307e-01 0.92 0.96248 0.27223 4.0840e-03 0.42 0.49303 0.37548 5.6552e-01 0.93 0.98050 0.25941 C .1172e-03 0.43 0.49738 0.37551 5.2228e-01 0.94 0.99904 0.24471 2.2840e-03 0.44 0.50183 0.37554 4.8288e-01 0.95 1.01811 0.22766 ,5825e-03 0.45 0.50641 0.37556 4.4690e-01 0.96 1.03773 0.20755 .0109e-03 0.46 0.51111 0.37558 4.1397e-01 0.97 1.05792 0.18324 !>.6777e-04 0.47 0.51593 0.37560 3.8379e-01 0.98 1.07867 0.15255 I!.5206e-04 0.48 0.52088 0.37561 3.5606e-01 0.99 1.10002 0.11000 6.2966e-05 0.49 0.52596 0.37561 3.3055e-01 1.00 1.12198 0.00000 0.0000e+00 0.50 0.53119 0.37561 3.0705e-01 33 Annex 2 Neutral axis position and stress intensity factor correction due to contact 1 -n. fl 2 (48) f y /i- (V f y y c- («c- y) (49) 16 [ y2 f^tnVrf dy ac nc S ac nc S 1.00 0.000000e+00 1.000000e+00 0.69 1.983671e-01 6.200634e-01 0.99 6.366167e-03 9.879684e-01 0.68 2.048524e-01 6.078007e-01 0.98 1.273258e-02 9.759122e-01 0.67 2.113483e-01 5.955586e-01 0.97 1.909923e-02 9.638315e-01 0.66 2.178556e-01 5.833398e-01 0.96 2.546632e-02 9.517273e-01 0.65 2.243751e-01 5.711469e-01 0.95 3.183400e-02 9.396004e-01 0.64 2.309078e-01 5.589828e-01 0.94 3.820245e-02 9.274517e-01 0.63 2.374545e-01 5.468505e-01 0.93 4.457184e-02 9.152821e-01 0.62 2.440162e-01 5.347531e-01 0.92 5.094234e-02 9.030925e-01 0.61 2.505941e-01 5.226939e-01 0.91 5.731414e-02 8.908838e-01 0.60 2.571894e-01 5.106762e-01 0.90 6.368745e-02 8.786571e-01 0.59 2.638034e-01 4.987037e-01 0.89 7.006247e-02 8.664134e-01 0.58 2.704374e-01 4.867800e-01 0.88 7.643942e-02 8.541539e-01 0.57 2.77093 le-01 4.749092e-01 0.87 8.28185 le-02 8.418795e-01 0.56 2.837722e-01 4.630954e-01 0.86 8.920001e-02 8.295916e-01 0.55 2.904764e-01 4.513429e-01 0.85 9.558414e-02 8.172914e-01 0.54 2.972079e-01 4.396566e-01 0.84 1.019712e-01 8.049800e-01 0.53 3.039690e-01 4.28041 le-01 0.83 1.083614e-01 7.926589e-01 0.52 3.107621e-01 4.165020e-01 0.82 1.147551e-01 7.803294e-01 0.51 3.175901e-01 4.050447e-01 0.81 1.211525e-01 7.679930e-01 0.50 3.244562e-01 3.936754e-01 0.80 1.275541e-01 7.556512e-01 0.49 3.313640e-01 3.824007e-01 0.79 1.339601e-01 7.433054e-01 0.48 3.383176e-01 3.712276e-01 0.78 1.403750e-01 7.309724e-01 0.47 3.453216e-01 3.601643e-01 0.77 1.467916e-01 7.186259e-01 0.46 3.523816e-01 3.492196e-01 0.76 1.532139e-01 7.062808e-01 0.45 3.595039e-01 3.384035e-01 0.75 1.596422e-01 6.939387e-01 0.44 3.666962e-01 3.277277e-01 0.74 1.660770e-01 6.816017e-01 0.43 3.739679e-01 3.172065e-01 0.73 1.725189e-01 6.692716e-01 0.42 3.813308e-01 3.068575e-01 0.72 1.789682e-01 6.569506e-01 0.41 3.888022e-01 2.967069e-01 0.71 1.854257e-01 6.446407e-01 0.40 3.964035e-01 2.867871e-01 0.70 1.918918e-01 6.323442e-01 34 Annex 3 Stress intensity factor and compliance function of a precracked bar under tension F.(-) - G(-) y/(l - alb) (51) a/b G Fl H a/b G Fl H 0.00 0.50000 0.50000 +INF 0.51 0.67692 0.47384 ,6691e-01 0.01 0.50252 0.50000 2.4647e+01 0.52 0.68190 0.47244 .5847e-01 0.02 0.50507 0.50000 .2147e+01 0.53 0.68699 0.47097 .5039e-01 0.03 0.50766 0.49999 7.9803e+00 0.54 0.69217 0.46946 .4267e-01 0.04 0.51029 0.49998 .i.8971e+00 0.55 0.69747 0.46788 .3527e-01 0.05 0.51295 0.49996 4.6472e+00 0.56 0.70287 0.46623 .2819e-01 0.06 0.51564 0.49993 :!.8141e+00 0.57 0.70839 0.46452 .2140e-01 0.07 0.51837 0.49989 :1.2190e+00 0.58 0.71402 0.46274 .1490e-01 0.08 0.52112 0.49984 2.7728e+00 0.59 0.71978 0.46088 ,0867e-01 0.09 0.52391 0.49978 2.4259e+00 0.60 0.72566 0.45895 ,0269e-01 0.10 0.52673 0.49970 2.1484e+00 0.61 0.73168 0.45693 9.6963e-02 0.11 0.52958 0.49961 ,9214e+00 0.62 0.73783 0.45483 9.1467e-02 0.12 0.53246 0.49949 ,7324e+00 0.63 0.74411 0.45263 i?.6196e-02 0.13 0.53537 0.49936 .5725e+00 0.64 0.75054 0.45032 !?.1141e-02 0.14 0.53832 0.49921 ,4355e+00 0.65 0.75712 0.44792 7.6292e-02 0.15 0.54129 0.49905 ,3169e+00 0.66 0.76385 0.44540 7.1641e-02 0.16 0.54430 0.49886 .2131e+00 0.67 0.77073 0.44275 6.718U-02 0.17 0.54733 0.49864 .1217e+00 0.68 0.77778 0.43998 6.2905e-02 0.18 0.55040 0.49841 .0405e+00 0.69 0.78499 0.43707 .>.8806e-02 0.19 0.55350 0.49815 9.6787e-01 0.70 0.79238 0.43400 >.4878e-02 0.20 0.55663 0.49787 9.0260e-01 0.71 0.79994 0.43078 5.1116e-02 0.21 0.55980 0.49756 ii.4362e-01 0.72 0.80768 0.42738 4.7514e-02 0.22 0.56300 0.49723 7.9007e-01 0.73 0.81561 0.42380 4.4067e-02 0.23 0.56623 0.49687 '?.4124e-01 0.74 0.82373 0.42002 4.0772e-02 0.24 0.56950 0.49648 6.9655e-01 0.75 0.83204 0.41602 :!.7623e-02 0.25 0.57281 0.49607 (>.5550e-01 0.76 0.84056 0.41179 :!.4617e-02 0.26 0.57616 0.49563 (i.l768e-01 0.77 0.84929 0.40731 :!.1750e-02 0.27 0.57954 0.49516 i .8272e-01 0.78 0.85823 0.40255 :2.9020e-02 0.28 0.58296 0.49466 i .5032e-01 0.79 0.86739 0.39749 :2.6422e-02 0.29 0.58643 0.49413 ! .2021e-01 0.80 0.87678 0.39211 '2.3956e-02 0.30 0.58994 0.49358 4.9218e-01 0.81 0.88640 0.38637 'M617e-02 0.31 0.59349 0.49299 '1.660 le-01 0.82 0.89625 0.38025 .9404e-02 0.32 0.59709 0.49237 't.4155e-01 0.83 0.90635 0.37370 .7316e-02 0.33 0.60073 0.49172 't,1862e-01 0.84 0.91670 0.36668 .5349e-02 0.34 0.60443 0.49104 :!.9710e-01 0.85 0.92730 0.35914 .3504e-02 0.35 0.60817 0.49032 :!.7686e-01 0.86 0.93816 0.35103 ,1779e-02 0.36 0.61197 0.48958 :!.5781e-01 0.87 0.94929 0.34227 .0172e-02 0.37 0.61583 0.48880 :S.3985e-01 0.88 0.96070 0.33280 j.6828e-03 0.38 0.61974 0.48798 :!.2288e-01 0.89 0.97239 0.32251 '7.3112e-03 0.39 0.62371 0.48713 :!.0684e-01 0.90 0.98437 0.31128 (5.0566e-03 0.40 0.62774 0.48625 :2.9166e-01 0.91 0.99664 0.29899 't.9187e-03 0.41 0.63184 0.48532 :!.7727e-01 0.92 1.00921 0.28545 .!.8975e-03 0.42 0.63600 0.48436 :2.636 le-01 0.93 1.02209 0.27042 '2.9934e-03 0.43 0.64023 0.48337 :2.5O65e-Ol 0.94 1.03529 0.25359 2.2067e-03 0.44 0.64454 0.48233 :2.3833e-01 0.95 1.04881 0.23452 l.5380e-03 0.45 0.64892 0.48125 :2.2660e-01 0.96 1.06266 0.21253 ).8819e-04 0.46 0.65337 0.48013 :2.1544e-01 0.97 1.07684 0.18651 5.5817e-O4 0.47 0.65791 0.47897 :2.0480e-01 0.98 1.09137 0.15434 2.4917e-04 0.48 0.66253 0.47776 .9466e-01 0.99 1.10626 0.11063 S.2582e-05 0.49 0.66724 0.47650 .8498e-01 1.00 1.12150 0.00000 ).0000e+00 0.50 0.67203 0.47520 .7574e-01 35