A METHOD

OF INVESTIGATING

THE CRACK RESISTANCE

OF CONSTRUCTIONAL

STEELS AT THE MOMENT OF CRACK START AND ARREST V. G. Kaplunenko

UDC 539.4

For investigating the resistance of constructional alloys to developing of rapidly advancing cracks, double-cantilever beam type samples are used, making it possible to repeatedly reproduce crack start and arrest on a single sample. To increase the number of crack arrests, increasing the rigidity of the loading to the maximum is aimed for, for which wedging forces are applied to the sample.

system

In loading a double-cantilever beam sample with a wedging screw [i, 2] the value of the fracture toughness determined is limited as the result of the low strength of the threaded joint of the loading screw with the sample. In the case of loading of the sample with two wedges through pins [3], additional elastic energy is stored in the system as the result of bending of the pins, which leads to an increase in the length of the crack jump and the kinetic energy of the sample, complicates analysis of the process, and increases the error in determination of the stress intensity factor at the moment of crack arrest KIo. With high values of KQ(KIc) it is impossible to arrest the crack after starting of it as the result of the large reserve of elastic energy. A common disadvantage of loading of a double-cantilever beam sample by known methods [1-4] is indeterminacy of the load acting on the sample. Therefore, in finding the stress intensity factor the amount of movement of the cantilever sample measured with the use of a strain measuring bracket is used. This introduces an error into the value of the stress intensity factor determined by calculation equations based on the assumption of rigid fixing of the sample, which is not completely realized in testing of the latter [2]. The purpose of this work includes the development of a method of determination of the crack resistance of a broad class of materials at the moment of crack start and arrest making it possible to utilize the advantages of a sample loading system with wedging forces, eliminating the above mentioned disadvantages. Figure 1 shows the plan of a double-cantilevel beam sample of original design. A feature of the sample 1 is the presence of the notch 2 consisting of two portions, one of which has a variable width with an angle of opening 2B and contains on its sides the through cylindrical grooves 3. The angle 28 of the notch is greater than the angle 2~ of the wedge 6, and the arc of the circumference enveloping in plan view the profile of the cylindrical groove 3 exceeds ~ radians. The grooves 3 are used for placing in them the support rollers 7, which transmit the force from the loading wedge 6 to the centilevers 5 of the sample i. The rollers 7 project above the surfaces of the notch 2, a n d w i t h the angle 28 of the notch more than the angle 2~ of the wedge, c o n t a c t b e t w e e n the rollers 7 and the wedge 6 is provided on the generatrix of the rollers 7. In the test the wedge 6 is fastened to the upper support 8 and the sample 1 is placed on the lower support 9 of the test machine with control of movement of the supports. From Fig. 1 it may d e seen that the wedge 6 transmits the force to the sample 1 through the support rollers 7, which do not project beyond the side surfaces of the sample. This eliminates their deflection under the action of the loading wedge 6 and, in the case of proper ridigity of the loading wedge and the test machine, does not lead to additional accumulation of elastic energy. Preliminary investigations made on samples with a thickness of B = 25 mm of 15Kh3MA steel showed that with a relatively short sample length of W = 185 mm it is possible to obtain up to five arrests of the crack with a value of KQ = 123.5 ~'~a~mm determined Institute of Strength Problems, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Problemy Prochnosti, No. i0, pp. 59-63, October, 1984. Original article submitted January 26, 1983.

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0039-2316/84/1610-

1414508.50

9 1985 Plenum Publishing

Corporation

2

Il

Fig. i. Double-cantilever beam sample (a) and method of loading of it (b).

/

0

0 a

b'0

Fig. 2. Load P--crack opening on the line of action of the force V diagram in the absence (a) and presence (b) of the force of friction. at the moment of start of the crack (T= 153~ and up to 15 crack arrests with KQ = 69 MPa#~m (T= 93~ This is an indication of the high rigidity of this method of loading of a doublecantilever beam sample. Since according to this method the maximum possible force wedging the sample is determined by the power of the test machine, with the use of a powerful machine and large wedge dimensions it is possible to test samples of practically any thickness. The proposed method makes it possible to calculate, by a calculation method, the force Q wedging the sample since the contact between the wedge 6 and the support rollers is linear, the point of contact and the angle 2a of the wedge 6 are known, and the load P applied to the wedge 6 is recorded by the test machine. The preliminary fatigue crack may be grown with the use of the wedge 6, or the sample 1 may be subjected to off-center pulsating tension, using the cylindrical grooves 3 for placement of the pins.

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Q

Q

al

b

R

''

0

[~-

C

Fig. 3. Plans of the forces acting in loading of a double-centilever beam sample with a wedge. In determining the force Q it is necessary to take into consideration the forces of friction occurring in movement of the wedge on the rollers 7. Under conditions of free rotation of the rollers 7 in the cylindrical grooves 3 a force of rolling friction occurs which may be neglected without a large error because of its smallness. If as the result of technical reasons or in the case of tests at very low temperatures conditions of free rotation of the rollers 7 are not obtained, a quite high force of rubbing friction, which must be taken into consideration, appears. The method of taking into consideration the force of friction is clear from Figs. 2 and 3, which show load P--crack opening V diagrams and the plans of action of the forces in the absence and presence of the force of friction. If the forces of friction are absent (Fig. 2a), then within the limits of elasticity the P--V diagrams in loading and unloading of the sample coincide and to each value of Vo corresponds a single completely determined value of Po. In the case of the action of the force of friction (Fig. 2b) these diagrams differ in loading and unloading of the sample and to a single value of Vo in loading corresponds Po' > Po, and in unloading P"o < Po. This may be explained by the fact that in loading of the sample (downward the wedge) the force of friction hinders opening of the crack and in unloading (upward movement of the wedge) closing of it.

movement of of the sample

Consequently, having several times loaded and unloaded the sample to different levels of load instead of a single P--V diagram we obtain a series of triangles OAB, OCD, ONM, etc., which are similar under unchanged test conditions since the angle of friction is constant. If through the P--V diagram obtain the ratio of the forces

obtained

is drawn any line n--n parallel

to the Y axis, we

k=PJP~, acting on the wedge

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in loading

and unloading

of the sample,

(i) which corresponds

to Vo = const.

Using the plans of forces shown in Fig. 3, we find the calculation equations for determination of the force Q: P0

2tg~

Q(in the absence of the force of friction

(2)

(Fig. 3a));

P'o

(3)

Q - 2tg (a + r (in

the

presence

of

the

force

of

friction

and an increasing

load

on the

sample

(Fig.

3b));

P; Q--

2 tg(a--~)

(4)

(in the presence of the force of friction and a decreasing load on the sample where ~ is the angle of friction.

(Fig. 3c)),

Since the equality v = CQ

(C is the pliability of the sample) and (4) we have

is valid,

k =

For solution

of Eq.

for a fixed value of Vo from Eqs.

(i), (3),

tg ( a + ~) tg (~ - - q9 "

(5)

(5) relative to the angle of friction we use the equalities

tg(a+

r

=

tga+tgcp . l--tgcz.tgq~ '

tg(a--~)=

tga--tg~ l +tgcz.tgq~

After transformations we arrive at the equation

I~- bf + I = 0, where f = t g ~

(6)

is the coefficient of friction and b --- ( k + 1)(tg=a+ 1) (k-- 1) t g a

(7)

The physical sense has the solution

0
B > 2,5 ( K,~ /~ \00.

The samples were cooled with contact-type method described in [6].

(8)

2 / "

coolants

(position i0 in Fig. ib) using the

The stress intensity factor was calculated using the equations K, =

Knom

.

0.2.&/B+0.8

'

[7]

(9)

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de/~c

-..\

!

//

/

I 175 0 --I

150! m - 2 9 9

,o

--3

/

iii III

/

--~

125 IOC

/ i

75 50 77

g3

i

~1iO.4

I

o

'

i

7

153 ~"

Fig. 4

Fig. 5

Fig. 4. Temperature relationships of KQ (white points) and KIo (black points): i, 3) Bn/B = 0.4; 2, 4) Bn/B = 0.8; 1-4) double-cantilever beam sample; 5) compact sample. Fig. 5. Temperature relationships of relative length of the arrested crack do/d c and Klo/Kc: i, 3) Bn/B = 0.4; 2, 4) Bn/B = 0.8. t~a M p a ~ m

'"i / 77

93

123

153 T,

Fig. 6. Temperature relationships of the values of KQ taking into consideration (i) and not taking into consideration (2) the forces of friction: i, 2) double-cantilever beam sample; 3) conpact sample. TABLE i. Mechanical Properties of 15Kh3MA Steel

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rts t , K

ou, M P a

ao,2 . M P a

6, %

t'. %

153 123 77

1030 1084 1250

993 1060 1218

19,7 15,8

68,4 66,7 54,1

21,2

h2

Knom = I /

12Q2a2 [I B.Bnh3

+ 1 . 3 2 + 4- 0 . 5 4 2 - ~ )

w h e r e Q i s t h e f o r c e w e d g i n g t h e s a m p l e a n d B, Bn, h, of the sample (Fig. la).

and a a r e

(10) the geometric

parameters

The v a l u e o f t h e s t r e s s i n t e n s i t y f a c t o r a t t h e moment o f s t a r t o f t h e c r a c k KQ was d e t e r m i n e d f r o m t h e maximum l o a d on t h e P--V d i a g r a m , w h i c h was l i n e a r a l l t h e way to f a i l u r e in the whole investigated t e m p e r a t u r e r a n g e , a n d a t t h e moment o f c r a c k a r r e s t KIo a s t h e static approximation without dynamic analysis of the conditions of crack arrest. For the purpose of experimental verification d e t e r m i n e d on s t a n d a r d c o m p a c t s a m p l e s .

of this

method,

the values

To i n v e s t i g a t e t h e i n f l u e n c e o f n o t c h d e p t h ( p o s i t i o n 4 on F i g . KQ a n d Kio , s a m p l e s w i t h Bn/B = 0 . 8 and Bn/B = 0 . 4 w e r e t e s t e d .

la)

o f KQ w e r e a l s o

on t h e v a l u e s

The r e l a t i o n s h i p s o f KQ and KIo and a l s o o f r e l a t i v e length of the arrested a o / a c t o t e s t t e m p e r a t u r e a r e shown i n F i g s . 4 and 5, r e s p e c t i v e l y .

of

crack

The e x p e r i m e n t a l d a t a p r e s e n t e d i n F i g s . 4 and 5 i n d i c a t e t h a t w i t h an i n c r e a s e i n t e s t t e m p e r a t u r e t h e v a l u e s o f KQ and Kio and t h e a m o u n t o f t h e c r a c k jump i n c r e a s e f o r b o t h Bn/B r a t i o s . The v a l u e o f KIo i s l e s s s e n s i t i v e to temperature change. W h i l e a t 77~ t h e v a l u e s KQ a n d Kio p r a c t i c a l l y c o i n c i d e , w i t h an i n c r e a s e i n t e m p e r a t u r e t h e d i f f e r e n c e between them s t e a d i l y increases. is

The i n c r e a s e i n t h e d i f f e r e n c e b e t w e e n KQ a n d Kio w i t h an i n c r e a s e a c c o m p a n i e d by an i n c r e a s e i n t h e amount o f t h e c r a c k jump ( F i g . 5 ) .

in test

of

temperature

Verification of fulfillment of the conditions of plane strain according to criterion (8) showed that starting with a temperature of 123~ and lower the value of KQ corresponds to the fracture toughness in plane strain Kic. The test results presented in Fig. 4 make it possible to judge that at a temperature above 123~ when plane strain conditions are not fulfilled, the characteristics KQ and KIo obtained in testing of samples with Bn/B = 0.4 are higher than of samples with Bn/B = 0.8. In this case the P--V diagrams are linear all the way to failure in testing samples both with Bn/B = 0.8 and with Bn/B = 0.4. A comparison of the values of KQ obtained in tests and of standard compact samples (Fig. 4) indicates that compact samples and double cantilever beam samples with For double-cantilever beam samples with Bn/B = 0.4, the samples without notches.

of double-cantilever beam samples these values determined in tests of Bn/B = 0.8 practically coincide. value of KQ is more than for compact

Figure 6 presents the temperature relationships of the characteristic KQ obtained taking into consideration and not taking into consideration the forces of friction acting in loading with a wedge of a double-cantilever beam sample with a ratio of Bn/B = 0.8. It also shows the values of KQ determined in testing compact samples. As may be seen, in taking into consideration the forces of friction the values KQ for double-cantilever beam and compact samples agree while without taking into consideration the forces of friction these values are significantly higher in the first case then in the second. Therefore, not taking into consideration the forces of friction in loading of double-cantilever beam samples with a wedge leads to significant errors in the characteristics determined. LITERATURE CITED i.

2.

Calculations and Tests for Strength. Methods of Mechanical Testing of Metals. Determination of the Characteristics of Fracture Toughness (Crack Resistance) in the Crack Arrest Stage. Method Recommendations [in Russian], Vsesoyuz. Nauch.-Issled. Inst. Norm. Mashinostr., Moscow (1982), pp. 12-13. V. M. Markochev, A. G. Kraev, A. P. Bobrinskii, et al., "An investigation of the fracture toughness of 12Kh2MFA casing steel at the moment of crack start and arrest under isothermal test conditions," Fiz. Mekh. Deform. Razrush., No. 4, 41-47 (1977).

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3.

4.

5. 6.

7.

R. Khoaglend, A. Rozenfel'd, P. Gelen, and Dzh. Khan, "A method of measurement of Klm, KID, and Kid in retarding of cracks," Mekh. Razrush. Bystr. Razrush., Ost. Treshchin, No. 25, 42-73 (1981). Dzh. Khan, A. Rozenfel'd, K. Marshall, et al., "The concept of crack arrest and its use," ibid., 222-253. "E309-74. Standard method of test for plane-strain fracture toughness of metallic materials," in: Annual Book of ASTM Standards (1976), pp. 471-490. V. T. Troshchenko, V. V. Pokrovskii, P. V. Yasnii, et al., "An experimental investigation of the cyclic crack resistance of constructional steels," in: Mechanical Tests of Constructional Alloys at Cryogenic Temperatures [in Russian], Naukova Dumka, Kiev (1982), pp. 166-177. V. P. Naumenko, "The influence of sample geometry on the results of determination of the value of Klc," Probl. Prochn., No. i0, 81-88 (1973).

OPENING OF THE TIP OF A THROUGH LONGITUDINAL CRACK IN A CYLINDRICAL SHELL UNDER THE ACTION OF INTERNAL PRESSURE V. A. Osadchuk, V. I. Kir'yan, and M. M. Nikolishin

UDC 539.375

Broadening of the area of practical use of the criteria of fracture mechanics is related to the necessity of investigation of the stress--strain state near cracks in solids of different configuration and loading conditions. For constructional elements of low- and medium-strength steels, such investigations taking into consideration the developed plastic deformation of the metal in the zones of the defects acquire important value. Since consideration of elastoplastic problems for bodies with cracks in a strictly classical formulation, in which two systems of equations (one in the elastic area and the other in the plastic for unknown contour separating the given areas) are solved in combination, encounters significant difficulties, simplified models conforming to experimental data deserve attention. For example, in the case of thin-walled structural elements, failure of which is preceded by the development of significant zones of plastic deformations, the Leonov--Panasyuk--Dugdale 6k-model [i, 2] is quite effective. This was shown experimentally on large model samples of weld joints in low- and medium-strength constructional steels [3-5]. It is important to note that satisfactory agreement of the theoretical and experimental results is observed in the case when the configuration of the plastic zones in the test samples differed from that accepted in the ~k-model. In this work on the basis of an analog of the 6k-model (Fig. i) with the use of the original equations of the general moment theory of shells an investigation is made of the stressed and strained state of a closed cylindrical shell of an ideal elastoplastic material with a regular system of k longitudinal through cracks of the same length 2Z. It should be noted that such an approach to solution of problems for shells with cracks with the use of an analog of the ~k-model has been considered within the limits of the theory of sloping shells [6]. Let us assume that the stressed and strained state caused by external actions in the shell without cracks is axially symmetric and the edges of the crack are free of load. In the given case in investigation of the stressed state of a shell with cracks it is sufficient to consider (in view of cyclic symmetry) a cylindrical panel l ~ l ~ / k with a crack I ~ l ~ 0 , 6 = 0 (s0=I/R), where ~ is the relative (measured in fractions of the radius of the mean surface R) distance with respect to the generatrix and B is the relative distance with respect to the direction circumference, that is, the center angle. Subsequently, we will be Institute of Applied Problems of Mechanics and Mathematics,Academy of Sciences of the Ukrainian SSR, Lvov, Kiev. Translated from Problemy Prochnosti, No. i0, pp. 64-67, October, 1984. Original article submitted March 31, 1983.

1420

0039-2316/8411610- 1420508.50

9 1985 Plenum Publishing Corporation