Materials Science, Vol. 31, No. 4, 1995

ESTIMATION MATERIALS

OF THE CRACK-GROWTH

RESISTANCE OF STRUCTURAL

IN THE STAGE OF STABLE CRACK GROWTH

V. I. Smirnov

UDC 620.172: 620.174

We studied some methodical problems connected with the estimation of crack-growth resistance of structural materials in the stage of stable crack growth. We propose a new simplified method for the construction of JR-Curves, which enables one to attain a significant decrease in the laboriousness of the method of "partial unloading" and to improve the accuracy of the determination of the time of crack initiation. The algorithm of numerical calculations suggested in the present work can easily be realized in the form of a computer program. This method was successfully applied to the investigation of crackgrowth resistance of 15Kh2NMFA and 15Kh2MFA steels and their welded joints. It is shown that the method under consideration is applicable to the analysis of actual practical problems.

At present, the numerical analysis of crack resistance of structures is most often performed by using the methods of fracture mechanics according to the criteria of fracture initiation (Klc, 5c, Jlc) by actual or hypothetical defects. It is worth noting that, in this case, the possibility of significant subcritical crack growth in the elastoplastic region is neglected, which may sometimes result in noticeable underestimation of the crack-growth resistance of materials. Thus, in particular, in view of the possibility of stable crack growth, the choice of a pressure vessel subjected to high preliminary loading would significantly increase the crack-growth resistance of the material in the region of brittle fracture [ 1]. The most adequate estimation of the resistance of a material to the growth of brittle crack can be obtained by constructing the JR-Curve [2]. In most cases [3, 4], the construction of the JR-Curve is regarded only as a convenient methodical procedure for a more precise computation of the J-integral at the time of the crack initiation. To a certain extent, this is explained by numerous technical and methodical difficulties encountered in plotting the JR-curve with required accuracy. Thus, in particular, the computation of current values of the J-integral in the stage of stable crack growth according to the procedure suggested by the American Standard (ASTM E1152) [2] is a quite complicated problem connected with the necessity of repeated calculation of the J-integral for every small increment in the length of the crack. In this case, the exact evaluation of small increments in the crack length is the most difficult methodical problem encountered in plotting the JR-curve. The method suggested for the evaluation of the J-integral by GOST 25.506-85 (based on marking the front of a growing crack in the batch of specimens completely unloaded on attaining the prescribed displacement) proves to be absolutely unsuitable for these purposes. The "key curve" method [5] and its subsequent modifications [6, 7] are more universal. However, to plot the required curve, it is necessary to perform preliminary testing of specimens with blunt notch (without crack) under the same conditions as in the case of cracked specimens. The method of "partial" unloading is recommended for measuring crack length by the ASTM E1152 standard. However, multiple removals of the load lead to a significant increase in the laboriousness of the test and complicate data processing. In what follows, we suggest a rather simple method for plotting the JR-curve according to the experimental data obtained for a single specimen. This enables one to make the unloading method much less laborious and to improve the accuracy of determination of the time of crack initiation. The indicated computational algorithm can be realized as a special computer software. If the programs are written in BASIC, one can use standard subroutines that can be found in [8]. If the experiments are carried out in installations equipped with microprocessor systems of data accumulation and processing, then the entire test procedure can be made fully automatic. Note that, in "Prometheus" Central Scientific Research Institute of Structural Materials, St-Petersburg. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 31, No. 4, pp. 28-35, July - August, 1995. Original article submitted July 8, 1994. 1068-820X/95/3104-0433

$12.50

© 1997 Plenum Publishing Corporation

433

434

V.I. SMIRNOV

the course of automatic processing of experimental data, the loading diagram is "formed" in the RAM of a computer as separate arrays of loads and strains in the specimen. Hence, the algorithm under consideration must be complemented by criteria aimed at the determination of the lower and upper bounds of the linear parts of the strain diagram both in the stage of initial elastic loading and for the lines of partial loading-unloading [9]. The indicated test procedure can be realized without any additional restrictions imposed on equipment and specimens, except the requirements specified by GOST 25.506-85. One must only perform one or two procedures of partial unloading of the specimen and record strain diagrams, terminate the process of loading as soon as the load reaches its maximum value P~ and then unload the specimen. The process of loading can easily be interrupted if we use hydraulic servosystems and control the level of loading by moving the grips. If the tests are performed in ordinary hydraulic machines, one must install special displacement limiters to prevent the final fracture of the specimen [ 10].

P

e

-I

-

Po

/

!,

/ v0IV

_I

I_/

_i

-I..1/ v;

',

V

Fig. 1. Determinationof design-basis parameters from the load-displacementdiagram. It is recommended to mark the front of the crack by any available method (e.g., according to GOST 25.506-85) as soon as the specimen is unloaded (to record the increment in crack length acquired under loading). According to the ASTM E1152 standard, the front of a growing crack is marked to estimate tkc error of the values of crack length obtained by the method of partial unloading. If the error exceeds the maximum admissible value, the results of the test are rejected. In the method suggested in the present work, the actual length of the propagating crack is used to determine correction factors to the compliance of the specimen which, as a rule, differs from its theoretically predicted value as can be explained by the compliance of the testing machine used in measuring deflection, by the displacements of grips, by incorrect mounting of the gauge for measuring the crack opening displacement, and by the error of the estimation of the elasticity modulus of the investigated material. It is also desirable to record the time of crack initiation in the cou.'se of the tests by any available physical method (e.g., by the method of acoustic emission [11] or ultrasonic method [12]) parallel with relevant numerical calculations [13]. This enables one to decrease the number of load removals because the realization of the described method requires at least two load removals (one of them is final) in the stage of stable crack growth. To demonstrate the basic methodical and computational features of this method, we consider, as an example, compact specimens tested by eccentric tension. Computations according to the proposed algorithm for processing experimental data require the following input data: the thickness of the specimen B, the thickness of its notched part B~ the width of the specimen W, the initial and final lengths of the crack (a 0 and af), the values of the load and displacement at the beginning of the nonlinear section of the diagram (P0 and V0), the values of the load Pt

ESTIMATION OF THE CRACK-GROWTH RESISTANCE OF STRUCTURAL MATERIALS IN THE STAGE OF STABLE CRACK GROWTH

435

and displacements (total VI* and elastic Vl) in the course of the first unloading (Fig. I), the values of the load Pf and displacements (total Vf* and elastic V¢) at the end of the procedure of loading (Fig. 1), and the elasticity modulus E of the material at test temperature. In the case where displacements are measured along the loading line of the specimen, one can use (for numerical calculations) any known functional dependence for the compliance of compact specimens [ 14, 15]. In the present paper, we use the function recommended by the ASTM El 152 standard

C =

EBV

P

(1)

= F(x),

where a x = -W

and

I + X'] 2

E(x)

= \ l - - ~ x / (2.163 + 12.219x - 20.065x 2 - 0.995x 3 + 20.609x 4 - 9.9314x5).

To determine crack length after the first unloading according to the experimentally measured values of compliance, one must first estimate the error of the method. For this purpose, we compare numerical values of the compliance of the specimen C" [computed according to relation (!)] with its experimental values C e for the initial and final crack lengths and determine the correction factors Z0 = C~0/ C~ and Zf = C~ / C~. According to the ASTM El 152 standard, the maximum admissible difference between experimental and theoretical values of compliance is 10%. In our opinion, for the method under consideration, this difference can be as high 20% because compliance is corrected for the initial and final lengths of the crack. If we test specimens with lateral notches (as specified by the ASTM E813 standard), then it is necessary to perform the analysis of compliance by using the so-called effective thickness, i.e., Be = B

(B-

BN'2"~

(2)

B

Lateral notches increase the rigidity of the stressed state and, as a rule, guarantee the rectilinearity of the front of propagating cracks. With an accuracy sufficient for engineering purposes, we can assume that the correction factor is a linear function of crack length, i.e.,

(3) The constants A t and B t are determined for Z = Z 0 and Z = Zf. In this case, the experimental values of compliance are described by the formula Ce =

EBV

P

-

[A 1 + B l x ] F ( x

).

(4)

To determine the length of the crack after the first unloading (xl), it is necessary to solve the nonlinear equation (4) with respect to x. It can be solved, e.g., by the method of quadratic interpolation [8] guaranteeing fairly rapid convergence for the required accuracy of numerical solutions. The formal application of Eq. (4) in the elastoplastic region where compliance is computed according to the values of total displacement, enables one to estimate the effective length of the crack, i.e., its length corrected to take the influence of the plasticity region into account. According to the ideas of Irwin, this region can be represented as a circle. In this case, crack length is corrected by the value of the radius of the plasticity region R. Thus, the effective length of the crack is aeff = a + R o r Xeff = x + R for relative sizes.

436

V.I. SMIRNOV

According to the experimental data and the data of numerical analysis [16], the area of the zone of plastic deformation at the crack tip (in the stage of its stable propagation) is, in fact, a linear function of the increment in crack length Aa, namely,

F = A2 + Bz(a-ao). Then, for the relative radius of the zone of plastic deformation, we obtain

R W

-~

~ 4 W[A2 +B2(x r~

Xo)]

~!M+D(x-xo)

(5)

where M and D are constants determined in the course of the tests and x 0 and x are, respectively, the initial and final relative crack lengths. If the time of crack initiation is not recorded, one must partially unload the specimen one or two times (in addition to unloading performed at the end of the loading procedure). The length of the crack x I is determined according to the value of the elastic compliance of the unloaded specimen C I. The values of the elastoplastic compliance C~ and C~ are used to determine the effective lengths of the crack Xeff, and Xefff and the radii Rt and Rf of the plastic zones after the first unloading and at the end of the entire loading procedure, respectively. Thus, in view of relation (5), the constants M and D are given by the formulas

M = -R2-D(xl-xo),

(6)

(7) xf - x I

The correction for the size of the elasticity zone at the time of crack initiation is R0 = ~ and, for the effective crack length, we can write Xeff0 = X0 + R 0. The elastoplastic compliance of the specimen at the time of crack initiation C7 is determined according to relation (4). The magnitude of the load at the time of crack initiation Pi can be found as the intersection point of the line of elastoplastic compliance with the strain diagram. This problem can be solved numerically by the method given in [8], provided that the nonlinear section of the diagram is approximated by a polynomial. In the case where installation is equipped with automatic system of data accumulation, it suffices to indicate the bounds of the segment of the strain diagram to be approximated and apply a special subroutine for the selection of approximating functions [8]. If it is impossible to use automatic systems, then the values of the load and displacement are recorded directly from the strain diagram and entered into the computer from the keyboard. If the time of crack initiation can be accurately recorded by physical methods in the course of the tests and, hence, the corresponding values of the load Pi and displacement V i are known, then it suffices to perform only one unloading on passing the maximum value of the load. Further, in accordance with the arguments presented above, we determine effective lengths Xeff0 and Xefff and the radii of the plasticity zones R0 and Rf. The constants M and D are also given by relations (6) and (7). In order to reduce the laboriousness of manual processing of the strain diagrams, one should split the section of the diagram lying between the values Vi and Vf into n (n = 5-10) equal segments and record the relevant values of the load. The values of Pn and Vn obtained as a result are used for the approximation of the diagram by an interpolation polynomial [8] which, in turn, is used to calculate intermediate loads Pj, j = 0 . . . . . m. According to the ASTM E1152 standard, m must be not less than 10 and, moreover, two values of J) must belong to the interval bounded by the origin and the secant J = 4~0. 2 Aa to the JR-curve. Further, in the cycle from

ESTIMATION OF THE CRACK-GROWTH RESISTANCE OF STRUCTURAL MATERIALS IN THE STAGE OF STABLE CRACK GROWTH

437

j = I to j = m, the values of the load Pj and displacement Vj are used to determine the elastoplastic compliance C; which is substituted in Eq. (4), where the effective length of the crack is represented in the form xeft)= xj + Rj. Equations (4) and (5) are simultaneously solved with respect to xj. In each cycle, we check the validity of the condition Rj > W - a. If this inequality is true, calculations are terminated because this situation corresponds to the extensive yield of the material in the net section of the specimen. The value of the J-integral for j = 0, i.e., at the time of crack initiation, is given by the following formula: J0 = K2(I - p2) + E

flAp BN(W -

(8)

ao)'

where the first and second terms correspond to the contributions of the elastic and plastic components, respectively, Ae is the plastic part of the area under the P - V curve bounded by the displacement determined at the time of crack initiation Vi, and

1] = 2 +

0.522 ( 1 - - ~ - ) .

Under the conditions of plane deformation, the value of the J-integral given by expression (8) is equal to Jk. The running value of the Jj-integral of permanently propagating cracks is computed according to the formulas recommended by the ASTM El 152 standard. For the elastic component, we have

j; _ Ky(l - l~ 2)

(9)

E The plastic component is described by the expression

,.

(aj - aj_l) ],

;;

=

where rlj = 1.O+0.76bj/W, yj = 2.0+0.522bj/W, The quantity Apj is given by the formula

J

(lo)

and bj = W-aj.

Ap(j) = Apcs-_l)+ [ Pj + Pj_l ] [<3p()) -8P(J-I)] 2

(11)

where 8p is the plastic part of displacement along the loading line, .

c j6

and Cj is the elastic compliance of the specimen with crack of length aj [Eq. (4)]. Relation (10) was obtained by using the fact that the J-integral is independent of the history of loading as follows from the deformation theory of plasticity. This is why the J-integral computed according to these formulas is traditionally denoted in the literature by Jd. Note that relation (10) is applicable only in the case where the increment in crack length does not exceed 10% of the height of the net section of the specimen. For larger increments in

438

V . I . SMIRNOV

the crack length, one can consider the modified J-integral suggested in [17] and denoted by Jm" If the value of is already known, then the value of the modified quantity

Jm(i+l) =

Jd

"In for compact specimens is given by the formulas

Jd(j+l) + AJj+I,

(12) AJ./+j =

For j = 0, we set

aJj + Iq-jT)(aj+l-ai)JpO ).

AJj = O.

/

:o_ 400 -

./p/

200 ~ ~ ' ~

0

-"

I

I

I

1.0

2.0

I

3.0 act, mm

Fig, 2. Dependences of the characteristics of crack-growth resistance J on the increment in crack length. The symbols O, II, +, and A mark the experimental values of J for 15Kh2MFA steel (at T = 270°C), 15Kh2NMFA steel (at T = 20°C) and 10KhMFF (at T = 200°C) and 09KhGNTA (at T-- 20°C) weld metals, respectively; the curves correspond to theoretical functional dependences of J constructed according to the simplified method. If the length of the crack is measured continuously and its nearest recorded values differ at most by 0.001 W, then the ASTM E1152 standard recommends to compute by the formula

Jpj

r

J~j

Jp(j-I)

|1 -

L

Tj-I

bj_ 1

aj -

-1

aj_ 1 I

2

J

+ []]JPJ "qJ-IPj-I]ISpJ -Sp(J-l) bj + b j _ 1 "2BN l}I where rlj = 2.0 + 0.522

bj/W

and y) = 1 + 0.76

bj/W.

+ 7j (a) l

bj

j-I)

(13)

ESTIMATION OF THE CRACK-GROWTH RESISTANCE OF STRUCTURAL MATERIALS IN THE STAGE OF STABLE CRACK GROWTH

439

The algorithm outlined above was realized as a computing program written in Basic for DVK-3 personal computers. In order to examine the method proposed in the present work, we tested bend-type specimens (with a thickness of 50 mm and lateral notches) made of 15Kh2NMFA steel (~o.2 = 554MPa) and 09KhGNTA weld metal (Co.2 = 560MPa) and compact specimens (with a thickness of 25 mm and lateral notches) made of 15Kh2MFA steel (c0. ~ = 874MPa) and 10KhMFT weld metal (a0. 2 = 566 MPa) for fracture toughness. The tests were carried out in Shenk-100S machines with recording "load-deflection-crack opening displacement" diagrams. For bend-type specimens, we additionally detected the time of crack initiation by the method of acoustic emission [11]. As soon as the loading procedure was terminated, we fixed the front of the propagating crack by applying cyclic loading up to the complete fracture of the specimens. The comparative analysis of the JRcurves constructed by using two different methods was carried out for specimens characterized by the greatest increments in crack length attained for a sufficiently large number of partial removals of the load. For the analysis of experimental results, the values of the parameter rl for bend-type specimens with short cracks (a/W*, 0.2) were taken from the literature [18]. It was discovered that, for specimens of both types, the simplified method enables one to plot JR-curves with an accuracy sufficient for engineering purposes (Fig. 2). The error of determination of the load of crack initiation did not exceed 10% as compared to the error of the method of acoustic emission [ 11 ]. In conclusion, we note that the proposed simplified method is applicable to the experimental investigation of the crack-growth resistance of structural materials in the stage of stable crack propagation.

REFERENCES 1, V.I. Smirnov and B. T. Timofeev, "On the influence of preliminary loading at elevated temperatures on the crack-growth resistance of structural materials," Fi~-Khim. Mekh. Mater., 28, No. 3, 7-14 (1992). 2. ASTM El 152-87. Standard Test Method for Determining J-R Curves. Section 3, Annual Books of ASTM Standards (1988), 3. V.I. Prokhorov, "Realization of a single-specimen method for the construction of J-curves in the process of bending of a 5 x 5 × 5.5 specimen for the evaluation of changes in the crack-growth resistance of zirconium alloys exposed to radiation," in: Problems of Nuclear Science and Engineering. Series : Materials Science and New Materials [in Russian], Issue 2, (I 992), pp. 17-21. 4.

V.A. Rakovskii, V. P. Naumenko, V. I. Prokhorov, and A. V. Golubenko, "A method for the estimation of crack-growth resistance of metallic materials on small specimens," Probl. Prochn., No. 6, 30-38 (1993).

5.

H.A. Ernst, P. S. Paris, and J. D. Landes, "Key curve method for evaluation of crack growth and J-integral during ductile tearing," in: Proceedings of the 13th Conference on Fracture Mechanics. ASTM STP743, American Society for Testing and Materials, Philadelphia (1981), pp. 476-502. 6. J. Rebey and R. E. Roche, "Determination of J-R curve from only one experimental test on one sample," Int. J. Pressure Vessel Piping, 13, 33-49 (1983). 7. X. Li and Y. Liu, "A new method for measuring J-R curve of mild steel," Eng. Fract. Mech., 30, No. 4, 445-450 (1988). 8. V.P. D'yakonov, A Handbook of Algorithms and Programs in Basic for Personal Computers [in Russian], Nauka, Moscow (1987). 9.

A.D. Zotov, M. A. Afinogenova, and V. G. Kokorin, "Automatic determination of the conditional yield limit in testing metals by tension," Zavod. Lab., No. 7, 34-35 (1991). 10. Yu. A. Kashtalyan, V. M. Torop, and 1. V. Orynyak, A Method for the Determination of the Characteristics of Crack-Growth Resistance of Materials in the Presence of the Arrested Crack [in Russian], Inventor's Certificate No. 1478080 USSR. MKI 4G01N3/00, Published on May, 7 1989, Bulletin No. 17. l 1. V.I. Smirnov, "Estimation of the sizes of defects by the method of acoustic emission from the viewpoint of fracture mechanics," Defektoskopiya, No. 2, 45-50 (1979). 12. A.P. Androsov, A. Sh. Deich, and L. A. Kopel'man, "Application of ultrasonics to the detection of the time of crack initiation and determination of the critical value of the J-integral," Zavod. Lab., No. 3, 266--268 (1979). 13. Y. Liu, X. Li, and X. Tao, °'A method for determining J of mild steel by means of one single specimen," Int. J. Pressure Vessel Piping, 31, No. 2, 240-245 (1988). 14. A. Saxena and S. J. Hudak, "Review and extension of compliance information for common crack growth specimens," Int. J. Fracture, 14, No. 5,453-467 (t978).

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15. J.A. Kapp, G. S. Leger, and B. Gross, "Wide-range displacement expression for standard fracture mechanics specimens," in: Proceedings of the 16th Symposium on Fracture Mechanics. ASTM STP 868 (1985), pp. 27--44. 16. G. Zhong-Hin, W. De-Ming, and Sheng Nan, "Deformation analysis of the local field in the vicinity of stable growing crack-tip," Eng. FractMech., 30, No. 4, 415-434 (1988). 17. H.A. Ernst, "Material resistance and instability beyond J-controlled crack growth," in: Elastic-Plastic Fracture: Second Symposium, Vol. I: Inelastic Crack Analysis. ASTM STP 803 (1983), pp. 1-191, pp. 1-213. 18. D.Z. Zhang and J. M. Lin, "A general formula for thee-point bend specimen J-integral calculation," Eng. Fract. Mech., 36, No. 5, 789-793 (1990).