A modified S theory

Engineering

Fracture

Printedin U.S.A.

Mechanics

0013-7!%4/85 $3.00+ .C!O 0 1985Pergamon Press Ltd.

Vol. 22, No. 4, pp. 579-584, 1985

A MODIFIED S THEORY Beijing Institute of Aeronautics

MAO-HUA WANG and Astronautics, Beijing, People’s Republic of China

Abstract-The S-theory of mixed fracture is analysed, and the analysis reveals that the main results of this theory are acceptable. But some problems still exist, and in this paper two modified criteria are presented.

1. INTRODUCTION BEFORE the fifties people always dealt with fracture problems in terms of crack growth in the original crack direction. In 1951 Yoffee [l] and in 1962 Barenblatt [2] first observed the phenomenon of crack branching when ZI> 0.7CR (where v is the velocity of crack propagation and CR the Rayleigh wave velocity), and they proposed that at r = const a crack propagates in the direction of maximum ue. In 1962 Carlsson [3] observed that, for steel sheets, when v = O.lC,, there occurred the branching phenomenon of crack propagation. In 1959 Schardin [4] experimented with glass and found that the branching propagation was associated with applied stress. In 1963 Erdogan and Sih [5] proposed the first criterion for mixed fracture, that is, the criterion of maximum circle stress. Thereafter, a number of theories were developed regarding stress. In 1974 Hussain [6] obtained the first expression for the energy release rate, pioneering the way for a number of subsequent theories. In addition, some criteria have been developed in the area of displacements. The development of the theories of mixed fracture may be found in Ref. [7]. Recently, M.-H. Wang 18, 93, Chang [lo] and Fan [ll] independently proposed the strain theory of mixed fracture, and M.-H. Wang [121 applied this theory to solving the problems of the crack growth of mixed fatigue. As of now, there still exist some disagreements among the different theoretical approaches to mixed fracture. The energy release rate theories of Hussain [6], Nuismer [13] and T. C. Wang [14] have been reviewed by Ichikawa and Tanaka 115, 161, T. C. Wang [171 and M.-H. Wang [18]. In this paper the main focus is on the theory of strain energy density factors of mixed fracture. Two modified criteria are presented.

2. REVIEW OF THE S THEORY The criterion for strain energy density factors proposed by Sih [19, 201 is as follows (see Fig. 1): (i) The direction of crack propagation coincides with the direction (Yof the minimum strain energy density around the crack tip (r = const), i.e. (Yshould satisfy

as

ds

eya

a2s

a82 (ii) A crack extension

0,

(la)

> 0.

(lb)

=

8=P

starts in the OLdirection when S reaches the critical value S,,, where (2)

and a11

=

J[(l 1611,

- J-

f cos 9) (K - cos ($1, a12 = &sin0[2cose

[(K + 1) (1 -

‘*’ - 16p.

- (K - l)], (3)

cos e> + (1 f cos e) (3 cos 8 - l)], a33 = 6. 579

580

M.-H. WANG

Y

$

Fig. 1

In eqn (3) p = E/2(1 + V) is the shear modulus, K = 3 - 4~ the plane strain, K = (3 - v)l (1 + v) the plane stress and v Poisson’s ratio. The energy density is

and

After this theory was proposed, it was found that if there was only one relative minimum value of S along the arc of a circle for all loading configurations, the validity of the S theory was clear; however, when two or more relative minimum values of S exist, it should be used in conjunction with a corollary. In 1976 Swedlow [21] discussed this problem. He thought that because there exist two or more relative minimum values of S, two discrepancies would occur. One of them is that, although Smincan be found with increasing load the largest relative minimum S will reach the critical value S,, first, which will lead to a discrepancy with prediction (ii) of the S theory. The second is that if a crack extension starts when the largest relative minimum S reaches the critical value S,, this will lead to a discrepancy with prediction (i) of the S theory. To overcome these dilemmas, Swedlow suggested that the S theory be used in conjunction with the corollary that the critical minimum S must be associated with a tensile hoop stress, oe > 0. This corollary is confirmed for most loading configurations. However, in 1982 Chang [22] pointed out that Swedlow’s corollary fails in certain cases as, for example, in Fig. 2. Under a uniaxial compressive load with b/a = 0, rola = 0.01, u = 0.25 and p = 89”, there exist three relative minimum values of S, and two of them are associated with a tensile hoop stress. Under uniaxial tension with b/a = 0.2 da = 0.15, v = 0.25 and p = lo”, there exists only one minimum value of S. But it is associated with erg < 0. In addition, there do not exist any minimum values of S for the biaxial tension cases in which b/a = 0.1, A = 8, ro/a = 0.01, p = 72”-84” and bfa = 0, X. = 0.25, &a = 0.01, f3 = O”-2”. In 1982Fan [ 101pointed out that, according to the S theory, the direction of crack propagation coincides with the direction 01and crack extension starts when Stii, reaches the critical value S,,. But, in fact, all values of S at other points will have reached the critical value S,, prior to reaching Stii,. Therefore, the S theory favours the equal points of a circle, so the discrepancies of predictions (i) and (ii) will occur. In 1982 Theocaris and Andrianopoulos [231 pointed out that S,,, should reach the critical value S,, first and Sdi, last. Therefore, there is a physical peculiarity in the S theory. The aforementioned criticisms are correct. There exist two problems with the S theory. (a) The physical meaning is not clear. There exists a discrepancy between predictions (i) and (ii). (b) For some cases, the prediction approach may fail.

A modified S theory

581

Fig. 2.

However, it is this author’s opinion that the main results of the S theory are valid because its rationality has been clearly shown in a lot of experimental comparisons as is well known. 3. THE MODIFIED THEORY Because the main results of the S theory are valid but there are some discrepancies in its application, this theory should be modified. Swedlow [21] suggested as a corollary that the relative minimum value of S be associated with ue > 0, which cannot overcome difficulty (a), that is, the discrepancy between predictions (i) and (ii). Meanwhile, it is impossible to remove discrepancy (b). Thus Swedlow’s modification should be discarded. The modified theory proposed by Theocaris and Andrianopoulos [23] is as follows: (1) The direction of crack propagation coincides with the direction CYof the minimum strain energy density around the crack tip (r = const); i.e. a should satisfy eqns (la) and (lb). (2) When the average strain energy density factor ?? reaches the critical value s,,, the crack extension begins:

s=l 2n

I aaS(e) de. e

It is seen that the initiation of the crack extension is determined by the energy concentration around the crack tip instead of by Smin at a special point. So the discrepancy between predictions (i) and (ii) of the S theory can be removed. But with regard to the determination of the direction of the crack extension, discrepancy (b) still cannot be removed. The preceding discussion was based on a circle with its center oriented at the crack tip. The radius Yis shown in Fig. 1. The geometrical sense of the preceding discussion is clear, but its physical sense is not. To remove the two discrepancies of the S theory and retain its main results, we shall make the determination along an isoenergy density line around the crack tip, as shown in Fig. 3. The following criterion is now proposed: Along an isoenergy density line (i.e. W = const): (1) The direction of the crack extension coincides with the direction (Yof the minimum value rmi, of r, which is the distance from the points along an isoenergy density line to the crack tip. (Yshould satisfy the following conditions:

a?

as

fJca

d’?l

ae2

> e=Cl

0,

(6d

0.

(6b)

=

M.-H. WANG

582

Fig. 3

(2) The crack extension starts when the circumference A of an isoenergy density line reaches the critical value A,, where A=

rdQ.

(7)

Along an isoenergy density line all of the points have the same energy density, i.e. an equal physical state. Therefore, the physical sense is clear, because at the point which is in direction (Yalong an isoenergy density line, r gets the minimum value rmin; then in this direction the average energy density change rate is maximum. Therefore, the direction should coincide with the direction cz. Since the circumference of an isoenergy density line with a constant value gradually increases, as shown in Fig. 3, when the real line advances to the position of the dotted line, the energy density at a point inside the original real line is larger than the original constant value. Thus, more and more energy will gather, with the dotted line advancing on the outside. When the dotted line advances to a certain position, i.e. the circumference of the dotted line reaches a certain value and hence the accumulated energy also reaches a certain value, the crack will begin to propagate. It is seen that the method for determining the starting point of the crack extension with the circumference of an isoenergy density line is reasonable. When W = C, from eqn (4), it follows that

c = ;. From eqns (6)~(8) we may obtain

(9b) and A=

Jo

rd0

=

Jd”;

de = ;

I,‘” S &

= g

3.

Therefore, this criterion can turn out to be of the same form as that proposed by Theocaris and Andrianopoulos. But it must be borne in mind that, in the present theory, the determination

A modified S theory

583

is made along an isoenergy density line instead of on a circle with its center oriented at the crack tip. Because, in practice, rti, always exists, discrepancy (b) of the S theory can be removed. With this criterion the two discrepancies of the S theory can be removed and the determination of crack extension with Sdi, can be retained. It is similar to the criterion of Theocaris and Andrianopoulos; i.e. the initiation of the crack extension is determined by the average quantity instead of by the directional state of the fields. To remove the two discrepancies of the S theory and to describe the theory exactly, a second modified criterion is proposed: Along an isoenergy density line ( W = const): (1) The direction of the crack extension coincides with the direction CYof the minimum value rmin of r, which is the distance from a point of an isoenergy density line to the crack tip; i.e. cx should satisfy eqns (6a) and (6b). (2) When the quantity W - re=, in the direction (Y reaches the critical value, the crack extension starts. Prediction (1) is the same as the author’s first modified criterion, but prediction (2) must be interpreted as follows: In direction OL,from eqn (9), we know that (W * rt3=a)e=u = (W * Y)eza = (S)e=,.

(11)

Therefore, prediction (2) can turn out to be that when S in direction 01reaches the critical value, the crack extension starts. After direction (Yhas been determined by prediction (l), then re =a * We=, will increase with an increase in loading, until it reaches the critical value, at which point the crack extension starts. Because all points, including point re = a, have the same value W, in other directions the W - re =a cannot reach the critical value prior to W * recu. in the direction cr. From the previous discussion, it can be seen that the second modified criterion proposed in this paper can not only remove the two discrepancies of the S theory, but can also retain its form and its main results. It must be borne in mind that the S theory should deal with an isoenergy density line, instead of a circle with its center oriented at the crack tip. 4. CONCLUSIONS (1) The S theory’s main results should be admitted. (2) In the S theory the following two problems exist: (a) The physical meaning is not clear. (b) For some cases, the prediction method may fail. (3) In this paper, two modified criteria are proposed. The second not only removes the two discrepancies within the S theory, but also retains its form and main results. It is important to point out that in order to retain the form and results of the S theory, it should deal with an isoenergy density line, instead of a circle with its center oriented at the crack tip. REFERENCES [l] E. H. Yoffee, Phil. Mug. 42, 739-750 (1951). [2] G. I. Barenblatt, Prikl. Mat. Mekh. 26, 328-334 (1962). [3] J. Carlsson, Trans. Royal Znt. Techen. Stockholm, 189 (1962). [4] H. Schardin, In Fracture, pp. 297-330 (1959). [5] F. Erdogan and G. C. Sih Troras, ASME J. Basic Engng 4, 519-525 (1963). [6] M. A. Hussain, S. L. Pu and J. Underwood, Fracture analysis. ASTM. STP 560, 2-28 (1974). [7] M.-H. Wang, PH.D. dissertation. Beijing Institute of Aeronautics and Astronautics, China (1982). [8] M.-H. Wang, Acta Mech. Sin., special issue, 281-285 (1981). [9] M.-H. Wang, Acta Mech. Solidu Sin. 4, 571-581 (1982). [lo] K. J. Chang, Engng Fracture Me&. 14, 107-124 (1981). Hll W.-N. Fan. Aoul. Math. Mech. 3. 211-224 (1982). [12] M.-H. Wang,-The strain fatigue of the mixed models. Submitted to Engng Fracture Mech. May 1983. [13] R. J. Nuismer, Znt. J. Fracture 11, 245-255 (1975). [14] T. C. Wang, Fracture 1977, University of Waterloo, Vol. 4, pp. 135-154 (1977). [IS] M. Ichikawa and S. Tanaka, Znt. J. Fracture 15, R49-R51 (1979).

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S. Tanaka and M. Ichikawa, In?. J. Fracture 15, Rl83-RI86 (1979). T. C. Wang, Int. J. Fracture 15, 117-119 (1979). M.-H. Wang, A Theory for the Mixed Energy Release Rate G. C. Sih, Znt. J. Fracfure 10, 305-321 (1974). G. C. Sih, Mechanics of Fracture II, pp. xv-liii. Noordhoff, The Netherlands J. L. Swedlow, Cracks and fracture. ASTM. STP 601, 506-521 (1976). K. J. Chang, J. Appl. Mech. 49, 377-382 (1982). [23] P. S. Theocaris and N. P. Andrianopoulos, J. Appl. Mech. 49, 81-86 (1982). [16] [17] [18] [19] [20] [21] [22]

(1975).