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Elliptic rule criterion for mixed mode crack propagation
Ertgineering Fracture Mechanics Vol. 37, No. 2, pp. 283-292, 1990 Printed in Great Britain.
ELLIPTIC
0013-7944/90 s3.00 + 0.00 Pergamon Press plc.
RULE CRITERION FOR MIXED CRACK PROPAGATION
MODE
ZHAO YISHU Department of M~hanics,
Huazhong University of Science and Technology, Wuhan, China
Abstract-As can be proved from theoretical analysis, some mixed mode fracture criteria may be converted into an elliptical or ellipsoidal formula with the aid of mathematical translation. Others are of themselves an elliptical or ellipsoidal formula. Hence, on the basis of Griffith-Irwin theory, the paper suggests to take (K,/K,,)‘+ A(KJK,,)* + B(KJK,,)* = 1 as the general criterion superseding all the mixed mode fracture criteria. A and B will be determined by the superseded criterion.
INTRODUCTION As EVERYBODY knows, energy release rate criterion (G~~te~on)~l-31, the energy density criterion (~~~te~on)~~] and the T&terion[7] are the very “hot” discussions and the competing three most popular criteria. Moreover, maximum tangential stress criterion& 91, maximum tangential principal stress criterion[lO] and maximum strain criterion[ll, 121 are also very attractive. Unfortunately, none of them have generally been accepted yet. They can not be applied widely to engineering practice because the structural evaluation from these criteria is a lengthy and difficult procedure to carry out. For a long time people attempted to take the formula (K l&C)” + (W&Y
= 1
(1)
as an empirical criterion of mixed mode fracture[l3-161, where the constants u and Y are experimentally determined for each material application. Someone suggests u = u = 2[14] or u = 2 and v = 0.475 - 2[15]. It is a pity that eq. (1) is not accepted because it has not been analysed strictly with mathematics and its physical significant is not clear. The use of generalized fracture toughness theory[ 17,181 offers one of the convenient mathematical means which expresses the mixed mode fracture criteria as a standard elliptical or ellipsoidal formula. Thus, expression of various mixed mode fracture criteria can be simplified and standardized and therefore a convenient way of generalizing mixed mode fracture criteria may be provided. The strain energy criterion for mixed mode crack propagation was published recently[l9]. Its Fracture Envelope formula is a ellipsoidal formula with high power b,lK:+b,2K:K:+b,3K:K:+b23K:K:+b22K~eb33K:=K~~
(2)
or b,1(rrr;/rsr,,)~+b,2(K,/Kt,)~(K*/K,,)*~
k;3(K,/K,,E)‘(K~/K,c)’ + b~~(K~/K*~)‘(~/K,~)* + b,,(&/JQ4
in which b,, = 1, b,* = (185 - 24v{ - 24~4 + 39/2)/(7{ - 12vt - 4v + I l/4), b,3 = (85 - 32v + 24)/(7t - 12vt - 4v + 1 l/4), b,, = (8< - 32v + 56)/(79 - 12~5 - 4v + 1l/4),
bz2=(11(
- 12vr -20~ i-99/4)/(7t
- 12v< -4v
b,, = 32/~7~ - 12v< - 4v -t 1l/4),
5 = 2(1 - 2v)ZjZ. EFM 37/2-C
283
+ 11/4),
+ ~~~(K~/K~~)4 = 1,
ZHAO YISHU
284
The strain energy criterion is different from the strain energy density criterion[4-61. The strain energy criterion takes the total strain energy in the region surrounded by the initial elastic-plastic boundary near the crack tip as the parameter dominating the crack propagation. The initial elastic-plastic boundary will be shrunk to an infinitesimal circular region for strongly brittle materials. Therefore, eq. (2) is rewritten to be a standard ellipsoidal formula C,,(K,I&)2
+
G2W21JGd2
+
G,wmz
=
1,
(3)
in which C,, = 1,
C,, = (9 - 8v)/(5 - Sv),
C,, = S/(5 - 8~).
Equation (2) will be transformed into a high power elliptical formula or a standard elliptical formula respectively under Mode I-II, I-III and II-III. Since mathematical expression of some fracture criteria may be transformed into a standard elliptical or ellipsoidal formula by use of the generalized fracture toughness theory[l7, 18], and others are of themselves an elliptical or ellipsoidal formula, therefore, the purpose of this paper lies in the fact that using a standard ellipsoidal formula represents all of the mixed mode fracture criteria. THEORY To date, fracture criteria under multiaxial loading is a twofold problem, (a) the crack propagation angle, specified through stationary conditions imposed on a prescribed functional, describing the stress intensity factors’ interaction for the specific application, and (b) the fracture load determined by the critical value that the same functional reaches during loading. For a two-dimensional condition, we have c~(e)Ki~j - Q, = 0,
(4)
C@)KiKj = 0,
(5)
in which i, j = 1,2. When using some fracture criteria, however, the procedure for finding the Cartesion equation of the Fracture Envelope by eliminating the argument 0 from eqs (4) and (5) is considerablely laborious. But it is known that eq. (4) describes a family of polarly symmetric tonics through varying the parameter 8, or rather, a family of the ellipses with their center on the origin of the K, K2 orthogonal reference system. Since eq. (5) is homogeneous, it represents a pair of lines passing through the origin 0 corresponding to a prescribed value 0,. This pair of lines intersect with the ellipse C~(e~)~i~~ - Qcr = 0 at least one point. The intersection is the critical point. The Fracture Envelope is the locus of the critical points of the ellipses of the family dete~ined by eq. (4). Dileonardo[l7, 181 has proved that any ellipse of the family is tangent in its critical point to the Envelope, its kinematics (in the K,g plane when 8 varies within the prescribed interval), represented by the ellipses of the family of eq. (4), make a rotation around its center 0 and a rolling motion inside the Envelope. Therefore, the Fracture Envelope may be constructed from the geometric properties of the rolling ellipses in the neighborhood of their critical point, namely r+)2+(+)1=,,
(6)
from which we defined a = K,, + m, b = aK2JK:,
(7)
+ 2&L,,) - I’*,
m = (K?, K$ - K:, K:, - K$, K:,)/2(&,
K:, f K,, K:, - K,&)
(8) (9)
where K,, and K,, can be calculated from following equations C,(8 ,)K,K, - Qcr = 0 C,i(d*)K,Kj=O
(10)
Elliptic rule criterion
285
in which @* is one of solutions satisfying the equation c,,(e)
= 0.
(11)
The geometrical graph of eq. (6) is shown in Fig. 1. It represents an ellipse of ~mi-diameters a and b, with center O(-HZ, 0) in iu, K, plane. If anti-plane shear is considered, the Fracture Envelope wilI be an ellipsoidal formula as follows
From which c = aK&.(Ki, + 2mK1,)- I’2.
(13)
The material constants Q, b and m in above equations are fuuctions of K,, and v and computed from eqs (7), (8) and (9) respectively. The energy release rate under critical state can be found in [ZO]
+ 4K, K2 sin 2y] + The coefficients Cii corresponding
to eq. (10) can be obtained from eq. (14)
and K% = 0.63K,, , K,, = (1 - v)‘~K~,,
in which y is the angle between the branch~ and the second of eqs (15), we obtain
crack and the main crack. According to eqs (lo), (11)
K,* = 0.61K,,, K,, = +0.63K,, .
06)
Substituting K, = 0.63K,, and eq. (16) into eq. (9), we obtain m = -0.30&.
(17)
‘6
Fig. 1. Fracture Envelope of two dimensions.
ZHAO YfSHU
286
According to eqs. (7), (8) and (17), m, a, b and c vs fracture toughness K,, are plotted in Fig. 2a-d. For each material, the Fracture Envelope formula (12) only specified by the fracture toughness K,, and Poisson’s ratio v. A convenient way of application for energy release rate criterion is provided by this approach. When K,, and v of the material are given, then m, a, b and c can be found out from Fig. 2a-d. Substituting m, a, b and c into eq. (12) the Fracture Envelope will be determined uniquely. Corresponding to the strain energy density criterion we have x;, = [3( 1 - 2v)/(2 - 2v - Y2)]*‘zK,c, K& = (1 - 2V)“%~~* Dileonardo
had given]1 71 J&l = [(l K,,
2v)3’2(1
= 2[v (1 -
-
v)]K,,,
2v)/( 1 - v)]“X,, .
Substituting above parameters into eqs (7)-(Q) and (13), m, a, b and c vs the fracture toughness K,, are plotted in Fig. 3a-d. Once the fracture toughness K,, and Poisson’s ratio v are assigned, then the Fracture Envelope will be determined solely. Geometrical graph of eq. (12) is shown in Fig. (4). It is an ellipsoid of semi-axes a, b and c, with its center at (-m, 0,O).
Fig. 2. Energy release rate criterion+ (a) tn vs fracture toughness R,,. (b) a vs fracture toughness K,=. (c) b vs fracture toughness K,,. (d) c vs fracture toughness I& and Poisson’s ratio v.
287
Elliptic rule criterion
70 80
60
40 30 20 10
020
(4
30
80
70
80
1201
W
I
02--I 1oy 2030
1
I
I
I
I
I
40
60
80
70
80
Kk eo
Kk Fig. 3. Strainenergy density criterion. (a) m vs fracture toughness K,, and Poisson’s ratio v+.(b) a vs fracture toughness K,, and Poisson’s ratio Y. (c) b vs fracture toughness K,, and Poisson’s ratlo v. (d) c vs fracture toughness K,, and Poisson’s ratio v.
From the preceding analysis it can be found that the Fracture Envelope of both the G-criterion and the S-criterion all are a part of the ellipse perimeter or the ellipsoid surface. This important conclusion is worth notice. ELLIPTIC
RULE CRITERION
Assume that the crack extension is coplanar, then the relationship between the energy release rate G of Mode I crack and the stress intensity factor K, under the case of plane strain has been given by Irwin[21] Gi = (1 - v2)K;/E.
(18)
288
ZHAO YISHU
Fig. 4. Fracture Envelope of three dimensions.
The strictly mathematical analysis of eq. (18) has been given by Bueckner[22]. Similarly, for Mode II and III crack, we have Gz = (1 - v*)K:/E, G3 = (1 + v)K:/E.
(19)
Therefore, for a combination of Mode I, II and III crack, the relationship between the energy release rate and the stress intensity factors is given as follows G = (1 - v*)K:/E + (1 - v*)K:/E + (1 + v)K:/E. When reaching the critical state, that is G = GCo,above equation is explicitly rewritten as (1 - v*)K,/E + (1 - v*)K:,‘E + (1 + v)K:/E = G,,.
(20)
For pure Mode I crack we obtain from eq. (20) G,, = (1 - v*)K:,/E.
(21)
Generally, it is assumed that the critical energy release rate under combined loading is the same as that of Mode I crack for that material. Therefore, we have from eqs (21) and (20)
From eq. (22), we find lu,, = Ki, and Kk = 6 obtain (K, /K,, )* + (Kz /kl;c
K,,. Substituting these results into eq. (22), we
I*
+
UG
l&c
I2 =
1,
or (K,lK,,)*+A(1Y2/k;E)‘+BtK~iK,Ktc)*=
1,
(23)
where A = K,,/K,,, B = KIE/K3,. Equation (23) is a typical formula ellipsoid. For Mode I-II, I-III and II-III, it will transform to be a typical formula of ellipse and is called the elliptic rule fracture criterion. Equation (23) is derived on the basis of Griffith-Irwin theory and under the hypothesis that the mixed mode crack extension is coplaner. In fact, however, the mixed crack extension will depart from its original straight trajectory or branch, then the energy release rate of the branched crack extension must be considered. Therefore, strictly speaking, the elliptic rule fracture criterion is an approximate criterion. Equation (14) is an expression of the energy release rate considering branched crack extension. It is equivalent to eq. (23) if y = 0.
Elliptic rule criterion
289
We suggest to take eq. (23) as a general criterion superseding all the mixed mode crack fracture criteria. A and B in eq. (23) will be determined from that superseded criterion. The G-criterion may be simplified by the use of eqs (23) and (15) as follows (Kl /K,C)2+ 2.52(K2/K~,)2 + l.42fKJK~,)2 = 1.
(24)
As to the precise expression of the G-criterion, it may be obtained (v = 0.30) from Fig. 2 and eq. (12) 2.5O(K, /&)* - 1.50(& /Klc) + 2.52(K2/&)2 + 1.42(K,/K,J2 = 1.
(25)
The Fracture Envelopes predicted by eqs (24) and (25) under mixed Modes I-II, I-III, II-III and I-II-III fracture are plotted in Fig. 5a-d respectively. Similarly, the S-criterion may be simplified by the use of eq. (23) as follows (K, /K,J2 + 1.09(KJK,J2 + 2.52(K,/K,,)2 = 1. According to eq. (12) the precise expression of the S-criterion
(26)
may be obtained (v = 0.30)
0.48(~~ /‘K,,)* + 0.52(K, /K,,) + 1.09(lu, /&C)2 + 2.52(K, /lytC)’ = 1.
(27)
The Fracture Envelopes predicted by eqs. (26) and (27) under mixed Mode I-II, I-III, II-III and I-II-III fracture are plotted in Fig. 6ad respectively.
I0
*.8y----p.
4r
, G /
\’
-Y--i 1’
I
5.f
z
\
0
0.8C
0.4 -
\
0
0.4
.
‘\
\
Elliptic /
0.2 -
\ \
0.2
*i
0.2
0.4
0.B
KI f K,,
0.8 .I
I
0
I
I
0.2
0.4
,\ I
0.0
\
I
0.8
b/he Fig. 5. Predicted by energy release rate criterion and elliptic rule criterion in comparison with test results. (a) Mode I-II fracture. (b) Mode I-III fracture. (c) Mode II-III fracture. (d) Mode I-11-111 fracture.
( P \ \ I \
0
290
ZHAO YISHU
The experimental points in Figs 5 and 6 are average values of the test results of 188 60Si2Mn steel specimens. CONCLUSION The elliptic rule criterion is based on Griffith-Irwin theory. As was mentioned above, Dileonardo[l7, 181 has proved analytically that some mixed mode fracture criteria may be expressed by a typical elliptical or ellipsoidal formula. As for the others, such as, the expression of the strain energy criterion[l9] are of themselves an ellipsoidal formula. Therefore, taking eq. (23) as an approximately general criterion for mixed mode fracture has sufficient basis in both of physical meaning and the mathematics. We say that the elliptic rule criterion is approximate because coplanar extension of the crack is assumed, more specifically, because extension of the mixed mode crack involving Mode II is assumed to be still in the plane of the original crack. In fact, extension of a pure Mode II crack or the mixed mode crack involving Mode II will branch at the crack tip. Therefore, the first formula in eq. (19), G2 = (1 - v*)K:/E, has no practical meaning. Thus eq. (23) is not precise. Equation (23) is correct only when the Mode II crack is not involved under mixed mode fracture. In fact, not one of the fracture criterion is perfect. They are derived from a certain assumption because of the mathematical complexities associated with the elastic stress field analysis of the branched crack, and there are disputes to this day. For example, the S-criterion[4-61 postulates existence of a minimum Smin of the total strain energy density along an artificially defined (b)
(4
_._
0
I
\ 0.2
1
1
0.4
0.6
01
I
0.2
0.4
0.8
0.8
KI IKI,
0.8
KI /KI,
(4
\ 0 0’
I 0.2
I
I
0.4
0.8
0
I 0.8
I 0.2
I 0.4
I 0.8
\ I 0.8
K,IKI,
K2lK1c Fig. 6. Predicted by strain energy density criterion and elliptic rule criterion in comparison with test results. (a) Mode I--II fracture. (b) Mode I-III fracture. (c) Mode II-III fracture. (d) Mode I-II-III fracture.
Elliptic rule criterion
291
locus of equal distance from the crack tip (circular region), the crack extends in a radial direction corresponding to the minimum Sminand the extension occurs when this minimum reaches a critical value. But there is, as yet, no physical basis to decide on the magnitude of this radius of circular region and S-criterion is suflicient when there is only one minimum Smi,, in the event of more than one minimum S,i, at a particular radius confusion arises. IJead et ~1. pointed out that the physical meaning of the S-criterion is not clear[24,27,28]. Eshelb~[2~31] and Rice1321 elaborated the fundamental fact that the total potential energy rather than just the strain energy density should be used for the derivation of the crack extension force. As for the G-criterion, because the expression for the energy release rate is derived by every author, each is different from the other, thus the predicted rate is also different[ 1,21,24]. Recently Theocaris proposes the T-criterion[7]. Though it appears to be a flexible fracture criterion and agrees with the test results quite well, a rigorous elastic-plastic analysis was not made for initial yield locus at the crack tip and the initial yield locus was defined by the elastic components of stress. Therefore, elastic part of the strain energy density only was considered. These simpli~cations are more appropriate for a brittle type of fracture. From the above it can be shown that all the criteria for mixed mode fracture introduced up to now are hypotheses, not final conclusions as are other failure theorems for mechanics of materials. As one of the hypotheses for the mixed mode crack extension, the elliptic rule criterion made only one assumption of coplanar crack extension. Figs 5 and 6 show that the differences between the Fracture Envelope predicted by eq. (23) and S- and G-criterion are not significant, the maximum error is 18% and 10% respectively for Mode I-II, I-III and I-II-III. However, for Mode II-III, the elliptic rule criterion is equivalent to the superseded criterion” We cannot judge which one in both the elliptic rule criterion and the superseded criteria agrees well with the measured data. The predicted will be more satisfactory if the values of &, and K, (or A and B) in eq. (23) are the measured values instead of those defined by superseded criterion when using the elliptic rule criterion[33]. REFERENCES [I] M. A. Hussain, S. L. Pu and J. Unde~ood,
Strain energy release rate for a crack under combined Mode I and Mode II. ASTM STP 560, 2-28 (1973). [2] K. Palaniswamy and W. G. Knauss, On the problem of crack extension in brittle solids under general loading, in ~eeho~ic~ Today (Edited by S. Nemat-Nasser), Vol. 8, pp. 87-148. Pergamon Press, New York (1978). (31 K. Palaniswamy and W. G, Knauss, Propagation of a crack under general, in-plane tension. Inr. J. Frac. Mech. 8, 114-I 17 (1978). {4] G. C. Sih, Strain-energy-density factor applied to mixed mode crack problems. Int. J. Fruc, IO, 305-321 (1974). [5] G. C. Sih and B. Macdonald, Fracture mechanics applied to engineering problems-strain energy density fracture criterion. Engng Fracture Mech. 6, 361-368 (1974). [6] G. C. Sih and B. C. K. Cha, A fracture criterion for three dimensional crack problem. Engng Fracture Me&. 6,699-723 (1974). [7] P. S. Theocaris, A higher-order 19, 978-991 (1984).
approximate for the T-Criterion of fracture in biaxial gelds. Engng Fracture Mech.
[S] F. Erdogan and G. C. Sih, On the crack extension in plates under plane loading and transverse shear. Trans. ASME, J. Basic Engng 85, 519-527 (1963). [9] K. J. Chang, Further studies of the maximum stress criterion on the angle crack problem. Engng Fracture Mech. 14,
125-142 (1981). [IO] S. K. Maiti and R. A. Smith, Comparison of the criterion for mixed mode brittle fracture based on the prei~stability stress-strain field-I and II. Inr. J. Fracture 23, 281-295 (1983). [I I] H. C. Wu, Dual failure criterion for plain concrete. J. Engng Mech. Div., AX&E I#, 1167-l I81 (1974). [12] K. J. Chang, On the maximum strain criterion-a new approach to the angle crack problem, Engng Fracture Mech.
14, 107-124 (1981). [13] D. Brock, Eiernentary Engineering Fracture Mechanics, pp, 359-361. Martinus Nijhoff Publishers, The Hague (1982). 1141 P. C. Shah, Fracture under combined modes in 4340 steel. ASTM STP 560, 29-52 (1974). [IS] M. T. Miglin. J. P. Hirth and A, R. Rosenfield, Estimation of transverse-shear fracture toughness for an HSLA steel. lat. J. Fruc. 22, R65-R67 (1983). [16] C. Atkinson, R. E. Smelser and J. Sanchez, Combined mode fracture via the cracked braziliam disk test. fnt. J. Fracture 18, 279--291 (1982). [17] G. Dileonardo, Fracture toughness characterization of materials under multiaxial loading. Int. J. Fracture 15,537-552 (1979). [IS] G. Dileonardo, Generalized fracture toughness of brittle materials. Inl. J. Fracture 20, 313-323 (1982). [is] Zhao Yishu, A strain energy criterion for mixed mode crack propagation. Engng Fracture Me& 26, 533-539 (1987). [20] Zhao Yishu, Griffith’s criterion for mixed mode crack propagation. Engtzg Fracture Me&. 26, 683-689 (1987). [21] G. R. Irwin, Analysis of stresses and strain near the end of crack traversing a plate. J. a&. Meeh. 24,361-364 (1957). (221 H. F. Bueckner, The propagation of crack and the energy of elastic deformation. Trans. ASME80, 1225-1230 (1958). [23] T. C. Wang, Frarture criteria for combined mode crack. ICF4, 4, 135-154 (1977).
292
ZHAO YISHU
[24] Y. Ueda, K. Ikeda, T. Yao and M. Aoki, Characteristics of brittle fracture under general combined modes including those under bi-axial tensile load, Engng Fracture Mech. 18, 113I-I 158 (1983). [25] R. J. Nusmer, An energy release rote criterion for mixed mode fracture. Int. J. Fracfure 11, 245-250 (1975). [26] M. Ickikawa and S. Tanaka, A critical analysis of the relationship between the energy release rate and the stress intensity factors for non-coplanar crack extension under combined mode loading. In?. J. Fracture 18, 19-28 (1982). 1271Y. Ueda, K. Ikeda, T. Yao. M. Aoki and T. Shirakura, Brittle fracture initiation characteristics under bi-axial loading. ZCF4, 2, 173-182 (1977). [28] Y. Ueda, K. Ikeda, T. Yao, T. Yoskie and T. Shirakura, Characteristics of brittle fracture under bi-axial tensile load. Trans. JWRI (Japan Welding Research Institute of Osaka University) 5, 69-77 (1976). [29] J. D. Eshelby, The continuum theory of latfice defects-solids state physics (Edited by F. Seitz and D. Turnball) Vol. III, 79;144, Academic Press, New York (1956). 1301J. D. Eshelby, Energy relation and the energy momentum tensor in continuum mechanics, Inelastic Behavior of Soli& @dited by Kanninem et al.) pp. 77-115, McGraw-Hill, New York (1970). [31] J. D. Eshelby, The elastic ener~-moments tensor. J. E&t. 5, 321-335 (1975). f32] J. R. Rice, Mathematical analysis in the mechanics of fracture. Fracture, An Advanced Treatise (Edited by H. Liebowitz). Vol. II, 191-311, Academic Press, New York (1968). [33] Zhao Yishu, Experimental study on elliptic rule criterion. J. Huazhong University of Science and Technology 1, 59-62 (1986). (Received 6 Juiy 1989)
ELLIPTIC
0013-7944/90 s3.00 + 0.00 Pergamon Press plc.
RULE CRITERION FOR MIXED CRACK PROPAGATION
MODE
ZHAO YISHU Department of M~hanics,
Huazhong University of Science and Technology, Wuhan, China
Abstract-As can be proved from theoretical analysis, some mixed mode fracture criteria may be converted into an elliptical or ellipsoidal formula with the aid of mathematical translation. Others are of themselves an elliptical or ellipsoidal formula. Hence, on the basis of Griffith-Irwin theory, the paper suggests to take (K,/K,,)‘+ A(KJK,,)* + B(KJK,,)* = 1 as the general criterion superseding all the mixed mode fracture criteria. A and B will be determined by the superseded criterion.
INTRODUCTION As EVERYBODY knows, energy release rate criterion (G~~te~on)~l-31, the energy density criterion (~~~te~on)~~] and the T&terion[7] are the very “hot” discussions and the competing three most popular criteria. Moreover, maximum tangential stress criterion& 91, maximum tangential principal stress criterion[lO] and maximum strain criterion[ll, 121 are also very attractive. Unfortunately, none of them have generally been accepted yet. They can not be applied widely to engineering practice because the structural evaluation from these criteria is a lengthy and difficult procedure to carry out. For a long time people attempted to take the formula (K l&C)” + (W&Y
= 1
(1)
as an empirical criterion of mixed mode fracture[l3-161, where the constants u and Y are experimentally determined for each material application. Someone suggests u = u = 2[14] or u = 2 and v = 0.475 - 2[15]. It is a pity that eq. (1) is not accepted because it has not been analysed strictly with mathematics and its physical significant is not clear. The use of generalized fracture toughness theory[ 17,181 offers one of the convenient mathematical means which expresses the mixed mode fracture criteria as a standard elliptical or ellipsoidal formula. Thus, expression of various mixed mode fracture criteria can be simplified and standardized and therefore a convenient way of generalizing mixed mode fracture criteria may be provided. The strain energy criterion for mixed mode crack propagation was published recently[l9]. Its Fracture Envelope formula is a ellipsoidal formula with high power b,lK:+b,2K:K:+b,3K:K:+b23K:K:+b22K~eb33K:=K~~
(2)
or b,1(rrr;/rsr,,)~+b,2(K,/Kt,)~(K*/K,,)*~
k;3(K,/K,,E)‘(K~/K,c)’ + b~~(K~/K*~)‘(~/K,~)* + b,,(&/JQ4
in which b,, = 1, b,* = (185 - 24v{ - 24~4 + 39/2)/(7{ - 12vt - 4v + I l/4), b,3 = (85 - 32v + 24)/(7t - 12vt - 4v + 1 l/4), b,, = (8< - 32v + 56)/(79 - 12~5 - 4v + 1l/4),
bz2=(11(
- 12vr -20~ i-99/4)/(7t
- 12v< -4v
b,, = 32/~7~ - 12v< - 4v -t 1l/4),
5 = 2(1 - 2v)ZjZ. EFM 37/2-C
283
+ 11/4),
+ ~~~(K~/K~~)4 = 1,
ZHAO YISHU
284
The strain energy criterion is different from the strain energy density criterion[4-61. The strain energy criterion takes the total strain energy in the region surrounded by the initial elastic-plastic boundary near the crack tip as the parameter dominating the crack propagation. The initial elastic-plastic boundary will be shrunk to an infinitesimal circular region for strongly brittle materials. Therefore, eq. (2) is rewritten to be a standard ellipsoidal formula C,,(K,I&)2
+
G2W21JGd2
+
G,wmz
=
1,
(3)
in which C,, = 1,
C,, = (9 - 8v)/(5 - Sv),
C,, = S/(5 - 8~).
Equation (2) will be transformed into a high power elliptical formula or a standard elliptical formula respectively under Mode I-II, I-III and II-III. Since mathematical expression of some fracture criteria may be transformed into a standard elliptical or ellipsoidal formula by use of the generalized fracture toughness theory[l7, 18], and others are of themselves an elliptical or ellipsoidal formula, therefore, the purpose of this paper lies in the fact that using a standard ellipsoidal formula represents all of the mixed mode fracture criteria. THEORY To date, fracture criteria under multiaxial loading is a twofold problem, (a) the crack propagation angle, specified through stationary conditions imposed on a prescribed functional, describing the stress intensity factors’ interaction for the specific application, and (b) the fracture load determined by the critical value that the same functional reaches during loading. For a two-dimensional condition, we have c~(e)Ki~j - Q, = 0,
(4)
C@)KiKj = 0,
(5)
in which i, j = 1,2. When using some fracture criteria, however, the procedure for finding the Cartesion equation of the Fracture Envelope by eliminating the argument 0 from eqs (4) and (5) is considerablely laborious. But it is known that eq. (4) describes a family of polarly symmetric tonics through varying the parameter 8, or rather, a family of the ellipses with their center on the origin of the K, K2 orthogonal reference system. Since eq. (5) is homogeneous, it represents a pair of lines passing through the origin 0 corresponding to a prescribed value 0,. This pair of lines intersect with the ellipse C~(e~)~i~~ - Qcr = 0 at least one point. The intersection is the critical point. The Fracture Envelope is the locus of the critical points of the ellipses of the family dete~ined by eq. (4). Dileonardo[l7, 181 has proved that any ellipse of the family is tangent in its critical point to the Envelope, its kinematics (in the K,g plane when 8 varies within the prescribed interval), represented by the ellipses of the family of eq. (4), make a rotation around its center 0 and a rolling motion inside the Envelope. Therefore, the Fracture Envelope may be constructed from the geometric properties of the rolling ellipses in the neighborhood of their critical point, namely r+)2+(+)1=,,
(6)
from which we defined a = K,, + m, b = aK2JK:,
(7)
+ 2&L,,) - I’*,
m = (K?, K$ - K:, K:, - K$, K:,)/2(&,
K:, f K,, K:, - K,&)
(8) (9)
where K,, and K,, can be calculated from following equations C,(8 ,)K,K, - Qcr = 0 C,i(d*)K,Kj=O
(10)
Elliptic rule criterion
285
in which @* is one of solutions satisfying the equation c,,(e)
= 0.
(11)
The geometrical graph of eq. (6) is shown in Fig. 1. It represents an ellipse of ~mi-diameters a and b, with center O(-HZ, 0) in iu, K, plane. If anti-plane shear is considered, the Fracture Envelope wilI be an ellipsoidal formula as follows
From which c = aK&.(Ki, + 2mK1,)- I’2.
(13)
The material constants Q, b and m in above equations are fuuctions of K,, and v and computed from eqs (7), (8) and (9) respectively. The energy release rate under critical state can be found in [ZO]
+ 4K, K2 sin 2y] + The coefficients Cii corresponding
to eq. (10) can be obtained from eq. (14)
and K% = 0.63K,, , K,, = (1 - v)‘~K~,,
in which y is the angle between the branch~ and the second of eqs (15), we obtain
crack and the main crack. According to eqs (lo), (11)
K,* = 0.61K,,, K,, = +0.63K,, .
06)
Substituting K, = 0.63K,, and eq. (16) into eq. (9), we obtain m = -0.30&.
(17)
‘6
Fig. 1. Fracture Envelope of two dimensions.
ZHAO YfSHU
286
According to eqs. (7), (8) and (17), m, a, b and c vs fracture toughness K,, are plotted in Fig. 2a-d. For each material, the Fracture Envelope formula (12) only specified by the fracture toughness K,, and Poisson’s ratio v. A convenient way of application for energy release rate criterion is provided by this approach. When K,, and v of the material are given, then m, a, b and c can be found out from Fig. 2a-d. Substituting m, a, b and c into eq. (12) the Fracture Envelope will be determined uniquely. Corresponding to the strain energy density criterion we have x;, = [3( 1 - 2v)/(2 - 2v - Y2)]*‘zK,c, K& = (1 - 2V)“%~~* Dileonardo
had given]1 71 J&l = [(l K,,
2v)3’2(1
= 2[v (1 -
-
v)]K,,,
2v)/( 1 - v)]“X,, .
Substituting above parameters into eqs (7)-(Q) and (13), m, a, b and c vs the fracture toughness K,, are plotted in Fig. 3a-d. Once the fracture toughness K,, and Poisson’s ratio v are assigned, then the Fracture Envelope will be determined solely. Geometrical graph of eq. (12) is shown in Fig. (4). It is an ellipsoid of semi-axes a, b and c, with its center at (-m, 0,O).
Fig. 2. Energy release rate criterion+ (a) tn vs fracture toughness R,,. (b) a vs fracture toughness K,=. (c) b vs fracture toughness K,,. (d) c vs fracture toughness I& and Poisson’s ratio v.
287
Elliptic rule criterion
70 80
60
40 30 20 10
020
(4
30
80
70
80
1201
W
I
02--I 1oy 2030
1
I
I
I
I
I
40
60
80
70
80
Kk eo
Kk Fig. 3. Strainenergy density criterion. (a) m vs fracture toughness K,, and Poisson’s ratio v+.(b) a vs fracture toughness K,, and Poisson’s ratio Y. (c) b vs fracture toughness K,, and Poisson’s ratlo v. (d) c vs fracture toughness K,, and Poisson’s ratio v.
From the preceding analysis it can be found that the Fracture Envelope of both the G-criterion and the S-criterion all are a part of the ellipse perimeter or the ellipsoid surface. This important conclusion is worth notice. ELLIPTIC
RULE CRITERION
Assume that the crack extension is coplanar, then the relationship between the energy release rate G of Mode I crack and the stress intensity factor K, under the case of plane strain has been given by Irwin[21] Gi = (1 - v2)K;/E.
(18)
288
ZHAO YISHU
Fig. 4. Fracture Envelope of three dimensions.
The strictly mathematical analysis of eq. (18) has been given by Bueckner[22]. Similarly, for Mode II and III crack, we have Gz = (1 - v*)K:/E, G3 = (1 + v)K:/E.
(19)
Therefore, for a combination of Mode I, II and III crack, the relationship between the energy release rate and the stress intensity factors is given as follows G = (1 - v*)K:/E + (1 - v*)K:/E + (1 + v)K:/E. When reaching the critical state, that is G = GCo,above equation is explicitly rewritten as (1 - v*)K,/E + (1 - v*)K:,‘E + (1 + v)K:/E = G,,.
(20)
For pure Mode I crack we obtain from eq. (20) G,, = (1 - v*)K:,/E.
(21)
Generally, it is assumed that the critical energy release rate under combined loading is the same as that of Mode I crack for that material. Therefore, we have from eqs (21) and (20)
From eq. (22), we find lu,, = Ki, and Kk = 6 obtain (K, /K,, )* + (Kz /kl;c
K,,. Substituting these results into eq. (22), we
I*
+
UG
l&c
I2 =
1,
or (K,lK,,)*+A(1Y2/k;E)‘+BtK~iK,Ktc)*=
1,
(23)
where A = K,,/K,,, B = KIE/K3,. Equation (23) is a typical formula ellipsoid. For Mode I-II, I-III and II-III, it will transform to be a typical formula of ellipse and is called the elliptic rule fracture criterion. Equation (23) is derived on the basis of Griffith-Irwin theory and under the hypothesis that the mixed mode crack extension is coplaner. In fact, however, the mixed crack extension will depart from its original straight trajectory or branch, then the energy release rate of the branched crack extension must be considered. Therefore, strictly speaking, the elliptic rule fracture criterion is an approximate criterion. Equation (14) is an expression of the energy release rate considering branched crack extension. It is equivalent to eq. (23) if y = 0.
Elliptic rule criterion
289
We suggest to take eq. (23) as a general criterion superseding all the mixed mode crack fracture criteria. A and B in eq. (23) will be determined from that superseded criterion. The G-criterion may be simplified by the use of eqs (23) and (15) as follows (Kl /K,C)2+ 2.52(K2/K~,)2 + l.42fKJK~,)2 = 1.
(24)
As to the precise expression of the G-criterion, it may be obtained (v = 0.30) from Fig. 2 and eq. (12) 2.5O(K, /&)* - 1.50(& /Klc) + 2.52(K2/&)2 + 1.42(K,/K,J2 = 1.
(25)
The Fracture Envelopes predicted by eqs (24) and (25) under mixed Modes I-II, I-III, II-III and I-II-III fracture are plotted in Fig. 5a-d respectively. Similarly, the S-criterion may be simplified by the use of eq. (23) as follows (K, /K,J2 + 1.09(KJK,J2 + 2.52(K,/K,,)2 = 1. According to eq. (12) the precise expression of the S-criterion
(26)
may be obtained (v = 0.30)
0.48(~~ /‘K,,)* + 0.52(K, /K,,) + 1.09(lu, /&C)2 + 2.52(K, /lytC)’ = 1.
(27)
The Fracture Envelopes predicted by eqs. (26) and (27) under mixed Mode I-II, I-III, II-III and I-II-III fracture are plotted in Fig. 6ad respectively.
I0
*.8y----p.
4r
, G /
\’
-Y--i 1’
I
5.f
z
\
0
0.8C
0.4 -
\
0
0.4
.
‘\
\
Elliptic /
0.2 -
\ \
0.2
*i
0.2
0.4
0.B
KI f K,,
0.8 .I
I
0
I
I
0.2
0.4
,\ I
0.0
\
I
0.8
b/he Fig. 5. Predicted by energy release rate criterion and elliptic rule criterion in comparison with test results. (a) Mode I-II fracture. (b) Mode I-III fracture. (c) Mode II-III fracture. (d) Mode I-11-111 fracture.
( P \ \ I \
0
290
ZHAO YISHU
The experimental points in Figs 5 and 6 are average values of the test results of 188 60Si2Mn steel specimens. CONCLUSION The elliptic rule criterion is based on Griffith-Irwin theory. As was mentioned above, Dileonardo[l7, 181 has proved analytically that some mixed mode fracture criteria may be expressed by a typical elliptical or ellipsoidal formula. As for the others, such as, the expression of the strain energy criterion[l9] are of themselves an ellipsoidal formula. Therefore, taking eq. (23) as an approximately general criterion for mixed mode fracture has sufficient basis in both of physical meaning and the mathematics. We say that the elliptic rule criterion is approximate because coplanar extension of the crack is assumed, more specifically, because extension of the mixed mode crack involving Mode II is assumed to be still in the plane of the original crack. In fact, extension of a pure Mode II crack or the mixed mode crack involving Mode II will branch at the crack tip. Therefore, the first formula in eq. (19), G2 = (1 - v*)K:/E, has no practical meaning. Thus eq. (23) is not precise. Equation (23) is correct only when the Mode II crack is not involved under mixed mode fracture. In fact, not one of the fracture criterion is perfect. They are derived from a certain assumption because of the mathematical complexities associated with the elastic stress field analysis of the branched crack, and there are disputes to this day. For example, the S-criterion[4-61 postulates existence of a minimum Smin of the total strain energy density along an artificially defined (b)
(4
_._
0
I
\ 0.2
1
1
0.4
0.6
01
I
0.2
0.4
0.8
0.8
KI IKI,
0.8
KI /KI,
(4
\ 0 0’
I 0.2
I
I
0.4
0.8
0
I 0.8
I 0.2
I 0.4
I 0.8
\ I 0.8
K,IKI,
K2lK1c Fig. 6. Predicted by strain energy density criterion and elliptic rule criterion in comparison with test results. (a) Mode I--II fracture. (b) Mode I-III fracture. (c) Mode II-III fracture. (d) Mode I-II-III fracture.
Elliptic rule criterion
291
locus of equal distance from the crack tip (circular region), the crack extends in a radial direction corresponding to the minimum Sminand the extension occurs when this minimum reaches a critical value. But there is, as yet, no physical basis to decide on the magnitude of this radius of circular region and S-criterion is suflicient when there is only one minimum Smi,, in the event of more than one minimum S,i, at a particular radius confusion arises. IJead et ~1. pointed out that the physical meaning of the S-criterion is not clear[24,27,28]. Eshelb~[2~31] and Rice1321 elaborated the fundamental fact that the total potential energy rather than just the strain energy density should be used for the derivation of the crack extension force. As for the G-criterion, because the expression for the energy release rate is derived by every author, each is different from the other, thus the predicted rate is also different[ 1,21,24]. Recently Theocaris proposes the T-criterion[7]. Though it appears to be a flexible fracture criterion and agrees with the test results quite well, a rigorous elastic-plastic analysis was not made for initial yield locus at the crack tip and the initial yield locus was defined by the elastic components of stress. Therefore, elastic part of the strain energy density only was considered. These simpli~cations are more appropriate for a brittle type of fracture. From the above it can be shown that all the criteria for mixed mode fracture introduced up to now are hypotheses, not final conclusions as are other failure theorems for mechanics of materials. As one of the hypotheses for the mixed mode crack extension, the elliptic rule criterion made only one assumption of coplanar crack extension. Figs 5 and 6 show that the differences between the Fracture Envelope predicted by eq. (23) and S- and G-criterion are not significant, the maximum error is 18% and 10% respectively for Mode I-II, I-III and I-II-III. However, for Mode II-III, the elliptic rule criterion is equivalent to the superseded criterion” We cannot judge which one in both the elliptic rule criterion and the superseded criteria agrees well with the measured data. The predicted will be more satisfactory if the values of &, and K, (or A and B) in eq. (23) are the measured values instead of those defined by superseded criterion when using the elliptic rule criterion[33]. REFERENCES [I] M. A. Hussain, S. L. Pu and J. Unde~ood,
Strain energy release rate for a crack under combined Mode I and Mode II. ASTM STP 560, 2-28 (1973). [2] K. Palaniswamy and W. G. Knauss, On the problem of crack extension in brittle solids under general loading, in ~eeho~ic~ Today (Edited by S. Nemat-Nasser), Vol. 8, pp. 87-148. Pergamon Press, New York (1978). (31 K. Palaniswamy and W. G, Knauss, Propagation of a crack under general, in-plane tension. Inr. J. Frac. Mech. 8, 114-I 17 (1978). {4] G. C. Sih, Strain-energy-density factor applied to mixed mode crack problems. Int. J. Fruc, IO, 305-321 (1974). [5] G. C. Sih and B. Macdonald, Fracture mechanics applied to engineering problems-strain energy density fracture criterion. Engng Fracture Mech. 6, 361-368 (1974). [6] G. C. Sih and B. C. K. Cha, A fracture criterion for three dimensional crack problem. Engng Fracture Me&. 6,699-723 (1974). [7] P. S. Theocaris, A higher-order 19, 978-991 (1984).
approximate for the T-Criterion of fracture in biaxial gelds. Engng Fracture Mech.
[S] F. Erdogan and G. C. Sih, On the crack extension in plates under plane loading and transverse shear. Trans. ASME, J. Basic Engng 85, 519-527 (1963). [9] K. J. Chang, Further studies of the maximum stress criterion on the angle crack problem. Engng Fracture Mech. 14,
125-142 (1981). [IO] S. K. Maiti and R. A. Smith, Comparison of the criterion for mixed mode brittle fracture based on the prei~stability stress-strain field-I and II. Inr. J. Fracture 23, 281-295 (1983). [I I] H. C. Wu, Dual failure criterion for plain concrete. J. Engng Mech. Div., AX&E I#, 1167-l I81 (1974). [12] K. J. Chang, On the maximum strain criterion-a new approach to the angle crack problem, Engng Fracture Mech.
14, 107-124 (1981). [13] D. Brock, Eiernentary Engineering Fracture Mechanics, pp, 359-361. Martinus Nijhoff Publishers, The Hague (1982). 1141 P. C. Shah, Fracture under combined modes in 4340 steel. ASTM STP 560, 29-52 (1974). [IS] M. T. Miglin. J. P. Hirth and A, R. Rosenfield, Estimation of transverse-shear fracture toughness for an HSLA steel. lat. J. Fruc. 22, R65-R67 (1983). [16] C. Atkinson, R. E. Smelser and J. Sanchez, Combined mode fracture via the cracked braziliam disk test. fnt. J. Fracture 18, 279--291 (1982). [17] G. Dileonardo, Fracture toughness characterization of materials under multiaxial loading. Int. J. Fracture 15,537-552 (1979). [IS] G. Dileonardo, Generalized fracture toughness of brittle materials. Inl. J. Fracture 20, 313-323 (1982). [is] Zhao Yishu, A strain energy criterion for mixed mode crack propagation. Engng Fracture Me& 26, 533-539 (1987). [20] Zhao Yishu, Griffith’s criterion for mixed mode crack propagation. Engtzg Fracture Me&. 26, 683-689 (1987). [21] G. R. Irwin, Analysis of stresses and strain near the end of crack traversing a plate. J. a&. Meeh. 24,361-364 (1957). (221 H. F. Bueckner, The propagation of crack and the energy of elastic deformation. Trans. ASME80, 1225-1230 (1958). [23] T. C. Wang, Frarture criteria for combined mode crack. ICF4, 4, 135-154 (1977).
292
ZHAO YISHU
[24] Y. Ueda, K. Ikeda, T. Yao and M. Aoki, Characteristics of brittle fracture under general combined modes including those under bi-axial tensile load, Engng Fracture Mech. 18, 113I-I 158 (1983). [25] R. J. Nusmer, An energy release rote criterion for mixed mode fracture. Int. J. Fracfure 11, 245-250 (1975). [26] M. Ickikawa and S. Tanaka, A critical analysis of the relationship between the energy release rate and the stress intensity factors for non-coplanar crack extension under combined mode loading. In?. J. Fracture 18, 19-28 (1982). 1271Y. Ueda, K. Ikeda, T. Yao. M. Aoki and T. Shirakura, Brittle fracture initiation characteristics under bi-axial loading. ZCF4, 2, 173-182 (1977). [28] Y. Ueda, K. Ikeda, T. Yao, T. Yoskie and T. Shirakura, Characteristics of brittle fracture under bi-axial tensile load. Trans. JWRI (Japan Welding Research Institute of Osaka University) 5, 69-77 (1976). [29] J. D. Eshelby, The continuum theory of latfice defects-solids state physics (Edited by F. Seitz and D. Turnball) Vol. III, 79;144, Academic Press, New York (1956). 1301J. D. Eshelby, Energy relation and the energy momentum tensor in continuum mechanics, Inelastic Behavior of Soli& @dited by Kanninem et al.) pp. 77-115, McGraw-Hill, New York (1970). [31] J. D. Eshelby, The elastic ener~-moments tensor. J. E&t. 5, 321-335 (1975). f32] J. R. Rice, Mathematical analysis in the mechanics of fracture. Fracture, An Advanced Treatise (Edited by H. Liebowitz). Vol. II, 191-311, Academic Press, New York (1968). [33] Zhao Yishu, Experimental study on elliptic rule criterion. J. Huazhong University of Science and Technology 1, 59-62 (1986). (Received 6 Juiy 1989)