Int. Journ. of. Fracture 20 (1982) 313-323. 00376--9429/82/04/0313-11500.20/0 © 1982 Martinus Nijhoff Publishers, The Hague. Printed in the Netherlands
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Generalized fracture toughness of brittle materials G. DI L E O N A R D O Knolls Atomic Power Laboratory, General Electric Company, Schenectady, New York, 12301, USA* (Received August 28, 1980; in revised form September 9, 1981)
ABSTRACT A generalization of the concept of fracture toughness of brittle materials, when subjected to multiaxial loadings, is presented. The theory characterizes the fracture strength of materials under any combination of the three basic modes of crack surfaces displacement. With reference to three-dimensional loading systems, the fracture toughness may be represented, in the KmK2K3Cartesian orthogonal space, by a surface Fracture Envelope characteristic for a specified material, whose equation is determined by the (symmetric) fracture toughness Kmc and Poisson's ratio v. It is shown that the most general fracture process, resulting from the combination of the opening mode of the tangential stress component and the tearing mode of the antiplane shear, may be conveniently analyzed with the aid of the generalized fracture toughness concept. From the knowledge of the Fracture Envelope relative to a structural material, a simple fracture criterion permits forecasting crack propagation for any combination of loads and geometries. The theory is applied to mixed-mode problems to define the analytic threshold of fatigue crack growth.
1. Introduction
Studies have shown [1] that the fracture criterion for brittle structural materials operating in biaxial loading fields can be conveniently replaced by a criterion based on a two-dimensional Fracture Envelope, geometrically constructed from the experimentally determined values Klc and v relative to the particular material under consideration. In the same reference system, a structural configurationt with a crack is represented by a point of coordinates kl, k2 indicating the stress intensity factors combination corresponding to the component geometry and applied tensile and in-plane shear loadings. Crack propagation ensues if the point is on or outside the Fracture Envelope, which, therefore, represents the generalized fracture toughness of the given material under biaxial loadings. Hence, the Fracture Envelope yields a fracture behavior characterization more general than the symmetric fracture toughness Kic. Superposing Mode III, generated by the tearing action of the antisymmetric shear loading, and the Mode I and II combination completes the fracture characterization process. The result is sufficient to describe the most general fracture phenomenon due to three-dimensional loadings, characterized by any combination of the basic stress intensity factors. It will be shown that, in the same reference system, the appropriate fracture criterion may be formulated in terms of the orthogonal
* Operated for the U.S. Department of Energy, Contract No. DE-AC12-76N00052. t Structural configuration is a component identified by geometric parameters and applied loadings.
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representation of the point of coordinates k~, k2, k3 and of a three-dimensional Fracture Envelope describing the generalized fracture toughness of the material under the applied loadings.
2. Theory In the KIK2K3 orthogonal reference system, it is convenient to postulate, for any material, a Fracture Envelope surface E**, locus of points of coordinates (kl, k2, k3) representing combinations of the three basic stress intensity factors describing those fracture onset events distinguished by the least amount of dissipated energy. Assume that the Fracture Envelope is generated by a family of three-dimensional surfaces, defined by:
F(kl, k2, k3, 0 ) = 0,
(1)
where 0 is a parameter varying between prescribed limits. In correspondence to a value 00, Eqn. (1) defines a specific surface of the family F(00), tangent to the Envelope at least at one critical pointt P(00) (Fig. 1). Its coordinates ks(Oo)(S = 1, 2, 3) satisfy (1) which, therefore, identifies the crack-growth criterion governing the most general fracture process. It will be shown that the geometrical envelope of the family identified by (1) is the locus of critical points of the surfaces F(O) corresponding to the values of 0 within the prescribed interval and, consequently, is the Fracture Envelope representing the generalized fracture toughness of materials subjected to multiaxial loadings. A generic surface of the family:
F(kl, k2, k3, 0) =F(ks, 0) = 0 (s = 1, 2,3) intersects the surface
F(k, 0 + A0) = 0
K3 *
FXX
P K2
KI
Figure 1. Critical point P(00) contact of the surface F(Oo)with the Fracture Envelope E**. t Critical point indicates a set of stress intensity factors leading to fracture.
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along the curve (F(ks, 0) = 0 F(ks, 0 + A0) = 0.
(2)
If 0 is within the interval 0, 0 + A0, the last equation is rewritten, according to the mean value theorem, as aF F(k,, O) + AO - ~ ( k s , #) = O,
and, for A 0 ~ 0 , as OF c~--O(ks, O) -- O.
It follows that Curve (2) becomes, at the limit, F ( k , O) -- 0
t
~0F (ks, 0) = 0,
(3)
which represents the characteristic curve of the family (1), locus of points describing critical stress intensity factor combinations corresponding to a specific value of the parameter. In fact, if a point of Curve (3) of coordinates k* identifying 0* through the condition aF 0-o (k*, 0") = 0
is critical, it must satisfy F ( k * , 0") = O.
Consequently, the system F(ks, 0") = 0
I
~o (ks, o*) =
0
(4)
represents the curve of critical sets of stress intensity factors corresponding to 0". Furthermore, it should be noted that there is a one-to-one correspondence between the values of the parameter and the curves of critical sets. By varying 0* within the prescribed interval, the curve of critical sets generates the three-dimensional Fracture Envelope that describes the generalized fracture toughness of materials. Its Cartesian equation is determined by eliminating 0 from the previous system of (3). However, the explicit form of F must first be determined. 3. Fracture criterion
From [1] it was shown that, in two-dimensional stress fields, the fracture process is associated with crack-surface displacement normal to the direction of crack propagation. Along the same direction, the elastic potential of the volume element in the crack-tip region is an absolute minimum, and the fracture criterion may be defined in terms of the critical value, invariant for a specific material, reached by the strength of this potential. This same conclusion was reached by Sih [2, 3] who, in a series of papers,
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G. D / L e o n a r d o
proposed a fracture criterion based on the strength of the same potential, extending the theory to the three-dimensional case. The form of this functional represents the parametric equation of a family of ellipses with centers at the origin of the KIK2 reference system [1], each containing at least one critical point characterizing the fracture event associated with the ellipse. The locus of critical points is the twodimensional Fracture Envelope, i.e., the geometrical envelope of the ellipses of the family. The corresponding Cartesian equation is
indicating an ellipse in canonical form, with its center at ( - m, 0) (Fig. 2). The material constants a, b, and m are functions of the conventional fracture toughness K~c and Poisson's ratio, v. Furthermore, the fracture toughness corresponding to skewsymmetric displacement of the crack surfaces is not an independent material constant: 2v) " i/2 K2c = (23(1~ _ vQ K,~. (6) The crack growth criterion is formulated in terms of the orthogonal representation of a fracture event in the Fracture Envelope plane as a point of coordinates klk:. Accordingly, the crack propagates if the point is located on or outside the Fracture Envelope.
/
/
b
I
/ I
I m C
\
o
K1
i /
\
/
\
Figure 2. Two-dimensional Fracture Envelope.
Y
o
x Z Figure 3. Crack border region reference system.
Generalized fracture toughness of brittle materials
317
Superposition of Mode III, characterized by the tearing motion of the crack surfaces due to antisymmetric shear shresses, and the mixed mode [1] yields the total description of the fracture process. The resulting elastic potential of the volume element in the proximity of the crack front (Fig. 3) may be written as: W ( r , 0) =
Q(O), r
where
Q(O)
cij(O)kik; + c33k~
{i=l,2
j=l,2
I
\Cij = Cii
is the strength of the potential. From [2] the coefficients are, explicitly: Cll = 1 - ~
[(1 + cos 0)(3 - 4v - cos 0)]
c,2=,~l
[2sinO(cosO-l+2v)]
C22 =
1
1 - - ~ [4(1 - v)(1 - cos O) + (1 + cos 0)(3 cos 0 - 1)] 1
c33 = 4~r/~ and, with the results of [1], the fracture criterion may be formulated as: Crack propagation ensues when the form Q reaches a critical value, Qcr, constant for a specific material, and in a direction O* where the functional W and, consequently, Q, is minimum. Hence, if (k~, k~, k~) indicates a critical set, it must satisfy:
cii(O*)k~,k~ + c33k~ = Qc,,
(7)
where 0", the crack propagation direction, is the root of
clj(O)k*,k~ = 0
(a)
and the prime denotes differentiation with respect to O. Furthermore, O* satisfies:
c'ij(O*)k*k* > 0.
(9)
Since Qcr is assumed to be invariant relative to the specific fracture mode, it may be computed from Mode I only. The result [1] is: Qcr =
1 -2VK2c. 4¢rp.
According to the Fracture Envelope theory, the fracture criterion may be written as:
ai~(O)kiki + k~ = (1 - 2v)K~c a~j(O)k,k~ = 0 ks=k*,O=O * s = 1,2,3
(10) (11)
where a~j(O)= 4¢r/~c~j(0), and the starred values indicate a critical condition. Equation (10) represents a family of ellipsoids with their center in the origin of the KIK2K3 orthogonal reference system, each defined by a particular value of 0 and
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intersecting the K3-axis at two fixed points of coordinates
(12)
P1,2 = [0, 0, -+ (1 - 2v)l/2Klc].
4. Three-dimensional fracture envelope
Since (11) subjected to the auxiliary condition (9) represents the family of parametric planes ka = f(O)kl passing through the K3-axis, the intersection of each plane with the corresponding ellipsoid defined by the same value 00 constitutes the characteristic curve of the family indicated by (10) [cf. (3)]. Since this ellipse is the locus of the critical points describing fracture events with the same crack propagation direction 00, it represents the ellipse of critical sets. By varying the value of the parameter, the ellipse rotates about the K3-axis and generates the three-dimensional Fracture Envelope (Fig. 4). Kinematically, (10) represents a variable ellipsoid rotating about the K3-axis and rolling inside the (fixed) Fracture Envelope. Hence, the ellipse of critical sets is the curve-contact of the two surfaces. Furthermore, since the Fracture Envelope is generated by the ellipse of critical sets, it describes the generalized fracture toughness of materials subjected to the most general loading system. From this representation, the mixed-mode Fracture Envelope [1] corresponding to plane loadings is the section of the three-dimensional Envelope with the plane k3 = 0. The intersections of the Fracture Envelope with the K3-axis correspond to the positive and negative antisymmetric stress intensity factors, representing the fracture toughness K3c: K3c = (1 - 2v)l/2Klc. The direct approach to obtain the Cartesian equation of the Fracture Envelope is to eliminate the parameter 0 between (10) and (11). However, following the method K3
K3c
LK 2
i \
2-/
i
K) Figure 4. The three-dimensional Fracture Envelope E**. e*--Ellipse of critical sets E*--Two-dimensional Fracture Envelope (Ref. [1]).
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319
outlined in [1], its equation may be determined from the geometrical properties of the rolling ellipsoid in the neighborhood of the ellipse of critical sets (Fig. 4). The result is:
The material constants a, b, c, and m are functions of Klc and v [1]:
a=Klc+m b
3(1 - 2v)Kl~
~ t12
(Klc + m)\(2[ - 2v - v2)(Kl~ + 2m))
,/ 1-2v )1/2 c = (Ktc + m)~Klc + 2m Klc 1 4 v ( 2 - 2v - v 2)- 3(1 - v)(1
m=-~4v(2_2v_vE)_3(l_v)(
~-1~ v---~-}K(1 - 2v)3~
1 (1_2v)312~1__~ ] to.
The constant c was determined from the condition that the coordinates (12) of the point P~ satisfy (13). Hence, the Fracture Envelope in presence of the three basic modes of fracture is an ellipsoid of semi-axes a, b, and c, with its center at ( - m, 0, 0), and the previous fracture criterion may be stated as follows: Crack growth occurs if the point of coordinates k~ ), representing the stressintensity factors pertaining to the applied loads and geometry characterizing the structural configuration, is on or outside the Fracture Envelope surface (13), i.e., if the set k~°) satisfies: (k~°)+ m)~ + / k ~2°)\2 (k~°'~
,,4>
Different structural components of the same material are characterized by the constant values of a, b, c, and m and, consequently, by the same Fracture Envelope, which, therefore, represents the generalized fracture toughness of that material. 5. Results
From the calculation of the material parameters a, b, c, and m, which are functions of the experimentally determined values of Klc and v, the equation of the Fracture Envelope is completely defined by expression (13). The numerical results for several TABLE 1 Fracture envelope parameters Alloy
K~
v
K:c/Klc
K3dKlc
m*
a*
b*
c*
4340 H-11 PH 13-8Mo 410 718 Ti 6-4 7075-T651
58 37 59 55 87 97 33
0.32 0.36 0.30 0.28 0.30 0.35 0.33
0.93 0.85 0.96 0.98 0.96 0.87 0.91
0.60 0.53 0.63 0.66 0.63 0.55 0.58
29.9 16.4 32.4 32.0 47.8 44.8 16.4
87.9 53.4 91.4 87.0 134.8 141.8 49.4
57.2 33.2 60.4 58.2 89.1 89.4 31.8
37.0 20.6 39.0 39.2 58.8 56.0 20.4
* Units in MPaX/m.
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320
K 3j
R ' _ _ ~ \
/
/°
\\\
\/
• ¥
IL
K2
/K
!
Figure 5. Fracture Enveloperepresentation. alloys are shown in Table 1. It may be noticed that Mode II and III critical stress intensity factors due to the presence of skew-symmetric and antisymmetric loadings, respectively, are generally lower than K~c. Crack growth may be investigated, for any structural configuration characterized by k~°), k~°), kg°) combinations, with the fracture criterion (14). Figure 5 illustrates the geometrical analog of the material parameters in the Fracture Envelope reference system. The constants a, b, and c represent the semidiameters of the ellipsoid whose center C is located at ( - m, 0, 0), with symmetry planes k2 = 0 and k3 = 0. Since this surface describes the generalized fracture toughness in the presence of multi-axial loadings, its representation was limited to the first octant of the reference space KlK2K3. Its intersections with the reference coordinate axes represent the values of the stress intensity factors corresponding to fracture conditions due to the presence of each basic mode only. In particular, the intersection with the Kl-axis is the value of the (conventional) fracture toughness K~c. The center of the ellipsoid is the origin of the parallel reference system K~YZ, translated toward the negative direction of the Kraxis by the amount m. Stress intensity factor combinations described by points inside the surface represent non-critical events. The opposite is true for events characterized by points on or outside the Fracture Envelope surface.
6. Analytical fatigue threshold in multiaxial loadings
The determination of the generalized threshold condition for materials subjected to complex cyclic loadings is a significant development based on the Fracture Envelope theory. Within the crack tip region, the strain energy density associated with the stress field range Acrij is defined as:
W(Acrij)- T(O), r
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321
where, corresponding to the mixed-mode case,
T(O) = cll(O)Ak~ +t 2cl2(O)AklAk2 + c22(O)Ak~. Postulating crack growth when
T(O) = To, i.e., the existence of a set Aki of fundamental stress intensity factor ranges corresponding to each value of the angle 0 such that threshold conditions prevail, the requirement for the envelope of the family of parametric curves,
F = cij(O)Ak~Akj - To = O, is mathematically established. This envelope is the locus of the threshold events described by the sets Akl, Ak2 of the stress intensity factor ranges corresponding to the mixed mode. Its equations, in parametric form, are:
( ci~(O)AkiAkj - To = 0 c~j( O)AkiAk i = O. Elimination of 0 between them yields the Cartesian equation of the envelope, i.e., the generalized threshold condition ._ \(Ak' +
2 {Ak2~2= 1, + \ bo ]
where a0, bo, and m0, functions of Poisson's ratio v and of the Mode I threshold range AK~°), are computed by multiplying the analogous constants of the Fracture Envelope by AK~°)/KIc. Experimentally determined near-threshold Akl, Akz combinations from aluminum plates with cracks inclined to the cyclic load direction were reported by Tanaka [5] and are summarized in Table 2. Figure 6 shows the near-threshold test points defined by the coordinates Ak~, Ak2, and the analytic envelope describing the generalized threshold conditions for any TABLE 2
Near-threshold test results [5] Initial c r a c k angle, deg
Crack growth Akl Ak2 rate, d a / d N M P a . m½ M P a . m~ (mm/cycle)
Crack grow t h angle, deg
90 90 90 90 72 72 45 45 30 30 30 30 30
2.02 1.72 1.86 1.98 1.66 1.76 1.12 1.20 0.74 0.77 0.85 0.74 0.78
0 --0 ------
* Units in MPaX/m.
0 0 0 0 0.54 0.57 1.12 1.20 1.28 1.34 1.47 1.29 1.34
5.8 0 0 7.9 0 7.4 0 7.2 0 0 1.4 0 2.1
x 10-7
x 10 -7 × 10 -7 × 10-7
x l 0 -6 × 10 -7
28 49
22 52
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322
combination of these stress intensity factor ranges. The envelope parameters ao, b0, and m0 have been computed from v = 1/3, and the estimated AK~°~= 1.90 MPaX/m. 7. Conclusions
The theory presented here introduces the novel concept of generalized fracture toughness of structural materials operating in multiaxial stress fields and extends the characterization process of [1]. The fracture strength of brittle materials is represented by a three-dimensional Fracture Envelope, which, in the KtK2K3 orthogonal reference system, is an ellipsoid whose diameters are functions of the conventional fracture toughness K,c and of Poisson's ratio, v. This representation completes the fracture characterization process for any combination of the basic crack-surface displacement modes. According to Sih [4], fracture behavior is totally described by superposing the mixed mode from the plane case [1] and the antiplane tearing mode due to out-of-plane shear loading. The crack growth onset direction remains unaffected by the presence of antiplane shear stresses, i.e., it is the same as that due to the presence of the mixed mode only of [1] (k3 = 0). It may be recalled that, along this direction, the fracture process is characterized by the symmetric displacement of the crack surfaces due to the action of the crack-tip tangential stress. A significant outgrowth of the theory is the analytic determination of the generalized threshold conditions for materials subjected to complex fatigue loadings. The Fracture Envelope concept distinctly simplifies the criterion governing the conditions of crack propagation due to complex loadings: from the computation of the stress intensity factors k~°~(s= 1, 2, 3) characterizing the response of the structure under the applied load, crack propagation ensues if the point of coordinates k~~is on or
1.6
0.8
O. ~
0
Nongrowth
\
o
0
0.4
0.8
1.2
l AK~ - MPa.m 2
1.6
2.0
Figure 6. Threshold envelope and test results [5] for aluminum plates subjected to mixed-mode fatigue loading.
Generalized .fracture toughness of brittle materials
323
outside the Fracture Envelope surface, i.e., if:
The values a, b, c, and m are constant for a given material. Therefore, the analysis of structures of the same material is performed with the aid of a single Fracture Envelope that, consequently, represents the generalized fracture toughness of that material.
REFERENCES [1] G. Di Leonardo, International Journal of Fracture 15 (1979) 537-552. [2] G.C. Sih, Mechanics of Fracture I, Noordhoff International Publishing, Leiden, Holland (1973) XXI-XLV. [3] G.C. Sih, International Journal of Fracture l0 (1974) 305-321. [4] G.C. Sih and H. Liebowitz, Fracture II 0968) 100-103. [5] K. Tanaka, Engineering Fracture Mechanics 6 (1974) 493-507.
RI~SUMI~ On pr6sente une g6n6ralisation du concept de t6nacit6 h la rupture de mat6riaux fragiles soumis ~ des contraintes multiaxiales. La th6orie propos6e caract6rise la r6sistance ~ la rupture des mat6riaux sous toutes les combinaisons possibles des trois modes de base des d6placements des surfaces d'une fissure. Par rapport ~tun syst~me de mise en charge ~ deux dimensions, la t6nacit6 h la rupture peut 6tre repr6sent6e dans un espace orthogonal cart6sien K1KjK3 par une Enveloppe de Rupture caract6ristique d'un mat6riau donn6 dont l'6quation est d6termin6e par la t6nacit6 h la rupture sym6trique K~c et le module de Poisson ~,. On montre que le processus de rupture le plus g6n6ral qui r6sulte de la combinaison d'une ouverture sous l'effet de la composante tangentielle de la contrainte et d'un arrachement sous l'effet du cisaillement antiplanaire peut ~tre analys6 d'une mani~re satisfaisante ~ l'aide du concept de t6nacit6 ~ la rupture g6n6ralis6e. A partir de la connaissance de l'Enveloppe de Rupture relative ~ un mat6riau de construction d6termin6, un crit~re simple de rupture permet de pr6voir la propagation d'une fissure pour toutes combinaisons de contraintes et de g6om6tries. La th6orie est appliqu6e ~ des probl~mes de fissure suivant des modes mixtes en vue de d6finir de mani~re analytique le seuil de propagation d'une fissure de fatigue.
