J O U R N A L O F M AT E R I A L S S C I E N C E L E T T E R S 1 6 ( 1 9 9 7 ) 1 5 3 3 ± 1 5 3 5
Strain criterion of fracture in brittle materials Y I W A N G B A O , R . W . S T E I N B R E C H Institut fuÈr Werkstoffe der Energietechnik, Forschungszentrum JuÈlich GmbH, 52425 JuÈlich, Germany
The strength and fracture behaviours of brittle materials are usually evaluated on the basis of stress concepts [1±3]. This approach provides a reliable failure characterization in all cases of uniaxial static stressing within the boundaries of linear elastic fracture. However, it is also known that it has limitations when dynamic [4, 5] and biaxial [6, 7] stress situations are given for brittle materials. It seems reasonable to consider the strain caused by applied stress rather than the stress itself as the relevant crack opening parameter, so that the effect of applied stress is embodied in a more general strain approach. However, as a consequence, the stress criterion will not be valid in all cases of brittle fracture, e.g. it will not apply when the stress-strain relationship is not simply proportional. Experimental arguments are presented below to support the understanding of failure as a strain controlled process. Indeed, there is evidence in theory and experimental practice that the fracture of a brittle solid is dominated by strain. According to the basic assumption of the theoretical strength analysis [8], the fracture of a solid occurs when the interatomic distance attains a critical value, i.e. an interatomic bond must be stretched critically to cause fracture. Macroscopically the increase of interatomic distance is re¯ected by strain. Bilby et al. [9] reported that a crack under static tensile load will only propagate catastrophically when the displacement caused by the load at the crack tip reaches a critical value. Finite element analysis [10] also showed that the determination of the stress intensity of a crack by displacement extrapolation (with an error less than 1%) is much more accurate than by stress extrapolation (up to 40% error). Also, the elastic link model [11] indicates that a prerequisite to crack growth in the regime of linear elasticity is the attainment of a critical strain at the crack tip. For brittle materials, the stress to failure is strongly in¯uenced by many factors, such as loading speed, stress state, specimen geometry, etc. The strain to failure, however, is only slightly in¯uenced by these factors [4, 5, 7, 10]. Hatano [4], after having investigated various types of failure in compression experiments with concrete and mortar for several years, came to the conclusion that failure of a brittle solid in compression can be formulated in terms of an ultimate strain theory. His experimental results showed that the stress to
Guest scientist supported by A.v. Humboldt Foundation, Bonn, Germany
0261-8028 # 1997 Chapman & Hall
fracture varies markedly with loading speed but ultimate strain remains nearly constant irrespective of loading speed. Furthermore, the Weibull modulus of ultimate strain was much higher than that of stress to failure for concrete and mortar. Similar experimental results can also be found in [12]. The variation of bending strength with loading rate was studied in the range from static fatigue to high speed impact [5] for different ceramic materials. The experiments revealed that the fracture strength increased greatly with loading rate but the ultimate strain did not vary by the same large amount. Deviation from the linear stress-strain relation under high speed loading was considered as an explanation. It was predicted that the transient peak stress under high speed impact load could approach the inherent theoretical strength of the material [5]. Subcritical crack growth experiments in ceramics [13, 14] showed that the stress to critical strain can degrade with loading time and fracture resistance is actually the strain resistance at the crack tip. Experimental studies [15, 16] indicate that the failure in many cases can be described by the strain intensity factor which is de®ned by p (1) Kå å y . Y . a where åy is the strain normal to the crack plane, Y is a geometrical factor and a is the crack length. Considering the effect of Poisson's ratio í, it was shown that the strain intensity factor is affected by the stress parallel to the crack plane, i.e. p (2) K å (ó y ÿ í . ó x ) . Y . a where ó y , ó x are the stresses normal and parallel to the crack plane, respectively. Fracture occurs when the strain intensity factor attains a critical value. Note that, in the uniaxial stress state, the strain intensity and stress intensity criteria are identical. If the critical strain intensity is taken as K IC =E, the fracture condition for biaxial plane stress has the form p (3) ó y ÿ í . ó x > K IC =(Y a) Equation 3 indicates that K IC will increase with the biaxial stress ratio, as has been demonstrated by experiments with foil specimens [7, 17]. The fact that a crack propagates along the direction normal to the maximum strain was observed in many cases. Under uniaxial compression, for example, a visible crack in brittle material extends parallel to the loading direction [18, 19]. All these phenomena can not be explained by the 1533
which indicates that crack growth should not be affected by a multiaxial load, but this conclusion was not con®rmed in tests of biaxial plane stress as follows. A thermomechanical testing method was developed to accomplish biaxial load for brittle foil materials [7]. Experiments with over 50 glass sheet specimens show that, for different loading rates, the critical stress intensity in biaxial tension is higher than that in uniaxial tension (Fig. 1). This result agrees with the experimental investigation of Kibler and Roberts [17] on elastic-plastic sheet specimens. The fracture toughness increasing with the tensile stress parallel to the crack plane re¯ects the strain dependence of fracture in the plane stress state. Since the strain at the crack tip is different for an elastic body under uniaxial and biaxial loading, the crack opening displacement will also be different even when the stresses normal to the crack in both cases are the same. The ®nite element method (FEM) conveniently provides the demonstration of the in¯uence of biaxial load on crack opening. Fig. 2 shows the calculation results with FEM for the biaxiality effect on crack opening of glass sheet. The fracture criterion of crack opening displacement is actually the strain criterion because the crack opening is proportional to the strain at the crack tip. Fig. 2 also provides an explanation for the biaxiality in¯uence displayed in Fig. 1. Another example demonstrated that the direction of crack growth is normal to the minimum strain resistance direction. For a homogeneous brittle material, a crack always propagates perpendicular to the direction of minimum strain restraint. The above statement can be con®rmed by indentation trials on small ceramic specimens with different cross-section (Fig. 3). This shows that the indentation crack size introduced by the same load varies with the strain restraint. At the location of the impression, the strain resistance in the x-direction,
1.0
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0
1
2 3 6 Loading rate, (8C min21)
10
Figure 1 Comparison of fracture toughness of glass foil between equibiaxial (j j) and uniaxial (j j) tension in thermomechanical tests for different loading rate. 1 8C minÿ1 strain rate of 11.2 3 10ÿ6 minÿ1 for this test.
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1.6 1.4 1.2
0.8 0.6 0.4 0.2 0
0.1
0.2 0.3 0.4 Position of crack edge, (mm)
Crack opening displacement, ( µm)
1.8
Fracture toughness, (MPa m½)
traditional fracture theory but are in agreement with the strain criterion. The stress intensity factor depends on only the stress normal to the crack plane p (4) KI ó y . Y . a
0 0.5
Figure 2 The calculation results of the crack opening displacement in glass sheet by FEM for uniaxial (r) (ó 10 MPa) and biaxial (j j) (ó1 ó2 10 MPa) tension, K I 0:4 MPa m1=2 ó 10 MPa, í 0.22, E 73 GPa, 2a 1 mm, plane stress model.
x
y
H
L
W
Figure 3 Schematic of indentation on small H-P Si3 N4 specimen and the indentation crack varying with different strain restraint at the indentations.
Råx , is smaller than that in the y-direction, Rå y , (Råx , Rå y ) due to a smaller stiffness in the xdirection (W , L). This experimental result reveals that the crack resistance is proportional to the strain resistance. As conclusions from the above results the following aspects should be emphasized. Fracture means material separation, and the separation depends on the value of strain. The ultimate strain represents a material property. The way to generate the strain is not unique and also not important for ®nal fracture. Of course, the failure causing strain must be non-equilibrium, the strain from free expansion has no effect on fracture. Since the measurement of stress is easier and more convenient than that of strain, the fracture community usually evaluate the fracture behaviour of solids with stress concepts. However, this is appropriate only for the uniaxial stress state within the con®nes of static mechanics and linear elasticity because a simple linear relationship between stress and strain is available in this case. The stress
approach is thus a special case of the strain concept for critical fracture of brittle materials. For mode I fracture, no matter how complex the stress state, the crack growth is controlled solely by the normal strain at the crack tip. When an applied stress does not cause any strain normal to a crack plane (due to the Poisson's ratio effect in the multiaxial stress state), the stress has little in¯uence on crack growth. Under complex stress conditions, and without an analytical expression of the stress intensity factor, it is possible to utilize the strain value to evaluate the stress intensity factor or failure analysis, if the strain can be calculated or measured.
5. Y. B AO and Z . J I N , Nucl. Eng. Des. 150 (1994) 323. 6. J. L . S W E D L OW, Int. J. Fract. Mech. 1 (1965) 210. 7. Y. B AO and R . W. S T E I N B R E C H , Proceedings of the
Symposium on Materials and Test, (Germany, 1996) p. 325.
8. M . B O R N and K . H UA N G , ``Dynamical Theory of Crystal
Lattices'' (Clarendon Press, Oxford, 1954).
9. B . A . B I L B Y, A . H . C O T T R E L L and K . H . S W I N D E N ,
Proc. R. Soc. Lond. A 272 (1963) 304.
10. W. X . Z H U and D. J. S M I T H , Eng. Fract. Mech. 51 (1995)
391.
11. Y. B AO and R . W. S T E I N B R E C H , ``Plane stress fracture 12. 13.
Acknowledgement The authors thank Dr J. H. You for his computation with the ®nite element method.
14.
References
17.
1. A . A . G R I F F I T H , in Proceedings of the First International
Congress on Applied Mechanics, Waltmen, Delft, (1924) p. 55. 2. J. M E N C I K , ``Strength and Fracture of Glass and Ceramics'', (Elsevier Science Publishing, 1992) p. 103. 3. D. B R O E K , ``Elementary Engineering Fracture Mechanics'', (Kluwer Academic Publisher, USA, 1986) p. 33. 4. T. H ATA N O , Int. J. Fract. Mech. 5 (1969) 73.
15. 16.
18. 19.
toughness of glass under biaxial tensile stress'', accepted by Ninth International Conference on Fracture, (Sydney, 1997). R . W. D AV I D G E and D. C . P H I L I P S , J. Mater. Sci. 7 (1972) 1308. R . W. S T E I N B R E C H , ``Fracture Mechanics of Ceramics'', Vol. 9 (1992) 187. D. S . JA C O B S and I . - W. C H E N , J. Am. Ceram. Soc. 78 (1995) 513. E . H . J O R D A N and G . J. M E Y E R S , J. Eng. Mater. Technol. 111 (1989) 307. A . O H TA , N . S U Z U K I and T. M AWA R I , Int. J. Fatigue 14 (1992) 224. J. J. K I B L E R and R . R O B E RT S , J. Eng. Ind. 92 (1970) 727. E . Z . WA N G and N . G . S H R I V E , Eng. Fracture Mech. 52 (1995) 1107. I . H AW K E S and M . M E L L O R , Eng. Geol. 4 (1970) 177.
Received 15 November 1996 and accepted 11 March 1997
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