A GENERALIZED GRIFFITH CRITERION FOR ... Succeeding investigators have generalized Griffith's analysis, but still employing .... sign to the applied shear.
0013-7944184 13.00+ .w Pergamon Press Ltd.
Engineering Frocfun Mechanics Vol. 19. No. 3. pp. 539-543. 19W Printed in Great Britain.
A GENERALIZED GRIFFITH CRITERION FOR CRACK PROPAGATION? L. G. MARGOLIN Earth and Space Science Division, Los Alamos National Laboratory,
Los Alamos, NM 87545, U.S.A.
Abstract-Griffith’s criterion for crack propagation is generalized to consider penny-shaped cracks in a spatially uniform, but otherwise arbitrary, state of stress. A further generalization incorporates the effect of interfacial friction for cracks closed in normal compression. Various criteria for crack propagation are discussed in terms of recent experimental evidence for process zones in which the region around a crack tip is seen to be permeated by smaller microcracks.
INTRODUCTION A CRITERIONfor deciding when cracks may grow is essential to a theory of fracture. Griffith[l] first stated such a criterion-that a crack will grow whenever that growth reduces the potential energy of the crack and the material that contains it. Griffith applied this criterion to the case of a two-dimension crack (slit) in normal tension. He found that the crack could grow if the rate of release of elastic strain energy from its growth exceeded the rate at which surface energy of the crack is increased. Succeeding investigators have generalized Griffith’s analysis, but still employing his criterion. Mossakovskii and Rybka[2] extended the analysis to two-dimensional cracks in plane stress or plane strain. Keer[3] further extended the analysis to three-dimensional cracks (penny-shaped cracks) in a biaxial state of stress. In this paper, we offer two generalizations, still based on Griffith’s criterion. First, we will consider a penny-shaped crack in a triaxial state of stress. This is the general case of a three-dimensional crack in a spatially uniform, but otherwise arbitrary stress field. The analysis shows that the presence of a shear stress enhances the instability of cracks in normal tension and makes crack growth possible under normal compression. McClintock and Walsh[4] have pointed out that friction is important for cracks closed in normal compression. The effect of friction may be included by adding the frictional work to the energy balance equation. The presence of friction increases the stability of the cracks. More important, friction introduces hysteresis into the stress-strain relations of a cracked body. TRIAXIAL STATE OF STRESS The condition that a crack may extend, according to Griffith[l], is
$(W-U)ZO,
(1)
where W is the strain energy, U is the surface energy, and c is the crack radius. The surface energy is given in terms of the surface tension T, r_J= 2PCZT.
(2)
The strain energy is found by integrating the local strain energy density over the body B
W =i
IB
d3XUii$.
(3)
The strain can be written in terms of the displacements Ui (4)
*Work supported by U.S. Department of Energy. 239
540
L. G. MARGOLIN
Equation (3) can be written
where we have made use of the equilibrium equation aoir/axi = 0. Finally, eq (5) can be written in terms of a surface integral W=5
dS[oiluini],
fs
(6)
where the surface S includes both the exterior surface of the body Sg and the crack surface Sc. If we apply a uniform stress - oij to the entire body, we have the equivalent problem [5] of an applied stress oii inside the crack and zero stress on the outer surface. Then we can write W=k
dS[oijuiur], f SC
(7)
where nj is the outward pointing normal to the crack. Since oijai is the force per unit area, oil”jUiis the work per unit area, and the integral is the work done in displacing the crack surfaces. Let us now refer to the coordinate system shown in Fig. 1. In this system, nj = 2 on the upper surface, and ttj = -2 on the lower surface. Thus, only the stress components a,,, uyyz,and u,, can contribute to the energy. Now the strain energies due to each of these components have been computed separately. The normal component can be found in Sneddon[S] and the shear components in Segedin[6]. All that remains is to show that these energies can be added. Let us write the displacement field due to an internal stress T,, as Ui’,that due to r,, as uf and that due to 7yZas u:. There are two ingredients to proving the additivity of the energies. First, the elastic equations are linear, thus allowing superposition of the elastic solutions for the individual internal stresses. That is, the displacement due to the internal stress rij is Ui =
Ui’
+
U'
+ Ui3.
(8)
Second, there is the detailed form of the displacements Ui’,~2, and u: on the crack surface. These can be calculated from Bell[7] and are, in the upper half plane, ui’ = 4uZZ(1- “*)dc’?rE
x2 - y2i (if a,, > 0),
ui2= 2uxfE- ‘) d/c2 - x2 - y2i, and v) Vc’_ x2 - #ji, ui3= 2417TE where v is Poisson’s ratio and E is Young’s modulus.
Fig. I. The crack-oriented
coordinate
system.
(9)
A generalized Griffith criterion for crack propagation
541
The displacements are orthogonal to each other, and so the total strain energy, from eqn (7) is W = ; s dS(u,,u,’ t a,,~:+ uyru;). f c
(10)
The integrals are easily performed, leading to
w = $1
-
v3&‘+&(1-
v)(a2,, + u$)c3.
(11)
This is in agreement with the results of Sneddon[6] and Segedin[7]. Substituting eqn (11) and eqn (2) into the Griffith criterion, we derive the criterion for crack growth in triaxial stress and normal tension,
(12)
EFFECT OF FRICTION IN CLOSED CRACKS
When the normal stress is compressive, the cracks are closed and friction plays an important role [4]. We assume that the frictional stress has the form
where r. is a cohesion and F is the dynamic coefficient of friction. The frictional stress has the opposite sign to the applied shear. For convenience, we now choose a coordinate system in which a,, is positive and uYZvanishes (this is always possible). The extra strain in the body due to the crack is proportional to the displacement of the crack faces, which is proportional to the effective stress on the crack faces. The effective stress is
The crack displacement, from eqn (O),is
uf=2(1+ dc’ - x2- y$.rxz 7TE
70 t
/ur,,)a.
(1%
The strain energy in this case cannot be calculated from the surface integral [eqn (7)]. The reasons are discussed in the appendix where it is shown that w=4(1-u)c3 3E
2 bXZ
T21’
(16)
- T2].
(17)
-
It is also shown that the work done against friction is F
=
8(1-
4c3[UJ
3E
Now we can generalize the Griffith criterion to include the work done against friction
(18) The effect of friction is to reduce the effective stress in the Griffith criterion which now can be written
(19)
542
L.G.MARGOLIN DISCUSSION
Several criteria have been devised to predict crack propagation. One example is the criterion described in this paper that is based on energy considerations. An alternate approach consists of looking at the stress field at the tip of a crack and asking whether the force is sufficient to break interatomic bonds[8]. Still a third approach is based on a combination of these two ideas. A stress intensity factor is defined in terms of a generalized force derived from the strain energy release rate[9]. The energy criterion corresponds to a macroscopic approach. The details of bond rupture and microscopic dissipation are hidden in the energy to create new surface area (the surface tension T in Section 2). Clearly, the energy criterion must represent a necessary condition for propagation if we are to respect the first law of thermodynamics. The sufficiency of the energy criterion has been shown in some cases. The calculation by Orowan[lO] is closely related to the stress intensity approach. Recent experimental work[ll, 121 has focused on the crack tip. In this region, one finds tiny microcracks which are nucleated and continue to grow due to the presence of the large crack. If one accepts this picture, two important points must be considered with respect to crack propagation. First, the region of energy dissipation is not only at the crack tip, but in a finite zone (the process zone). This implies the linear elastic theory breaks down not because of the failure of the continuum hypothesis, on interatomic length scales, but because of the failure of the assumption of linear elasticity, and on a much larger length scale. If one assumes an elastic plastic medium, then the plastic work scales [13] as the crack length cubed and the process zone is some (possibly large) fraction of crack length. Then the force calculated from a stress intensity approach may not be localized at the crack tip and may not represent a sufficient condition for rupture. The second point to be considered is that if one accepts the process zone picture, then crack growth should be viewed as a coalescence process rather than a rupture process. The calculation of force at the crack tip may not represent a necessary condition for fracture. To summarize, calculation of the force at a crack tip may not lead to a necessary or a sufficient condition for crack propagation. Focusing on a microscopic picture of crack growth is misleading if we focus down on the wrong picture. The macroscopic approach of the energy criterion may be its strength as well as its weakness. By not focusing on a microscopic process of extension, it is applicable to either the rupture or the coalescence picture. In particular, energy dissipation that does not assume a constant surface tension, as suggested by Nichols[l4], is easily incorporated into the criterion. The purpose of this paper is to demonstrate that the energy criterion for crack growth is a powerful tool. Our result for propagation in general stress states is equivalent to saying that the stress intensity factors for modes I, II, and III loading can be added. Cracks in tension are more unstable, that is, have a smaller minimum size for growth in the presence of shear. When cracks are in normal compression, one must consider the effect of friction. Andrews[lS] showed that the effect of friction in shear cracks was to simply reduce the effective loading shear stress. Our results of Section 3 verify this. However, we see that this result depends upon including the frictional work in the energy balance. The results of Section 3 show that cracks can propagate in normal compression if the shear stress is sufficiently large. However, in the presence of large compressional stresses, the cracks may be locked in place in spite of large shear. This phenomenon is probably related to the brittle-ductile transition observed in many rocks and metals. APPENDIX
In this appendix we consider the calculation of the strain energy associated with a shear crack in the presence of friction. There are two results. First we give mathematical and physical reasons to illustrate that one cannot use eqn (7) (rewritten here as eqn Al) to calculate the strain energy. Second, we present a more general calculation of the strain energy. The alternate method reduces to the surface integral in the absence of friction. Consider again the surface integral of eqn (7)
W =1
dS[oirUinr].
(Al)
The integral of force through displacement represents the work done by external forces on the body. On the surface, this must include the work done against friction, and so eqn Al could not be equal to the strain energy.
A generalized
Griffith
criterion
for crack
propagation
543
In fact, eqn Al is not the total work either. Equation (3) in the text is derived as the integral of force through displacement in a case where stress is proportional to strain. In the medium this assumption is valid, but on the surface the strain is proportional to the net stress which is the difference of the applied stress (u,..) and the frictional stress (7). This is, in fact, eqn (15). Integrating the force through displacement here leads to the total work (Q) done by the external stress
W) where we define a= f s,
8(1 - v)c3 -2(1-v)~(c2-,2)rdr= 3E . 7rE
(A3)
To derive A2, we integrate the external stress from T to a,,. This represents a particular assumption of loading where displacement begins when the external stress exceeds the frictional stress, and that the frictional stress is constant during subsequent loading. With this loading path, the work done against friction is
T2](Y.
(A4)
;rui, -T2](Y
(A9
F = [f&,7 -
Thus, the strain energy W is the difference Q - F or
w=
The reason eqn Al cannot be used is due to the nature of the frictional stresses. Physically, the stresses of elasticity arise from relative displacement (i.e. strain) of points in the medium. Friction, on the other hand, represents a stress that arises in the absence of displacement. It is precisely because there is no displacement that the work done by the frictional stresses is not recoverable. Mathematically, the problem lies in the use of the divergence theorem to convert the volume integral of eqn (5) into the surface integral of eqn (6). The use of the divergence theorem requires that the integrand (UiiUj)be continuous and have a continuous derivative. However, the strain is not continuous as one approaches the crack surface. Again, this is because the stress in the body is elastic, and so due to strain, whereas part of the stress on the surface is due to friction. The problem really lies in the breakdown of the model. One simply cannot deal with friction in the framework of elasticity. However, the arguments leading to A5 are more general, being based on the mechanical definition of work and on the conservation of energy. REFERENCES [I] A. [2] [4 [4] [5] [6] [7] [8] [9] [IO]
A. Griffith, Phenomena of rupture and flow in solids. Phil. Trans. Roy. Sot. A. 221, 163-198 (1920). Y. A. Mossakovskii and M. T. Rybka, An attempt to construct a theory of fraction for brittle materials based on Griffith’s criterion. Prih. Mat. 1. M&h. 29, 326-332 (1965). L. M. Keer, A note on shear and combined loading for a penny-shaped crack. J. Me& Phys. Solids 14, l-6 (1%6). F. A. McClintock and J. B. Walsh, Friction on Griffith cracks in rocks under pressure. Proc. 4th U.S. Nat. Cong. Appl. Mech. pp. 1015-1021 Berkeley, California (1%2). 1. N. Sneddon and M. Lowengrub, Crack Problems in the CIassical Theory of Elusticity. Wiley, New York (1969). C. M. Segedin, Note on a penny-shaped crack under shear. Proc. Camb. Phil. Sot. 47, 396-400 (1951). J. C. Bell, Stresses from arbitrary loads on circular cracks. Int. J. Fracture 15, 85-104 (1979). G. I. Barenblatt, V. M. Entov and R. L. Salganik, Some problems of the kinetics of crack propagation. In Inelastic Behavior of Solids (Edited by M. F. Kanninen et al.), pp. 559-584. McGraw-Hill, New York (1970). G. R. Irwin, Analysis of stresses and strains near the end of a crack. J. App. Mech. 24, 361-364 (1957). E. Orowan, Energy criteria of fracture. Welding J. Supp. 34, 157-160 (1955).
R. G. Hoagland and J. D. Emburg, A treatment of inelastic deformation around a crack tip due to microcracking. J. Amer. Cer. Sot. 63, 404-410 (1980). 1121 R. G. Hoagland, G. T. Hahm and A. R. Rosenfield. Influence of micro-structure on the fracture propagation in rock. Rock Mech. 5, 77-106 (1973). ]l3] Z. Olesiak and M. Wnuk, Plastic energy dissipation due to a penny-shaped crack. Inf. J. Fracture Mech. 4, 383-396 (1968). [I41 F. A. Nichols, How brittle is brittle fracture? Engng Fracf. Mech. 12, 307-316 (1979). ]I51 D. J. Andrews, Rupture velocity of plane strain cracks. J. Geophys. Res. 81, 5679-5687 (1976). [Ill
(Received
21 June 1982; receiued for publication
25 November 1982)
