The effect of the stress far field on the crack tip behaviour

The boundary value problem of that the The


-INTRODUCTION
The knowledge of stresses and displacements near a fault is one of the main purposes of Geophysics. However, for a fault model which allows a mathematical description we must concern ourselves with fairly idealized situations for the present. In this paper we shall start with the two-dimensional model of a crack which has received much attention since the work of Inglis (1913) and has been the main source of progress in fracture mechanics. As far as the crack tip behaviour is concerned some other problems can be of geophysical interest as the real effect of the far field on the crack tip stress and displacement distributions and on the crack propagation criteria.
In recent publications (Carpinteri, 1968;di Tommaso ed al., 1976;Viola, 1978) it has been shown that the singular for a plane cracked body subjected to biaxial tension at infinity are insufficient for a correct description of the crack tip behaviour.
The arbitrary omission of non singular terms depending on the biaxial load parameter K causes, in general, incorrect predictions on physical quantities such as the maximum tensile stress, the elastic strain energy density and the local elastic strain energy rate.
It follows that the crack propagation criteria such as the maximum tensile stress criterion and the minimum strain energy density criterion will fail in their predictions (Carpinteri, 1968;Di Tommaso et al., 1976;Viola, 1978). Apparently surprising is the independence of the biaxial load parameter, shown in  of the //-integral which is connected with a crack extension criterion, along the direction of the crack, through the attainment of a critical value 7i c . This independence was justified by  considering that the //-integral is connected with the global strain energy rate which appears to be independent of the horizontal load applied parallel to the plane of the crack. It seems then that the biaxial load affects only the local quantities.
The aims of this paper are to describe the elastic fields of stresses and displacements near the tip of a central crack in an infinite elastic sheet loaded with a uniform shear and biaxial tension at infinity. Some considerations are made on the influence of non singular terms on crack propagation criteria the independence of the J vector from the biaxial load parameter is proved.

-STATEMENT AND SOLUTION OF THE ELASTIC PROBLEM
We consider an infinite purely elastic body with a plane crack loaded at infinity, as shown in fig. 1. The solution of the elastic problem under plane strain or plane stress conditions is reduced to the evolution of the potential functions Q (z) and 0 (z) holomorphic in a plane parallel to the xy plane and cut along the trace of the crack. These functions are related to stresses and displacement functions by the well-known relations [7] (Muskelishvili et al., 1953) <3XX + Vyy = Z rnYz) + «'(!)] 3 -4v for plan strain 3 -v for plane stress 1 + v where D = u + iv is the complex displacement, |x is the shear modulus and v is the Poisson ratio. We have for the stress distribution the following far field conditions: where K is a real constant which gives a measurement of the biaxiality of the stress field at infinity. Moreover, we have the boundary conditions on the traction-free crack surface: Moreover, after substitution of [2.1] 2 into [2.3] we obtain the Hilbert problem: Making use of [2.4] we obtain the solution of [2.5] in the following form: where X(z) = (z 2 -a 2 )Vi is the Plemelej function of the problem (Muskelishvili, 1953), whose branch is selected in such a way that X (z)-> z for large | z | • By integration of [2.6] we have also: n .
(K -l)r + 2is with: where K t = rVraz and K 2 = SVTOI are the stress intensity factors for the opening and sliding mode respectively. For the complex displacement function we obtain: where: where, We see that, as pointed out also in [3,4] for the case of simple biaxial tension, the stress and displacement functions 13.3] and [3.4] depend on the far field also through terms which are non singular at the crack tip. The arbitrary omission of ihese terms can lead to incorrect predictions on stress and displacement distributions and on crack propagation criteria.

THE ELASTIC STRAIN ENERGY DENSITY CRITERION
For the plane problem the elastic strain energy density at any point of the plane body is given by: According to Sih's theory (Sih, 1973) on the prediction of the direction of initial crack extension, crack growth will start along the radial direction for which the local elastic strain energy density attains a minimum. In view of this hypothesis the growing direction will be a solution of the following conditions: where the radial distance r D is very small, but unspecified.
Recently, an estimation of the range of r 0 was given in the case of pure tension applied at infinity [2, 6], As shown in [4,5] for the case of a biaxially loaded crack the value of cp for which the elastic strain energy density attains a minimum is a function of K, as confirmed by experimental data. Choosing t'o = 10" 2 a in agreement with the condition on the distance r a from the crack border, the results of plotting the expression [4.5] vs. cp in the case of plane stress are shown in figs. 3-6 with K --1, 0, 3, 5, 10, the Poisson ratio v = 0.1, 0.5 and n = 0, 1.
For n = 0 we obtain the same results ans in [2,3,4,5,5,6] corresponding to the biaxial tension-compression loads. The angle of initial crack extension corresponding to the minimum of the strain energy density occurs at cp = 0° for K = -1 (and in general for K < -1) for all values of the Poisson ratio. For K = 0 this angle occurs at cp = 0° or cp = 90° depending on the values of the Poisson ratio and for K -3 at cp = 90°. The graphs show also that if K is very large (K ^ 5) the angle of initial extension occurs at cp = 90° for v = 0.1 and an increase of the Poisson ratio shifts this angle at cp -80° ( fig. 4). For n -1 the angle of initial crack extension is predicted to be greater than 90° for any value of K and v, that is the crack should bend back towards its centre. Since behaviour of this kind as never been reported this portion of the curves are shown by dashed lines. It seems that in the above circumstance the Sih criterion is inadequate.

THE MAXIMUM NORMAL STRESS CRITERION
The maximum normal stress criterion proposed firstly by Yoffe (1951) and reviewed by Erdogan and Sih (1963) is concerned with the direction of crack extension too and maintains that a crack will begin to extend radially along the plane on which the stress normal to this plane attains a positive maximum value. If cp Q is the angle of the tangent to this plane, the criterion requires that, near the tip of the crack, we should have: The variation of the angle cp" of initial crack extension is shown in figs. 7 and 8 where the function erg /T is plotted for different values of K. We have considered that S = nT with n = 0,1 and r 0 /a -0.07. This last condition satisfies [5.1]j and the requirement tha 0 < rja 1. It. can be seen ( fig. 7) that for n = 0 and K -2 the maximum occurs at cp Q = 0". As far as K increases tp 0 increases up to cp" ^ 80 u for K = 6. In fig. 8 can be seen the influence of the shear stress field. We note that for n = 1 the maximum of the normal stress occurs from cp 0 --30° to cp 0 --70° depending on the value of K.
Here again we find evidence of the influence of non singular terms contribution on the prediction of the angle of initial crack extension.

THE LOCAL ELASTIC STRAIN ENERGY RATE AND THE /-VECTOR
The elastic strain energy W due to the presence of the crack is independent of the horizontal stress applied at infinity. In fact, we have: Moreover, the local elastic strain energy W c obtained by integratio of [4.3] over a circular region centred at the crack tip with radius 0 < r c « 1 has the following expression: (2x -1) T 2 + (2x + 3) 2 (S 2 ) la 2 T 2 (r a 2 ) I + r a (2x-l)T 2 +(2x+3)S 2 T 2 + + 16 (K -1) (5x -7) \T 0 15ti V 2" [3.4] We see that the global strain energy rate is independent of the biaxial stress at infinity which otherwise has a significant influence on the local elastic strain energy rate. This behaviour was pointed out also in [4,5] for the case of simple biaxial tension at infinity. Rice (1968) showed that the energy release rate for a twodimensional crack extending in its plane in a homogeneous material was equal to a path-independent integral Ji formulated by Eshelby (1956) in the theory of lattice defects, and applied to crack problems by Sanders (1960) and Cherepanov (1969). Knowles and Stenberg (1972) generalized Ji to a vector Jk corresponding to the energy release rate for a movement in any direction of the crack edge. Budiansky and Rice (1973) have shown how the formula may be simplified for a homogeneous isotropic material by using the theory of functions of a complex variable. Recently some authors (Bui, 1976); Carpinteri, 1978) showed that the energy release due to the extension of the crack along some direction is equal to the component of the J vector along the direction of extension.
Here we formulate an expression for the /-integral in terms of the fundamental combinations of stresses and the derivatives of the complex displacement. The two components of the J vector are: Thus, the value of the J (/,, / 2 ) is independent of the non singular terms appearing in the expressions of stresses and displacement, as was shown for /i-integral by  in the case of a simple biaxial load.

CONCLUSION
Assuming a two-dimensional crack as a model of a fault some results are obtained on the basis of recent progress in fracture mechanics which could be of interest in Geophysics.
As far as the crack tip behaviour is concerned the influence of the stress far field through terms which are non singular is pointed out. The arbitrary omission of these terms should cause an incorrect prediction on stress and displacement distributions and on local crack propagation criteria. It appears that the biaxiality of the stress far field affects only the local strain energy release rate and the independence of the /-vector from the biaxial load parameter is shown.