International Journal of Fracture 99: 1–11, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands.
Intersonic crack propagation in an orthotropic material K.B. BROBERG Department of Mathematical Physics, University College Dublin, Belfield, Dublin 4, Ireland Received 19 February 1997; accepted in final form 22 August 1997 Abstract. Intersonic crack propagation is found to exhibit essentially the same features in orthotropic and isotropic materials, provided that the crack propagates along a plane of elastic symmetry. Thus the stress and strain singularity at the crack edge is weaker than the inverse square root singularity in the sub-Rayleigh case, except at one distinct velocity. The energy flux into the process region is determined by using the Barenblatt model. It depends on the crack velocity and on the size of the process region, approaching zero with this size. Key words: Intersonic crack propagation, mode II crack, Barenblatt region, energy flux.
1. Introduction It can be demonstrated that mode II crack propagation is possible in elastic isotropic materials at intersonic velocities, i.e., at velocities between S and P wave speeds (Freund, 1979 to Broberg, 1995). On the other hand, the velocity region between Rayleigh and S wave speeds is forbidden, because a solution associated with positive energy flow to the crack edge cannot be found. A formal calculation results in an energy flow away from the crack edge. The energy flux into the crack edge at intersonic mode II crack propagation in an isotropic material depends critically on the size and properties of the process region. This fact is dependent on the weaker singularity in stresses and displacement gradients compared to the inverse square root singularity at sub-Rayleigh crack velocities. Thus, if the singularity is expressed by r −g , 0 < g < 1/2, then the energy flux is proportional to (d/a)1−2g , where a is a crack length parameter and d expresses the linear extension of the process region (Broberg, 1994; 1995). This contrasts to sub-Rayleigh crack propagation, in which the energy flux is very insensitive to the size and properties of the process region, provided that it is small compared to the crack length. This fact is sometimes taken for granted, and the energy flux is then calculated by assuming a priori vanishing process region. However, by using a Barenblatt model (Barenblatt, 1959a, b; 1962), the energy flux can be easily calculated for a finite process region (Broberg, 1964; 1967). A Barenblatt region preserves the linearity of the problem, which, for instance, allows a solution to be obtained by superpositions of solutions for cases of vanishing process regions. For intersonic velocities, the introduction of a Barenblatt model appears to be the only way for determining the energy flux to the process region. However, in √ the isotropic case, at the curious crack velocity 2cS , where cS is the velocity of S waves, an inverse square root singularity appears for stresses and displacement gradients, which implies a finite energy flow to the crack edge, even to an infinitesimally small process region. In the present work, intersonic mode II crack propagation is studied for an orthotropic material, assuming that the crack propagates along a plane of elastic symmetry. From a physical point of view it is noted that such crack propagation is likely if this plane of elastic
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symmetry is also a weak plane. Note also that mode II can be completely decoupled from modes I and III. 2. Basic relations and statement of the problem Under conditions of plane strain, the generalized Hooke’s law for the in-plane stresses and strains in an orthotropic solid, reads σ11 = c11 u1,1 + c12 u2,2
(1)
σ12 = c66 (u1,2 + u2,1 )
(2)
σ22 = c12 u1,1 + c22 u2,2 ,
(3)
where cαβ , α, β = 1, 2, 6, are the elastic stiffnesses in contracted notation, satisfying the symmetry conditions cβα = cαβ . Because the stress-strain energy is positive, if not all strains are zero, the matrix cαβ is positive definite. This implies, for instance, that all diagonal 2 elements are non-zero and positive, and so are the principal minors. Thus, c11 c22 − c12 > 0. It is further assumed that c66 < c11 . There is also a transverse stress, σ33 = c13 u1,1 + c23 u2,2 .
(4)
Note that any plane strain solution for in-plane stresses and strains is also valid for plane stress after substituting cαβ by cαβ − cα3 c3β /c33 . The transverse strain 33 = −(c13 11 + c23 22 )/c33 for plane stress. The equations of motions are σij,j = %
∂ 2 ui , ∂t 2
(5)
where % is the density of the solid and t is time. Consider the neighbourhood of a mode II crack edge, travelling with velocity V . The crack faces are assumed to be traction free, and approximative steady state conditions are assumed within a region of radius L from the crack edge. A process region, according to the Barenblatt model (Barenblatt, 1959a, b; 1962) with length d L is assumed. Stresses and displacements in the steady state region can then be found by considering the problem of a semi-infinite crack with its edge along x1 = Vt − d, x2 = 0, and the front end of the Barenblatt region along x1 = Vt, x2 = 0. The boundary conditions at x2 = 0 are σ22 = 0 for all x1 0 σ21 = σ21 (x1 − Vt)
(6) for x1 < Vt,
(7)
0 where σ21 (x1 − Vt) equals zero for x1 < Vt − d and increases from zero to τD in the interval 0 Vt − d < x1 < Vt. Thus the given function σ21 (x1 − Vt) expresses the shear stress distribution along the Barenblatt region. In addition to the boundary conditions for x2 = 0 there is a condition of vanishing stresses at infinity: q σ22 , σ21 → 0 as (x1 − Vt)2 + x22 → ∞. (8)
Intersonic crack propagation in an orthotropic material
3
3. Solution of the problem Introduce the Galilean transformation, X1 = x1 − Vt,
X2 = x2 .
(9)
The equations of motions then take the form (c11 − %V 2 )u1,11 + c66 u1,22 + (c12 + c66 )u2,12 = 0
(10)
(c12 + c66 )u1,12 + (c66 − %V 2 )u2,11 + c22 u2,22 = 0.
(11)
A solution is now written in the general form u1 = a1 f (X1 + pX2 ),
u2 = a2 f (X1 + pX2 )
(12)
which, inserted into the equations of motion leads to f 00 (X1 + pX2 )[(c11 − %V 2 + c66 p 2 )a1 + (c12 + c66 p)a2 ] = 0
(13)
f 00 (X1 + pX2 )[(c12 + c66 )pa1 + (c66 − %V 2 + c22 p 2 )a2 ] = 0.
(14)
For non-trivial solutions, the determinant must vanish, which gives 2 c22 c66 p 4 + [c11 c22 − c12 − 2c12 c66 − (c22 + c66 )%V 2 ]p 2
+c11 c66 − (c11 + c66 )%V 2 + %2 V 4 = 0.
(15) (16)
If V = 0 all four roots are imaginary. This√also holds for increasing V as long as √ √the constant term is positive. It becomes negative for c /% < V < c /%. Note that c66 /% is the 66 11 √ lowest and √c11 /% the highest wave propagation velocity in the x1 direction. For convenience their ratio, √ c66 /c11 will be denoted k in the continuation, and the dimensionless velocity β = V / c11 /% is introduced. In the intersonic interval, k < β < 1, two roots are imaginary, and two roots are real. The root with positive imaginary part is denoted p1 , and the other imaginary root is then equal to p1 = −p1 , where a bar expresses complex conjugation. The positive real root is denoted p2 , and the other real root is then equal to −p2 . p1 approaches zero as β → 1 and p2 approaches zero as β → k. After the roots have been determined, the eigenvectors are found. The following notations are used: For p = ±p1 :
a2 /a1 = ±A21 (imaginary)
(17)
For p = ±p2 :
a2 /a1 = ±A22 (real).
(18)
Use of (13) gives A2i = −
c11 − %V 2 + c66 pi2 , (c12 + c66 )pi
i = 1, 2.
(19)
Note that the solutions (12) represent a wave with the front along the plane X1 + pX2 = 0 if p is real. Thus, as expected, two head-waves originate from the crack edge, one, represented by f (X1 + p2 X2 ), in the upper half-plane for X1 + p2 X2 6 0, and the other, represented by
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K.B. Broberg
f (X1 − p2 Xq 2 ), in the lower half-plane for X1 − p2 X2 6 0. The propagation velocity of these
waves is V / 1 + p22 . Due to the symmetry of the problem it suffices to consider X2 > 0. A general solution can be written in the form u+ 1 = f1+ (X1 + p1 X2 ) + f1+ (X1 + p1 X2 ) + f2+ (X1 + p2 X2 )
(20)
u+ 2 = A21 [f1+ (X1 + p1 X2 ) − f1+ (X1 + p1 X2 )] + A22 f2+ (X1+ + p2 X2 )
(21)
+ which ensures that u+ 1 and u2 are real. Index plus indicates that the functions are defined in the upper half-plane. Note that f2+ (·) = 0 for positive values of its argument. From the generalized Hooke’s law it follows that + 0 0 σ21 = c66 {(p1 + A21 )[f1+ (X1 + p1 X2 ) − f1+ (X1 + p1 X2 )] 0 +(p2 + A22 )f2+ (X1 + p2 X2 )}
(22)
+ 0 0 σ22 = (c12 + c22 p1 A21 )[f1+ (X1 + p1 X2 ) + f1+ (X1 + p1 X2 )] 0 +(c12 + c22 p2 A22 )f2+ (X1 + p2 X2 ).
(23)
The boundary conditions (6)–(7) give 0 0 0 (c12 + c22 p1 A21 )[f1+ (X1 ) + f1+ (X1 )] + (c12 + c22 p2 A22 )f2+ (X1 ) = 0
for all X1 ,
(24)
0 0 0 (p1 + A21 )[f1+ (X1 ) − f1+ (X1 )] − (p2 + A22 )f2+ (X1 ) =
for X1 < 0.
0 (X1 ) σ21 c66
(25)
Introduce now the sectionally analytic function, F (Z), where Z = X1 + iX2 and defined 0 0 so that F+ (X1 ) = f1+ (X1 ) for X2 = +0 and F− (X1 ) = −f1+ (X1 ) for X2 = −0. From (24) it follows that 0 f2+ (X1 ) = −
c12 + c22 p1 A21 0 0 [f (X1 ) + f1+ (X1 )], c12 + c22 p2 A22 1+
(26)
whereupon insertion into (25) gives (Q + iP )F+ (X1 ) − (Q − iP )F− (X1 ) =−
c12 + c22 p2 A22 0 σ21(X1 ) c66
for X1 < 0,
(27)
where P = i(p1 + A21 )(c12 + c22 p2 A22 )
(28)
Q = (p2 + A22 )(c12 + c22 p1 A21 ).
(29)
Note that both P and Q are real.
Intersonic crack propagation in an orthotropic material
5
Figure 1. The exponent g as a function of crack velocity in the intersonic region. The velocity is given as (β − k)/(1 − k), where β is a dimensionless crack velocity, equalling k at the lower end of the intersonic interval and unity at the upper end. The elastic stiffnesses were chosen so that c11 /c66 = 3.2, c12 /c66 = 0.8 and c22 /c66 = 1.8. This implies that the ratio k between the lowest and highest velocities for wave propagation in the crack direction is about 0.511. The exponent g reaches its maximum, 0.5, at β = β∗ ≈ 0.784, corresponding to approximately 0.511 on the abscissa in the diagram.
Equation (27) can be written in the form F+ (X1 ) − e−2iπg F− (X1 ) = −
p
0 (X1 ) (c12 + c22 p2 A22 )σ21
c66 Q2 + P 2 [cos(πg) + i sin(πg)] 0 P σ21 (X1 ) e−iπg p = i for X1 < 0, c66 (p1 + A21 ) Q2 + P 2
(30)
where
1 P g = atan . π Q
(31)
The velocity dependence of g is shown in Figure 1. Investigation of the argument P /Q for the arcustangens function shows that Q → −∞ as β → k and that P → 0 as β → 1, i.e., at the ends of the intersonic interval. The factors of P , i(p1 + A21 ) and (c12 + c22 p2 A22 ), are both negative, and thus P is positive in the intersonic region. The factors of Q, (p2 + A22 ) and (c12 + c22 p1 A21 ), are both increasing functions of V , reaching zero simultaneously at s 2 c11 c22 − c12 β = β∗ = . (32) c11 (c12 + c22 )
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K.B. Broberg
Thus, Q is non-negative in the intersonic region and reaches zero at β = β∗ . Hence, g does not change sign, it is positive in the whole intersonic interval, and it increases from zero to 12 when β increases from k to β∗ , whereupon it decreases to zero when β increases to 1. In the investigation of the character of P /Q, a program for symbolic mathematics is very helpful – here Maple V has been used. Now the following Hilbert problem is obtained F+ (X1 ) − e−2iπg F− (X1 ) = i
0 (X1 ) sin(πg) e−iπg σ21 c66 (p1 + A21 )
for X1 < 0
F+ (X1 ) − F− (X1 ) = 0 for X1 > 0,
(33) (34)
where the last equation follows from (24), recalling that f2+ (X1 ) = 0 for X1 > 0. Introduce the sectionally analytic function G(Z) = Z g−1 , defined in the Z-plane cut along the negative real axis, with the branch chosen so that G(Z) is real and positive for Z = X1 > 0. Then, −2πig G− (X1 ) for X1 < 0 e (35) = 1 for X1 > 0 G+ (X1 ) and thus, for all X1 , G+ (X1 )F+ (X1 ) − G− (X1 )F− (X1 ) = i
0 (X1 ) sin(πg) eiπg σ21 2, c66 (p1 + A21 )
where 2 = 1 for X1 < 0 and 2 = 0 for X1 > 0. The solution is Z 0 0 sin(πg) σ21 (w) G(Z)F (Z) = − dw + P (Z), 1−g 2π c66 (p1 + A21 ) −d |w| (w − Z)
(36)
(37)
where P (Z) is a polynomial of finite degree. In order to satisfy (8), P (Z) must be zero. Thus, Z 0 0 sin(πg)Z 1−g σ21 (w) F (Z) = − dw. (38) 1−g 2π c66 (p1 + A21 ) −d |w| (w − Z) In particular, for X2 = 0, 1−g
Z
d
0 σ21 (−w) dw for X1 > 0 1−g (w + X ) w 1 0 Z 0 σ21 (−w) sin(πg) e∓iπg |X1 |1−g d F± (X1 ) = − C dw 1−g 2π c66 (p1 + A21 ) 0 w (w + X1 )
sin(πg)X1 F± (X1 ) = 2π c66 (p1 + A21 )
±
0 i sin(πg) e∓iπg σ21 (X1 ) 2c66 (p1 + A21 )
for X1 < 0,
(39)
(40)
where C on the integral sign denotes the Cauchy principal value. Stresses and displacements can now be determined from 0 f1+ (X1 + p1 X2 ) = F+ (X1 + p1 X2 )
(41)
0 f1+ (X1 − p1 X2 ) = −F− (X1 − p1 X2 )
(42)
0 f2+ (X1 + p2 X2 ) = −
c12 + c22 p1 A21 [F+ (X1 + p2 X2 ) − F− (X1 + p2 X2 )]. c12 + c22 p2 A22
(43)
Intersonic crack propagation in an orthotropic material
7
0 Note that f2+ (X1 + p2 X2 ) = 0 for β = β∗ , because then c12 + c22 p1 A21 = 0. Thus no head√ wave appears when β = β∗ ! This corresponds to the absence of the S wave at β = k 2 in the isotropic case. In particular, for X2 = 0,
σ21 = c66 (p1 + A21 )[F+ (X1 ) + F− (X1 )] Z 0 sin(πg) 1−g d σ21 (−w) = X1 dw 1−g π (w + X1 ) 0 w
for X1 > 0,
(44)
∂u+ c22 (p2 A22 − p1 A21 ) 1 = [F+ (X1 ) − F− (X1 )] ∂x1 c12 + c22 p2 A22 ( Z d 0 c22 YII (β) σ21 (−w) 1−g = sin(πg)·|X1 | C dw 1−g 2π c66 (c12 + c66 ) (w + X1 ) 0 w ) 0 +π cos(πg)σ21 (X1 )
for X1 < 0,
(45)
where YII (β) =
2c66 (p22 − p12 ) sin(πg) . P
(46)
The function YII (β) approaches zero as β → k and it is finite and positive for β > k. For |X1 | d the expressions reduce to Z 0 (−w) sin(πg) d σ21 σ21 = dw for X1 > 0 g w 1−g π X1 0 Z d 0 ∂u+ σ21 (−w) c22 YII (β) sin(πg) 1 =− dw for X1 < 0. g ∂x1 2π c66 (c12 + c66 )|X1 | 0 w 1−g
(47)
(48)
Note that the transverse stress, σ33, can be determined from (4) once the in-plane displacement gradients u1,1 and u2,2 are found. This stress might cause mode I cracking if planes parallel to x3 = 0 are weak planes. 4. The energy flux into the process region The energy flux into the process region can be calculated as (Broberg, 1967) Z 0 ∂u+ 0 G = −2 σ21 (X1 ) 1 dX1 ∂x1 −d " Z d Z 0 σ21 (−w) c22 YII (β) sin(πg) 0 0 = − σ21 (X1 ) |X1 |1−g C dw 1−g π c66 (c12 + c66 ) −d (w + X1 ) 0 w # 0 +π cot(πg) σ21 (X1 ) dX1 .
(49)
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K.B. Broberg
This expression will be written in a form more suitable for interpretations and numerical calculations. To this end the stress distribution in the Barenblatt region is written as 0 σ21 (X1 ) = τD ·D(X1 /d),
(50)
where τD is the cohesive strength and D(X1 /d) a shape function, which equals +1 for X1 /d = 0, is zero for X1 /d < −1 and X1 /d > 0 and is increasing in the interval −1 < X1 /d < 0. One simple example is D(X1 /d) = 1 + X1 /d, −1 < X1 /d < 0. The term π cot(πg) in (49) is now substituted by an expression involving the integral Z ∞ dw (51) = −π cot πg |X1 |g−1 if X1 < 0. 1−g w (w + X ) 1 0 Then, after some rearrangement and redefinition of the integration variable w, it follows that G=
c22 τD2 d ·YII (β) sin(πg)·wD (g), π c66 (c12 + c66 )
where
Z
"Z
1
wD (g) =
D(−v)v 1−g 0
(52)
D(−v) − D(−w) dw w 1−g (w − v) 0 # Z ∞ dw +D(−v) dv. w 1−g (w − v) 1 1
(53)
Note that both integrands of the inner integrals are positive, so that wD (g) > 0. The function sin(πg) wD (g) → constant as g → 0, i.e., as β → k or β → 1. Obviously, then, G → 0 as β → k and G approaches a finite positive value as β → 1. The dependence on d, the length of the process region model, becomes more clear for a situation with a given crack-length involved (Broberg, 1994; 1995). Here a length parameter is arbitrarily chosen as L, which was taken as an approximate measure of the steady state region in the original problem. Assume that the shear stress σ21 at X1 = L, X2 = 0, is known and L equal to σ21 . Then, because L d, it follows from (47) that Z 0 (−w) sin(πg) d σ21 L σ21 = dw (54) g πL w 1−g 0 which, after redefinition of the integration variable, is written as !g Z 1 d D(−w) sin(πg) L σ21 = dw. τD π L w 1−g 0 Thus, after elimination of τD , L 2 c22 π(σ21 )L d G= ·YII (β)·0D (g)· 2c66 (c12 + c66 ) L
(55)
!1−2g ,
(56)
where 0D (g) =
2wD (g) "Z #2 . 1 D(−w) sin πg dw w 1−g 0
(57)
Intersonic crack propagation in an orthotropic material
9
Figure 2. Normalized energy flux into the process region as a function of crack velocity in the intersonic region. The velocity is given as (β − k)/(1 − k), where β is a dimensionless crack velocity, equalling k at the lower end of L )2 L/c . the intersonic interval and unity at the upper end. The energy flux is normalized through division by π(σ21 66 L Note that in each given situation, involving a defined length parameter, such as the crack length, σ21 is a specific function of the crack velocity. The elastic stiffnesses are chosen so that c11 /c66 = 3.2, c12 /c66 = 0.8 and c22 /c66 = 1.8. The size of the process region is given by d/L = 0.001.
The function 0D (g) approaches a finite positive value as β → k or β → 1. Note that G in the expression (56) is composed by four factors, the first one dependent on the outer field, the second one on the crack speed, the third one on the shape of the process region and the velocity-dependent constant g, and the fourth one on the size of the process region. It is immediately seen that the assumption of a point size process region, d = 0 results in zero energy flux into the crack edge, if g < 12 , but a non-zero flux, if g = 12 , i.e. at β = β∗ . On the other hand, for most velocities in the intersonic region, g is not much different from 1 , and then the factor (d/L)1−2g is not very small compared to unity until d/L becomes 2 extremely small. The dependence of the energy flux into the process region is shown in Figure 2. Note further, from (47) and (54), √ that the ordinary stress intensity factor appears in the 1 L case g = 2 and equals KII = σ21 2π L. Then, because it can be shown that 0D (g) → 1 as g → 1/2, the energy flux becomes G→
c22 KII2 YII (β∗ ) 4c66 (c12 + c66 )
(58)
which is the same expression as for the subsonic √ case! The significance of the curious velocity 2cS for mode II crack propagation in isotropic materials was discussed in (Freund, 1979; Burridge et al., 1980; Broberg, 1980; 1989). Formal calculations for mode I result in an association of this velocity with an inverse square root singularity, but also with a negative energy flux into the crack edge, which is physically
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K.B. Broberg
unacceptable (Broberg, 1989). Similar calculations for moving indentation of a plane surface of an isotropic elastic body, √a problem which is related to mode I crack propagation, also show a specific role of velocity 2cS (Georgiadis, 1993). 5. Specialization to transversely isotropic and isotropic materials For a transversely isotropic material, c22 = c11 , c12 = c11 − 2c66 p p p1 = i 1 − β 2 , p2 = β 2 /k 2 − 1,
(59) A21 = p1 ,
A22 = −1/p2 .
(60)
For an isotropic material c11 =
2(1 − ν)µ µ = 2, 1 − 2ν k
c66 = µ,
where µ is the modulus of rigidity and ν is Poisson’s ratio. Then the following specializations to transversely isotropic materials are obtained: p p 1 4k 3 1 − β 2 β 2 − k 2 g = atan π (β 2 − 2k 2 )2
(61)
(62)
c22 1 = c66 (c12 + c66 ) (1 − k 2 )c66
(63)
(1 − k 2 )β 2 sin(πg) p 2k 2 c66 1 − β 2
(64)
YII (β) =
√ β∗ = k 2
(65)
and then specialization to an isotropic solid is obtained by substituting c66 by µ. 6. Summary and discussion As expected, intersonic mode II crack propagation in an orthotropic material, along a plane of elastic symmetry, exhibits the same main features as in an isotropic material. Assumption of an infinitesimally small process region leads to weaker singularities in stresses and strains than the inverse square root singularity in the sub-Rayleigh case, except at one curious distinct velocity. A Barenblatt model of the process region has therefore been assumed, and this enables a finite energy flux into the process region. The expression for the energy flux depends upon the outer field, the crack velocity and the shape of the Barenblatt region. The energy flux increases from zero at β = k to a maximum at some intermediate intersonic crack velocity, whereupon it decreases toward a small value when the maximum wave propagation velocity in the crack direction is reached. How a crack can bypass the forbidden subsonic super-Rayleigh region to reach the intersonic region was discussed in an earlier paper, concerning intersonic crack propagation in isotropic materials (Broberg, 1995). The question about directional stability of a running mode II crack is difficult to answer. It is known from sub-Rayleigh crack propagation that mode I in
Intersonic crack propagation in an orthotropic material
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general takes over, after kinking if the original crack is subjected to purely mode II loading. In general a high ambient pressure is needed to prevent the occurrence of mode I instead of mode II propagation (Melin, 1986). References Barenblatt, G.I. (1959a). Concerning equilibrium cracks forming during brittle fracture. The instability of isolated cracks, relationship with energetic theories. Journal of Applied Mathematics and Mechanics 23, 1273–1282. English translation from PMM 23. Barenblatt, G.I. (1959b). The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axially-symmetric cracks. Journal of Applied Mathematics and Mechanics 23, 622–636. English translation from PMM 23, 434–444. Barenblatt, G.I., Salganik, R.L. and Cherepanov, G.P. (1962). On the nonsteady motion of cracks. Journal of Applied Mathematics and Mechanics 26, 469–477. English translation from PMM 26, 328–334. Broberg, K.B. (1980). Velocity Peculiarities at Slip Propagation. Report from the Division of Engineering, Brown University, Providence, R.I. Broberg, K.B. (1994). Intersonic bilateral slip. Geophysical Journal International 119, 706–714. Broberg, K.B. (1995). Intersonic mode II crack expansion. Archives of Mechanics 47, 859–871. Broberg, K.B. (1964). On the speed of a brittle crack. Journal of Applied Mechanics 31, 546–547. Broberg, K.B. (1967). Discussion of fracture from an energy point of view. Recent Progress in Applied Mechanics (Edited by B. Broberg, J. Hult and F. Niordson), Almqvist and Wiksell, Stockholm, 125–151. Broberg, K.B. (1989). The near-tip field at high crack velocities. International Journal of Fracture 39, 1–13. Burridge, R., Conn, G. and Freund, L.B. (1979). The stability of a plane strain shear crack with finite cohesive force running at intersonic speeds. Journal of Geophysical Research 84, 2210–2222. Freund, L.B. (1979). The mechanics of dynamic shear crack propagation. Journal of Geophysical Research 84, 2199–2209. Georgiadis, H.G. and Barber, J.R. (1993). On the super-Rayleigh/subseismic elastodynamic indentation problem. Journal of Elasticity 31, 141–161. Melin, Solveig. (1986). When does a crack grow under mode II conditions? International Journal of Fracture 30, 103–114.