International Journal of Fracture 112: 69–85, 2001. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.
On the intersonic crack propagation in an orthotropic medium L. FEDERICI1, L. NOBILE1, A. PIVA2 and E. VIOLA1
1 Department DISTART, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy 2 Department of Physics, University of Bologna, Via Irnerio 46, 40126 Bologna, Italy
Received 21 July 2000; accepted in revised form 12 June 2001 Abstract. Motivaded by recent theoretical studies the elastodynamic response of an orthotropic material with a semi-infinite line crack, which propagates intersonically. is revisited through an approach which differs from those used in previous studies. The near tip stress and displacement fields are obtained for Mode I and Mode II of steady state crack propagation. The strain energy release rate analysis confirms that the Mode I is physically impossible due to the order of stress singularity, which is larger then one half. For Model II the order of stress is less than one half and it is shown that a steady state intersonic propagation is allowed only for a particular crack tip velocity which is a function of the material orthotropy. Key words: Elastodynamic response, intersonic crack propagation, near tip displacement field, near tip stress field, orthotropic material, semi-infinite line crack.
1. Introduction The theoretical study of intersonic crack propagation, i.e. when the crack tip velocity lies between the shear and dilational wave speeds of the material, is drawing the efforts of many investigators. Particular attention has been devoted to study the intersonic crack propagation along elastic-rigid material interfaces. Theoretical analyses performed by Liu et al.(1995), Yu and Yang (1995) and Huang et al. (1996) among others, have shown that the asymptotic elastic fields for intersonic interfacial crack propagation is predominantly of shear nature, the power of stress singularity is always less than one half and propagates into the more compliant material as a shear shock wave following the crack tip. In addition, the above mentioned studies showed that a pure Mode I steady state, intersonic crack propagation is impossible because the corresponding energy release rate takes an unbounded negative value. Recently, the theoretical investigation has been extended to orthotropic materials as well as to unidirectional fiber reinforced composites. Piva and Hasan (1996) obtained the singular asymptotic stress field of intersonic shear crack growth in orthotropic materials. Huang et al. (1999) were able to predict the same physical features pointed out in previous papers. In particular, they showed that there is only one velocity at which the crack tip energy release rate can assume a finite positive value so that a steadily Mode II intersonic crack propagation can occur. In this topic, Broberg (1999) developed a remarkable cohesive zone analysis of shear intersonic crack propagation in an orthotropic medium. Assuming the Barenblatt model for the process region, he has shown that the near tip elastic solution is strictly related to the details of the cohesive model.
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The main result of Broberg’s analysis is the rationalization for predicting a finite energy flux into the propagating process region through the whole intersonic regime. In fact, he bas shown that the energy flux is a positive smooth function of the crack tip velocity with a maximum in a neighbouroud of a critical velocity c = c∗ , also predicted by singular elastic solutions, which reduces to an impulsive like function with support c∗ when the length of the cohesive region tends to zero. The purpose of the present work is to reanalyze the problem of intersonic crack propagation in an orthotropic medium by using an approach proposed by Piva (1987) and Piva and Viola (1988) which differs from those used in the above mentioned papers. The method of solution is particularly powerful in simplifying the basic elastodynamic equations for an orthotropic medium as well as in solving the boundary value problem related to intersonic crack propagation. At first, the two coupled elastodynamic equations for an orthotropic medium are reduced to a first order system involving a four dimensional vector field. A similarity transformation applied to the matrix of elastodynamic coefficients allows to recognize that the elastodynamic problem consists of two distinct decoupled problems. The first one is an elliptic problem represented by a system of Cauchy–Riemann type whose solution is related to the elastic field following the crack tip. Although the general solution of the system is an analytic function, a complex variable formulation is avoided by the justified assumption that the local elastic fields following the crack tip may be represented in a separated variable form. The second problem is a hyperbolic first order problem whose general solution is a plane wave which, due to the Cauchy data on the crack line, assumes the aspect of a backward propagating shock wave. The asymptotic near-tip expressions of the elastic fields are obtained for Mode I and Mode II of intersonic crack propagation. It is shown that for Mode I the order of stress singularity is larger than one half and, using the concept of crack closure energy, it is confirmed that a steadily Model I crack propagation is physically forbidden. For Mode II the order of stress singularity is less than one half and the crack closure energy concept allows to demonstrate that there is a limiting velocity, corresponding to the square root singularity, at which intersonic crack propagation is physically admissible. It is also shown that when the limiting velocity is reached by the crack tip the shock wave and the near tip tensile stress distribution, vanish. This particular feature of intersonic crack propagation has been notified also by Broberg (1999) and corresponds to the so called radiation-free solution whose existence has been recently shown by Gao et al. (1999) in a unified analysis of intersonic motion of cracks and dislocations. 2. Foundation Consider an infinite orthotropic elastic medium with the axes of elastic symmetry coinciding with the axes of a Cartesian coordinate system O(X, Y, Z). By assuming plane stress conditions, the system of equations of motion governing elastodynamic problems in the X-Y plane are c11 uXX + c66 uY Y + (c12 + c66 )νXY = ρut t ,
(2.1a)
On the intersonic crack propagation in an orthotropic medium c66 νXX + c22 νY Y + (c12 + c66 )uXY = ρνt t ,
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(2.1b)
in which u = u(X, T , t), ν = ν(X, Y, t) are the displacement components in X and Y directions respectively, t is the time, ρ is the mass density of the material and cij are the elastic coefficients. The in-plane stress-strain equations may be written as follows σXX = c11 uX + c12 νY ,
(2.2a)
σY Y = c12 uX + c22 νY ,
(2.2b)
τXY = c66 (uY + νX ).
(2.2c)
By setting x = X − ct = Y , where c is a constant speed, Equations (2.1a,b) become uxx + auyy + 2βνxy = 0,
(2.3a)
νxx + α1 νyy + 2β1 uxy = 0,
(2.3b)
where α= α1 =
c11
c66
c66 , 1 − M12
c22 , 1 − M22
2β =
c12 + c66 , c11 1 − M12
2β1 =
c12 + c66 . c66 1 − M22
(2.4a,b)
(2.4c,d)
The quantities M1 = c/ce and M2 = c/cs are the Mach numbers, where ce = (c11 /)1/2 and cs = (c66 /)1/2 are the longitudinal wave speed and the shear wave speed of the material, respectively. According to Piva and Hasan (1996), the system of Equations (2.3a,b) may be rewritten as x + Ay = 0,
(2.5)
where is a 4 × 1 matrix valued function defined as T = (1 , 2 , 3 , 4 ) = (ux , uy , νx , νy ) and A is a 4 × 4 constant matrix, given by 0 α 2β 0 −1 0 0 0 . A= 0 0 α 2β 1 1 0
(2.6)
(2.7)
0 −1 0
The characteristic Equation of (2.7) is m4 + 2a1 m2 + a2 = 0,
(2.8)
in which 2a1 = α + α1 − 4ββ1 ,
a2 = αα1 .
(2.9)
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In what follows the intersonic regime, 0 < M1 < 1 and M2 > 1, will be assumed so that Equation (2.8) provides the four eigenvalues m1 = p, m2 = −p, m3 = iq, m4 = −iq, with
1/2
1/2 1/2 1/2 − a1 and q = a12 − a2 + a1 positive numbers. p = a12 − a2 According to Piva and Hasan (1996) the system (2.5) may be transformed to the following form x + B y = 0,
(2.10)
where = P and
(2.11)
2βp 2 2βq 2 2βp 2 − 0 − α + p2 α + p2 α − q2 2βp 2βq 2βp − 0 2 α − q2 P = α + p2 α + p −p p −q 0 1
1
B=P
−1
p
0
0 0
0
,
1
0 −p 0 0 . AP = 0 0 0 −q 0
0
q
(2.12a)
(2.12b)
0
Equation (2.10) leads to the first-order systems ψ1x + pψ1y = 0,
ψ2x − pψ2y = 0
(2.13a,b)
ψ3x − qψ4y = 0,
ψ4x + qψ3y = 0
(2.14a,b)
and
It should be noted that the system (2.13) is of hyperbolic type whereas Equations (2.14) represent a Cauchy–Reimann system. By setting ζ = ψ1 + ψ2 ,
η = ψ1 − ψ2 ,
(2.15a,b)
it may be shown that the above functions satisfy the following system ζx + pηy = 0,
ηx + pζy = 0,
(2.16a,b)
which leads to satisfy the same wave equation ζxx − p 2 ζyy = 0,
ηxx − p 2 ηyy = 0.
(2.17a,b)
Keeping in mind Equations (2.6), (2.11), (2.12a) and (2.15a,b) the stress-strain relations (2.2) may be rewritten as
On the intersonic crack propagation in an orthotropic medium σxx = l1 ψ4 + l2 ζ, c66 where
τxy = l5 ψ3 − l6 η, c66
(2.18a,b,c)
2βp 2 , , α + p2 c22 c12 2βq 2 c22 c12 2βp 2 + = − , l , l3 = 4 c66 c66 α − q 2 c66 c66 α + p 2 2β 2β −1 , l6 = p 1 − . l5 = q α − q2 α + p2
c12 c11 + l1 = c66 c66
σyy = l3 ψ4 + l4 ζ, c66
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2βq 2 α − q2
c12 c11 − l2 = c66 c66
The functions ψ3 and ψ4 , which enter into the Cauchy–Riemann system (2.14), deal with the elastic field following the propagating crack tip. Thence, in what follows, Equations (2.14) will be integrated by referring to the system of moving polar coordinates defined by: x = r cos θ, y = r sin θ, r > 0, −π < θ < π , and assuming a separated variables representation for the displacement field, i.e., ν = r γ V (θ),
u = r γ U (θ),
(2.19)
where the exponent γ will be determined through appropriate conditions. By using (2.6), (2.11), (2.12a) and (2.19) gives ψ3 (r, θ) = r γ −1 h3 (θ),
ψ4 (r, θ) = r γ −1 h4 (θ),
(2.20a,b)
where
2k 2β γ U (θ) sin θ + U (θ) cos θ + h3 (θ) = q α + p2
2k 2β γ U (θ) cos θ + U (θ) sin θ + h4 (θ) = α α + p2 and
γ V (θ) cos θ − V (θ) sin θ) γ V (θ) sin θ + V (θ) cos θ)
, (2.21) (2.22)
α + p2 α − q 2 . k= 4β p 2 + q 2
3. The general solution of the Cauchy–Riemann system The general solution to the Cauchy–Riemann system (2.14) may be obtained by introducing the following polar-coordinates transformation tg θ (3.1a,b) θ1 = tg−1 r1 = r[g(θ)]1/2 , q with g(θ) = cos2 θ + (sin2 θ)/q 2 . In view of (3.1) the functions (2.20) become γ −1
ψ3 (r1 , θ1 ) = r1
H3 (θ1 ),
γ −1
ψ4 (r1 , θ1 ) = r1
H4 (θ1 ),
(3.2a,b)
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where H3 (θ1 ) = [g(θ(θ1 )](1−γ )/2h3 [θ(θ1 )],
(3.3a)
H4 (θ1 ) = [g(θ(θ1 )](1−γ )/2h4 [θ(θ1 )].
(3.3b)
By using the rule of differentiation sin θ1 ∂ ∂ ∂ = cos θ1 − , ∂x ∂r1 r1 ∂θ1
(3.4a)
sin θ1 ∂ cos θ1 ∂ ∂ = + . ∂x q ∂r1 qr1 ∂θ1
(3.4b)
Equations (2.14) lead to the following system in the unknown H3 (θ1 ) and H4 (θ1 ) cos θ1 − sin θ1 cos θ1 H3 (θ1 ) H3 (θ1 ) sin θ1 = (γ − 1) , − cos θ1 sin θ1 sin θ1 cos θ1 H4 (θ1 ) H4 (θ1 ) which may be reduced to the system with constant coefficients 0 −(γ − 1) H3 (θ1 ) H3 (θ1 ) = . (γ − 1) 0 H4 (θ1 ) H4 (θ1 )
(3.5)
(3.6)
By noting that the matrix: 0 −(γ − 1) A= (γ − 1) 0 has complex coniugates eigenvalues λ1 = i|γ − 1|, λ2 = λ1 , one obtains the general solution to (3.6) in the following form H3 (θ1 ) = C1 cos(γ − 1)θ1 + C2 sin |γ − 1|θ1 ,
(3.7a)
H4 (θ1 ) = ε(γ )[C1 sin |γ − 1|θ1 − C2 cos(γ − 1)θ1 ],
(3.7b)
where C1 and C2 are arbitrary constants, and ε(γ ) = sgn(γ − 1). Thence, the required general solution to Equations (2.14) is γ −1
ψ3 = r1
[C1 cos(γ − 1)θ1 (θ) + C2 sin |γ − 1|θ1 (θ)],
γ −1
ψ4 = εr1
[C1 sin |γ − 1| θ1 (θ) − C2 cos(γ − 1)θ1 (θ)],
(3.8) (3.9)
in which θ1 (θ) is defined by (3.1b). 4. Statement of problem – Particular solutions for Mode I and Mode II crack propagation Consider the elastodynamic problem of a traction-free semi-infinite crack situated along the fixed X-axis and propagating with a constant velocity c, with cs < c < cl , i.e., in the intersonic regime which may be equivalently defined by 1 < M2 < (c11 /c66 )1/2 . By using (2.16), the simmetry conditions
On the intersonic crack propagation in an orthotropic medium u(x, y) = u(x, −y),
ν(x, y) = −ν(x, −y),
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(4.1a,b)
required for Mode I fracture, lead to the following conditions ahead of the crack tip U (0) = V (0) = 0.
(4.2)
In the same way, from the skew-symmetry conditions u(x, y) = −u(x, −y),
ν(x, y) = ν(x, −y)
(4.3a,b)
valid for Mode II fracture, one obtains U (0) = V (0) = 0.
(4.4)
Applying conditions (4.2) to Equations (2.20) gives ψ3 (x, 0) = 0, ψ4 (x, 0) = (x)
(4.5a) γ −1 2k
α
2β γ U (0) + V (0) , α + p2
x > 0.
(4.5b)
In order for the solution (3.8) to satisfy condition (4.5a) it has to be valid that C1 = 0 and C2 ε(γ = −kA2 with 2β 2 γ U (0) + V (0) . A2 = α α + p2 Thence, the solution to the Cauchy–Riemann system (2.14) for Mode I fracture reduces to ψ31 = −
kA2 γ −1 r sin |γ − 1|θ1 (θ), ε(γ ) 1 γ −1
ψ41 = kA2 r1
(4.6a)
cos(γ − 1)θ1 (θ).
(4.6b)
Conditions (4.4) applied to (2.20) gives ψ4 (x, 0) = 0,
(4.7a)
ψ3 (x, 0) = (x)γ −1
2βγ 2k U (0) + V (0) , q α + p2
x > 0,
(4.7b)
therefore, for Mode II of fracture, the following solution is obtained: γ −1
cos(γ − 1)θ1 (θ),
(4.8a)
γ −1
sin(γ − 1)θ1 (θ),
(4.8b)
ψ3u¯ = kA1 r1 ψ4u¯ = kA1 r1 in which A1 =
2 2βγ U (0) + V (θ) . q α + p2
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5. The wave-type solutions The general solutions to wave Equations (2.17a,b) are y y + f1 x − , ζ(x, y) = f x + p p y y + g1 x − . η(x, y) = g x + p p
(5.1a)
(5.1b)
The two functions f1 (x − y/p) and g1 (x − y/p) represent signals travelling to the right of the crack tip which is physically meaningless as pointed out also in previous papers, see for example Huang et al. (1999), Piva and Hasan (1996). Therefore, it will be stated that f1 = g1 = 0 so that y , (5.2a) ζ(x, y) = f x + p y . (5.2b) η(x, y) = g x + p For Mode I of fracture the stress-simmetry condition τxy (x, 0±) = 0,
|x| < ∞
(5.3)
and the traction-free condition on the crack faces σyy (x, 0±) = 0,
x<0
(5.4)
applied to (2.17c,b) lead, respectively, to l5 1 l5 ψ3 (x, 0±) = ± kA2 (−x)γ −1 sin γ π, l6 l6
x < 0,
(5.5a)
l3 l3 ζ I (x, 0±) = − ψ41 (x, 0±) = kA2 (−x)γ −1 cos γ π, l4 l4
x < 0,
(5.5b)
ηI (x, 0±) =
where Equations (4.6a,b) have been used. The Cauchy data (5.5) allow to obtain the solutions (5.2a) and (5.2b) in the following form |y| γ −1 |y| l3 I , (5.6a) H −x − ζ (x, y) = kA2 cos γ π −x − l4 p p |y| γ −1 |y| l5 , H −x − η (x, y) = kA2 sin γ π sgn(y) −x − l6 p p I
(5.6b)
where H (•) is the Heaviside step function. For Mode II of fracture the stress-symmetry condition σyy (x, 0±) = 0,
|x| < ∞
and the traction-free condition on the crack faces
(5.7)
On the intersonic crack propagation in an orthotropic medium
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Figure 1. The angle of shock wave front vs. M2 .
τxy (x, 0±) = 0,
x<0
(5.8)
give ηII (x, 0±) =
l5 II l5 ψ3 (x, 0±) = − kA1 (−x)γ −1 cos γ π, l6 l6
l3 l3 ζ II (x, 0±) = − ψ4II (x, 0±) = ± kA1 (−x)γ −1 sin γ π, l4 l4
x < 0, x < 0.
Thence, for Mode II of fracture the required solutions are |y| γ −1 |y| l3 II , H −x − ζ (x, y) = kA1 sin γ π sgn(y) −x − l4 p p ηII (x, y) = −kA1
|y| γ −1 |y| l5 . cos γ π −x − H −x − l6 p p
(5.9a) (5.9b)
(5.10a)
(5.10b)
Both solutions (5.6) and (5.10) are defined in the Mach cone, |y| < −px, x < 0, and the half-lines |y| = −px, x < 0 represent the front of a shock wave following the propagating crack tip. In Figure 1 is represented the angle 7 = tg−1 (p) between the shock front and the upper crack face as a function of the Mach number M2 , for two orthotropic materials whose parameters are outlined in Table 1.
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L. Federici et al. Table 1. Material parameters. Composite type
c11 /c66
c22 /c66
c12 /c66
Graphite epoxy Boron epoxy I
27.385 51.114
2.239 2.886
0.716 0.750
6. Power of stress singularity The exponent γ is determined by substituting (5.6) and (5.10) into Equations (2.16). One obtains γI = 8I + N1 ,
(6.1)
for Mode I, and γII = 8II + N2 ,
(6.2)
for Mode II, where 1 l3 l6 , 8I = − tg−1 π l4 l5
8II =
1 −1 l4 l5 tg π l3 l6
(6.3a,b)
and N1 , N2 are arbitrary integers. By noting that 0 < 81 <
1 2
(6.4)
boundedness of displacement at the crack tip requires to chose N1 = 0 in (6.1) and N2 = 1 in (6.2). Therefore, the order of stress singularity is 1 l3 l6 − 1, µI = γI − 1 = − tg−1 π l4 l5
−1 < µI < − 12 ,
(6.5)
for Mode I of fracture, and µII = γII − 1 =
1 −1 l4 l5 tg , π l3 l6
− 12 < µII < 0,
(6.6)
for Mode II of fracture. In Figures 2 and 3 the quantities µI and µII are represented as functions of M2 for orthotropic materials described in Table 1. 7. Mode-I analysis By using (4.6a,b) and (5.6a,b), the stress field for Mode I of fracture is obtained from (2.18) as follows l2 l3 |y| µI |y| σxx µI , (7.1a) = kA2 l1 r1 cos µI θ1 + cos γI π −x − H −x − c66 l4 p p
On the intersonic crack propagation in an orthotropic medium
Figure 2. The order of stress singularity µI vs. M2 .
Figure 3. The order of stress singularity µII vs. M2 .
79
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L. Federici et al. |y| µI |y| σyy µI , = kA2 l3 r1 cos µI θ1 + cos γI π −x − H −x − c66 p p |y| µI |y| τxy µI . = −kA2 l5 r1 sin µI θ1 + sin γI π sgn(y) −x − H −x − c66 p p
(7.1b) (7.1c)
Combining (2.11) with (2.12a) and substituting the appropriate functions, the following displacement field is obtained p2 |y| γI |y| q2 l3 2βkA2 a γI , r cos γI θ1 + cos γI π −x − H −x − u= γI α − q2 1 α + p 2 l4 p p
ν=
kA2 l3 |y| γ qr1 I sin γI θ1 − p cos γI π sgn(y) −x − γI l4 p
γI
|y| . H −x − p
(7.2a) (7.2b)
It can be verified that .the normal stress ahead of the crack tip is negative and the normal displacement behind the crack tip is positive. This situation leads to a negative strain energy release rate as may be demonstrated by assuming that for intersonic crack growth this function may be calculated in the same way as for the subsonic regime, i.e., through the expression for the crack-closure energy. For Mode I one has 7l 1 σyy (x; 0)7ν(x; 7l) dx. (7.3) GI = lim 7l→0 27l 0 By using (7.1b), (7.2b) and performing the integration gives GI = lim A(γI )K(γI , γI + 1)(7l)2γI −1 , 7l→0
(7.4)
where
2k 2 A22 c66 l3 l3 q sin γI π − p cosγI π A(γI ) = γI l4 and K(γI , γI + 1) is the beta function defined as 1 t γI −1 (1 − t)γI dt. K(γI , γI + 1) =
(7.5)
(7.6)
0
For the reason that 0 < γI < for which one obtains G∗I = (2k 2 A22 c66 ql3 )
γI =
1 2
1 2
the limit (7.4) is infinite except for the limiting value γI =
1 2
(7.7)
However, the above limiting value is forbidden in the intersonic range because the function l4 l5 in (6.5) is always negative. In addition, the function A(γI ) too is negative in the intersonic range, so that the limit (7.4) leads to a negative infinite value which is physically meaningless. Thence, it seems conclusive that the steady-state intersonic regime is forbidden for Mode I crack propagation. This result has been provided, through different approaches, by Liu et al. (1995) and Huang et al. (1996). 8. Mode II analysis For Mode II of fracture, Equations (4.7a,b) and (5.10a,b) substituted into (2.18a,b,c) lead to:
On the intersonic crack propagation in an orthotropic medium
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Figure 4. Shear stress for two orthotropic materials.
l2 l3 |y| µII |y| σxx µII , (8.1a) = kA1 l1 r1 sin µII θ1 + sin γII π sgn(y) −x − H −x − c66 l4 p p |y| µII |y| σyy µII , (8.1b) = kA1 l3 r1 sin µII θ1 + sin γII π sgn(y) −x − H −x − c66 p p |y| µII |y| τxy µII . (8.1c) = kA1 l5 r1 cos µII θ1 + cos γII π −x − H −x − c66 p p The displacement field is given by p2 |y| γII q2 l5 2βkA1 γII r sin γ θ + cos γ π sgn(y) −x − u= II 1 II γII α − q2 1 α + p 2 l6 p (8.2a) |y| , H −x − p l5 |y| γII |y| kA1 γII . (8.2b) H −x − qr1 cos γII θ1 + p cos γII π −x − ν=− γII l6 p p In Figures 4 and 5 the dimensionless shear stress τxy /A1 c66 and the tangential displacement u/A1 , evaluated at the upper crack face, are represented respectively as functions of the distance from the crack tip, for some values of M2 . It is worth noting that the above illustrated functions are positive in the intersonic range. The expression for the crack closure energy is 7l 1 τxy (x; 0)7u(x; 7l) dx. (8.3) GII = lim 7l→0 27l 0 By using (81.c) and (8.2a) one obtains
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Figure 5. Tangential displacement for two orthotropic materials.
GII = lim B(γII )K(γII , γII + 1)(7l)2γII −1 ,
(8.4)
7l→0
in which 2k 2 A21 βc66 l5 B(γII ) = γII
p 2 l5 q2 sin γ π + cos γII π II α − q2 α + p 2 l6
(8.5)
and K(γII , γII + 1) is again the beta function defined as in (7.6). For Mode II it is 12 < γII < 1 so that the limit (8.4) is zero except for the limiting value γII = 12 for which one obtains the positive limit 2 2 2 2k A1 βq c66 l5 ∗ . (8.6) GII = 1 α − q2 γII = 2
In Figure 6 the limiting behaviour of the crack closure energy (8.4), normalized to the factor 2k 2 A21 βc66 K(γII , γII + 1) is represented vs. M2 , for two orthotropic materials. It is shown that the above function tends to an impulsive function centered at some value M2∗ which depends on the degree of material orthotropy. It should be pointed that the value γII = 12 is reached when the term l3 l6 in (6.6) , which is a function of M2 , vanishes. In Figure 7 where the above function is represented for two orthotropic materials, it is shown that it has a zero of order two in correspondence of a value M2∗ which lies in the intersonic range. In fact, both equation l3 (M2 ) = 0 and l6 (M2 ) = 0 have the same solution 2 c11 c22 − c12 . (8.7) M2∗ = c66 (c12 + c22 ) The limiting value (8.7) is the support of the crack closure energy, identified in Figure 6 and corresponds to the unique steady state crack tip velocity
On the intersonic crack propagation in an orthotropic medium
Figure 6. Normalized crack closure energy behaviour vs M2 for two orthotropic materials.
Figure 7. Function l3 l6 vs. M2 for two orthotropic materials.
83
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L. Federici et al. c∗ = M2∗ cs ,
(8.8)
at which the strain energy release rate assumes the finite positive value (8.6). It is noted that the relation (8.7) coincides with that obtained by Huang et al. (1996), through a different approach. As a consequence of the above results, when the crack tip velocity is c = c∗ the near tip stress field (8.1) is in the following form ∗ sin θ ∗ /2 σxx = −k ∗ A∗1 l1∗ 1∗ , c66 r1 ∗ σyy
c66 ∗ τxy
c66
= 0, = k ∗ A∗1 l5∗
(8.9a)
(8.9b) cos θ1∗ /2 ∗ . r1
The corresponding displacement field (8.2) simplifies as ∗ ∗ θ1∗ q2 ∗ ∗ ∗ ∗ , r sin u = −4β k A1 1 α − q2 2 θ∗ ν ∗ = −2k ∗ q ∗ A∗1 r1∗ cos 1 . 2
(8.9c)
(8.10a)
(8.10b)
It is worth noting that at the intersonic crack tip velocity c = c∗ the shock wave type component and the near tip tensile stress vanish. The behaviour of the near tip elastic fields is the same as that shown in the subsonic regime. This result is related to the so called radiation-free solution whose consistency with intersonic crack propagation was recently proved by Gao et al. (1999) through an indirect approach. Acknowledgments The financial support by the Italian Ministry of University (MURST) is acknowledged. Authors are very grateful to the Reviewers for having drawn attention to the works of Gao et al. (1999) and of Broberg (1999) of which they were unaware. References Broberg, K.B. (1999). Intersonic crack propagation in an orthotropic material. International Journal of Fracture 99, 1–11. Gao, H., Huang, Y., Gumbsch, P. and Rosakis, A.J. (1999). On radiation-free transonic motion of cracks and dislocations. Journal of Mechanics, Physics and Solids 47, 1941–1961. Huang, Y., Liu, C. and Rosakis, A.J. (1996). Transonic crack growth along a bimaterial interface: an investigation of the asymptotic structure of near-tip fields. International Journal of Solids and Structures 33, 2625–2645. Huang, Y., Wang, W., Liu, C. and Rosakis, A.J. (1999). Analysis of Intersonic Crack Growth in Unidirectional Fiber-Reinforced Composites. Journal of Mechanics, Physics and Solids 47, 1893–1916. Liu, C., Huang, Y. and Rosakis, A.J. (1995). Shear Dominated Transonic Crack Growth in a Bimaterial- II. Asymptotic Fields and Favorable Velocity Regimes. Journal of Mechanics, Physics and Solids 43, 189–206.
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Piva, A. and Hasan, W. (1996). Effect of orthotropy on the intersonic shear crack propagation. Journal of Applied Mechanics 63, 933–938. Piva, A. (1987). An alternative approach to elastodynamic crack problems in an orthotropic medium. Quarterly of Applied Mathematics 45, 97–104. Piva, A. and Viola, E.L. (1988). Crack propagation in an orthotropic medium. Engineering Fracture of Mechanics 29, 535–548. Yu, H. and Yang, W. (1995). Mechanics of Transonic Debonding of a Bimaterial Interface: the In-Plane Case. Journal of Mechanics, Physics and Solids 43, 207–232.