http://mms.sagepub.com/ Mathematics and Mechanics of Solids http://mms.sagepub.com/content/early/2012/12/20/1081286512462182 The online version of this article can be found at: DOI: 10.1177/1081286512462182 published online 20 December 2012 Mathematics and Mechanics of Solids YA Antipov and AV Smirnov Subsonic propagation of a crack parallel to the boundary of a half-plane Published by: http://www.sagepublications.com can be found at: Mathematics and Mechanics of Solids Additional services and information for http://mms.sagepub.com/cgi/alerts Email Alerts: http://mms.sagepub.com/subscriptions Subscriptions: http://www.sagepub.com/journalsReprints.nav Reprints: http://www.sagepub.com/journalsPermissions.nav Permissions: What is This? - Dec 20, 2012 OnlineFirst Version of Record >> at LOUISIANA STATE UNIV on January 16, 2013 mms.sagepub.com Downloaded from
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http://mms.sagepub.com/Mathematics and Mechanics of Solids
http://mms.sagepub.com/content/early/2012/12/20/1081286512462182The online version of this article can be found at:
DOI: 10.1177/1081286512462182
published online 20 December 2012Mathematics and Mechanics of SolidsYA Antipov and AV Smirnov
Subsonic propagation of a crack parallel to the boundary of a half-plane
Published by:
http://www.sagepublications.com
can be found at:Mathematics and Mechanics of SolidsAdditional services and information for
YA Antipov and AV SmirnovDepartment of Mathematics, Louisiana State University, USA
Received 19 June 2012; accepted 20 June 2012
AbstractA two-dimensional steady-state model problem on a half-plane with a semi-infinite crack propagating at constant speedparallel to the boundary of the half-plane is considered. The crack faces are subjected to normal and tangential loads,while the boundary of the half-plane is free of traction. The problem is formulated as an order-2 vector Riemann–Hilbertproblem and then reduced to a system of singular integral equations in a semi-infinite segment with respect to thederivatives of the displacement jumps. The solution to the system of integral equations is represented in a series form interms of an orthonormal basis of the associated Hilbert space, the orthonormal Jacobi polynomials. The coefficients ofthe expansions solve an infinite system of linear algebraic equations of the second kind. The stress intensity factors andthe weight functions are determined and computed. The Griffith energy criterion is applied to derive a crack growthcriterion in terms of the KI - and KII -stress intensity factors, the crack speed, the shear and dilatational waves speedsand the Griffith material constant.
KeywordsGriffith energy criterion, orthogonal polynomials, subsonic crack propagation, systems of singular integral equations
1. IntroductionThe main motivation of this work is to study the boundary effects on a crack propagating parallel to the boundaryof a solid. The static problem for a semi-infinite crack parallel to the boundary of a half-plane was analyzed byZlatin and Khrapkov [1]. They transformed the model problem into an order-2 vector Riemann–Hilbert (RH)problem and managed to derive a closed-form solution by exact factorization of the matrix coefficient of theRH problem. The steady-state problem for a plane with a semi-infinite crack −∞ < x1 < 0, x2 = 0 driven bymoving normal and tangential loads applied to the crack faces was considered by Craggs [2]. Because of thesymmetry, the Craggs problem admits decoupling and can be solved in closed form by a variety of methodsincluding the factorization method for a scalar RH problem, the Mellin transform method, which bypasses theRH problem, and the method of orthogonal polynomials. Many researchers analyzed different aspects of theCraggs model problem and considered its generalizations. Surveys of the results were given by Freund [3]and Broberg [4]. To our knowledge, no analytical solution for the steady-state problem on a coupled mode-I,IIsemi-infinite crack propagating parallel to the boundary of a half-plane is available in the literature.
In Section 2, we state the model problem for a half-plane {−∞ < x1 < ∞, −∞ < x2 < δ} (δ > 0) with acrack extending from x1 = −∞, x2 = 0 to x1 = Vt, x2 = 0 and driven by normal and tangential componentsof traction applied to the crack faces. It is assumed that the loads move at the same speed V . By employing themethod of integral transformations, we map the boundary-value problem for the governing system of partial
Corresponding author:YA Antipov, Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA.Email: [email protected]
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differential equations to a vector RH problem with the matrix coefficient G(p) = a(p)I + b(p)S(p), where a(p),b(p) are Hölder functions, I = diag{1, 1},
S(p) =(
1 −iα1 tanhα0piα2 tanhα0p −1
), (1)
and α0,α1 and α2 are real nonzero constants. At this stage, all our efforts availed us no results in factorizing thematrix G(p).
In Section 3, we transform the RH problem into a system of two singular integral equations first on thesemi-infinite segment (−∞, 0) and then on the finite segment (−1, 1). Its solution is found in the Hilbert spaceL2,ρ(−1, 1), ρ = (1 + ζ )1/2(1 − ζ )−1/2. By expanding the unknown functions in terms of an orthonormal basisof this space, we reduce the system of integral equations to an infinite system of linear algebraic equations ofthe second kind. Its solution is found by the reduction method.
In Section 4, we derive the stress intensity factors (SIFs) and the weight functions. In addition, we determinethe energy released when the crack extends from x to x + r, and r is small. Then we apply the Griffith criterionand derive a Willis-type formula for a mode-I,II semi-infinite crack propagating along the boundary of a half-plane. The criterion of steady-state propagation of a semi-infinite crack in a plane was derived by Willis [5]. Wealso discuss some numerical results obtained for the SIFs, the weight functions and the Griffith criterion.
2. Formulation and a vector RH problemA semi-infinite crack that occupies, at time t, the surface S2(t) = {x : −∞ < x1 < Vt, x2 = 0, |x3| < ∞}propagates in the direction parallel to the boundary of an elastic half-space R3
+ = {x : |x1| < ∞, −∞ < x2 <δ, |x3| < ∞} (δ > 0). The boundary of the half-space is assumed to be free of traction. The speed V is constantand V < cR, where cR is the Rayleigh wave speed for the elastic solid whose density and the Lame constantsare ρ, λ and μ, respectively. The banks of the crack are subjected to plane-strain loading
The system of equations for two-dimensional dynamic elasticity is the following
c2l�φ − ∂2φ
∂t2= 0, c2
s�ψ − ∂2ψ
∂t2= 0, x ∈ R2
+ \ S1(t), (3)
where cl and cs are the longitudinal and shear wave speeds
cl =√λ+ 2μ
ρ, cs =
√μ
ρ, (4)
S1(t) = {x : −∞ < x1 < Vt, x2 = 0}, R2+ = {x : |x1| < ∞, −∞ < x2 < δ}, and φ and ψ are dynamic
potentials such that the displacements u1, u2 and the stresses σ11, σ12 are
u1 = ∂φ
∂x1+ ∂ψ
∂x2, u2 = ∂φ
∂x2− ∂ψ
∂x1,
σ12 = μ
(2∂2φ
∂x1∂x2− ∂2ψ
∂x21
+ ∂2ψ
∂x22
),
σ22 = λ�φ + 2μ
(∂2φ
∂x22
− ∂2ψ
∂x1∂x2
). (5)
The problem is steady state, and the advantage of introducing the moving coordinates x = x1 − Vt, y = x2 isevident. The governing equations (3) are simplified as
α2 ∂2φ
∂x2+ ∂2φ
∂y2= 0, β2 ∂
2ψ
∂x2+ ∂2ψ
∂y2= 0, (6)
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are to be determined. Six out of the eight equations in (11) determine the unknown functions Cj(p), Dj(p)(j = 0, 1, 2). The other two equations constitute the following vector RH problem(
It can be shown that the eigenvalues of the matrix G(p), λ1(p) and λ2(p), do not satisfy the conditionλ1(p)λ2(p) > 0 for all p ∈ (−∞, ∞), and the matrix G(p) cannot be factorized such that G(p) = g0(p)G0(p),where g0(p) is a scalar, and the matrix G0(p) is positive definite for all real p. The vector RH problem may berewritten as (
�+1 (p)
�+2 (p)
)= G(p)
(β−1X −
1 (p)α−1X −
2 (p)
)−(�−
1 (p)�−
2 (p)
), −∞ < p < +∞, (17)
where the new matrix coefficient has the structure
G(p) = a(p)I + b(p)S(p),
a(p) = e−(α+β)δ|p|
2(1 − β2)
[R1 sinh(α + β)δp + 2d(p) sgn p
R1
],
b(p) = − R2
2(1 − β2)e−(α+β)δ|p| sinh(α − β)δp,
S(p) =(
1 − 4iαR1
(1 + β2) tanh 12 (α − β)δp
4iβR1
(1 + β2) tanh 12 (α − β)δp −1
), (18)
and I is the unit 2 × 2 matrix. The matrix G(p) does not admit a Wiener–Hopf factorization by the methodscurrently available in the literature.
3. System of integral equationsIn this section we aim to map the RH problem into a system of singular integral equations on a semi-infinitesegment and develop an efficient numerical scheme for its solution. By applying the inverse Fourier transform tothe RH problem, using the convolution theorem and evaluating the integrals, we find the following expressionsvalid for −∞ < x < ∞
σxy(x, 0) = − γ
πβ
{∫ 0
−∞
[1
ξ − x+ k11(ξ − x)
]χ1(ξ )dξ +
∫ 0
−∞k12(ξ − x)χ2(ξ )dξ
},
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For −∞ < x < 0, the traction vector components σxy(x, 0) and σyy(x, 0) are known (see (10)). Slightlyrearranged, the expressions (19) give the resulting system of singular integral equations
∫ 0
−∞
[1
ξ − x+ k11(ξ − x)
]χ1(ξ )dξ +
∫ 0
−∞k12(ξ − x)χ2(ξ )dξ = −π f1(x), −∞ < x < 0,
∫ 0
−∞k21(ξ − x)χ1(ξ )dξ +
∫ 0
−∞
[1
ξ − x+ k22(ξ − x)
]χ2(ξ )dξ = −π f2(x), −∞ < x < 0, (21)
where
f1(x) = β
γσ ◦
12(x), f2(x) = α
γσ ◦
22(x). (22)
For δ = ∞ (the half-plane becomes a plane), the kernels kij(x) (i, j = 1, 2) vanish, the system decouples, and itsclosed-form solution is
χj(x) = 1
π√−x
∫ 0
−∞
√−ξ fj(ξ )dξ
ξ − x, −∞ < x < 0, j = 1, 2. (23)
In the case of finite values of δ we propose an approximate scheme based on the method of orthogonal poly-nomials. First, we transform the system into another one on the segment (−1, 1). Introduce new variables andfunctions
As in the case δ = ∞, we seek the solution, the functions χ1(x) and χ2(x), in the space L2,ρ(−1/2,−3/2)(0, ∞). Atzero and infinity, the functions behave as
χj(x) ∼ C0jx−1/2, x → 0; χj(x) ∼ C1jx
−3/2, x → ∞ (27)
(C0j and C1j are nonzero constants). When written for the new functions χ1(ζ ) and χ2(ζ ), this results
Show finally that if δ → ∞, then the solution of the infinite system tends to the closed-form solution (23)for the whole plane. When δ → ∞, then c(j,l)
nm → 0 and a(j)m → b(j)
m , that is if δ = ∞, then
χj(ζ ) = −√
1 − ζ
1 + ζ
∫ 1
−1fj(ξ )
√1 + η
(1 − η)3�(η, ζ )dη, (42)
where
�(η, ζ ) =∞∑
m=0
P−1/2,1/2m (η)P1/2,−1/2
m (ζ ). (43)
To summarize this series, we employ the relations (39) and also the formula
limq→1−
∞∑m=0
q2m+1 sin(2m + 1)x = 1
2 sin x. (44)
This gives us
�(η, ζ ) = 1
π(η − ζ ), (45)
and therefore,
χj(x) = − 1
π
√(1 − ζ )3
1 + ζ
∫ 1
−1fj(ξ )
√1 + η
(1 − η)3
dη
η − ζ, (46)
where x = (ζ + 1)/(ζ − 1), ξ = (η + 1)/(η − 1). This formula, when rearranged, coincides with (23).
4. Near-tip field
4.1. SIFs
To evaluate the SIFsKI = lim
x→0+
√2πxσyy(x, 0), KII = lim
x→0+
√2πxσxy(x, 0), (47)
we analyze the asymptotics of the stresses σxy(x, 0), σyy(x, 0) as x → 0+. From (19), we have
σyy(x, 0) ∼ − γ
πα
∫ 0
−∞
χ2(ξ )dξ
ξ − x, x → 0+. (48)
On making the substitutions (24) from (29), we derive as x → 0+ (ζ → −1−)
σyy(x, 0) ∼ γ (1 − ζ )
πα
∞∑m=0
a(2)m
∫ 1
−1
√1 − η
1 + ηP1/2,−1/2
m (η)dη
η − ζ. (49)
We now employ the following relation for the Jacobi polynomials∫ 1
0τα1 (1 − τ )α2Pα1,α2
n (1 − 2τ )dτ
τ − t= �(α1)�(α2 + n + 1)
�(α1 + α2 + n + 1)F(1 + n, −α1 − α2 − n, 1 − α1; t)
+�(−α1)�(1 + α1 + n)
n!(−t)α1F(1 + α1 + n, −α2 − n, 1 + α1; t), t /∈ (0, 1). (50)
For the orthonormal Jacobi polynomials P1/2,−1/2n (η) this relation can be simplified, and we obtain
σyy(x, 0) ∼ 2γ x−1/2
√πα
∞∑m=0
(−1)ma(2)m , x → 0+. (51)
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Figure 1. The SIFs versus δ for V/cR = 0.5: K(1)I and K
(1)II are the SIFs for the case σ ◦
22(x) = 1, −1 < x < 0, σ ◦22(x) = 0, x < −1,
and σ ◦12(x) = 0, x < 0; K
(2)I and K
(2)II are the SIFs for the case σ ◦
12(x) = 1, −1 < x < 0, σ ◦12(x) = 0, x < −1, and σ ◦
22(x) = 0, x < 0.
On comparing this asymptotic relation with the definition of the SIF KI , we derive the final formula for KI ,
KI = 2√
2γ
α
∞∑m=0
(−1)ma(2)m . (52)
Similarly,
KII = 2√
2γ
β
∞∑m=0
(−1)ma(1)m . (53)
For computations, we take the Poisson ratio ν = 0.3. The Rayleigh speed cR is defined explicitly [7] as
cR = cs√
s∗, s∗ = 1
3(8 − R+ − R−), R± =
(45κ0
2− 404 ± 3
√3R∗2
)1/3
,
R∗ = −14656 + 2768κ0 − 181κ20 + 4κ3
0 , κ0 = 8(2 − ν)/(1 − ν) ∈ (16, 24). (54)
The way in which the SIFs approach their asymptotic values as δ → ∞ is seen in Figure 1: the factors K(1)I
and K(2)II tend to K◦
I and K◦II , respectively, while the other two factors, K(2)
I and K(1)II tend to zero. Here,
K◦I = 2
√2γ
α
∞∑m=0
(−1)mb(2)m , K◦
II = 2√
2γ
β
∞∑m=0
(−1)mb(1)m . (55)
In Figure 1, the crack propagation speed V is chosen to be V = 0.5cR. The factors K(1)I and K(1)
II denote theSIFs for the case when σ ◦
22(x) = 1 for −1 < x < 0, and σ ◦22(x) = 0 otherwise, and σ ◦
12(x) = 0 for all x < 0.
The factors K(2)I and K(2)
II are the SIFs for the case σ ◦12(x) = 1 for −1 < x < 0, and σ ◦
12(x) = 0, x < −1, and
σ ◦22(x) = 0 for all x < 0. As δ → 0, the absolute values of all the factors except for K(2)
II are growing. The SIF
K(2)II approaches zero as the distance between the crack and the boundary of the half-plane tends to zero.
When the crack is close to the surface (δ = 2) and the crack propagation speed V is growing toward theRayleigh speed cR, the magnitudes of all the SIFs apart from K(1)
I are growing. The factor K(1)I is decreasing as
V/cR approaches 1. This is shown in Figure 2.
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while the elevation of the function WI ,I (x) and also the function
WII ,I (x) = 2√
2γ
β
∞∑m=0
(−1)ma(1)m , (64)
requires the solution of the infinite system
a(j)n +
∞∑m=0
[c(j,j)nm a(j)
m + c(j,3−j)nm a(3−j)
m ] = b(j)n , n = 0, 1, . . . ; j = 1, 2, (65)
where
b(1)n = 0, b(2)
n = −∞∑
m=0
c(2,2)nm b(2)
m . (66)
Similarly, the pair WI ,II (x), WII ,II (x) may be written as
WI ,II (x) = 2√
2γ
α
∞∑m=0
(−1)ma(2)m ,
WII ,II (x) = −√
2
−πx+ 2
√2γ
β
∞∑m=0
(−1)ma(1)m , (67)
where the coefficients a(1)m , a(2)
m form the solution of the infinite system (65) with the right-hand side given by
b(1)n = −
∞∑m=0
c(1,1)nm b(1)
m , b(2)n = 0. (68)
For computations of the weight functions, we choose ν = 0.3 and x = −1. When V is fixed (V < cR) andδ → ∞, the weight functions WI ,I and WII ,II tend to −√
2/π , while the other two functions, WI ,II and WII ,I ,vanish, see Figure 3. The magnitudes of all the weight functions grow as δ → 0 and V is fixed (V = 0.5cR inFigure 3).
The weight function curves in Figure 4 for δ = 1, ν = 0.3, x = −1 show that the weight function WI ,Idecreases as V → cR. The other functions may also decrease when the normalized speed V/cR is close to 1.Our numerical scheme becomes less reliable when V approaches the critical speed cR.
4.3. Griffith criterion
Following Willis [5] consider the potential energy δU released when the crack S1(t) = {x : −∞ < x1 <Vt, x2 = 0} extends to S1(t) + δS1(t) = {x : −∞ < x1 < Vt + r, x2 = 0}, where r is small. The energy δU maybe expressed as
δU = 1
2
∫ r
0{σxy(x, 0)δ[u](x) + σyy(x, 0)δ[v](x)}dx. (69)
Here, [u] + δ[u], [v] + δ[v] are the displacement jumps related to the extended crack, and
σxy ∼ KII√2πx
, σyy ∼ KI√2πx
, x ∈ (0, r), r → 0+. (70)
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Figure 3. The weight functions Wj,l(x) (j, l = I , II) versus δ for V/cR = 0.5, x = −1.
0.0 0.2 0.4 0.6 0.8-4
-3
-2
-1
0
1
W I,II
W II,I
W II,II
W I,I
Figure 4. The weight functions Wj,l(x) (j, l = I , II) versus V/cR for δ = 1, x = −1.
To find asymptotic expansions for δ[u], δ[v], we integrate (30) and fix the constant of integration by assuringthat the displacement jumps vanish at the crack tip. For the displacement jumps [u] and [v], we have
(u(x, 0+) − u(x, 0−)v(x, 0+) − v(x, 0−)
)= 2
∞∑m=0
(a(1)
ma(2)
m
)∫ x
0P1/2,−1/2
m
(ξ + 1
ξ − 1
)dξ
(1 − ξ )√−ξ , x < 0. (71)
The displacement jumps vanish at the point x = 0 and tend to the values
−2∞∑
m=0
(a(1)
ma(2)
m
)∫ 0
−∞P1/2,−1/2
m
(ξ + 1
ξ − 1
)dξ
(1 − ξ )√−ξ (72)
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as x → −∞. These values are finite and, in general, nonzero.Since P1/2,−1/2
m (−1) = π−1/2(−1)m, for small negative x,
[v](x) = −4
√−x
π
∞∑m=0
(−1)ma(2)m + O(|x|3/2), x → 0−. (73)
When the crack extends to x = r, due to the relation (52),
[v](x) = −√
2(r − x)
π
αKI
γ+ O{(r − x)3/2}, x → r. (74)
Similarly,
[u](x) = −√
2(r − x)
π
βKII
γ+ O{(r − x)3/2}, x → r. (75)
Now, on substituting formulas (70), (74) and (75) into (69), we obtain
δU ∼ − 1
4γ(αK2
I + βK2II )r, r → 0+. (76)
This formula can be written in terms of the Rayleigh function
D(V ) = 4
√(1 − V 2
c2l
)(1 − V 2
c2s
)−(
2 − V 2
c2s
)2
(77)
as
δU ∼ V 2r
2csμD(V )(αK2
I + βK2II ), r → 0+. (78)
According to the Griffith criterion, the crack starts propagating if the energy δU is equal to or greater than theincrease in the surface energy 2Tr, δU ≥ 2Tr, where T is the Griffith material constant. This criterion may berepresented in terms of the SIFs in the form√
1 − V 2
c2l
K2I +
√1 − V 2
c2s
K2II ≥ 4Tc2
sμD(V )
V 2. (79)
Notice, that if δ = ∞, and σ ◦12(x) = 0 for all x < 0, then the inequality (79) coincides with the criterion
K2I ≥ 4Tc2
sμD(V )
V 2√
1 − V 2/c2l
(80)
obtained by Willis [5]. For finite values of δ, even when the tangential component of loading vanishes, the SIFKII = 0, and both factors, KI and KII , are involved in the Griffith crack propagation criterion.
Another way to represent the crack propagation criterion is to rewrite inequality (79) as
H(KI , KII , V/cs, V/cl) ≥ μT , (81)
where
H = αK2I + βK2
II
4(cs/V )2D(V ). (82)
Figure 5 shows the results of calculations of the function H versus δ for V/cR = 0.5 and some loads. It isseen that H rapidly advances as the distance between the crack and the half-plane boundary decreases.
The dependence of H on the normalized crack speed V/cR when δ = 2 is plotted in Figure 6. The functionH → ∞ as V/cR → 0 and it grows as the crack speed V approaches the Rayleigh speed. The curves in Figure6 are reminiscent of the graph of modulus of cohesion kc(V ) versus V/cs [9] in the Barenblatt-type criterion forintersonic shear crack propagation.
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Figure 5. The function H versus δ for V/cR = 0.5, x = −1 and loading σ ◦22 = δ(ξ − x), σ ◦
12 = 0: H (1); σ ◦22 = 0, σ ◦
12 = δ(ξ − x):
H (2); σ ◦22 = δ(ξ − x), σ ◦
12 = δ(ξ − x): H (3).
0.2 0.4 0.6 0.80
50
100
150
200
H (3)
H (1)
H (2)
Figure 6. The function H versus V/cR for δ = 2, x = −1 and loading σ ◦22 = δ(ξ − x), σ ◦
12 = 0: H (1); σ ◦22 = 0, σ ◦
12 = δ(ξ − x): H (2);
σ ◦22 = δ(ξ − x), σ ◦
12 = δ(ξ − x): H (3).
5. ConclusionsWe have analyzed a two-dimensional steady-state problem on propagation of a semi-infinite crack in a half-plane. The crack is subjected to normal and tangential loads applied to its faces, and it propagates at speedV along the half-plane boundary free of traction. The boundary of the half-plane breaks the symmetry of theproblem, and, in contrast to the problem for a plane, the modes I and II are coupled. We have deduced an order-2vector RH problem associated with the model. The coefficient is a Hermitian matrix that cannot be factorizedin a closed form. We have reduced the problem to a system of two singular integral equations with respect tothe derivatives of the displacement jumps. The method of orthogonal polynomials has been employed for its
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solution. The unknown functions have been expanded in terms of the orthonormal Jacobi polynomials. Thecoefficients of the expansions have been determined from an infinite system of linear algebraic equations of thesecond kind.
We have derived formulas for the SIFs KI and KII and the weight functions WI ,I , WI ,II , WII ,I and WII ,II . Bydetermining the energy δU released when the crack extends to a small distance, we applied the Griffith criterionand established that the crack starts propagating when H ≥ μT , where
H =√
1 − (V/cl)2)K2I +
√1 − (V/cs)2)K2
II
4(cs/V )2D(V ),
D(V ) is the Rayleigh function, cs, cl are the shear and longitudinal waves speeds, μ is the shear modulus andT is the Griffith material constant. We have computed the SIFs, the weight functions and the function H fordifferent V/cR and δ (cR is the Rayleigh speed, and δ is the distance between the half-plane boundary and thecrack). It has been found that H grows to infinity when the distance δ between the crack and the half-planeboundary decreases while the crack speed does not vary. The function H monotonically decreases as δ grows.When the distance δ is fixed, H , as a function of V/cR, attains its minimum in the interval (0, 1) and grows asV/cR approaches the points 0 and 1.
Funding
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
Conflict of interest
None declared.
References
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[2] Craggs, JW. On the propagation of a crack in an elastic-brittle material. J Mech Phys Solids 1960; 8: 66–75.[3] Freund, LB. Dynamic fracture mechanics. Cambridge: Cambridge University Press, 1990.[4] Broberg, KB. Cracks and fracture. San Diego: Academic Press, 1999.[5] Willis, JR. A comparison of the fracture criteria of Griffith and Barenblatt. J Mech Phys Solids 1967; 15: 151–162.[6] Abramowitz, M, and Stegun, IA (eds). Handbook of mathematical functions. Washington: Dover Publications, 1965.[7] Antipov, YA. Subsonic semi-infinite crack with a finite friction zone in a bimaterial. J Mech Phys Solids 2009; 57: 1934–1957.[8] Antipov, YA. Weight functions of a crack in a two-dimensional micropolar solid. Quart J Mech Appl Math 2012; 65: 239–271.[9] Antipov, YA, Obrezanova, O, and Willis, JR. A fracture criterion of “Barenblatt” type for a transonic shear crack. Mathematics
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