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Crack initiation behaviour of orthotropic solids as predicted by the strain energy density theory
Theoretical and Applied Fracture Mechanics 38 (2002) 109–119 www.elsevier.com/locate/tafmec
Crack initiation behaviour of orthotropic solids as predicted by the strain energy density theory C. Carloni, L. Nobile
*
DISTART Department, University of Bologna, Viale risorgimento 2, 40136 Bologna, Italy
Abstract The strain energy density theory is applied to determine crack initiation in orthotropic solids. An approach is used to derive the complex variable expressions of the elastic fields. The formulation is used to solve the boundary value problem by superimposing of Mode-I and Mode-II crack problems. It is shown that asymptotic expressions of the crack tip stress and displacement fields are affected by non-singular terms related to biaxial loading. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction Natural materials, such as soils, rocks and wood, as well as man-made materials such as fibre-reinforced composites and laminates possess microstructures with preferred directions. They are either inherent or introduced in service. The well-known ‘strain energy density theory’ [1–3] has been used to predict crack growth in isotropic solids. It makes use of the so-called strain energy density function dW =dV which varies from location to location in a solid. This criterion can be extended to orthotropic materials. In this case, the critical strain energy density function is assumed to have a polar variation. Crack initiation direction can be determined by minimizing the normalized strain energy density with reference to the critical strain energy density. Investigations related with the problem of finding the elastic fields around a crack in homogenized anisotropic media have been made in [4–7]. The analytical tool used for solving crack problems of anisotropic plates is the complex function approach for plane anisotropic elasticity. The foundation of this theoretical approach can be found in [6] which extends the theory in [4] by reducing the equilibrium equations for plane anisotropic elasticity to canonical form such that the displacement components can be expressed in terms of two independent and distinct complex variables. One of the main objectives of this work is to obtain a complex variable formulation of plane orthotropic elasticity. It is an extension of previous works in [8,9]. The equilibrium equations of an orthotropic medium
*
Corresponding author. Tel.: +39-51-209-3519; fax: +39-51-209-3495. E-mail address: [email protected] (L. Nobile).
0167-8442/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 0 2 ) 0 0 0 8 9 - 7
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are reduced to a first order system involving a four-dimensional vector field. A similarity transformation is then introduced to obtain a canonical form as a couple of independent Cauchy–Riemann systems. This leads to the use of variable notation. Another objective of this work is to point out the effects of orthotropy and load biaxiality on the crack tip fields and the angle of incipient crack extension. The influence of the load biaxiality for cracked isotropic bodies has been considered in [10–13]. It was pointed out that load biaxiality could affect the elastic fields, strain energy density, near tip maximum shear stress and the angle of crack initiation. In contrast to the isotropic case, biaxial load effects in cracked anisotropic media have not received sufficient attention. Complex potentials for an horizontal crack in an orthotropic homogeneous plate subjected at infinity to a biaxial load have been derived [14] from the elliptic hole solution as a limit. The effects of load parallel to the crack are also investigated using the strain energy density criterion [1–3] for the angles of crack initiation under Mode I. 2. Basic equations of orthotropic elasticity Consider an unbounded orthotropic homogeneous continuum with the axes of elastic symmetry coinciding with rectangular coordinates axes x, y, z. The displacement component along the z-axis with their derivatives with respect to z are assumed to vanish. The constitutive equations are ou ov ou ov ou ov rx ¼ C11 þ C12 ; ry ¼ C12 þ C22 ; sxy ¼ C66 þ ð1Þ ox oy ox oy oy ox where uðx; yÞ and vðx; yÞ are the x and y components of the displacement vector, respectively. The parameters in Eq. (1) are related to the elastic constants according to plane strain or plane stress. The equilibrium equations are o2 u o2 u o2 v ¼ 0; þ a þ 2b ox2 oy 2 ox oy
o2 v o2 v o2 u ¼0 þ a þ 2b 1 1 ox2 oy 2 ox oy
ð2Þ
In which a¼
C66 ; C11
2b ¼
C12 þ C66 ; C11
a1 ¼
C22 ; C66
2b1 ¼
C12 þ C66 2b ¼ a C66
ð3Þ
2.1. Vector function Introduce the four-dimensional vector function ou ou ov ov T Uðx; yÞ ¼ ðU1 ; U2 ; U3 ; U4 Þ ; ; ; ox oy ox oy
ð4Þ
Such that the equilibrium equations can be written in a compact form oU oU þA ¼0 ox oy where A is the matrix: 0 0 a 2b B 1 0 0 A¼B @ 2b1 0 0 0 0 1
ð5Þ 1 0 0C C a1 A 0
ð6Þ
C. Carloni, L. Nobile / Theoretical and Applied Fracture Mechanics 38 (2002) 109–119
111
The eigenvalue characteristic equation of (6) takes the form k4 þ 2a1 k2 þ a2 ¼ 0
ð7Þ
with 2a1 ¼ a þ a1 4bb1 ;
a2 ¼ aa1
ð8Þ
In what follows, attention will be focused on the cases where a21 a2 > 0 and a1 > 0. Hence, Eq. (7) gives the imaginary eigenvalues k1 ¼ ip1 ;
k2 ¼ ip2
ð9Þ
and their complex conjugates. Note that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2 p1 ¼ a1 a21 a2 ; p2 ¼ a1 þ a21 a2
ð10Þ
Turning to the approach already used in [8,9], Eq. (5) simplifies to oW oW þB ¼0 ox oy
ð11Þ
The vector function W is defined by Wðx; yÞ ¼ U 1 Uðx; yÞ
ð12Þ
where 0 0
B B B 2bp1 U ¼B B a p2 B @ p1 1
2bp12 a p12 0 0 1
0
0 2bp2 a p22 p2 0
1 2bp22 a p22 C C C 0 C C C 0 A 1
ð13Þ
and 0
0 B p 1 B ¼ U 1 AU ¼ B @0 0
p1 0 0 0
0 0 0 p2
1 0 0 C C p2 A 0
ð14Þ
2.2. Decoupled Cauchy–Riemann system Make use of Eq. (11), two decoupled systems of the Cauchy–Riemann type are obtained by introducing the analytic functions y X1 ðz1 Þ ¼ w1 ðx; y1 Þ þ iw2 ðx; y1 Þ; z1 ¼ x þ iy1 ; y1 ¼ p1 ð15Þ y X2 ðz2 Þ ¼ w3 ðx; y2 Þ þ iw4 ðx; y2 Þ; z2 ¼ x þ iy2 ; y2 ¼ p2 Moreover, introduce the potentials K1 ðz1 Þ ¼ ip1 l1 ðp1 p2 ÞX1 ðz1 Þ;
K2 ðz2 Þ ¼ ip2 l2 ðp1 p2 ÞX2 ðz2 Þ
ð16Þ
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with l1 ¼
2b a þ p12 ; a p12
l2 ¼
2b a p22 a p22
ð17Þ
Using Eqs. (12) and (15), Eq. (1) become C66 Re½p1 K2 ðz2 Þ p2 K1 ðz1 Þ p1 p2 ðp2 p1 Þ C66 ry ¼ Re½p1 K1 ðz1 Þ p2 K2 ðz2 Þ p2 p1 C66 sxy ¼ Im½K1 ðz1 Þ K2 ðz2 Þ p2 p1 rx ¼
Eq. (12) further leads to the following displacement expressions: 2b p1 p2 uðx; yÞ ¼ k k Re ðz Þ ðz Þ 1 1 2 2 p1 p2 l1 ða p12 Þ l2 ða p22 Þ 1 k1 ðz1 Þ k2 ðz2 Þ vðx; yÞ ¼ Im p1 p2 l1 l2
ð18Þ
ð19Þ
Here, k1 ðz1 Þ and k2 ðz2 Þ are the primitives of K1 ðz1 Þ and K2 ðz2 Þ, respectively. The uniform stress field at infinity is given by rð1Þ ¼ T1 ; x
rð1Þ ¼ T2 ; y
sð1Þ xy ¼ T3
ð20Þ
This determines the potential functions p1 ðp22 T1 þ T2 Þ ðp2 p1 Þ þi T3 C66 ðp1 þ p2 Þ 2C66 p2 ðp12 T1 þ T2 Þ ðp2 þ p1 Þ i K2 ðz2 Þ ¼ K02 T3 C66 ðp1 þ p2 Þ 2C66 K1 ðz1 Þ ¼ K01
ð21Þ
where K01 ðz1 Þ and K02 ðz2 Þ are the analytic functions vanishing at infinity. 3. Inclined crack Consider the elastostatic plane problem of an infinite orthotropic medium, containing a traction-free central crack, of length 2l, which is inclined of an angle x with respect to the X-axis of a Cartesian orthogonal system OðX ; Y Þ. The crack lies on a plane of elastic symmetry (Fig. 1) that coincides with the xaxis of the plane orthogonal system Oðx; yÞ. The uniform biaxial stress field is rx ¼ kT ; ry ¼ T ; sxy ¼ 0 ð22Þ where k is a real positive parameter which gives the measure of biaxiality. The crack is traction free while a uniform stress field prevails far away. At infinity, the stress field is T ½ð1 þ kÞ ð1 kÞ cos 2x 2 T T2 ¼ ½ð1 þ kÞ þ ð1 kÞ cos 2x 2 T T3 ¼ ð1 kÞ sin 2x 2
rð1Þ T1 ¼ x rð1Þ y sð1Þ xy
ð23Þ
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Fig. 1. The inclined crack geometry.
Fig. 2. Boundary condition for Mode I and Mode II.
The problem is solved by determining the potential functions K01 ðz1 Þ and K02 ðz2 Þ for Mode-I and Mode-II problems with uniform self-equilibrating tractions applied to the edges of the crack and null stress field at infinity (Fig. 2). The potential functions in Eq. (21) are obtained by applying the superposition principle. 3.1. Mode I The boundary conditions for Mode-I crack problem are rþ y ðx; 0Þ ¼ T2 ; sþ xy ðx; 0Þ þ
¼ 0;
v ðx; 0Þ ¼ 0;
jxj < l
jxj < 1
ð24Þ
jxj > l
At infinity, there prevails ¼ rð1Þ ¼ sð1Þ rð1Þ x y xy ¼ 0
ð25Þ
By Eq. (12), the second and third of Eq. (18) and the symmetry properties Kj ðzj Þ ¼ Kj ðzj Þ (j ¼ 1; 2), the boundary conditions in the second and third of Eqs. (24) and (25) yield K01 ðxÞ ¼ K02 ðxÞ;
jxj < 1
ð26Þ
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and þ
ðK0j Þ ðxÞ ¼ ðK0j Þ ðxÞ;
jxj > l;
j ¼ 1; 2
ð27Þ
It follows that the first of Eq. (24) gives the Hilbert formulation: ðK0j Þþ ðxÞ þ ðK0j Þ ðxÞ ¼
2T2 ; C66
jxj < l;
j ¼ 1; 2
ð28Þ
whose solution is ½K0j ðzj ÞI ¼
T2 ½1 F ðzj Þ; C66
j ¼ 1; 2
ð29Þ
In which zj F ðzj Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 z j l2
ð30Þ
3.2. Mode II For Mode-II problem, the following formulation holds sþ xy ðx; 0Þ ¼ T3 ;
jxj < l
rþ jxj < 1 y ðx; 0Þ ¼ 0; þ ouðx; 0Þ ¼ 0; jxj > l ox
ð31Þ
with rð1Þ ¼ rð1Þ ¼ sð1Þ x y xy ¼ 0
ð32Þ
From Eq. (12), the second and third of Eq. (18) and the skew-symmetry conditions Kj ðzj Þ ¼ Kj ðzj Þ (j ¼ 1; 2), the conditions the second and third of Eqs. (31) and (32) give p1 K01 ðxÞ ¼ p2 K02 ðxÞ;
jxj < 1
ð33Þ
and þ
ðK0j Þ ðxÞ ¼ ðK0j Þ ðxÞ;
jxj > l; j ¼ 1; 2
ð34Þ
Taking into account Eqs. (33) and (34), the first condition Eq. (31) yields for jxj < l: þ
ðK01 Þ ðxÞ þ ðK01 Þ ðxÞ ¼
2ip2 T3 ; C66
þ
ðK02 Þ ðxÞ þ ðK02 Þ ðxÞ ¼
2ip1 T3 C66
ð35Þ
Eq. (35) can be satisfied by having ½K01 ðz1 ÞII ¼ for jxj < l.
ip2 T3 ½1 F ðz1 Þ; C66
½K02 ðz2 ÞII ¼
ip1 T3 ½1 F ðz2 Þ C66
ð36Þ
C. Carloni, L. Nobile / Theoretical and Applied Fracture Mechanics 38 (2002) 109–119
115
Substituting Eqs. (29) and (36) into Eq. (21) and applying the superposition principle, potential functions for the inclined crack problem are obtained: K1 ðz1 Þ ¼
T2 ip2 T3 p1 ðp22 T1 þ T2 Þ ðp2 þ p1 ÞT3 þi ½1 F ðz1 Þ C66 ðp1 þ p2 Þ C66 2C66
K2 ðz2 Þ ¼
T2 ip1 T3 p2 ðp12 T1 þ T2 Þ ðp2 þ p1 ÞT3 i ½1 F ðz2 Þ C66 ðp1 þ p2 Þ C66 2C66
ð37Þ
Integration yield k1 ðz1 Þ ¼
T2 ip2 T3 p2 ðT2 p1 p2 T1 Þ ðp2 þ p1 ÞT3 z1 i G1 ðz1 Þ z1 C66 ðp1 þ p2 Þ C66 2C66
T2 ip1 T3 p1 ðT2 p1 p2 T1 Þ ðp2 þ p1 ÞT3 z2 i k2 ðz2 Þ ¼ G2 ðz2 Þ þ z2 C66 ðp1 þ p2 Þ C66 2C66 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi in which Gj ðzj Þ ¼ z2j l2 (j ¼ 1; 2).
ð38Þ
4. The near tip elastic fields By substituting Eq. (37) into Eq. (18), it gives the stress components
T2 1 rx ¼ T1 T2 ½p2 ReF1 ðz1 Þ p1 ReF2 ðz2 Þ þ T3 ½p22 ImF1 ðz1 Þ p12 ImF2 ðz2 Þ þ p1 p2 p1 p2 ðp2 þ p1 Þ ry ¼
1 fT2 ½p2 ReF2 ðz2 Þ p1 ReF1 ðz1 Þ þ p1 p2 T3 ½ImF2 ðz2 Þ ImF1 ðz1 Þg p2 p1
sxy ¼
1 fT3 ½p2 ReF1 ðz1 Þ p1 ReF2 ðz2 Þ þ T2 ½ImF2 ðz2 Þ ImF1 ðz1 Þg p2 p1
ð39Þ
The displacement components are obtained by substituting Eq. (38) into Eq. (19). The results are 2b p2 p1 T2 ReG2 ðz2 Þ ReG1 ðz1 Þ uðx; yÞ ¼ C66 ðp1 p2 Þ l2 ða p22 Þ l1 ða p12 Þ 1 1 þ p1 p2 T3 ImG2 ðz2 Þ ImG1 ðz1 Þ l2 ða p22 Þ l1 ða p12 Þ 2
vðx; yÞ ¼
2bp1 p2 ðT2 p1 p2 T1 Þ bðp1 þ p2 Þ T3 x y 2 2 C66 l1 l2 ða p1 Þða p2 Þ C66 l1 l2 ða p12 Þða p22 Þ
ð40Þ
1 fT2 ½l1 ImG2 ðz2 Þ l2 ImG1 ðz1 Þ þ T3 ½l2 p2 ReG1 ðz1 Þ l1 p1 ReG2 ðz2 Þg C66 l1 l2 ðp1 p2 Þ ðT2 p1 p2 T1 Þ l2 p2 l1 p1 ðp1 þ p2 ÞT3 þ ðl1 l2 Þx yþ 2 2 p2 2C66 l1 l2 ðp1 p2 Þ C66 l1 l2 ðp1 p2 Þ p1
Expressing functions F ðzj Þ as power series expansion of complex variables nj ¼ zj l, (j ¼ 1; 2), originating at the right crack tip (Fig. 3) and retaining non-singular term, the following asymptotic stress components in terms of polar coordinates ðr; hÞ centred at crack tip are found:
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C. Carloni, L. Nobile / Theoretical and Applied Fracture Mechanics 38 (2002) 109–119
Fig. 3. Polar coordinates at the crack tip.
# " #) rffiffiffiffiffi( " T2 1 l cosðh1 =2Þ cosðh2 =2Þ 2 sinðh2 =2Þ 2 sinðh1 =2Þ T2 p2 pffiffiffiffiffiffiffiffiffiffiffi p1 pffiffiffiffiffiffiffiffiffiffiffi þ T3 p1 pffiffiffiffiffiffiffiffiffiffiffi p2 pffiffiffiffiffiffiffiffiffiffiffi r x ¼ T1 þ p1 p2 p1 p2 ðp2 p1 Þ 2r g1 ðhÞ g2 ðhÞ g2 ðhÞ g1 ðhÞ # " #) rffiffiffiffiffi( " 1 l cosðh2 =2Þ cosðh1 =2Þ sinðh1 =2Þ sinðh2 =2Þ T2 p2 pffiffiffiffiffiffiffiffiffiffiffi p1 pffiffiffiffiffiffiffiffiffiffiffi þ p1 p2 T3 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ry ¼ ðp2 p1 Þ 2r g2 ðhÞ g1 ðhÞ g1 ðhÞ g2 ðhÞ # " #) rffiffiffiffiffi( " 1 l cosðh1 =2Þ cosðh2 =2Þ sinðh1 =2Þ sinðh2 =2Þ T3 p2 pffiffiffiffiffiffiffiffiffiffiffi p1 pffiffiffiffiffiffiffiffiffiffiffi þ T2 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi sxy ¼ ð41Þ ðp2 p1 Þ 2r g1 ðhÞ g2 ðhÞ g1 ðhÞ g2 ðhÞ
where the following notations have been adopted: y tg h hj ¼ tg1 ; gj ðhÞ ¼ ¼ tg1 pj ðx lÞ pj
sin2 h cos h þ 2 pj
!1=2
2
ð42Þ
The same procedure when applied to functions Gj ðzj Þ leads the local displacements ( " pffiffiffiffiffiffiffiffiffiffiffi # pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi p2 g2 ðhÞ 2b h2 p1 g1 ðhÞ h1 2lr T2 cos cos u¼ C66 ðp1 p2 Þ l2 ða p22 Þ 2 l1 ða p12 Þ 2 " pffiffiffiffiffiffiffiffiffiffiffi #) pffiffiffiffiffiffiffiffiffiffiffi g2 ðhÞ g1 ðhÞ h2 h1 sin sin þ p1 p2 T3 2 l1 ða p12 Þ 2 l2 ða p22 Þ 2
2bp1 p2 ðT2 p1 p2 T1 Þ bT3 ðp1 þ p2 Þ ðl þ r cos hÞ r sin h C66 l1 l2 ða p12 Þða p22 Þ C66 l1 l2 ða p12 Þða p22 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 2lr 1 h2 h1 v¼ T2 l1 g2 ðhÞ sin l2 g1 ðhÞ sin C66 ðp1 p2 Þ l1 l2 2 2 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi h1 h2 þ T3 l2 p2 g1 ðhÞ cos l1 p1 g1 ðhÞ cos 2 2 T3 ðp1 þ p2 Þðl1 l2 Þ ðT2 p1 p2 T1 Þ p2 p1 ðl þ r cos hÞ þ þ r sin h 2C66 l1 l2 ðp1 p2 Þ C66 ðp12 p22 Þ l1 p1 l2 p2
ð43Þ
5. Strain energy density criterion The strain energy density criterion can be applied to predict the crack initiation and extension in isotropic materials [1–3]. This criterion is applied to orthotropic medium containing an inclined crack aligned with one of the axes of elastic symmetry of the body.
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117
Making use of the stress and strain components rij and eij , the strain energy density can be written as Z eij dW ¼ rij deij ð44Þ dV 0 For linear elasticity, Eq. (44) reduces to dW 1 ¼ rij eij dV 2
ð45Þ
For the elastostatic plane problem of an infinite orthotropic medium containing a traction-free central crack of length 2l inclined at an angle x with respect to the X-axis, Eq. (45) takes the specific form " # dW 1 r2x C22 þ r2y C11 2rx ry C12 s2xy ¼ þ ð46Þ 2 dV 2 C11 C22 C12 C66 In which Cij are related to elastic constants for plane problems. For isotropic materials, the relative minimum of dW =dV is assumed to be associated with the direction of crack initiation while the crack is assumed to grow when dW =dV reaches a critical value ðdW =dV Þc . In the orthotropic case, ðdW =dV Þc in the x–y-direction are obviously different. A simpler relation for ðdW =dV Þc is defined: h x y dW dW dW ¼ sin2 h þ cos2 h ð47Þ dV c dV c dV c In contrast to strength criterion, dW =dV accounts for both the stress and strain effects since it represents the x y area under the uniaxial stress and strain curve. In Eq. (47), ðdW =dV Þc and ðdW =dV Þc are, respectively, the critical energy density in the x–y-direction. They can be found from tests. Their substitution into Eq. (47) determines ðdW =dV Þhc .
Fig. 4. Crack extension angle h0 vs. crack inclination angle x for various values of biaxial load parameter, for graphite-epoxy with strength ratio Yt =Xt ¼ 0:04.
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C. Carloni, L. Nobile / Theoretical and Applied Fracture Mechanics 38 (2002) 109–119
Fig. 5. Crack extension angle h0 vs. crack inclination angle x for various values of biaxial load parameter, for E glass-epoxy with strength ratio Yt =Xt ¼ 0:2.
Referring to the graphite-epoxy and E glass-epoxy [15], with a given ðdW =dV Þxc =ðdW =dV Þyc ratio, a corresponding crack initiation angle h0 can be obtained for different crack inclination angle x, for different values of the biaxial load parameter k. The results are in Figs. 4 and 5. Note that when k ¼ 1 the crack inclination effect disappears. 6. Conclusions The problem of crack initiation in orthotropic medium with a crack aligned along one of the three orthogonal axes of elasticity is analyzed. A uniform biaxial load applied at infinity is taken. Using a complex variables formulation, the crack tip stress components and the displacement fields are obtained by considering the non-singular terms that influence by biaxial load effect. The strain density energy theory is been used to predict the crack initiation. The critical strain energy density is determined by including the directional dependency of the tensile strength for orthotropic materials. Crack initiation is found to depend not only on the biaxiality parameter but also on the anisotropic character of the material. Acknowledgements This work has been supported by Italian Ministry for Education, University and Research MIUR (40% and 60%). References [1] G.C. Sih, A special theory of crack propagation: methods of analysis and solutions of crack problems, in: Mechanics of Fracture I, Noordhoff, Leyden, 1973, pp. 21–45.
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[2] G.C. Sih, Strain density factor applied to mixed mode crack problems, Int. J. Fract. 10 (1974) 305–321. [3] G.C. Sih, A three dimensional strain density factor theory of crack propagation: three-dimensional crack problems, in: Mechanics of Fracture I, Noordhoff, Leyden, 1975, pp. 15–53. [4] N.I. Muskelishvili, Some Basic Problems on the Mathematical Theory of Elasticity, Noordhoff, Groningen, 1952. [5] G.N. Savin, Stress Concentration Around Holes, Pergamon Press, Oxford, 1961. [6] S.G. Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day, San Francisco, 1963. [7] G.C. Sih, P.C. Paris, G.R. Irwin, On cracks in rectilinearly anisotropic bodies, Int. J. Fract. Mech. 1 (1965) 189–203. [8] A. Piva, E. Viola, Crack propagation in an orthotropic medium, Engng. Fract. Mech. 29 (1988) 535–548. [9] E. Viola, A. Piva, E. Radi, Crack propagation in an orthotropic medium under general loading, Engng. Fract. Mech. 34 (1989) 1155–1174. [10] J. Eftis, N. Subramonian, H. Liebowitz, Crack border stress and displacement equations revisited, Engng. Fract. Mech. 9 (1977) 189–220. [11] J. Eftis, N. Subramonian, The inclined crack under biaxial load, Engng. Fract. Mech. 10 (1978) 43–67. [12] J. Eftis, D.L. Jones, Influence of load biaxiality on the fracture load of centre cracked sheets, Int. J. Fract. 20 (1982) 267–289. [13] S.K. Maiti, R.A. Smith, Criteria for brittle fracture in biaxial tension, Engng. Fract. Mech. 19 (1984) 793–804. [14] W.K. Lim, S.Y. Choi, B.V. Sankar, Biaxial load effects on crack extension in anisotropic solids, Engng. Fract. Mech. 68 (2001) 403–416. [15] G.C. Sih, M. Arcisz, Effect of orthotropy on crack propagation, Theor. Appl. Fract. Mech. 1 (1984) 225–238.
Crack initiation behaviour of orthotropic solids as predicted by the strain energy density theory C. Carloni, L. Nobile
*
DISTART Department, University of Bologna, Viale risorgimento 2, 40136 Bologna, Italy
Abstract The strain energy density theory is applied to determine crack initiation in orthotropic solids. An approach is used to derive the complex variable expressions of the elastic fields. The formulation is used to solve the boundary value problem by superimposing of Mode-I and Mode-II crack problems. It is shown that asymptotic expressions of the crack tip stress and displacement fields are affected by non-singular terms related to biaxial loading. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction Natural materials, such as soils, rocks and wood, as well as man-made materials such as fibre-reinforced composites and laminates possess microstructures with preferred directions. They are either inherent or introduced in service. The well-known ‘strain energy density theory’ [1–3] has been used to predict crack growth in isotropic solids. It makes use of the so-called strain energy density function dW =dV which varies from location to location in a solid. This criterion can be extended to orthotropic materials. In this case, the critical strain energy density function is assumed to have a polar variation. Crack initiation direction can be determined by minimizing the normalized strain energy density with reference to the critical strain energy density. Investigations related with the problem of finding the elastic fields around a crack in homogenized anisotropic media have been made in [4–7]. The analytical tool used for solving crack problems of anisotropic plates is the complex function approach for plane anisotropic elasticity. The foundation of this theoretical approach can be found in [6] which extends the theory in [4] by reducing the equilibrium equations for plane anisotropic elasticity to canonical form such that the displacement components can be expressed in terms of two independent and distinct complex variables. One of the main objectives of this work is to obtain a complex variable formulation of plane orthotropic elasticity. It is an extension of previous works in [8,9]. The equilibrium equations of an orthotropic medium
*
Corresponding author. Tel.: +39-51-209-3519; fax: +39-51-209-3495. E-mail address: [email protected] (L. Nobile).
0167-8442/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 0 2 ) 0 0 0 8 9 - 7
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are reduced to a first order system involving a four-dimensional vector field. A similarity transformation is then introduced to obtain a canonical form as a couple of independent Cauchy–Riemann systems. This leads to the use of variable notation. Another objective of this work is to point out the effects of orthotropy and load biaxiality on the crack tip fields and the angle of incipient crack extension. The influence of the load biaxiality for cracked isotropic bodies has been considered in [10–13]. It was pointed out that load biaxiality could affect the elastic fields, strain energy density, near tip maximum shear stress and the angle of crack initiation. In contrast to the isotropic case, biaxial load effects in cracked anisotropic media have not received sufficient attention. Complex potentials for an horizontal crack in an orthotropic homogeneous plate subjected at infinity to a biaxial load have been derived [14] from the elliptic hole solution as a limit. The effects of load parallel to the crack are also investigated using the strain energy density criterion [1–3] for the angles of crack initiation under Mode I. 2. Basic equations of orthotropic elasticity Consider an unbounded orthotropic homogeneous continuum with the axes of elastic symmetry coinciding with rectangular coordinates axes x, y, z. The displacement component along the z-axis with their derivatives with respect to z are assumed to vanish. The constitutive equations are ou ov ou ov ou ov rx ¼ C11 þ C12 ; ry ¼ C12 þ C22 ; sxy ¼ C66 þ ð1Þ ox oy ox oy oy ox where uðx; yÞ and vðx; yÞ are the x and y components of the displacement vector, respectively. The parameters in Eq. (1) are related to the elastic constants according to plane strain or plane stress. The equilibrium equations are o2 u o2 u o2 v ¼ 0; þ a þ 2b ox2 oy 2 ox oy
o2 v o2 v o2 u ¼0 þ a þ 2b 1 1 ox2 oy 2 ox oy
ð2Þ
In which a¼
C66 ; C11
2b ¼
C12 þ C66 ; C11
a1 ¼
C22 ; C66
2b1 ¼
C12 þ C66 2b ¼ a C66
ð3Þ
2.1. Vector function Introduce the four-dimensional vector function ou ou ov ov T Uðx; yÞ ¼ ðU1 ; U2 ; U3 ; U4 Þ ; ; ; ox oy ox oy
ð4Þ
Such that the equilibrium equations can be written in a compact form oU oU þA ¼0 ox oy where A is the matrix: 0 0 a 2b B 1 0 0 A¼B @ 2b1 0 0 0 0 1
ð5Þ 1 0 0C C a1 A 0
ð6Þ
C. Carloni, L. Nobile / Theoretical and Applied Fracture Mechanics 38 (2002) 109–119
111
The eigenvalue characteristic equation of (6) takes the form k4 þ 2a1 k2 þ a2 ¼ 0
ð7Þ
with 2a1 ¼ a þ a1 4bb1 ;
a2 ¼ aa1
ð8Þ
In what follows, attention will be focused on the cases where a21 a2 > 0 and a1 > 0. Hence, Eq. (7) gives the imaginary eigenvalues k1 ¼ ip1 ;
k2 ¼ ip2
ð9Þ
and their complex conjugates. Note that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2 p1 ¼ a1 a21 a2 ; p2 ¼ a1 þ a21 a2
ð10Þ
Turning to the approach already used in [8,9], Eq. (5) simplifies to oW oW þB ¼0 ox oy
ð11Þ
The vector function W is defined by Wðx; yÞ ¼ U 1 Uðx; yÞ
ð12Þ
where 0 0
B B B 2bp1 U ¼B B a p2 B @ p1 1
2bp12 a p12 0 0 1
0
0 2bp2 a p22 p2 0
1 2bp22 a p22 C C C 0 C C C 0 A 1
ð13Þ
and 0
0 B p 1 B ¼ U 1 AU ¼ B @0 0
p1 0 0 0
0 0 0 p2
1 0 0 C C p2 A 0
ð14Þ
2.2. Decoupled Cauchy–Riemann system Make use of Eq. (11), two decoupled systems of the Cauchy–Riemann type are obtained by introducing the analytic functions y X1 ðz1 Þ ¼ w1 ðx; y1 Þ þ iw2 ðx; y1 Þ; z1 ¼ x þ iy1 ; y1 ¼ p1 ð15Þ y X2 ðz2 Þ ¼ w3 ðx; y2 Þ þ iw4 ðx; y2 Þ; z2 ¼ x þ iy2 ; y2 ¼ p2 Moreover, introduce the potentials K1 ðz1 Þ ¼ ip1 l1 ðp1 p2 ÞX1 ðz1 Þ;
K2 ðz2 Þ ¼ ip2 l2 ðp1 p2 ÞX2 ðz2 Þ
ð16Þ
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with l1 ¼
2b a þ p12 ; a p12
l2 ¼
2b a p22 a p22
ð17Þ
Using Eqs. (12) and (15), Eq. (1) become C66 Re½p1 K2 ðz2 Þ p2 K1 ðz1 Þ p1 p2 ðp2 p1 Þ C66 ry ¼ Re½p1 K1 ðz1 Þ p2 K2 ðz2 Þ p2 p1 C66 sxy ¼ Im½K1 ðz1 Þ K2 ðz2 Þ p2 p1 rx ¼
Eq. (12) further leads to the following displacement expressions: 2b p1 p2 uðx; yÞ ¼ k k Re ðz Þ ðz Þ 1 1 2 2 p1 p2 l1 ða p12 Þ l2 ða p22 Þ 1 k1 ðz1 Þ k2 ðz2 Þ vðx; yÞ ¼ Im p1 p2 l1 l2
ð18Þ
ð19Þ
Here, k1 ðz1 Þ and k2 ðz2 Þ are the primitives of K1 ðz1 Þ and K2 ðz2 Þ, respectively. The uniform stress field at infinity is given by rð1Þ ¼ T1 ; x
rð1Þ ¼ T2 ; y
sð1Þ xy ¼ T3
ð20Þ
This determines the potential functions p1 ðp22 T1 þ T2 Þ ðp2 p1 Þ þi T3 C66 ðp1 þ p2 Þ 2C66 p2 ðp12 T1 þ T2 Þ ðp2 þ p1 Þ i K2 ðz2 Þ ¼ K02 T3 C66 ðp1 þ p2 Þ 2C66 K1 ðz1 Þ ¼ K01
ð21Þ
where K01 ðz1 Þ and K02 ðz2 Þ are the analytic functions vanishing at infinity. 3. Inclined crack Consider the elastostatic plane problem of an infinite orthotropic medium, containing a traction-free central crack, of length 2l, which is inclined of an angle x with respect to the X-axis of a Cartesian orthogonal system OðX ; Y Þ. The crack lies on a plane of elastic symmetry (Fig. 1) that coincides with the xaxis of the plane orthogonal system Oðx; yÞ. The uniform biaxial stress field is rx ¼ kT ; ry ¼ T ; sxy ¼ 0 ð22Þ where k is a real positive parameter which gives the measure of biaxiality. The crack is traction free while a uniform stress field prevails far away. At infinity, the stress field is T ½ð1 þ kÞ ð1 kÞ cos 2x 2 T T2 ¼ ½ð1 þ kÞ þ ð1 kÞ cos 2x 2 T T3 ¼ ð1 kÞ sin 2x 2
rð1Þ T1 ¼ x rð1Þ y sð1Þ xy
ð23Þ
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113
Fig. 1. The inclined crack geometry.
Fig. 2. Boundary condition for Mode I and Mode II.
The problem is solved by determining the potential functions K01 ðz1 Þ and K02 ðz2 Þ for Mode-I and Mode-II problems with uniform self-equilibrating tractions applied to the edges of the crack and null stress field at infinity (Fig. 2). The potential functions in Eq. (21) are obtained by applying the superposition principle. 3.1. Mode I The boundary conditions for Mode-I crack problem are rþ y ðx; 0Þ ¼ T2 ; sþ xy ðx; 0Þ þ
¼ 0;
v ðx; 0Þ ¼ 0;
jxj < l
jxj < 1
ð24Þ
jxj > l
At infinity, there prevails ¼ rð1Þ ¼ sð1Þ rð1Þ x y xy ¼ 0
ð25Þ
By Eq. (12), the second and third of Eq. (18) and the symmetry properties Kj ðzj Þ ¼ Kj ðzj Þ (j ¼ 1; 2), the boundary conditions in the second and third of Eqs. (24) and (25) yield K01 ðxÞ ¼ K02 ðxÞ;
jxj < 1
ð26Þ
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and þ
ðK0j Þ ðxÞ ¼ ðK0j Þ ðxÞ;
jxj > l;
j ¼ 1; 2
ð27Þ
It follows that the first of Eq. (24) gives the Hilbert formulation: ðK0j Þþ ðxÞ þ ðK0j Þ ðxÞ ¼
2T2 ; C66
jxj < l;
j ¼ 1; 2
ð28Þ
whose solution is ½K0j ðzj ÞI ¼
T2 ½1 F ðzj Þ; C66
j ¼ 1; 2
ð29Þ
In which zj F ðzj Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 z j l2
ð30Þ
3.2. Mode II For Mode-II problem, the following formulation holds sþ xy ðx; 0Þ ¼ T3 ;
jxj < l
rþ jxj < 1 y ðx; 0Þ ¼ 0; þ ouðx; 0Þ ¼ 0; jxj > l ox
ð31Þ
with rð1Þ ¼ rð1Þ ¼ sð1Þ x y xy ¼ 0
ð32Þ
From Eq. (12), the second and third of Eq. (18) and the skew-symmetry conditions Kj ðzj Þ ¼ Kj ðzj Þ (j ¼ 1; 2), the conditions the second and third of Eqs. (31) and (32) give p1 K01 ðxÞ ¼ p2 K02 ðxÞ;
jxj < 1
ð33Þ
and þ
ðK0j Þ ðxÞ ¼ ðK0j Þ ðxÞ;
jxj > l; j ¼ 1; 2
ð34Þ
Taking into account Eqs. (33) and (34), the first condition Eq. (31) yields for jxj < l: þ
ðK01 Þ ðxÞ þ ðK01 Þ ðxÞ ¼
2ip2 T3 ; C66
þ
ðK02 Þ ðxÞ þ ðK02 Þ ðxÞ ¼
2ip1 T3 C66
ð35Þ
Eq. (35) can be satisfied by having ½K01 ðz1 ÞII ¼ for jxj < l.
ip2 T3 ½1 F ðz1 Þ; C66
½K02 ðz2 ÞII ¼
ip1 T3 ½1 F ðz2 Þ C66
ð36Þ
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115
Substituting Eqs. (29) and (36) into Eq. (21) and applying the superposition principle, potential functions for the inclined crack problem are obtained: K1 ðz1 Þ ¼
T2 ip2 T3 p1 ðp22 T1 þ T2 Þ ðp2 þ p1 ÞT3 þi ½1 F ðz1 Þ C66 ðp1 þ p2 Þ C66 2C66
K2 ðz2 Þ ¼
T2 ip1 T3 p2 ðp12 T1 þ T2 Þ ðp2 þ p1 ÞT3 i ½1 F ðz2 Þ C66 ðp1 þ p2 Þ C66 2C66
ð37Þ
Integration yield k1 ðz1 Þ ¼
T2 ip2 T3 p2 ðT2 p1 p2 T1 Þ ðp2 þ p1 ÞT3 z1 i G1 ðz1 Þ z1 C66 ðp1 þ p2 Þ C66 2C66
T2 ip1 T3 p1 ðT2 p1 p2 T1 Þ ðp2 þ p1 ÞT3 z2 i k2 ðz2 Þ ¼ G2 ðz2 Þ þ z2 C66 ðp1 þ p2 Þ C66 2C66 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi in which Gj ðzj Þ ¼ z2j l2 (j ¼ 1; 2).
ð38Þ
4. The near tip elastic fields By substituting Eq. (37) into Eq. (18), it gives the stress components
T2 1 rx ¼ T1 T2 ½p2 ReF1 ðz1 Þ p1 ReF2 ðz2 Þ þ T3 ½p22 ImF1 ðz1 Þ p12 ImF2 ðz2 Þ þ p1 p2 p1 p2 ðp2 þ p1 Þ ry ¼
1 fT2 ½p2 ReF2 ðz2 Þ p1 ReF1 ðz1 Þ þ p1 p2 T3 ½ImF2 ðz2 Þ ImF1 ðz1 Þg p2 p1
sxy ¼
1 fT3 ½p2 ReF1 ðz1 Þ p1 ReF2 ðz2 Þ þ T2 ½ImF2 ðz2 Þ ImF1 ðz1 Þg p2 p1
ð39Þ
The displacement components are obtained by substituting Eq. (38) into Eq. (19). The results are 2b p2 p1 T2 ReG2 ðz2 Þ ReG1 ðz1 Þ uðx; yÞ ¼ C66 ðp1 p2 Þ l2 ða p22 Þ l1 ða p12 Þ 1 1 þ p1 p2 T3 ImG2 ðz2 Þ ImG1 ðz1 Þ l2 ða p22 Þ l1 ða p12 Þ 2
vðx; yÞ ¼
2bp1 p2 ðT2 p1 p2 T1 Þ bðp1 þ p2 Þ T3 x y 2 2 C66 l1 l2 ða p1 Þða p2 Þ C66 l1 l2 ða p12 Þða p22 Þ
ð40Þ
1 fT2 ½l1 ImG2 ðz2 Þ l2 ImG1 ðz1 Þ þ T3 ½l2 p2 ReG1 ðz1 Þ l1 p1 ReG2 ðz2 Þg C66 l1 l2 ðp1 p2 Þ ðT2 p1 p2 T1 Þ l2 p2 l1 p1 ðp1 þ p2 ÞT3 þ ðl1 l2 Þx yþ 2 2 p2 2C66 l1 l2 ðp1 p2 Þ C66 l1 l2 ðp1 p2 Þ p1
Expressing functions F ðzj Þ as power series expansion of complex variables nj ¼ zj l, (j ¼ 1; 2), originating at the right crack tip (Fig. 3) and retaining non-singular term, the following asymptotic stress components in terms of polar coordinates ðr; hÞ centred at crack tip are found:
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Fig. 3. Polar coordinates at the crack tip.
# " #) rffiffiffiffiffi( " T2 1 l cosðh1 =2Þ cosðh2 =2Þ 2 sinðh2 =2Þ 2 sinðh1 =2Þ T2 p2 pffiffiffiffiffiffiffiffiffiffiffi p1 pffiffiffiffiffiffiffiffiffiffiffi þ T3 p1 pffiffiffiffiffiffiffiffiffiffiffi p2 pffiffiffiffiffiffiffiffiffiffiffi r x ¼ T1 þ p1 p2 p1 p2 ðp2 p1 Þ 2r g1 ðhÞ g2 ðhÞ g2 ðhÞ g1 ðhÞ # " #) rffiffiffiffiffi( " 1 l cosðh2 =2Þ cosðh1 =2Þ sinðh1 =2Þ sinðh2 =2Þ T2 p2 pffiffiffiffiffiffiffiffiffiffiffi p1 pffiffiffiffiffiffiffiffiffiffiffi þ p1 p2 T3 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ry ¼ ðp2 p1 Þ 2r g2 ðhÞ g1 ðhÞ g1 ðhÞ g2 ðhÞ # " #) rffiffiffiffiffi( " 1 l cosðh1 =2Þ cosðh2 =2Þ sinðh1 =2Þ sinðh2 =2Þ T3 p2 pffiffiffiffiffiffiffiffiffiffiffi p1 pffiffiffiffiffiffiffiffiffiffiffi þ T2 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi sxy ¼ ð41Þ ðp2 p1 Þ 2r g1 ðhÞ g2 ðhÞ g1 ðhÞ g2 ðhÞ
where the following notations have been adopted: y tg h hj ¼ tg1 ; gj ðhÞ ¼ ¼ tg1 pj ðx lÞ pj
sin2 h cos h þ 2 pj
!1=2
2
ð42Þ
The same procedure when applied to functions Gj ðzj Þ leads the local displacements ( " pffiffiffiffiffiffiffiffiffiffiffi # pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi p2 g2 ðhÞ 2b h2 p1 g1 ðhÞ h1 2lr T2 cos cos u¼ C66 ðp1 p2 Þ l2 ða p22 Þ 2 l1 ða p12 Þ 2 " pffiffiffiffiffiffiffiffiffiffiffi #) pffiffiffiffiffiffiffiffiffiffiffi g2 ðhÞ g1 ðhÞ h2 h1 sin sin þ p1 p2 T3 2 l1 ða p12 Þ 2 l2 ða p22 Þ 2
2bp1 p2 ðT2 p1 p2 T1 Þ bT3 ðp1 þ p2 Þ ðl þ r cos hÞ r sin h C66 l1 l2 ða p12 Þða p22 Þ C66 l1 l2 ða p12 Þða p22 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 2lr 1 h2 h1 v¼ T2 l1 g2 ðhÞ sin l2 g1 ðhÞ sin C66 ðp1 p2 Þ l1 l2 2 2 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi h1 h2 þ T3 l2 p2 g1 ðhÞ cos l1 p1 g1 ðhÞ cos 2 2 T3 ðp1 þ p2 Þðl1 l2 Þ ðT2 p1 p2 T1 Þ p2 p1 ðl þ r cos hÞ þ þ r sin h 2C66 l1 l2 ðp1 p2 Þ C66 ðp12 p22 Þ l1 p1 l2 p2
ð43Þ
5. Strain energy density criterion The strain energy density criterion can be applied to predict the crack initiation and extension in isotropic materials [1–3]. This criterion is applied to orthotropic medium containing an inclined crack aligned with one of the axes of elastic symmetry of the body.
C. Carloni, L. Nobile / Theoretical and Applied Fracture Mechanics 38 (2002) 109–119
117
Making use of the stress and strain components rij and eij , the strain energy density can be written as Z eij dW ¼ rij deij ð44Þ dV 0 For linear elasticity, Eq. (44) reduces to dW 1 ¼ rij eij dV 2
ð45Þ
For the elastostatic plane problem of an infinite orthotropic medium containing a traction-free central crack of length 2l inclined at an angle x with respect to the X-axis, Eq. (45) takes the specific form " # dW 1 r2x C22 þ r2y C11 2rx ry C12 s2xy ¼ þ ð46Þ 2 dV 2 C11 C22 C12 C66 In which Cij are related to elastic constants for plane problems. For isotropic materials, the relative minimum of dW =dV is assumed to be associated with the direction of crack initiation while the crack is assumed to grow when dW =dV reaches a critical value ðdW =dV Þc . In the orthotropic case, ðdW =dV Þc in the x–y-direction are obviously different. A simpler relation for ðdW =dV Þc is defined: h x y dW dW dW ¼ sin2 h þ cos2 h ð47Þ dV c dV c dV c In contrast to strength criterion, dW =dV accounts for both the stress and strain effects since it represents the x y area under the uniaxial stress and strain curve. In Eq. (47), ðdW =dV Þc and ðdW =dV Þc are, respectively, the critical energy density in the x–y-direction. They can be found from tests. Their substitution into Eq. (47) determines ðdW =dV Þhc .
Fig. 4. Crack extension angle h0 vs. crack inclination angle x for various values of biaxial load parameter, for graphite-epoxy with strength ratio Yt =Xt ¼ 0:04.
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Fig. 5. Crack extension angle h0 vs. crack inclination angle x for various values of biaxial load parameter, for E glass-epoxy with strength ratio Yt =Xt ¼ 0:2.
Referring to the graphite-epoxy and E glass-epoxy [15], with a given ðdW =dV Þxc =ðdW =dV Þyc ratio, a corresponding crack initiation angle h0 can be obtained for different crack inclination angle x, for different values of the biaxial load parameter k. The results are in Figs. 4 and 5. Note that when k ¼ 1 the crack inclination effect disappears. 6. Conclusions The problem of crack initiation in orthotropic medium with a crack aligned along one of the three orthogonal axes of elasticity is analyzed. A uniform biaxial load applied at infinity is taken. Using a complex variables formulation, the crack tip stress components and the displacement fields are obtained by considering the non-singular terms that influence by biaxial load effect. The strain density energy theory is been used to predict the crack initiation. The critical strain energy density is determined by including the directional dependency of the tensile strength for orthotropic materials. Crack initiation is found to depend not only on the biaxiality parameter but also on the anisotropic character of the material. Acknowledgements This work has been supported by Italian Ministry for Education, University and Research MIUR (40% and 60%). References [1] G.C. Sih, A special theory of crack propagation: methods of analysis and solutions of crack problems, in: Mechanics of Fracture I, Noordhoff, Leyden, 1973, pp. 21–45.
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[2] G.C. Sih, Strain density factor applied to mixed mode crack problems, Int. J. Fract. 10 (1974) 305–321. [3] G.C. Sih, A three dimensional strain density factor theory of crack propagation: three-dimensional crack problems, in: Mechanics of Fracture I, Noordhoff, Leyden, 1975, pp. 15–53. [4] N.I. Muskelishvili, Some Basic Problems on the Mathematical Theory of Elasticity, Noordhoff, Groningen, 1952. [5] G.N. Savin, Stress Concentration Around Holes, Pergamon Press, Oxford, 1961. [6] S.G. Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day, San Francisco, 1963. [7] G.C. Sih, P.C. Paris, G.R. Irwin, On cracks in rectilinearly anisotropic bodies, Int. J. Fract. Mech. 1 (1965) 189–203. [8] A. Piva, E. Viola, Crack propagation in an orthotropic medium, Engng. Fract. Mech. 29 (1988) 535–548. [9] E. Viola, A. Piva, E. Radi, Crack propagation in an orthotropic medium under general loading, Engng. Fract. Mech. 34 (1989) 1155–1174. [10] J. Eftis, N. Subramonian, H. Liebowitz, Crack border stress and displacement equations revisited, Engng. Fract. Mech. 9 (1977) 189–220. [11] J. Eftis, N. Subramonian, The inclined crack under biaxial load, Engng. Fract. Mech. 10 (1978) 43–67. [12] J. Eftis, D.L. Jones, Influence of load biaxiality on the fracture load of centre cracked sheets, Int. J. Fract. 20 (1982) 267–289. [13] S.K. Maiti, R.A. Smith, Criteria for brittle fracture in biaxial tension, Engng. Fract. Mech. 19 (1984) 793–804. [14] W.K. Lim, S.Y. Choi, B.V. Sankar, Biaxial load effects on crack extension in anisotropic solids, Engng. Fract. Mech. 68 (2001) 403–416. [15] G.C. Sih, M. Arcisz, Effect of orthotropy on crack propagation, Theor. Appl. Fract. Mech. 1 (1984) 225–238.