Eshelby tensor for a crack in an orthotropic elastic medium

C. R. Mecanique 333 (2005) 467–473 http://france.elsevier.com/direct/CRAS2B/

Eshelby tensor for a crack in an orthotropic elastic medium Cosmin Gruescu, Vincent Monchiet, Djimedo Kondo Laboratoire de mécanique de Lille-UMR CNRS 8107, université de sciences et technologies Lille, cité scientifique, boulevard Paul-Langevin, 59655 Villeneuve d’Ascq cedex, France Received 1 April 2005; accepted after revision 12 April 2005 Available online 5 May 2005 Presented by André Zaoui

Abstract In the present Note, we provide new analytical expressions of the components of Hill tensor P (or equivalently the Eshelby tensor S) associated to an arbitrarily oriented crack in orthotropic elastic medium. The crack is modelled as an infinite cylinder along a symmetry axis of the matrix, with low aspect ratio. The three dimensional results obtained show explicitly the interaction between the primary (structural) anisotropy and the crack-induced anisotropy. They are validated by comparison with existing results in the case where the crack is in a symmetry plane. To cite this article: C. Gruescu et al., C. R. Mecanique 333 (2005).  2005 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Tenseur d’Eshelby pour une fissure en milieu élastique orthotrope. On présente une méthode permettant de déterminer analytiquement le tenseur d’Eshelby S (ou, de manière équivalente, du tenseur de Hill P) associé à une fissure arbitrairement orientée dans un milieu orthotrope. La fissure est modélisée comme un cylindre infini suivant un des axes d’orthotropie, de faible rapport d’aspect. Les résultats analytiques obtenus mettent en évidence l’intéraction entre l’anisotropie structurale du matériau et l’anisotropie induite par la fissuration. Ils sont validés par comparaison avec des résultats existants dans le cas où la fissure se trouve dans un plan de symétrie. Pour citer cet article : C. Gruescu et al., C. R. Mecanique 333 (2005).  2005 Académie des sciences. Published by Elsevier SAS. All rights reserved. Keywords: Damage; Homogenization; Micromechanics; Cracks; Anisotropy; Eshelby tensor Mots-clés : Endommagement ; Homogénéisation ; Micromécanique ; Fissuration ; Anisotropie ; Tenseur d’Eshelby

E-mail addresses: [email protected] (C. Gruescu), [email protected] (V. Monchiet), [email protected] (D. Kondo). 1631-0721/$ – see front matter  2005 Académie des sciences. Published by Elsevier SAS. All rights reserved. doi:10.1016/j.crme.2005.04.005

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1. Introduction Microcracking is one of the basic mechanisms of inelastic deformation for a large class of materials ranging from geologic materials (Sedimentary Rocks) to man-made materials (Concrete, Ceramics Matrix Composites). The modelling of the mechanical behavior for such deteriorating materials is classically performed in the framework of Continuum Damage Mechanics, using either a purely macroscopic approach or a micromechanics technique. In the latter, the determination of the overall properties of the cracked material requires primarily the calculation of the Eshelby tensor S (or equivalently the Hill tensor P) associated to a crack. Existing works concern linear isotropic solid matrix containing penny-shaped cracks (see for instance [1–3]). For materials exhibiting a primary (structural) anisotropy, only few results exist and correspond to the case when the crack lies in one plane of symmetry of the solid matrix (see [4–6]). The main difficulty comes from the lack of an analytical expression for P for arbitrarily oriented cracks. The objective of this note is to provide new results for the case of arbitrarily oriented cracks embedded in an orthotropic solid matrix. Summation convention on repeated indices is adopted.

2. Determination of P for an arbitrarily oriented crack in an orthotropic medium 2.1. Introduction Let us consider an orthotropic solid matrix (with a stiffness tensor Cs ), weakened by a crack. The geometrical modelling of the crack in it’s associated local frame is given by: z22 z12 + = 1, a 2 b2

−∞ < z3 < ∞

(1)

That is, the crack is modelled as an infinite cylinder in the direction 3, having an elliptical section and low aspect ratio X = b/a (see Fig. 1). Let us first note that in the local frame of the crack the components of the stiffness s , with: tensor are obtained from usual transformation rules: Cij kl = Uip Uj q Ukr Uls Cpqrs 

cos θ U =  sin θ 0

− sin θ cos θ 0

 0 0. 1

Fig. 1. Crack modelled as an infinite cylinder with elliptical section; the aspect ratio is X = b/a.

(2)

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469

We note that the stiffness tensor of the crack depends on the angle θ which defines the crack plane and presents an apparent monoclinic symmetry. The determination of P is based on the work of Kinoshita and Mura [7] (see also Faivre [8], Laws [4] or Willis [9]) who proved that: ab Pij kl = π

2π

Tij kl (ξ1 , ξ2 ) a 2 ξ12 + b2 ξ22

0

dψ

(3)

which implies an integration over the unit circle centered at the origin, in the plane (ξ1 , ξ2 ): | ξ |= 1, i.e. for ξ = cos ψe1 + sin ψe2 . The components of the fourth order tensor T can be expressed: Tij kl (ξ1 , ξ2 ) = Dij kl (ξ1 , ξ2 , 0), with: 1 −1 −1 −1  Dij kl = ξi Kj−1 k ξl + ξj Kik ξl + ξi Kj l ξk + ξj Kil ξk 4

(4) (5)

K = ξ · C · ξ represents the acoustic tensor associated with C and the ξ direction: Kik (ξ ) = Cij kl ξj ξl . Note that the anisotropy of the elastic solid matrix affects the determination of P through the acoustic tensor K. Besides, the dependence of P on the crack’s orientation is understood from the fact that the above expressions imply calculation in the local frame of the crack. We note that the components of P can be put in the following form: 1 Pij kl = [Mij kl + Mj ikl + Mij lk + Mj ilk ], 4 2π ξ K −1 ξ i jk l ab Mij kl = dψ 2 π a ξ12 + b2 ξ22

where:

(6)

(7)

0

2.2. Developed methodology for the determination of P Based on the recent study of Suvorov and Dvorak [10] and following the procedure described by Ting and Lee [11],1 we consider the two fixed unit orthogonal vectors e1 and e2 in the plane ξ3 = 0. Recalling that any unit vector in this plane is defined as ξ = cos ψe1 + sin ψe2 , K = ξ · C · ξ takes the form: K = (cos ψ)2 Q + cos ψ sin ψ(R + RT ) + (sin ψ)2 T

(8)

By the substitution z = cot ψ, (8) can be rewritten in the following form:  K(ψ) = (sin ψ)2 Qz2 + z(R + RT ) + T = (sin ψ)2 K(z),

with:

K(z) = z Q + z(R + R ) + T T

2

(9) (10)

The second order tensors Q, R, and T are defined as: Q = e1 · C · e1 ; R = e1 · C · e2 ; T = e2 · C · e2 . In order

to evaluate P (Eq. (4)) one needs to invert K(z). Denoting |K(z)| the determinant of K(z) and K(z) it’s adjoint,

= |K(z)|1; it follows that: one has: K(z) · K(z) K−1 (z) =

K(z) |K(z)|

1 Note that these authors have not studied the case of an arbitrarily oriented inclusion in an orthotropic solid matrix.

(11)

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Similarly, for the tensor ∆ = ξ ⊗ ξ which, with K−1 , enters in definition (4), it can be verified that: ∆(ψ) = (sin ψ)2 ∆(z) with ∆(z) = z2 e1 ⊗ e1 + z(e1 ⊗ e2 + e2 ⊗ e1 ) + e2 ⊗ e2 . Moreover, we note that a 2 ξ12 + b2 ξ22 = (sin ψ)2 (a 2 z2 + b2 ). By taking into account the variable change, z = cot ψ, (7) can be written: ab Mij kl = π

∞ −∞

∞

j k (z)∆il (z)

j k (z)∆il (z) XK K 1 dz = dz 2 2 2 2 π (a z + b )|K(z)| (z + X 2 )|Q|f (z)

(12)

−∞

where Eq. (9), identity |K(z)| = |Q|f (z) and the aspect ratio X = ab were used. The interest of (12) lies in the fact that it allows the evaluation of Mij kl with the help of the residues theorem, the function to be integrated being holomorphic out of the poles that have to be determined. An interesting observation is that z = ±iX are two of these poles. It is also useful to note that in the general case f (z) is a polynomial function of degree 6, with three ‘pairs’ of complex conjugate roots. Obviously, these are the roots of K(z) = 0. Noting zp (p = 1, 2, 3), the roots whose imaginary part is positive, one has: f (z) = (z − z1 )(z − z¯ 1 )(z − z2 )(z − z¯ 2 )(z − z3 )(z − z¯ 3 )

(13)

The application of the residues theorem for the evaluation of M tensor is detailed in the next section. 3. Analytical expression of the P tensor and validation 3.1. Analytical expression of the components of Hill’s tensor P In the case of a crack with an arbitrary orientation (represented by an angle θ ) with respect to the symmetry axis 1 of the material, the calculation performed in the local frame of the crack leads to a polynomial function (of degree 6) which allows to write f (z) as the product of a polynomial function of degree 4 (f1 (z)) and a polynomial function of degree 2 (f2 (z)). Therefore:   K(z) = |Q|f (z) = f1 (z) · f2 (z) (14) The determinant |Q| which appears in this expression takes the form: s s s s |Q| = C1111 C1212 (cos2 θ + α sin2 θ )(cos2 θ + β sin2 θ )(C3232 sin2 θ + C3131 cos2 θ )

The analytical expressions of f1 (z), f2 (z) are presented in Appendix A.1. The solutions of f1 (z) = 0 are: √ √ i α cos θ − sin θ i α cos θ + sin θ ; z1 = z1 = √ √ cos θ + i α sin θ − cos θ + i α sin θ √ √ i β cos θ − sin θ i β cos θ + sin θ z2 = ; z2 = √ √ cos θ + i β sin θ − cos θ + i β sin θ

(15)

(16)

with α and β the complex conjugate roots of the characteristic equation of the orthotropic 2D solid: 2 s s s s s s s s s C1111 C1212 x 2 − (C1111 C2222 − C1122 − 2C1122 C1212 )x + C2222 C1212 =0

(17)

The roots of f2 = 0 are: z3 = − z3 = −

s s s s sin(2θ )(C3232 + C3131 ) + 2i C3232 C3131 cos(2θ ) s s 2(C3131 cos2 θ + C3232 sin2 θ ) s s s s sin(2θ )(C3232 + C3131 ) − 2i C3232 C3131 cos(2θ ) s s 2(C3131 cos2 θ + C3232 sin2 θ )

(18)

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Then, using (16) and (18), |K(z)| can be rewritten in the form:   K(z) = |Q|(z − z1 )(z − z1 )(z − z2 )(z − z2 )(z − z3 )(z − z3 )

(19)

The poles being assumed to be different, the expression of M provided by (12) can be expressed:

3

j k (zi )∆il (zi )X  Kj k (iX)∆il (iX)  K + Mij kl = 2i 2i|Q|f (iX) |Q|f  (zi )(X 2 + zi2 ) i=1

j k are the components of the adjoint of K. It is recalled here that K Considering only the approximation to first order in X at the value X = 0, (20) gives:  3

j k (0)∆il (0) 2iX  K˜ j k (zi )∆il (zi ) K Mij kl = + f (0)|Q| |Q| f  (zi )zi2 i=1   1 f  (0)  



+ K (0)∆il (0) + Kj k (0)∆il (0) − Kj k (0)∆il (0) 2f (0) j k f (0) It is shown that the imaginary part of M is null:  

 





Kj k (z2 )∆il (z2 ) Kj k (z3 )∆il (z3 ) Kj k (z1 )∆il (z1 ) 2X + +  (Mij kl ) = |Q| f  (z1 )z12 f  (z2 )z22 f  (z3 )z32

   X

j k (0)∆il (0) − K

j k (0)∆il (0) f (0) = 0

j k (0)∆il (0) + K + K f (0)|Q| f (0) and its real part gives: Mij kl = (Mij kl ) =

471

3

K˜ j k (0)∆il (0) 2X  K j k (zi )∆il (zi ) −  f (0)|Q| |Q| f  (zi )zi2

(20)

(21)

(22)

(23)

i=1

These results are associated to the local frame of the crack. A last change of basis allows to express the tensor P in the global frame defined by the symmetry axes of the solid matrix. The detailed expressions of the nine components of P in this global frame are given in Appendix A (Eqs. (A.4)). 3.2. Validation of obtained results As a first validation of the results presented in Appendix A we consider the case of a parallel crack whose orientation coincides with the symmetry axis 1 of the solid matrix. This configuration has been studied by Laws [4], who gives the following non zero components of P: √ √ s s s s C2222 C2222 + C1212 αβ − C1212 (α + β + αβ) 1 P2222 = s + s P1111 = s √ X √ √ √ X; √ √ s s C2222 C1111 C1212 αβ( α + β) C2222 C1212 αβ( α + β) s s s C3131 C1122 + C1212 1 − X (24) P1122 = P2211 = − s √ X; P3232 = √ √ s s 3/2 s 4C3232 C1111 C1212 αβ( α + β) 4C3232 1 X; P3131 = s s 4 C3232 C3131

P1212 =

2 s s s C1111 C2222 − C1122 1 − √ √ X √ s 2 s s 4C1212 αβ( α + β) 4C1111 C1212

which corresponds to the particular case (θ = 0) of the general expressions (A.4) established in the present Note. The general analytical new expressions provided are easily incorporated into various homogenization schemes (dilute scheme, Mori–Tanaka model, self-consistent scheme) in order to determine the macroscopic properties of an orthotropic solid matrix weakened by cracks. For application purposes in [12] a more general validation of pure 2D solutions (see [13]) is also provided.

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Appendix A. Analytical expressions of |K(z)| and the components of the P tensor A.1. Expressions of f1 (z), f2 (z) in |K(z)| The two polynomial functions f1 (z), f2 (z) entering in (14) are respectively:  s s s s f1 (z) = C1111 C1212 (cos θ )4 + C2222 C1212 (sin θ )4 + χ(cos θ )2 (sin θ )2 z4   s s s s + 2C1111 C1212 (cos θ )2 − 2C2222 C1212 (sin θ )2 − χ cos 2θ sin 2θ z3    s s s + χ 1 − 6(cos θ )2 (sin θ )2 + 6C1212 (C1111 + C2222 )(cos θ )2 (sin θ )2 z2  s s s s + 2C1111 C1212 (sin θ )2 + 2C2222 C1212 (cos θ )2 + χ cos 2θ sin 2θ z s s s s C1212 (sin θ )4 + C2222 C1212 (cos θ )4 + χ(sin θ )2 (cos θ )2 + C1111 2 s s s s s where: χ = (C1111 C2222 − C1122 − 2C1122 C1212 ), and: s s s s sin2 θ + C3131 cos2 θ )z2 + (C3232 + C3131 ) sin 2θ f2 (z) = (C3232

s s z + C3232 cos2 θ + C3131 sin2 θ

(A.1)

(A.2)

Note that, in order to make the search of the roots of f1 (z) = 0 easier, we introduced the variable change z = (u cos θ − sin θ )/(cos θ + u sin θ ), which allowed (A.1) to be written as: f1 (z) =

2 s s s s s s s s s C1111 C1212 u4 + (C1111 C2222 − C1122 − 2C1122 C1212 )u2 + C2222 C1212 (cos θ + u sin θ )4

(A.3)

A.2. Components of P The determination of the components Pij kl is performed by combination of (6) and (23). In the frame defined by the material symmetry axes, the following expressions are obtained: √ s (cos θ )2 + C s (sin θ )2 ] s − C s ) − q √β(βC s − C s )] X[p α(αC66 (sin θ )2 [C22 66 22 66 22 + P1111 = s C s (α − β) n C11 66 s + C s )(cos θ )2 (sin θ )2 s + C s )(p √α − q √β) X(C (C12 66 12 66 + P1122 = − s C s (α − β) n C11 66 √ s (sin θ )2 + C s (cos θ )2 ] s − αC s ) − (q/√β)(C s − βC s )] X[(p/ α)(C66 (cos θ )2 [C11 66 11 66 11 + P2222 = s C s (α − β) n C11 66 √

s s (sin θ )2 s + C s ) − s √β(βC s + C s )]  X[r α(αC12 C22 (cos θ )2 − C12 22 12 22 + P1112 = sin 2θ s C s (α − β) 4n C11 66

s s (cos θ )2 ] s + C s ) − s √β(βC s + C s )]  X[r(αC11 [C11 (sin θ )2 − C12 12 11 12 − (A.4) P2212 = sin 2θ s C s (α − β) 4n C11 66 s (sin θ )4 + C s (cos θ )4 − 2C s (sin θ )2 (cos θ )2 ] s C s − C s 2 )(p √α − q √β) X(C11 [C11 22 12 22 12 P1212 = + s C s 2 (α − β) 4n 4C11 66  s 2 X C44  s (sin θ ) 2 s 2 + 2 P3131 = s C44 (cos θ ) − C55 (sin θ ) m m C55  s  X C55 (cos θ )2 s s − 2 P3232 = C44 (cos θ )2 − C55 (sin θ )2 s m m C44 s s X C55 C44 sin 2θ sin θ cos θ P3132 = − m m2

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In previous expressions the following notations were used for the elastic coefficients:  s  2 s m = 4 C44 (cos θ )2 + C55 sin2 θ ; n = C11 C66 (sin θ )4 + αβ(cos θ )4 + (α + β)(cos θ )2 (sin θ )2   2  2  p = α(cos θ )2 − (sin θ )2 α(cos θ )2 + (sin θ )2 ; q = β(cos θ )2 − (sin θ )2 β(cos θ )2 + (sin θ )2  2 2 √  r = α α(cos θ )2 + (sin θ )2 ; s = β β(cos θ )2 + (sin θ )2

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