Cosmin Gruescu

Djimedo Kondo

Vincent Monchiet

Comptes Rendus Mécanique (CRM)

/ycylindereractionexistinginerntropie, deturale due cas où laC. R. Mecanique 333 (2005) 467–473http://france.elsevier.com/direct/CRAS2BEshelby tensor for a crack in an orthotropic elastic mediumCosmin Gruescu, Vincent Monchiet, Djimedo KondoLaboratoire de mécanique de Lille-UMR CNRS 8107, université de sciences et technologies Lille, cité scientifique,boulevard Paul-Langevin, 59655 Villeneuve d’Ascq cedex, FranceReceived 1 April 2005; accepted after revision 12 April 2005Available online 5 May 2005Presented by André ZaouiAbstractIn the present Note, we provide new analytical expressions of the components of Hill tensorP (or equivalently the EshelbtensorS) associated to an arbitrarily oriented crack in orthotropic elastic medium. The crack is modelled as an infinitealong a symmetry axis of the matrix, with low aspect ratio. The three dimensional results obtained show explicitly the intbetween the primary (structural) anisotropy and the crack-induced anisotropy. They are validated by comparison withresults 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.RésuméTenseur d’Eshelby pour une fissure en milieu élastique orthotrope.On présente une méthode permettant de détermanalytiquement le tenseur d’EshelbyS (ou, de manière équivalente, du tenseur de HillP) associé à une fissure arbitrairemeorientée dans un milieu orthotrope. La fissure est modélisée comme un cylindre infini suivant un des axes d’orthotfaible rapport d’aspect. Les résultats analytiques obtenus mettent en évidence l’intéraction entre l’anisotropie strucmatériau et l’anisotropie induite par la fissuration. Ils sont validés par comparaison avec des résultats existants dans lfissure 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 tensorMots-clés :Endommagement ; Homogénéisation ; Micromécanique ; Fissuration ; Anisotropie ; Tenseur d’EshelbyE-mail addresses:ic.gruescu@ed.univ-lille1.fr (C. Gruescu), vincent.monchiet@ed.univ-lille1.fr (V. Monchiet), kondo@univ-lille1.fr(D. Kondo).1631-0721/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crme.2005.04.005468 C. Gruescu et al. / C. R. Mecanique 333 (2005) 467–473ng fromes). Thework ofue. In theof theopicuctural)ry of thecrackscalaspectiffness1. IntroductionMicrocracking is one of the basic mechanisms of inelastic deformation for a large class of materials rangigeologic materials (Sedimentary Rocks) to man-made materials (Concrete, Ceramics Matrix Compositmodelling of the mechanical behavior for such deteriorating materials is classically performed in the frameContinuum Damage Mechanics, using either a purely macroscopic approach or a micromechanics techniqlatter, the determination of the overall properties of the cracked material requires primarily the calculationEshelby tensorS (or equivalently the Hill tensorP) associated to a crack. Existing works concern linear isotrsolid matrix containing penny-shaped cracks (see for instance [1–3]). For materials exhibiting a primary (stranisotropy, only few results exist and correspond to the case when the crack lies in one plane of symmetsolid matrix (see [4–6]). The main difficulty comes from the lack of an analytical expression forP for arbitrarilyoriented cracks. The objective of this note is to provide new results for the case of arbitrarily orientedembedded in an orthotropic solid matrix.Summation convention on repeated indices is adopted.2. Determination of P for an arbitrarily oriented crack in an orthotropic medium2.1. IntroductionLet us consider an orthotropic solid matrix (with a stiffness tensorCs ), weakened by a crack. The geometrimodelling of the crack in it’s associated local frame is given by:z21a2+ z22b2= 1, −∞ < z3 < ∞ (1)That is, the crack is modelled as an infinite cylinder in the direction 3, having an elliptical section and lowratio X = b/a (see Fig. 1). Let us first note that in the local frame of the crack the components of the sttensor are obtained from usual transformation rules:Cijkl = UipUjqUkrUlsCspqrs , with:U =cosθ −sinθ 0sinθ cosθ 00 0 1 . (2)Fig. 1. Crack modelled as an infinite cylinder with elliptical section; the aspect ratio isX = b/a.C. Gruescu et al. / C. R. Mecanique 333 (2005) 467–473 469anlsosionsng andtWe note that the stiffness tensor of the crack depends on the angleθ which defines the crack plane and presentsapparent monoclinic symmetry. The determination ofP is based on the work of Kinoshita and Mura [7] (see aFaivre [8], Laws [4] or Willis [9]) who proved that:Pijkl = abπ2π∫0Tijkl(ξ1, ξ2)a2ξ21 + b2ξ22dψ (3)which implies an integration over the unit circle centered at the origin, in the plane (ξ1, ξ2): | ξ |= 1, i.e. forξ = cosψe1 + sinψe2. The components of the fourth order tensorT can be expressed:Tijkl(ξ1, ξ2) = Dijkl(ξ1, ξ2,0), with: (4)Dijkl = 14(ξiK−1jk ξl + ξjK−1ik ξl + ξiK−1j l ξk + ξjK−1il ξk)(5)K = ξ · C · ξ represents the acoustic tensor associated withC and theξ direction:Kik(ξ) = Cijklξj ξl .Note that the anisotropy of the elastic solid matrix affects the determination ofP through the acoustic tensorK .Besides, the dependence ofP on the crack’s orientation is understood from the fact that the above expresimply calculation in the local frame of the crack. We note that the components ofP can be put in the followingform:Pijkl = 14[Mijkl +Mjikl +Mij lk +Mjilk], where: (6)Mijkl = abπ2π∫0ξiK−1jk ξla2ξ21 + b2ξ22dψ (7)2.2. Developed methodology for the determination ofPBased on the recent study of Suvorov and Dvorak [10] and following the procedure described by TiLee [11],1 we consider the two fixed unit orthogonal vectorse1 ande2 in the planeξ3 = 0. Recalling that any univector in this plane is defined asξ = cosψe1 + sinψe2, K = ξ · C · ξ takes the form:K = (cosψ)2Q + cosψ sinψ(R + RT ) + (sinψ)2T (8)By the substitutionz = cotψ , (8) can be rewritten in the following form:K(ψ) = (sinψ)2[Qz2 + z(R + RT ) + T] = (sinψ)2K(z), with: (9)K(z) = z2Q + z(R + RT ) + T (10)The second order tensorsQ, R, andT are defined as:Q = e1 · C · e1; R = e1 · C · e2; T = e2 · C · e2. In orderto evaluateP (Eq. (4)) one needs to invertK(z). Denoting|K(z)| the determinant ofK(z) andK̃(z) it’s adjoint,one has:K(z) · K̃(z) = |K(z)|1; it follows that:K−1(z) = K̃(z)|K(z)| (11)1 Note that these authors have not studied the case of an arbitrarily oriented inclusion in an orthotropic solid matrix.470 C. Gruescu et al. / C. R. Mecanique 333 (2005) 467–473,eralse areytion (oflSimilarly, for the tensor∆ = ξ ⊗ ξ which, with K−1, enters in definition (4), it can be verified that:∆(ψ) =(sinψ)2∆(z) with ∆(z) = z2e1 ⊗ e1 + z(e1 ⊗ e2 + e2 ⊗ e1) + e2 ⊗ e2. Moreover, we note thata2ξ21 + b2ξ22 =(sinψ)2(a2z2 + b2). By taking into account the variable change,z = cotψ , (7) can be written:Mijkl = abπ∞∫−∞K̃jk(z)∆il(z)(a2z2 + b2)|K(z)| dz =1π∞∫−∞XK̃jk(z)∆il(z)(z2 + X2)|Q|f (z) dz (12)where Eq. (9), identity|K(z)| = |Q|f (z) and the aspect ratioX = bawere used.The interest of (12) lies in the fact that it allows the evaluation ofMijkl with the help of the residues theoremthe function to be integrated being holomorphic out of the poles that have to be determined.An interesting observation is thatz = ±iX are two of these poles. It is also useful to note that in the gencasef (z) is a polynomial function of degree 6, with three ‘pairs’ of complex conjugate roots. Obviously, thethe roots ofK(z) = 0. Notingzp (p = 1,2,3), the roots whose imaginary part is positive, one has:f (z) = (z − z1)(z − z̄1)(z − z2)(z − z̄2)(z − z3)(z − z̄3) (13)The application of the residues theorem for the evaluation ofM tensor is detailed in the next section.3. Analytical expression of theP tensor and validation3.1. Analytical expression of the components of Hill’s tensorPIn the case of a crack with an arbitrary orientation (represented by an angleθ ) with respect to the symmetraxis 1 of the material, the calculation performed in the local frame of the crack leads to a polynomial funcdegree 6) which allows to writef (z) as the product of a polynomial function of degree 4 (f1(z)) and a polynomiafunction 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:|Q| = Cs1111Cs1212(cos2 θ + α sin2 θ)(cos2 θ + β sin2 θ)(Cs3232sin2 θ + Cs3131cos2 θ) (15)The analytical expressions off1(z), f2(z) are presented in Appendix A.1. The solutions off1(z) = 0 are:z1 = i√α cosθ − sinθcosθ + i√α sinθ ; z1 =i√α cosθ + sinθ−cosθ + i√α sinθz2 = i√β cosθ − sinθcosθ + i√β sinθ ; z2 =i√β cosθ + sinθ−cosθ + i√β sinθ(16)with α andβ the complex conjugate roots of the characteristic equation of the orthotropic 2D solid:Cs1111Cs1212x2 − (Cs1111Cs2222− Cs11222 − 2Cs1122Cs1212)x + Cs2222Cs1212= 0 (17)The roots of 2 = 0 are:z3 = −sin(2θ)(Cs3232+ Cs3131) + 2i√Cs3232Cs3131cos(2θ)2(Cs3131cos2 θ + Cs3232sin2 θ)z3 = −sin(2θ)(Cs3232+ Cs3131) − 2i√Cs3232Cs3131cos(2θ)s 2 s 2(18)2(C3131cos θ + C3232sin θ)C. Gruescu et al. / C. R. Mecanique 333 (2005) 467–473 471ensorponentswhosey LawsNote.schemeserties ofof pureThen, 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 ofM provided by (12) can be expressed:Mijkl = 2i{K̃jk(iX)∆il(iX)2i|Q|f (iX) +3∑i=1K̃jk(zi)∆il(zi)X|Q|f ′(zi)(X2 + z2i )}(20)It is recalled here that̃Kjk are the components of the adjoint ofK .Considering only the approximation to first order inX at the valueX = 0, (20) gives:Mijkl = K̃jk(0)∆il(0)f (0)|Q| +2iX|Q|{3∑i=1K̃jk(zi)∆il(zi)f ′(zi)z2i+ 12f (0)[K̃ ′jk(0)∆il(0) + K̃jk(0)∆′il (0) − K̃jk(0)∆il(0)f ′(0)f (0)]}(21)It is shown that the imaginary part ofM is null:(Mijkl) = 2X|Q|{[K̃jk(z1)∆il(z1)f ′(z1)z21]+ [K̃jk(z2)∆il(z2)f ′(z2)z22]+ [K̃jk(z3)∆il(z3)f ′(z3)z23]}+ Xf (0)|Q|{[K̃ ′jk(0)∆il(0) + K̃jk(0)∆′il(0) − K̃jk(0)∆il(0)f ′(0)f (0)]}= 0 (22)and its real part gives:Mijkl = (Mijkl) = K̃jk(0)∆il(0)f (0)|Q| −2X|Q|3∑i=1K̃jk(zi)∆il(zi)f ′(zi)z2i(23)These results are associated to the local frame of the crack. A last change of basis allows to express the tP inthe global frame defined by the symmetry axes of the solid matrix. The detailed expressions of the nine comof P in this global frame are given in Appendix A (Eqs. (A.4)).3.2. Validation of obtained resultsAs a first validation of the results presented in Appendix A we consider the case of a parallel crackorientation coincides with the symmetry axis 1 of the solid matrix. This configuration has been studied b[4], who gives the following non zero components ofP:P1111= Cs2222+ Cs1212√αβCs1111Cs1212√αβ(√α + √β)X; P2222=1Cs2222+ Cs2222− Cs1212(α + β +√αβ)Cs2222Cs1212√αβ(√α + √β) XP1122= P2211= − Cs1122+ Cs1212Cs1111Cs1212√αβ(√α + √β)X; P3232=14Cs3232−√Cs31314Cs32323/2X (24)P3131= 14√Cs3232Cs3131X; P1212= 14Cs1212− Cs1111Cs2222− Cs112224Cs1111Cs12122√αβ(√α + √β)Xwhich corresponds to the particular case (θ = 0) of the general expressions (A.4) established in the presentThe general analytical new expressions provided are easily incorporated into various homogenization(dilute scheme, Mori–Tanaka model, self-consistent scheme) in order to determine the macroscopic propan orthotropic solid matrix weakened by cracks. For application purposes in [12] a more general validation2D solutions (see [13]) is also provided.472 C. Gruescu et al. / C. R. Mecanique 333 (2005) 467–473nedAppendix A. Analytical expressions of|K(z)| and the components of theP tensorA.1. Expressions off1(z), f2(z) in |K(z)|The two polynomial functionsf1(z), f2(z) entering in (14) are respectively:f1(z) =[Cs1111Cs1212(cosθ)4 + Cs2222Cs1212(sinθ)4 + χ(cosθ)2(sinθ)2]z4+ {[2Cs1111Cs1212(cosθ)2 − 2Cs2222Cs1212(sinθ)2] − χ cos2θ}sin2θz3+ {χ[1− 6(cosθ)2(sinθ)2] + 6Cs1212(Cs1111+ Cs2222)(cosθ)2(sinθ)2}z2+ [2Cs1111Cs1212(sinθ)2 + 2Cs2222Cs1212(cosθ)2 + χ cos2θ]sin 2θ z+ Cs1111Cs1212(sinθ)4 + Cs2222Cs1212(cosθ)4 + χ(sinθ)2(cosθ)2 (A.1)where:χ = (Cs1111Cs2222− Cs11222 − 2Cs1122Cs1212), and:f2(z) = (Cs3232sin2 θ + Cs3131cos2 θ)z2 + (Cs3232+ Cs3131)sin 2θ z + Cs3232cos2 θ + Cs3131sin2 θ (A.2)Note that, in order to make the search of the roots off1(z) = 0 easier, we introduced the variable changez =(ucosθ − sinθ)/(cosθ + usinθ), which allowed (A.1) to be written as:f1(z) = Cs1111Cs1212u4 + (Cs1111Cs2222− Cs11222 − 2Cs1122Cs1212)u2 + Cs2222Cs1212(cosθ + usinθ)4 (A.3)A.2. Components ofPThe determination of the componentsPijkl is performed by combination of (6) and (23). In the frame defiby the material symmetry axes, the following expressions are obtained:P1111= (sinθ)2[Cs22(cosθ)2 + Cs66(sinθ)2]n+ X[p√α(αCs66 − Cs22) − q√β(βCs66 − Cs22)]Cs11Cs66(α − β)P1122= − (Cs12 + Cs66)(cosθ)2(sinθ)2n+ X(Cs12 + Cs66)(p√α − q√β)Cs11Cs66(α − β)P2222= (cosθ)2[Cs11(sinθ)2 + Cs66(cosθ)2]n+ X[(p/√α)(Cs66 − αCs11) − (q/√β)(Cs66 − βCs11)]Cs11Cs66(α − β)P1112= sin 2θ{Cs22(cosθ)2 − Cs12(sinθ)24n+ X[r√α(αCs12 + Cs22) − s√β(βCs12 + Cs22)]Cs11Cs66(α − β)}P2212= sin 2θ{ [Cs11(sinθ)2 − Cs12(cosθ)2]4n− X[r(αCs11 + Cs12) − s√β(βCs11 + Cs12)]Cs11Cs66(α − β)}(A.4)P1212= [Cs11(sinθ)4 + Cs22(cosθ)4 − 2Cs12(sinθ)2(cosθ)2]4n+ X(Cs11Cs22 − Cs122)(p√α − q√β)4Cs11Cs662(α − β)P3131= (sinθ)2m+ Xm2√Cs44Cs55[Cs44(cosθ)2 − Cs55(sinθ)2]P3232= (cosθ)2m− Xm2√Cs55Cs44[Cs44(cosθ)2 − Cs55(sinθ)2]P = sinθ cosθ − X√Cs55Cs44sin2θ3132m m2C. Gruescu et al. / C. R. Mecanique 333 (2005) 467–473 473ete Sci.Nether-d issues,25 (1977)triceh. Phys.In previous expressions the following notations were used for the elastic coefficients:m = 4[Cs44(cosθ)2 + Cs55sin2 θ]2; n = C11C66[(sinθ)4 + αβ(cosθ)4 + (α + β)(cosθ)2(sinθ)2]p = [α(cosθ)2 − (sinθ)2]/[α(cosθ)2 + (sinθ)2]2; q = [β(cosθ)2 − (sinθ)2]/[β(cosθ)2 + (sinθ)2]2r = √α/[α(cosθ)2 + (sinθ)2]2; s = √β/[β(cosθ)2 + (sinθ)2]2References[1] L. Dormieux, E. Lemarchand, D. Kondo, E. Fairbairn, Elements of poro-micromechanics applied to concrete, ConcrEngrg. 37 (265) (2004) 31–42.[2] S. Nemat-Nasser, M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials, North-Holland, Amsterdam, Thelands, 1993.[3] V. Pensée, D. Kondo, L. Dormieux, Three-dimensional micromechanical approach of brittle anisotropic damage and relateJ. Engrg. Mech. ASCE 128 (8) (2002) 889–897.[4] N. Laws, A note on interaction energies associated with cracks in anisotropic media, Philos. Magazine 36 (1977) 367–372.[5] N. Laws, A note on penny-shaped cracks in transversely isotropic materials, Mech. Mater. 4 (1985) 209–212.[6] T. Mura, Micromechanics of Defects in Solids, Nijhoff, The Hague, The Netherlands, 1987.[7] N. Kinoshita, T. Mura, Elastic field of inclusions in anisotropic media, Phys. Status Solidi A 5 (1971) 759–769.[8] G. Faivre, Hétérogénéités ellipsoïdales dans un milieu élastique anisotrope, J. Phys. 32 (252) (1971) 325–331.[9] J.R. Willis, Bounds and self-consistent estimates for the overall properties of anisotropic composites, J. Mech. Phys. Solids185–202.[10] A.P. Suvorov, G.J. Dvorak, Rate form of the Eshelby and Hill tensors, Int. J. Solids Structures 39 (2002) 5659–5678.[11] T.C.T. Ting, Anisotropic Elasticity: Theory and Applications, Oxford Science Publications, New York, 1996.[12] C. Gruescu, Approche par changement d’échelle de l’endommagementdes matériaux anisotropes: application aux composites à mafragile, Thèse de doctorat, Université des Sciences et Technologies de Lille, 2004.[13] C. Mauge, M. Kachanov, Effective elastic properties of an isotropic material with arbitrarily oriented interacting cracks, J. MecSolids 42 (4) (1994) 561–584.

Eshelby tensor for a crack in an orthotropic elastic medium

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Eshelby tensor for a crack in an orthotropic elastic medium

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