Vincent Monchiet

Cosmin Gruescu

Oana Cazacu

Djimedo Kondo
email: djimedo.kondo@upmc.fr

A micromechanical approach of crack-induced damage in orthotropic media : application to a brittle matrix composite

Keywords: Homogenization, Eshelby tensor, Anisotropy, Damage, Brittle materials, Ceramic Matrix Composites, Micromechanics, Cracks

Coupling between initial and damage-induced anisotropies in 3D elastic damaged materials has been so far addressed by homogenization techniques only for particular microcracks configurations. The main difficulty in developing a general 3D micromechanical model lies in the lack of a closed-form solution of the Eshelby tensor corresponding to cracks in an elastic anisotropic medium. In this study, we begin to present closed form expressions of the Eshelby tensor S that we derived for a cylindrical crack embedded in an orthotropic material. The 3D obtained expressions reduce to existing results in the 2D case or in particular 3D cracks configurations. The effective compliance of an orthotropic medium containing arbitrarily oriented cracks is then derived by using the new Eshelby tensor. Finally, a damage model is formulated by combining the above results with classical thermodynamics approach. The ability of the model to capture coupling between initial orthotropy and damage induced anisotropy is demonstrated through a comparison with experimental data available for a ceramic matrix composite (unidirectional SiC-SiC).

Introduction

In the last few decades the study of inelastic behavior and failure characteristics of man-made or natural brittle materials under mechanical loading has been the object of numerous studies. Both phenomenological (i.e. continuum damage mechanics -CDM) or micromechanical based approaches have been considered (see for instance the reference works of [START_REF] Krajcinovic | Damage mechanics[END_REF], [START_REF] Nemat-Nasser | Micromechanics : overall properties of heterogenous materials[END_REF][START_REF] Dragon | Damage mechanics. some modelling challenges[END_REF] where some of the main contributions are reviewed). Since the mechanical behavior and some other microstructure-sensitive properties of heterogeneous media are strongly influenced by damage mechanisms occurring at different scales, the micromechanical approach of damage modeling appears to be very powerful. In particular, for quasi brittle materials considered in the present study, the inelastic deformation or other various complex effects observed at the macroscopic level by means of standard mechanical tests (e.g. deterioration of elastic properties, damage-induced anisotropy, volumetric dilatancy, unilateral effects, etc) are commonly attributed to matrix microcracking.

The present study addresses the problem of estimating the effective elastic properties of anisotropic bodies permeated by cylindrical cracks in the framework of Eshelby-like methods and the formulation of a micro-macro model able to account for the anisotropy of crack-induced damage. Despite the progresses concerning the mathematical formulation of constitutive laws and the evaluation of overall properties of engineering materials ( [START_REF] Lekhnitski | Stresses in an infinitely large anisotropic plate which is weakened by an elliptical hole[END_REF][START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF][START_REF] Eshelby | The elastic field outside an ellipsoidal inclusion[END_REF][START_REF] Wang | Mixed-mode crack analysis of rectilinear anisotropic solids using conservation laws of elasticity[END_REF][START_REF] Hutchinson | Crack tip shielding by micro-cracking in brittle solids[END_REF][START_REF] Wu | Analysis of finite anisotropic media containing multiple cracks using superposition[END_REF][START_REF] Mauge | Anisotropic materials with interacting arbitrarily oriented cracks -stress intensity factors and crack-microcrack interactions[END_REF][START_REF] Pan | Elastic moduli of heterogeneous solids with ellipsoidal inclusions and elliptic cracks[END_REF][START_REF] Kachanov | Mechanics of anisotropic materials with multiple cracks[END_REF][START_REF] Ting | Anisotropic elasticity: Theory and applications[END_REF] and many others, especially in the 2D context), the 3D modelling of interaction between initial and damage-induced anisotropies remains a difficult task, even in the context of phenomenological models (see, for example, [START_REF] Ladeveze | Inelastic strains and damage (chap. 4)[END_REF][START_REF] Talreja | Damage mechanics of composite materials[END_REF][START_REF] Halm | An modular damage model for quasi-brittle solids-interaction between initial and induced anisotropy[END_REF][START_REF] Cazacu | On Modeling the Interaction between Initial and Damage-Induced Anisotropy in Transversely Isotropic Solids[END_REF]). In this paper, this coupling is addressed by means of Eshelby-type homogenization techniques. The general case of arbitrarily oriented cylindrical cracks with respect to the symmetry axis of an orthotropic medium is considered.

It is worth emphasizing that the analysis of a plastic matrix containing microcracks is beyond the scope of the present work. Such an analysis can be conducted following the classic approach of [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part i. yield criteria and flow rules for porous ductile materials[END_REF]. The reader is referred to yielding of plastic microcracked materials (see [START_REF] Gologanu | Approximate models for ductile metals containing non-spherical voids -case of axisymmetric oblate ellipsoïdal cavities[END_REF][START_REF] Gologanu | Recent extensions of Gurson's model for porous ductile metals[END_REF], the recent studies of [START_REF] Monchiet | Approximate yield criteria for anisotropic metals with prolate or oblate voids[END_REF][START_REF] Monchiet | Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids[END_REF] and the recent review by [START_REF] Benzerga | Ductile fracture by void growth to coalescence[END_REF]).

In the context of upscaling techniques, the determination of the overall elastic properties of a cracked material requires the computation of the Eshelby tensor S associated to any crack. Many studies have addressed the case of a linearly elastic isotropic matrix containing penny-shaped and/or elliptical cracks (see for instance [START_REF] Horii | Overall moduli of solids with microcracks: Load induced anisotropy[END_REF][START_REF] Mura | Micromechanics of defects in solids[END_REF][START_REF] Nemat-Nasser | Micromechanics : overall properties of heterogenous materials[END_REF][START_REF] Kachanov | Elastic solids with many cracks and related problems[END_REF]) for which the Eshelby tensor is well known. In the case of materials with matrix exhibiting structural (primary) anisotropy, few analytical results exist in literature. In a three dimensional case, only particular crack configurations have been considered, i.e. when the crack lies in one of the symmetry planes of the solid matrix (see [START_REF] Laws | A note on interaction energies associated with cracks in anisotropic solids[END_REF][START_REF] Laws | A note on penny-shaped cracks in transversely isotropic materials[END_REF][START_REF] Mura | Micromechanics of defects in solids[END_REF] and the more recent work of [START_REF] Kanaun | Elliptical cracks arbitrarily oriented in 3d-anisotropic elastic media[END_REF]). For an arbitrarily oriented crack in an anisotropic matrix, the results existing in literature are restricted to planar elasticity. The principal difficulty in conducting for a 3D analysis is that the Green function associated to anisotropic solids (from which is classically determined the analytical expression of the Eshelby tensor S) is generally not known in closed form (see, for example, [START_REF] Ting | Anisotropic elasticity: Theory and applications[END_REF], [START_REF] Martin | On green's function for a three-dimensional exponentially graded elastic solid[END_REF], [START_REF] Kachanov | Handbook of elasticity solutions[END_REF]). In this paper, we seek to address these limitations. The objectives of these study are:

(i) to obtain the closed-form expression of the effective compliance of an orthotropic matrix containing cylindrical cracks. To this end we will recall and use analytical expressions of the Eshelby tensor S that we briefly exposed in [START_REF] Gruescu | Eshelby tensor for a crack in an orthotropic elastic medium[END_REF] 1 based on the general methodology introduced by [START_REF] Kinoshita | Elastic field of inclusions in anisotropic media[END_REF] and [START_REF] Faivre | Hétérogénéités ellipsoïdales dans un milieu élastique anisotrope[END_REF] (see also the works of [START_REF] Ting | The three-dimensional elastostatic Green's function for general anisotropic linear elastic solids[END_REF][START_REF] Suvorov | Rate form of the Eshelby and Hill tensors[END_REF]).

(ii) to propose an anisotropic damage model developed by the implementation of these new results in an homogenization scheme, namely the Mori-Tanaka model, in order to determine the macroscopic properties of the orthotropic material weakened by an arbitrarily oriented system of cracks. The anisotropic damage model is further applied for the prediction of the inelastic behavior of a ceramic matrix composite.

The paper is organized as follows. We begin with a brief description of the Mori-Tanaka homogenization scheme, that will be further used to obtain the effective compliance [START_REF] Barthelemy | Compliance and hill polarization tensor of a crack in an anisotropic matrix[END_REF]. The focus of section 3 is the derivation of an analytic expression of the Eshelby tensor for 3D microcracks systems. in particular, we present the basic methodology in subsection 3.1 while section 3.2 is devoted to the validation of the results and to the presentation of their connection with existing results in planar elasticity. The determination of the effective compliance and the illustration of combined effects of microcracks-induced anisotropic damage and initial anisotropy are performed in section 4. These results are then used in section 5 where is presented a micromechanical damage model able to describe the inelastic behavior of orthotropic composites. The proposed model is applied and tested on a ceramic matrix composite (SiC-SiC), in order to illustrate it's ability to account for the interaction ef-fects between the initial anisotropy and cracks orientation.

Notations: Standard tensorial notations will be used throughout. Lower underlined case letters will describe vectors, bold script capital letters will be associated to second-order tensors and mathematical double-struck capital letters will denote fourth-order tensors respectively. The following vector and tensor products are exemplified:

(A.b) i = A ij b j , (A.B) ij = A ik B kj , (A : B) ij = A ijkl B kl , (A : B) ijkl = A ijpq B pqkl .
Einstein summation convention, applied for the repeated indices and Cartesian coordinates are used. As usual, small respectively large characters refer to microscopic (resp. macroscopic) quantities. The reference configuration corresponds to the natural state (undamaged material); 1 and I are, respectively, the second and fourth order identity tensors, the former represents the Kronecker symbol (δ ij ) and the latter is defined as:

I ijkl = (1/2)(δ ik δ jl + δ il δ jk ).

Mori-Tanaka scheme for orthotropic matrix weakened by arbitrary oriented cracks

As already mentioned, in contrast to the continuum macroscopic damage approaches, the micromechanically based approaches to modeling microcraksinduced anisotropy present the advantage of incorporating damage mechanisms occurring at microscopic scales. In addition to having a physical basis, the resulting damage models provide an explicit way of modeling the coupling between the structural and microcracks-induced anisotropies. This section begins with a short description of the general concepts of the homogenization of disordered media using the classical Eshelby's equivalent inclusion method and how the Eshelby results can be further used in the Mori-Tanaka scheme to derive the effective properties of the cracked media.

Basic principles of Eshelby-based homogenization techniques

Consider a representative volume element (RVE) Ω of a material consisting of an orthotropic solid matrix Ω s (of stiffness tensor C s and volume fraction φ s ) and a system of microcracks occupying a domain Ω f . The orientation of each family r of microcracks (r = 1 to N) is characterized by a normal unit vector denoted by n. Uniform strain conditions are prescribed on the outer boundary ∂Ω of the RVE (see Figure 1). The crack is considered as an elastic material of stiffness tensor C f . ζ = E.x The microscopic stress and strain fields are respectively denoted by σ and ε whereas x and ζ denote respectively the position vector in the RVE and the microscopic displacement. A crucial step of the homogenization techniques consists in finding the fourthorder localization tensor A(x) which relates the microscopic strain field to the macroscopic one, i.e. ε(x) = A(x) : E. It follows that the homogenized stiffness tensor C hom is (see, for instance, [START_REF] Zaoui | Continuum micromechanics: Survey[END_REF]):

C hom = ⟨C(x) : A(x)⟩ Ω = C s + N ∑ r=1 φ f r ( C f r -C s ) : A f r (1)
where φ f r is the volume fraction of the r th family of microcracks (r = 1 to N), and the identity ⟨A(x)⟩ Ω = I (which results from the strain average rule) is used. The localization problem is solved by using the fundamental result provided by [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF]: for an ellipsoidal crack (defining the r th family) embedded in a linear elastic solid matrix. In the dilute homogenization scheme, the localization tensor reads:

A f r = (I + P r : δC) = (I -S r ) -1 (2)
In (2), S r and P r denote the Eshelby and the Hill tensors respectively (S r = P r : C s ). Since only opened cracks are considered in the present study, C f = 0 and δC = C f r -C s (= -C s for opened cracks). Let us recall that the dilute scheme is restricted to infinitesimal concentrations of the inhomogeneous inclusions.

The Mori-Tanaka scheme applied to cracked media

The Mori-Tanaka homogenization scheme (see [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF]) accounts for interaction effects. It is classically formulated by adapting the Eshelby result to this situation. The resulting localization tensor for the r th family of microcracks (r = 1 to N) takes then the form (see for instance [START_REF] Deudé | Micromechanical approach to nonlinear poroelasticity : application to cracked rocks[END_REF] or [START_REF] Dormieux | Poroelasticity and damage theory for cracked media[END_REF]):

A f r = (I -S r ) -1 :   φ s I + N ∑ j=1 φ f j (I -S j ) -1   -1 (3)
which, when substituted in (1), leads to:

C hom = C s - N ∑ r=1 φ f r C s : (I -S r ) -1 :   φ s I + N ∑ j=1 φ f j : (I -S j ) -1   -1 (4)
Note that the determination of the homogenized stiffness (or compliance) tensor reduces to the computation of S r . For orthotropic media, this is not a straightforward task and is the focus of the next subsection.

3 Closed-form expression of Eshelby tensor S r for an arbitrarily oriented cylindrical crack in an orthotropic medium

Methodology and results

Consider first that the solid matrix is orthotropic (of stiffness tensor C s ) and is weakened by a set of parallel microcracks. Let (z 1 , z 2 ,z 3 ) be the rectangular cartesian coordinates system and (k 1 , k 2 , k 3 ) the local frame associated to the microcracks. Let also define by (x 1 , x 2 , x 3 ) the global coordinate system and by (e 1 , e 2 , e 3 ) the global frame associated to the axes of symmetry (orthotropy) of the matrix. We model the crack as an infinite cylinder aligned along the direction e 3 ≡ k 3 and elliptical cross-section of low aspect ratio X = b/a (see Figure 2) geometrically prescribed by: Let us also recall that the components of the solid matrix stiffness tensor in the local frame (the angle θ gives the orientation of the crack with respect to the axes of orthotropy of the matrix), are obtained using classical tensor transformation rules as:

z 2 1 a 2 + z 2 2 b 2 = 1, -∞ < z 3 < ∞ (5)
C ijkl = U ip U jq U kr U ls C s pqrs .
The second-order orthogonal transformation tensor U used for the local frame coordinate system associated to the crack (z 1 , z 2 , z 3 ) is:

U =        cos θ sin θ 0 -sin θ cos θ 0 0 0 1        (6)
Note that, for an arbitrary orientation of the cracks plane, θ, the stiffness tensor in the local frame displays a matrix of 13 non-zero components. The determination of the interaction tensor P, which is based on the work of [START_REF] Kinoshita | Elastic field of inclusions in anisotropic media[END_REF], (see also [START_REF] Faivre | Hétérogénéités ellipsoïdales dans un milieu élastique anisotrope[END_REF], [START_REF] Laws | A note on interaction energies associated with cracks in anisotropic solids[END_REF] or [START_REF] Willis | Bounds and self-consistent estimates for the overall properties of anisotropic composites[END_REF]), consists in evaluating the integral:

P ijkl = ab π ∫ 2π 0 N ijkl (ξ 1 , ξ 2 ) (a 2 ξ 2 1 + b 2 ξ 2 2 ) dψ ( 7
)
The integration is performed on the unit circle centered at the origin, in the plane (ξ 1 , ξ 2 ): ∥ξ∥ = 1 (e.g. ξ = cos ψe 1 + sin ψe 2 ). The components of the fourth order tensor N read:

N ijkl (ξ 1 , ξ 2 ) = D ijkl (ξ 1 , ξ 2 , 0) with: (8)
D ijkl = 1 4 ( ξ i K -1 jk ξ l + ξ j K -1 ik ξ l + ξ i K -1 jl ξ k + ξ j K -1 il ξ k ) (9)
In equation( 9), K = ξ.C.ξ represents the acoustic tensor associated to C and the vector ξ. It can be noted that the anisotropy of the elastic solid matrix affects P through the acoustic tensor K. Since the above expressions (equation 7, together with 8 and 9) imply computations in the local frame of the crack, P depends a priori on the orientation of the latter. Equivalently, the components of the tensor P can be expressed as:

P ijkl = 1 4 (M ijkl + M jikl + M ijlk + M jilk ) (10)
where:

M ijkl = ab π ∫ 2π 0 ξ i K -1 jk ξ l (a 2 ξ 2 1 + b 2 ξ 2 2 ) dψ ( 11
)
As mentioned before, note that the analytical evaluation of the interaction tensor P is not an easy task. An interesting and elegant method for estimating this integral which is based on the residue theorem was used by [START_REF] Gruescu | Eshelby tensor for a crack in an orthotropic elastic medium[END_REF] (see appendix A). The analytical expressions of the components of the tensor P are given in the appendix B.

It is worth noting that only the orientation and the geometry of the elliptical inhomogeneity (i.e. the angle θ and the aspect ratio X = b/a) as well as the initial anisotropy of the solid matrix (the stiffness tensor C s ) affects the components of the interaction tensor P. Thus, the influence of the initial anisotropy on the overall elastic properties of the material (which are now obtainable by incorporating the previous results in various homogenization schemes) and the coupling between initial and crack induced anisotropy is described. To the best of our knowledge (especially for the validation of the obtained results) the closed form expressions of the interaction tensor P of ellipsoidal inhomogeneities in orthotropic media are known only for cylindrical cracks normal to one of the material symmetry axis (see [START_REF] Laws | A note on interaction energies associated with cracks in anisotropic solids[END_REF][START_REF] Faivre | Hétérogénéités ellipsoïdales dans un milieu élastique anisotrope[END_REF]). The only 3D analytical expressions were obtained by [START_REF] Laws | A note on penny-shaped cracks in transversely isotropic materials[END_REF] for penny-shaped cracks normal to the symmetry axis, i.e. lying in the isotropy plane of the transversely isotropic matrix. Thus, for validation purposes, we can only consider these particular cases and show that the new results reduce to the corresponding existing ones.

Connection with existing results

In the present section are presented several validations of the previous results implying directly either the P or equivalently Q tensor (introduced in section 3.2.2).

Aligned cracks normal to a symmetry axis of the orthotropic matrix

A first validation of the new results is done by considering the particular case of a system of aligned cracks normal to the symmetry axis e 1 of the solid matrix. This problem has been studied by [START_REF] Laws | A note on interaction energies associated with cracks in anisotropic solids[END_REF], who obtained the following non-zero components for the fourth-order interaction tensor P:

P 1111 = C s 2222 + C s 1212 √ αβ C s 1111 C s 1212 √ αβ( √ α + √ β) X P 2222 = 1 C s 2222 + C s 2222 -C s 1212 (α + β + √ αβ) C s 2222 C s 1212 √ αβ( √ α + √ β) X P 1122 = - C s 1122 + C s 1212 C s 1111 C s 1212 √ αβ( √ α + √ β) X P 2323 = 1 4C s 2323 - √ C s 1313 4C s 2323 3/2 X; P 1313 = 1 4 √ C s 2323 C s 1313 X P 1212 = 1 4C s 1212 - C s 1111 C s 2222 -C s 1122 2 2C s 1111 C s 1212 2 √ αβ( √ α + √ β) X ( 12
)
where α and β are the complex conjugate roots of the characteristic equation for the orthotropic solid matrix:

C s 1111 C s 1212 x 2 - [ C s 1111 C s 2222 -C s 1122 (C s 1122 + 2C s 1212 ) ] x + C s 2222 C s 1212 = 0 (13)
It can be easily verified that for θ = 0, the expressions (B.1) that we established coincide with the expressions (12).

Validation through comparison with classic results obtained from the study in planar elasticity of orthotropic cracked media

Rigorous results for planar elasticity problems2 involving elliptical inclusions (of aspect ratio X = b/a) embedded in an anisotropic matrix were available long time ago by using the complex potentials approach (see [START_REF] Lekhnitski | Stresses in an infinitely large anisotropic plate which is weakened by an elliptical hole[END_REF]). Interestingly, they are recently applied by [START_REF] Tsukrov | Effective moduli of an anisotropic material with elliptical holes of arbitrary orientational distribution[END_REF] for the homogenization of cracked materials with orthotropic matrix. At the difference of the complex potentials method, the Eshelby-based results obtained in the present study allows to investigate 3D cracks configurations. The purpose of this second level of validation is to show that the results derived in the present study reduce to existing ones when 2D cracks are considered. The starting point is the relationship already established by [START_REF] Tsukrov | Effective moduli of an anisotropic material with elliptical holes of arbitrary orientational distribution[END_REF] between the strain induced by the cavity, (ε c ), and the macroscopic stress field Σ applied to the RVE:

ε c = H : Σ (14)
in which, following [START_REF] Laws | A note on interaction energies associated with cracks in anisotropic solids[END_REF], the H tensor is related to the Hill tensor by:

H = dXQ -1 with Q = C s -C s : P : C s ( 15
)
where d is the crack density parameter (d = Ma 2 , M being the number of cracks per unit volume) introduced by [START_REF] Budiansky | Elastic moduli of a cracked solid[END_REF] and Q the second Hill tensor.

As a validation, it is convenient to compute tensor Q (corresponding to the arbitrarily oriented elliptical crack in the orthotropic matrix) from (B.1) and then its inverse Q -1 . It follows that:

XQ -1 1111 = t √ αβ( √ α + √ β) [ X 2 cos 2 (θ) + sin 2 (θ) + C s 1122 X C s 1111 √ αβ( √ α + √ β) ] XQ -1 1112 = XQ -1 1211 = t √ αβ( √ α + √ β) cos θ sin θ(X 2 -1)
XQ -1 1122 = XQ -1 2211 = -t √ αβX XQ -1 1212 = t( √ α + √ β) 4 • { X 2 [ cos 2 (θ) + √ αβ sin 2 (θ) ] + [ sin 2 (θ) + √ αβ cos 2 (θ) ] + X( √ α + √ β) } XQ -1 2212 = XQ -1 1222 = t( √ α + √ β) 2 (X 2 -1) cos θ sin θ XQ -1 2222 = t( √ α + √ β) { X 2 sin 2 (θ) + cos 2 (θ) + X √ α + √ β } ( 16
)
where

t = C s 1111 /(C s 111 C s 2222 -C s 1122 2
). The final verification requires to compare the components of XQ -1 given by [START_REF] Halm | An modular damage model for quasi-brittle solids-interaction between initial and induced anisotropy[END_REF] with that of H given by [START_REF] Tsukrov | Effective moduli of an anisotropic material with elliptical holes of arbitrary orientational distribution[END_REF] and reported in appendix (D) (in which the definition d = πa 2 /A must be adopted, A being the reference cell area).

The Mori-Tanaka estimate of the effective compliance: combined effects of initial and damage-induced anisotropies

Determination of the effective compliance

We aim at deriving the macroscopic elastic properties of an orthotropic solid matrix containing randomly oriented microcracks. To this end, a Mori-Tanaka estimate of the homogenized stiffness tensor of the microcracked medium is considered (4). Following [START_REF] Budiansky | Elastic moduli of a cracked solid[END_REF], let us recall the 2D cracks density parameter d r = Ma r 2 , where M denotes the number of cracks per unit area of the considered cracks system. Recalling the notation φ f r for the volume fraction of the r th family of microcracks (r = 1 to N), one has φ f r = πd r X r . Taking then the limit for low aspect ratio (lim Xr→0 ), it can be shown that the homogenized stiffness tensor, given by ( 4), takes the form (as in the isotropic case):

C hom = C s : ( I + r=N ∑ r=1 πd r T r ) -1 where T r = lim Xr→0 X r [ I -S r
] -1 [START_REF] Hayes | Connexions between the moduli for anisotropic elastic materials[END_REF] The components of tensor T r are given in Appendix D. Note that they depend on the cracks orientation defined by the θ angle (see figure 2). Moreover, the material orthotropy is only preserved when cracks are aligned along a material symmetry axis (θ = 0 or θ = π/2). This corresponds to the cancelation of some components of T r . The homogenized compliance tensor is obtained by inverting [START_REF] Hayes | Connexions between the moduli for anisotropic elastic materials[END_REF]:

S hom = ( I + r=N ∑ r=1 πd r T r ) : S s ( 18
)
where S s = (C s ) -1 represents the compliance tensor of the solid matrix. The components of S hom are given by:

S
√ α + √ β) [ (sin θ) 2 + √ αβ(cos θ) 2 ] S s 3333 (19)
Note that the components of the homogenized compliance tensor depends on the elastic properties of the solid matrix and strongly on the crack orientation (the θ angle). Such dependence accounts for the interaction between the initial anisotropy and the cracks orientation. The deviation from material orthotropy can be found in components S hom 1112 = S hom 1211 , S hom 2212 = S hom 1222 and S hom 1323 = S hom 2313 .

Effects of interaction between the initial and micro-cracks induced anisotropy on the elastic moduli

We propose to illustrate here the effect of open cracks systems on the homogenized elastic properties, the solid matrix being orthotropic. Following [START_REF] Welemane | Some remarks on the damage unilateral effect modelling for microcracked materials[END_REF], one can derive from S hom the expressions of the generalized elastic moduli: the Young modulus, E(m), corresponding to an arbitrary direction defined by a unit vector m, the Poisson ratio ν(m, p), and the shear modulus µ(m, p), associated to two orthogonal directions of respective unit vectors m and p.

These moduli are respectively defined as (see also [START_REF] Hayes | Connexions between the moduli for anisotropic elastic materials[END_REF]): As an example, the effect of cracks on the generalized Young modulus E(m) will be only presented here. To this end, we consider a set of parallel cracks whose unit normal is n, e.g. associated to the angle θ with respect to the symmetry axis e 2 of the matrix (see Figure 2). For each space direction given by a vector m, taken in the plane (e 1 , e 2 ), the value of the elastic moduli is normalized by the corresponding value of the solid matrix (plotted as the unit circle in dashed line). The analysis is done for the case of open cracks. The elastic moduli of the orthotropic solid matrix, which are assumed to correspond to the composite studied in section are: E s 1 = 320000 MPa, E s 2 = 170000 MPa for the Young moduli, G s 12 = 90000 MPa for the shear modulus and ν s 12 = 0.18 for the Poisson ratio. The results (see for instance Figure 3a corresponding to θ = 0) show a significant degradation of the Young modulus in the normal direction to the crack. The same observation can be done for a cracks family oriented with an angle θ = π/2 (see Figure 3b). However, it must be emphasized that this second curve is not obtained by a simple rotation of the previous one (θ = 0), as it would be the case for an isotropic solid matrix. This is clearly a consequence of the orthotropy of the solid matrix. The effect of the matrix anisotropy is even more pronounced for θ = π/3 (see Figure 3c which shows a notable distortion).

                   E(m) = [(m ⊗ m) : S hom : (m ⊗ m)] -1 ν(m, p) = - (p ⊗ p) : S hom : (m ⊗ m) (m ⊗ m) : S hom : (m ⊗ m) µ(m, p) = [4(m ⊗ p) : S hom : (m ⊗ p)] -1 (20)

A micromechanical damage model for initially orthotropic materials

In this section we propose a micromechanical damage model for initially orthotropic materials. The model is based on the results presented in the preceding section and on the choice of a damage criterion which will allow to describe the propagation of distributed microcracks. The predictive capabilities of the micromechanical model are demonstrated by application to a brittle matrix composite. Comparison with experimental data obtained by [START_REF] Aubard | Modélisation et identification du comportement mécanique des matériaux composites 2D SiC-SiC[END_REF] on a SiC-SiC ceramic composite shows the ability of the proposed model to describe the overall stress-response of this material when submitted to off axis loadings.

Free enthalpy. Damage propagation by cracks growth

A stress-based formulation of the damage is proposed. The corresponding macroscopic free enthalpy for an arbitrarily oriented cracks system depends on the macroscopic stress and on the damage state:

W * = 1 2 Σ : S hom : Σ (21)
S hom depends on the cracks density parameters d r (representing the damage variables) and is given by ( 18) when a Mori-Tanaka scheme is considered.

The first state law provides the macroscopic strain tensor E as a function of the macroscopic stress Σ:

E = ∂W * ∂Σ = S hom : Σ (22)
To derive the complete formulation of the model, one needs to propose a damage criterion and a damage evolution law. Let us recall that the quantities d r (for r = 1 to N) constitute the set of internal damage variables. The damage criterion being based on the intrinsic dissipation due to damage, the latter can be obtained from the expression:

D = r=N ∑ r=1 ∂W * ∂d r ḋr = r=N ∑ r=1 F dr ḋr ≥ 0 ( 23
)
in which, F dr is the thermodynamic force (energy release rate associated to the r th cracks family) associated to damage variable d r :

F dr = ∂W * ∂d r r = 1 to N ( 24
)
The previous expression constitutes the second state law which was obtained by adopting as damage variable the crack density parameter d r corresponding to a crack family denoted i. Based on the above thermodynamic arguments, the damage criterion for the considered cracks family can be put in the form (see also [START_REF] Marigo | Modelling of brittle and fatigue damage for elastic material by growth of microvoids[END_REF]):

f r (F dr , d r ) = F dr -R(d r ) r = 1 to N (25)
In [START_REF] Krajcinovic | Damage mechanics[END_REF] the function R(d r ) describes the crack resistance to damage propagation. In principle, the damage activation criterion can be determined from experimental investigations (see [START_REF] Mai | Failure characterization of fibre-reinforced cement composites with R-curve characteristics[END_REF] and [START_REF] Ch | An r-curve approach for fracture of quasi-brittle materials[END_REF]). It's dependency on d r confers to the latter the role of a hardening variables. For simplicity, the following form is chosen for R(d r ):

R(d r ) = k(1 + ηd r ) r = 1 to N ( 26
)
where k is a parameter which describes the damage threshold and η accounts for the hardening effect of damage. These two parameters can be identified from standard tensile tests. The damage evolution is obtained by assuming the normality rule:

ḋr = Λdr ∂f r (F dr , d r ) ∂F dr = Λdr ; Λdr ≥ 0 so ḋr =      0 if f r < 0 or f r = 0, ḟr < 0 Λdr if f r = 0 and ḟr = 0 (27)
The damage multiplier Λdr is derived from the classical Kuhn-Tucker consistency condition: ḟr = 0. From a physical point of view, this implies that the representing point of loading is placed on the loading surface, so:

Λdr = 1 R ′ (d r ) (B r : Σ) + (28)
with B r = ∂F dr /∂Σ. It should be stated that the adopted damage evolution law supposes implicitly that cracks are propagating in their own plane. The adopted framework (of thermodynamic irreversible processes) indicates also that the propagation speed ḋr of each variable is dependent of the macroscopic stress Σ only via the associated thermodynamic force. This framework is similar to the one adopted by [START_REF] Rice | Inelastic constitutive relations for solids: an internal variable theory and its application to metal plasticity[END_REF] for the micro-macro transition, by using the parameters describing the microstructural arrangement of the material as internal variables.

Rate formulation of the stress-based model

For a damage activation state unchanged by the application of a stress increment, the tangent compliance tensor is determined by the use of the damage evolution law (relations [START_REF] Laws | A note on interaction energies associated with cracks in anisotropic solids[END_REF][START_REF] Laws | A note on penny-shaped cracks in transversely isotropic materials[END_REF]). The rate form of the damage model can be then deduced as:

Ė = S hom t : Σ (29)
with

S hom t = S hom - N ∑ r=1 ϖ r 1 R ′ (d r ) G r B r ⊗ B r (30)
and

G r =      0 if f r < 0 or f r = 0, ḟr < 0 1 if f r = 0 and ḟr = 0 (31)
ϖ r denotes the associated weight of the r th integration point. Due to the symmetry of the second order tensor B r , the homogenized tangent compliance S hom t has the minor and major symmetries.

Numerical local integration of the constitutive damage law

Algorithmic aspects related with the numerical integration of the proposed model are presented in the following. The formulation is strain based. During a time step (t j , t j+1 ) an elastic trial state is computed:

f r (F dr j+1 , d j r ) ≤ 0 (32)
then the trial state is the final state (i.e. the solution to the problem):

     E j+1 = E j + ∆E d j+1 r = d j r (r = 1 to N) (33)
If the condition (32) is not satisfied, a linear correction should be imposed:

       ∆d r = 1 kη f r (F dr j+1 , d j r ) d j+1 r = d j r + ∆d r (34)

Application of the model to an unidirectional SiC-SiC composite

In this section, we apply the proposed model to study the response of a brittle matrix SiC-SiC ceramic composite subjected to uniaxial off-axis tensile loading in different directions φ with respect to the axis of symmetry of the material (see Figure 4). The experimental data were reported by [START_REF] Aubard | Modélisation et identification du comportement mécanique des matériaux composites 2D SiC-SiC[END_REF]. The elastic moduli of the orthotropic solid matrix, which are assumed to correspond to the initial elastic moduli of the composite are: E 1 = 320000 MPa, E 2 = 170000 MPa for the Young moduli, G 12 = 90000 MPa for the shear modulus and ν 12 = 0.18 for the Poisson ratio.

The model involves two material parameters k and η that appear in the expression of the damage criterion. The identification procedure consists in calibrating the model parameters k and η from the tensile test corresponding to φ = 0˚. The obtained numerical values are: k = 3.75J.m -2 and η = 140. No experimental data were available on the initial cracks density parameter. In all the simulations, we set this parameter value to 0.01. We also assumed that there are 60 distinct families of cracks randomly oriented in the solid matrix.

The validation of the model is done by simulating off-axis tests (corresponding to φ = 20 0 and φ = 45 0 ). Figure 5 shows the overall stress-strain response in tensile tests for φ = 0 0 , φ = 20 0 and φ = 45 0 . The agreement between the simulated response at φ = 0 0 confirms the relevance of the parameters calibration. The results predicted for the off-axis loading tests show also a good agreement with experimental data; this indicates the good predictive capabilities of the proposed micromechanical damage model. In particular, the relative positions of the three curves manifest the combined effect of the initial anisotropy and the evolving damage.

Conclusions

In the present study we investigate and evaluate the effective properties of an orthotropic matrix weakened by arbitrarily oriented microcracks. The determination of the macroscopic elastic properties of this class of materials has been performed in the framework of Eshelby's equivalent inclusion problems. To this end, we have derived original closed-form expressions of the Eshelby tensor (or equivalently, of the Hill tensor). The macroscopic elastic properties (compliance), deduced from these results, shows a coupling between the initial orthotropy of the solid matrix and the cracks-induced anisotropy. As a consequence, the deviation from elastic orthotropy can be characterized. A new damage model for initially orthotropic materials is then proposed and implemented by adopting a Mori-Tanaka homogenization scheme and a damage criterion based on the derived energy release rate. The numerical predictions of the damage model are in good agreement with experimental data reported by [START_REF] Aubard | Modélisation et identification du comportement mécanique des matériaux composites 2D SiC-SiC[END_REF] on a ceramic matrix composite (SiC-SiC), particularly for off-axis loadings experiments.

For the computation of P (equations (7-9)), one needs to invert K(y). By denoting | K(y) | the determinant of K(y) and by K(y) it's adjoint, it follows that: K(y). K(y) =| K(y) | 1, and respectively:

K -1 (y) = K(y) | K(y) | (A.5)
For the second-order tensor ∆ = ξ ⊗ ξ which, as for K -1 , appears in the definition (9) of the tensor N, it can be shown that: ∆(ψ) = (sin ψ) 2 ∆(y) with ∆(y) = y 2 e 1 ⊗ e 1 + y(e 1 ⊗ e 2 + e 2 ⊗ e 1 ) + e 2 ⊗ e 2 . Note also that a 2 ξ 2 1 + b 2 ξ 2 2 = (sin ψ) 2 (a 2 y 2 +b 2 ). Using then equation (A.2) and the change of variable y = cot ψ, the expression (11) becomes:

M ijkl = ab π ∫ ∞ -∞ Kjk (y)∆ il (y)dy (a 2 y 2 + b 2 ) | K(y) | = 1 π ∫ ∞ -∞ X Kjk (y)∆ il (y)dy (y 2 + X 2 ) | Q | f (y) (A.6)
where X = b/a denotes the microcrack aspect ratio. In general, f (y) is a polynomial of degree 6:

f (y) = (y -y 1 )(y -ȳ1 )(y -y 2 )(y -ȳ2 )(y -y 3 )(y -ȳ3 ). (A.7)
where y p (p = 1, 2, 3) are the roots with positive imaginary part, while ȳp their complex conjugate, respectively. The M tensor depends only on the shape of the inclusion and on the material's stiffness. Note also that the representation (A.6) allows the evaluation of M ijkl by the residues theorem. The first order approximation of M at X = 0 (X being the microcrack aspect ratio) gives:

M ijkl = Kjk (0)∆ il (0) f (0)| Q | + 2iX | Q | 3 ∑ i=1 Kjk (y i )∆ il (y i ) f ′ (y i )y 2 i + iX f (0) | Q | • [ K′ jk (0)∆ il (0) + Kjk (0)∆ ′ il (0) -Kjk (0)∆ il (0) f ′ (0) f (0) ] (A.8)
where Kjk are the components of the adjoint of K. The quantity | Q |, the determinant of Q (see (A.4)) is expressed as:

| Q |= C s 1111 C s 1212 (cos 2 (θ) + α sin 2 (θ))(cos 2 (θ) + β sin 2 (θ)) (C s 3232 sin 2 (θ) + C s 3131 cos 2 (θ)) (A.9)
with α and β are given by [START_REF] Gologanu | Recent extensions of Gurson's model for porous ductile metals[END_REF]. Thus,

α + β = C s 1111 C s 2222 -C s 1122 2 -2C s 1122 C s 1212 C s 1111 C s 1212 αβ = C s 2222 C s 1111 (A.10)
It can be shown that the imaginary part, ℑ(M), of M is null:

ℑ(M ijkl ) = 2X | Q | ℜ 3 ∑ i=1 Kjk (y i )∆ il (y i ) f ′ (y i )y 2 i + X f (0) | Q | • [ K′ jk (0)∆ il (0) + Kjk (0)∆ ′ il (0) -Kjk (0)∆ il (0) f ′ (0) f (0) ] = 0. (A.11)
Hence its real part, ℜ(M), is:

M ijkl = ℜ(M ijkl ) = Kjk (0)∆ il (0) f (0) | Q | - 2X | Q | ℑ 3 ∑ i=1 Kjk (y i )∆ il (y i ) f ′ (y i )y 2 i (A.12)
In summary, we have obtained the expression of M in the local frame of the crack. Its expression in the global frame is given by a simple change of frame.

B Expression of the components of Hill's tensor P

In the global frame, the components of Hill's tensor P are:

P 1111 = (C 2222 u 2 + C 1212 v 2 )v 2 C 1111 C 1212 (αu 2 + v 2 )(βu 2 + v 2 ) + X C 1111 C 1212 (α -β) • { √ α(αu 2 -v 2 )(αC 1212 -C 2222 ) (αu 2 + v 2 ) 2 - √ β(βu 2 -v 2 )(βC 1212 -C 2222 ) (βu 2 + v 2 ) 2 } P 1122 = - (C 1122 + C 1212 )u 2 v 2 C 1111 C 1212 (αu 2 + v 2 )(βu 2 + v 2 ) + X(C 1122 + C 1212 ) C 1111 C 1212 (α -β) { √ α(αu 2 -v 2 ) (αu 2 + v 2 ) 2 - √ β(βu 2 -v 2 ) (βu 2 + v 2 ) 2 } P 2222 = (C 1111 v 2 + C 1212 u 2 )u 2 C 1111 C 1212 (αu 2 + v 2 )(βu 2 + v 2 ) + X C 1111 C 1212 (α -β) • { (αu 2 -v 2 )(αC 1111 -C 1212 ) √ α(αu 2 + v 2 ) 2 - (βu 2 -v 2 )(βC 1111 -C 1212 ) √ β(βu 2 + v 2 ) 2 } P 1112 = (C 2222 u 2 -C 1122 v 2 )

D Components of the H tensor

The components of the H tensor obtained by [START_REF] Tsukrov | Effective moduli of an anisotropic material with elliptical holes of arbitrary orientational distribution[END_REF] in planar elasticity are as follows:

H 1111 = πL A √ E s 1 [ (b 2 -a 2 )
H 1212 = πL 4A √ E s 1 E s 2 [ (a 2 -b 2 ) ( √ E s 2 - √ E s 1 ) cos 2 (θ) + a 2 √ E s 1 + abL √ E s 1 E s 2 + b 2 √ E s

Figure 1 .

 1 Figure 1. The representative volume element (RVE) of the orthotropic cracked solid.

3 Figure 2 .

 32 Figure 2. Crack modelled as an infinite cylinder with elliptical section; the aspect ratio is X = b/a

c) θ = π/ 3 Figure 3 .

 33 Figure 3. Variation of the generalized Young's modulus produced by opened cracks systems. Three distinct families are respectively considered: a) = 0; b) comparison for orientations θ = 0 and θ = π/2; c) θ = π/3. The results are normalized with the initial values of the Young moduli, which are represented by the unit circle

ϕ e 1 e 2 ΣFigure 4 .

 124 Figure 4. Representation of the tensile test applied with the φ angle

Figure 5 .

 5 Figure 5. Comparison of the tensile tests of unidirectional SiC-SiC composite

(B. 1 )

 1 with u = cos(θ) and v = sin(θ).C Components of the T tensorComponentwise T ijkl terms deduced in the section 4.1 are as follows (using the notation t = C s 1111 /(C s 111 C s 2222 -

2 ](D. 1 )

 21 in which A is the representative area of the considered cell. The constant L is related to the elastic moduli of the orthotropic 2D matrix by: are the roots of the characteristic equation of the bidimensional orthotropic medium.

  uv 2C 1111 C 1212 (αu 2 + v 2 )(βu 2 + v 2 ) + Xuv C 1111 C 1212 (α -β) • { √ α(αC 1122 + C 2222 ) (αu 2 + v 2 ) 2 -√ β(βC 1122 + C 2222 ) (βu 2 + v 2 ) 2 } P 2212 = (C 1111 v 2 -C 1122 u 2 )uv 2C 1111 C 1212 (αu 2 + v 2 )(βu 2 + v 2 ) -Xuv C 1111 C 1212 (α -β) • C 1111 v 4 + C 2222 u 4 -2C 1122 v 2 u 2 4C 1111 C 1212 (αu 2 + v 2 )(βu 2 + v 2 ) C 2323 u 2 -C 1313 v 2 ) (C 3232 u 2 + C 3131 v 2 ) 2 2323 u 2 -C 1313 v 2 (C 3232 u 2 + C 1313 v 2 ) 2 P 1323 = uv 4(C 2323 u 2 + C 3131 v 2 ) -X √ C 1313 C 2323 uv 2(C 2323 u 2 + C 1313 v 2 ) 2

		{ √	α(αC 1111 + C 1122 ) (αu 2 + v 2 ) 2	-	√	β(βC 1111 + C 1122 ) (βu 2 + v 2 ) 2
							+	X(C 1111 C 2222 -C 2 1122 ) 4C 1111 C 2 1212 (α -β)	•
		{ √ (αu 2 + v 2 ) 2 -α(αu 2 -v 2 )	(βu 2 + v 2 ) 2 √ β(βu 2 -v 2 )	}
	P 1313 = P 2323 =	v 2 4(C 2323 u 2 + C 1313 v 2 ) u 2 4(C 3232 u 2 + C 1313 v 2 )	+ -	X 4 X 4	√ √	C 2323 C 1313 C 1313 C 2323

} P 1212 = (C

  cos 2 (θ) + a 2 +

					L √ ab E s 1	]
	H 1122 = -	A	πab √ E s 1 E s 2
	H 1112 = H 2222 =	π(b 2 -a 2 )L 2A √ E s 1 πL A √ E s 2 [ (a 2 -b 2 ) cos 2 (θ) + b 2 + sin(θ) cos(θ)	L √ ab E s 2	]
	H 2212 =	π(b 2 -a 2 )L 2A √ 2 E s	sin(θ) cos(θ)

We recall that, in the present study, the analytical expressions of the components of P tensor were obtained in a tridimensional context.

These authors have not studied the case of arbitrarily oriented inclusions in an orthotropic solid matrix

A Proposed methodology for the determination of P tensor

The starting point here is the recent study of [START_REF] Suvorov | Rate form of the Eshelby and Hill tensors[END_REF] which follows the procedure described by [START_REF] Ting | Anisotropic elasticity: Theory and applications[END_REF] 3 . Let us consider the two global unit orthogonal vectors e 1 and e 2 in the plane ξ 3 = 0; any unit vector in this plane reads then ξ = cos ψe 1 + sin ψe 2 . It follows that the acoustic tensor K = ξ.C.ξ takes the form:

Substituting y = cot ψ in this expression yields:

The second order tensors Q, R, and T are defined, respectively as:


Sofia Diamantopoulou
email: s.diamantopoulou@erasmusmc.nl

Frank C Verhulst

Jan Van Der Ende

The parallel development of ODD and CD symptoms from early childhood to adolescence

Keywords: Longitudinal study, disruptive behavior problems, Child Behavior Checklist, path-analyses, gender differences

This study examined the developmental relations between symptoms of Oppositional Defiant Disorder (ODD) and Conduct Disorder (CD) from early childhood to adolescence. Specifically we tested, according to parent reported problems, whether symptoms of ODD precede the development of CD symptoms, whether ODD and CD symptoms are reciprocally associated across time, or whether ODD and CD symptoms develop parallel to each other across time. Participants were a community-based sample (at time 1: N = 485, 48 % boys) assessed biannually five times from age 4-6 until age 12-14. The findings suggested that, with control for stability effects, baseline SES, and symptoms of Attention Deficit Hyperactivity Disorder (ADHD), ODD and CD symptoms develop parallel to each other. No gender differences were obtained. We conclude that without the initial presence of CD symptoms, ODD symptoms are not developmental precursors to CD symptoms.

Evidence suggests that Oppositional Defiant Disorder (ODD) and Conduct Disorder (CD) are closely related in taxonomic and developmental terms [START_REF] Loeber | Oppositional defiant disorder and conduct disorder: A review of the past 10 years, part I[END_REF]. Although ODD and CD are considered as two independent disorders, the behaviors that define ODD have long been hypothesized to be developmental precursors to serious conduct problems [START_REF] Burke | Developmental transitions among affective and behavioral disorders in adolescent boys[END_REF][START_REF] Harvey | Predicting attentiondeficit/hyperactivity disorder and oppositional defiant disorder from preschool diagnostic assessments[END_REF][START_REF] Whittinger | Clinical precursors of adolescent conduct disorder in children with attention-deficit/hyperactivity disorder[END_REF]. However, not all children with ODD will necessarily progress to CD [START_REF] August | Persistence and desistance of oppositional defiant disorder in a community sample of children with ADHD[END_REF][START_REF] Biederman | The longterm longitudinal course of oppositional defiant disorder and conduct disorder in ADHD boys: findings from a controlled 10-year prospective longitudinal follow-up study[END_REF][START_REF] Costello | Prevalence and development of psychiatric disorders in childhood and adolescence[END_REF][START_REF] Lahey | Are oppositional-defiant and hyperactive-inattentive symptoms developmental precursors to conduct problems in late childhood?: Genetic and environmental links[END_REF][START_REF] Mannuzza | Significance of childhood conduct problems to later development of conduct disorder among children with ADHD: A prospective follow up study[END_REF][START_REF] Rowe | The relationship between DSM-IV oppositional defiant disorder and conduct disorder: Findings from the Great Smoky Mountains Study[END_REF][START_REF] Speltz | Preschool boys with oppositional defiant disorder: Clinical presentation and diagnostic change[END_REF] suggesting some discontinuity between the disorders. Determining whether ODD is a risk factor for, or develops secondarily to CD is of particular theoretical and clinical importance as it could suggest different etiological models and intervention efforts.

Table 1 provides an overview of prospective studies that examined the developmental relations between ODD and CD. As shown in the table approximately half of the studies suggest that ODD and CD may represent the same underlying pathology as most children or youth with CD previously met diagnostic criteria for ODD or displayed elevated levels of ODD symptoms [START_REF] Burke | Developmental transitions among affective and behavioral disorders in adolescent boys[END_REF][START_REF] Harvey | Predicting attentiondeficit/hyperactivity disorder and oppositional defiant disorder from preschool diagnostic assessments[END_REF][START_REF] Whittinger | Clinical precursors of adolescent conduct disorder in children with attention-deficit/hyperactivity disorder[END_REF]. In support of the above one study has even reported reciprocal temporal associations between the two constructs across time in terms of ODD symptoms preceding CD symptoms and vice versa CD symptoms preceding ODD symptoms [START_REF] Lahey | Waxing and waning in concert: Dynamic comorbidity of conduct disorder with other disruptive and emotional problems over 7 years among clinic-referred boys[END_REF].

Nevertheless, several studies in Table 1 report either no or weak prospective relations between childhood ODD and later developing CD [START_REF] August | Persistence and desistance of oppositional defiant disorder in a community sample of children with ADHD[END_REF][START_REF] Biederman | The longterm longitudinal course of oppositional defiant disorder and conduct disorder in ADHD boys: findings from a controlled 10-year prospective longitudinal follow-up study[END_REF][START_REF] Costello | Prevalence and development of psychiatric disorders in childhood and adolescence[END_REF][START_REF] Lahey | Are oppositional-defiant and hyperactive-inattentive symptoms developmental precursors to conduct problems in late childhood?: Genetic and environmental links[END_REF][START_REF] Mannuzza | Significance of childhood conduct problems to later development of conduct disorder among children with ADHD: A prospective follow up study[END_REF][START_REF] Rowe | The relationship between DSM-IV oppositional defiant disorder and conduct disorder: Findings from the Great Smoky Mountains Study[END_REF][START_REF] Speltz | Preschool boys with oppositional defiant disorder: Clinical presentation and diagnostic change[END_REF]. Several methodological differences between the studies may explain the discrepant findings.

Insert Table 1 about here First, significant temporal associations between the two constructs are more likely in clinical-than in community-based samples [START_REF] August | Persistence and desistance of oppositional defiant disorder in a community sample of children with ADHD[END_REF][START_REF] Costello | Prevalence and development of psychiatric disorders in childhood and adolescence[END_REF]. However, because clinical samples are typically subject to higher rates of comorbidity, studies of clinical samples may have exaggerated the degree of the temporal relations between ODD and CD [START_REF] Goodman | Representativeness of clinical samples of youths with mental disorders: A preliminary population-based study[END_REF]. Second, in the DSM [START_REF]Diagnostic and statistical manual of mental disorders[END_REF] CD and ODD are not allowed to be comorbid diagnoses. As a result, the use of clinical cut-offs in assigning ODD and CD diagnoses in studies may have boosted the prediction of CD from previous levels of ODD when subclinical levels of CD have not been taken into account, whereas the appearance of more "severe" manifestations of CD may have overshadowed the stability of ODD. Only two studies in Table 1 utilized a dimensional approach assessing the two constructs as a continuum to avoid over-or underestimating their temporal relations and reached strikingly different conclusions: ODD is (2) and ODD is not [START_REF] Mannuzza | Significance of childhood conduct problems to later development of conduct disorder among children with ADHD: A prospective follow up study[END_REF] that ODD is more likely to be a precursor of CD in early (i.e., up to the age of 7 years) than in late childhood [START_REF] Loeber | Oppositional defiant disorder and conduct disorder: A review of the past 10 years, part I[END_REF][START_REF] Rutter | Continuities and discontinuities in psychopathology between childhood and adult life[END_REF], age effects may explain the studies" discrepant findings. For instance, two studies in Table 1 that measured ODD in early childhood [START_REF] Harvey | Predicting attentiondeficit/hyperactivity disorder and oppositional defiant disorder from preschool diagnostic assessments[END_REF][START_REF] Lahey | Are oppositional-defiant and hyperactive-inattentive symptoms developmental precursors to conduct problems in late childhood?: Genetic and environmental links[END_REF] reported significant prospective associations between ODD and CD, whereas two studies that measured ODD after the age of six reported weak or no prospective associations between ODD and CD [START_REF] August | Persistence and desistance of oppositional defiant disorder in a community sample of children with ADHD[END_REF][START_REF] Whittinger | Clinical precursors of adolescent conduct disorder in children with attention-deficit/hyperactivity disorder[END_REF]. Finally, discrepancies between the findings in the literature may also be due to the use of different conceptualizations of ODD and CD including diagnoses according to older versions of the DSM [START_REF] Mannuzza | Significance of childhood conduct problems to later development of conduct disorder among children with ADHD: A prospective follow up study[END_REF][START_REF] Speltz | Preschool boys with oppositional defiant disorder: Clinical presentation and diagnostic change[END_REF], diagnoses based on parental questionnaires [START_REF] August | Persistence and desistance of oppositional defiant disorder in a community sample of children with ADHD[END_REF][START_REF] Lahey | Are oppositional-defiant and hyperactive-inattentive symptoms developmental precursors to conduct problems in late childhood?: Genetic and environmental links[END_REF] versus diagnostic interviews [START_REF] Lahey | Waxing and waning in concert: Dynamic comorbidity of conduct disorder with other disruptive and emotional problems over 7 years among clinic-referred boys[END_REF][START_REF] Burke | Developmental transitions among affective and behavioral disorders in adolescent boys[END_REF]. Clearly, more research on the temporal relations between ODD and CD assessing the two constructs dimensionally from early childhood to adolescence is needed.

Certain limitations in the literature need also to be overcome if we are to understand the relations between ODD and CD across time. First, because of the higher prevalence of ODD and CD in boys compared to girls [START_REF] Maughan | Conduct disorder and oppositional defiant disorder in a national sample: developmental epidemiology[END_REF][START_REF] Moffitt | Sex differences in antisocial behaviour: conduct disorder, delinquency, and violence in the Dunedin longitudinal study[END_REF], little is known about the development of the two constructs in girls.

Interestingly, the one study that examined gender differences in the prospective associations between ODD and CD showed that ODD appears to be a developmental precursor to CD only among boys [START_REF] Rowe | The relationship between DSM-IV oppositional defiant disorder and conduct disorder: Findings from the Great Smoky Mountains Study[END_REF]. Second, because symptoms of Attention Deficit Hyperactivity Disorder (ADHD) often co-occur with both ODD and CD and may even partly account for the developmental progression of ODD into CD [START_REF] Waschbush | A meta-analytic examination of comorbid hyperactive-impulsive-attention problems and conduct problems[END_REF] studies need to control for ADHD symptoms when examining the developmental relations between ODD and CD.

This study aimed to overcome the above limitations in the literature and disentangle the developmental relations between ODD and CD symptoms from early childhood to adolescence assessing the two constructs dimensionally in a community-based sample.

Method

Participants and procedure

This study is based on data derived from a seven-wave, multi-cohort, longitudinal study of behavioral and emotional problems in children that started in 1983 [START_REF] Verhulst | Mental health in Dutch children (I) A crosscultural comparison[END_REF]

Measures

ODD, CD, and ADHD symptoms. We obtained parent ratings of children"s symptoms of ODD, CD, and ADHD over the past six months on the Child Behavior Checklist (CBCL; 20, 21). Ratings were made on a three-point scale (0 = not true, 1 = somewhat or sometimes true, 2 = very true or often true) and we used the summed score of items for all the scales. The CBCL has been translated and validated for use in the Netherlands [START_REF] Verhulst | Handleiding voor de CBCL/4-18 (Manual for the CBCL/4-18)[END_REF]. CBCL items reflecting a similar content to DSM-IV criteria for ODD, CD, and ADHD were used in the analyses (i.e., the DSM-oriented scales; 23). Due to the use of the earlier versions of the CBCL (i.e., the 1983 version at times 1 to 4 and the 1991 version at time 5) that did not include the DSM-oriented scales that were developed in the later version (i.e., 2001) of the instrument, not all items of the scales were included (i.e., at all time points the items "fails to finish" and "inattentive" for the ADHD scale, and the item "breaks rules" for the CD scale were not included). The items and the internal consistency of the scales measured as Cronbach"s alpha are presented in Table 2.

Although the DSM-oriented scales of the CBCL have not been used in the literature as much as the empirically-derived syndrome scales of the instrument, there is some evidence for their psychometric properties. For instance, in a Dutch clinical sample (N = 44), the correlations (phi) between the DSMoriented scales of ADHD, ODD, and CD and the equivalent diagnoses obtained by the Diagnostic Interview Schedule for Children (DISC-IV) ranged from .43, to .67, p < .01 [START_REF] Krol | From CBCL to DSM: A comparison of two methods to screen for DSM-IV diagnoses using CBCL data[END_REF]. In a large multi-ethnic, clinical sample (N = 673) the internal consistency of the scales was high (Cronbach"s alpha ≥ .85), children and youths diagnosed with ADHD, ODD, or CD scored higher on the respective scales compared to those without a diagnosis, and finally the correlation between parent ratings on the three DSM-oriented scales correlated significantly high (rs ≥ .56) with parental dimensional ratings of hyperactivity/impulsivity and inattention symptoms and oppositional and delinquent behavior [START_REF] Nakamura | A Psychometric analysis of the child behavior checklist DSM-oriented scales[END_REF].

To confirm that the ODD and CD items form two separate factors at times 1 to 5, we conducted Confirmatory Factor Analyses (CFAs) using Mpuls version 5.0 [START_REF] Muthén | Mplus user's guide[END_REF]. To account for the ordinal item responses, we applied Weighted Least Squares Estimation (WLSE) analyses. To avoid statistical risks associated with low frequency cells we applied WLS analyses to tetrachoric correlations between item scores dichotomized as 0 vs. 1 or 2. We compared a one-factor model where all items were hypothesized to measure a single underlying general factor with a correlated 2-factor model representing the two DSMoriented scales of the CBCL for ODD and CD where each item was assigned to the syndrome it was supposed to measure. Model fit was assessed with the Comparative Fit Index (CFI; 27) that is based on the average size of the correlations in the data. Models with a CFI close to 1 are considered to be fitting the observed data well. Model selection was based on the differences in the two models" Root Mean Square Error of Approximation (RMSEA; 27) which is a measure of the empirical validity of the tested models (i.e., how well the models fit the observed data). According to guidelines [START_REF] Mccallum | Testing differences between nested covariance structure models: Power analysis and null hypotheses[END_REF], better fitting models are accepted when the change in the RMSEA (ΔRMSEA) is equal or greater than 0.01. When the fit of a more complex model does not differ from the fit of a more parsimonious model, then the more parsimonious model is accepted. Finally, we conducted chi-square difference tests to compare the fit of the one-and twofactor models. The results showed that the model fit of the one-and the two-factor model was good at all time points (one-factor model: .69 ≤ CFI ≤ .92; two-factor model: .75 ≤ CFI ≤ .95) except for Time 5 for the one-factor model (CFI = .69). However, model comparison showed that, at each time point, the twofactor model fitted the data better than a one-factor model (ΔRMSEA ≥ .01) confirming the measurement quality of the ODD and CD scales. Results of the chi-square difference tests showed that at all time points the two-factor model fitted the data better (i.e., the two-factor model had a lower 2) than a one-factor model: 2 (1) ≥ 10.28, p < .01. The correlation between the two latent factors in the two-factor CFA ranged from r = .36 p < .01 at Time 5 to r = .81, p < .01 at Time 2.

SES. Data on SES were obtained at Time 1 according to a six-step scale of parental education with ascending numbers indicating higher SES [START_REF] Van Westerlaak | Beroepenklapper (Manual for occupational level)[END_REF].

Insert Table 2 about here

Statistical analyses

We computed Pearson"s correlation coefficients to assess the relations between all study variables and examined gender differences in all variables with two-tailed t-tests. We examined the developmental relations between symptoms of ODD and CD using path modeling with Mplus [START_REF] Muthén | Mplus user's guide[END_REF]. Because the data departed from multivariate normality (see Table 2) we used full information maximum likelihood parameter estimates with standard errors that are robust to non-normality (i.e., MLR). Maximum likelihood estimation procedure is particularly appropriate for longitudinal studies that often contain missing data [START_REF] Collins | A comparison of inclusive and restrictive strategies in modern missing data procedures[END_REF]. Chisquare difference analyses guided decisions concerning selection of the best fitting model (i.e., the Satorra-Bentler scaled chi-square difference test which is appropriate for models estimated with MLR; 31). The goodness-of-fit of the models was evaluated by using two indicators: the Root Mean Square Error of Approximation (RMSEA) where values below .06 indicate a good fit [START_REF] Brown | Alternative ways of assessing model fit[END_REF] and the Comparative Fit Index (CFI) where values larger than .90 or close to .95 indicate a good fit [START_REF] Hu | Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives[END_REF]. Figure 1 depicts the three competing models that we tested. The first model assumed reciprocal temporal associations between ODD and CD symptoms across time and therefore included, apart from stability paths and cross-sectional correlations, cross-lagged paths between the two constructs (i.e., the cross-lagged model). The second model assumed that only ODD symptoms predict subsequent CD symptoms and therefore included apart from stability paths and cross-sectional correlations, directional paths only from previous ODD symptoms to subsequent CD symptoms (i.e., the unidirectional model). The third model assumed parallel but associated development of ODD and CD symptoms across time and included, apart from stability paths, only cross-sectional correlation paths between the two constructs (i.e., the correlational model). In all models we controlled for SES at Time 1 and for ADHD symptoms at all time points (depicted as correlational paths in Fig. 1). Included in the model but not depicted in Fig. 1 are covariances between the residuals of ADHD, ODD, and CD symptoms from Time t to Time t+1. The models were tested on the whole sample and separately for boys and girls.

Insert Figure 1 about here

Results

The correlation matrix, the means, and the standard deviations for the measured variables are shown in Table 3. As seen in the table, both ODD and CD symptoms were always correlated with their previous levels across time. The concurrent correlations between ODD and CD symptoms, between ODD and ADHD, and between CD and ADHD symptoms were always positive and significant. At Time 1, SES was negatively correlated with ODD, CD, and ADHD symptoms. Examination of gender differences showed that boys compared to girls had higher levels of ODD symptoms at the first two time points and they had higher levels of ADHD and CD symptoms at all time points (2.15 ≤ ts ≤ 4.17; p < .01).

Insert Table 3 about here

All three models provided an equally good fit to the data: CFI = .97, RMSEA = .04 for all three models. However, in the cross-lagged model none of the cross-lagged paths (i.e., paths a and b in Fig. 1) were significant whereas in the unidirectional model none of the paths to subsequent CD symptoms from previous ODD symptoms (i.e., paths a) were significant. The chi-square difference tests indicated that: a) the unidirectional model did not yield a deteriorated model fit compared to the cross-lagged model (Δχ² [START_REF] Whittinger | Clinical precursors of adolescent conduct disorder in children with attention-deficit/hyperactivity disorder[END_REF] = 2.66, ns) and, b) the correlational model did not yield a deteriorated model fit compared both to the crosslagged model (Δχ² [8] = 5.54, ns) and to the unidirectional model (Δχ² [4] = 2.89, ns). When we examined the three models separately for boys" and girls" data results were replicated (Boys: CFI = .96, RMSEA = .05 for the three models; Girls: CFI = .97, RMSEA= .04 for the three models; none of the cross-lagged or the unidirectional paths between ODD and CD symptoms were significant for boys" or for girls" data). Chisquare difference tests indicated that the correlational model did not yield a deteriorated model fit compared both to the cross-lagged and the unidirectional model for both boys" and girls" data (3.73 ≤ Δχs² ≤ 6.85, df=4-8, ns).

Because the most parsimonious, correlation model fitted the data equally well as the cross-laggedand the univariate model, we selected it as the final model. The standardized estimates for the correlational model are depicted in Fig. 2. Considering time intervals of two years for the five assessments, temporal stabilities for ODD, CD, and ADHD symptoms were high. The concurrent associations between ODD and CD symptoms were high across time and so were the associations between ADHD symptoms and the two constructs.

Insert Fig. 2 about here

To further examine the influence of ADHD symptoms on the temporal relations between ODD and CD symptoms, we examined the cross-lagged, unidirectional, and correlational models without control for ADHD symptoms. These three models also fitted the data well: CFI = .99 for all three models and RMSEA = .02 for the cross-lagged and the unidirectional models and RMSEA= .03 for the correlational model. Comparison of the models indicated that although the model fit of the cross-lagged versus the unidirectional model did not differ significantly (Δχ² [4] = 6.09, ns), the cross-lagged and the unidirectional models fitted the data better than the correlational model (Δχ² [START_REF] Costello | Prevalence and development of psychiatric disorders in childhood and adolescence[END_REF] = 19.21, p < .05 and Δχ² [4] = 14.22, p < .05 respectively). However, none of the cross-lagged or unidirectional paths reached statistical significance indicating that even without control for ADHD symptoms ODD symptoms did not significantly predict subsequent CD symptoms and ODD and CD symptoms were not reciprocally associated across time.

Results of analyses conducted separately for boys and girls replicated the above findings for boys but for girls the three models fitted the data equally well.

Discussion

This study examined whether ODD symptoms should be viewed as precursors of later developing CD, whether ODD and CD are likely to show reciprocal temporal associations across time, or finally, whether ODD and CD develop parallel to each other. In line with the findings of previous studies of community-based samples, no temporal associations between ODD and CD symptoms were observed [START_REF] August | Persistence and desistance of oppositional defiant disorder in a community sample of children with ADHD[END_REF][START_REF] Costello | Prevalence and development of psychiatric disorders in childhood and adolescence[END_REF]. Instead, the strongest predictor of subsequent CD symptoms was initial CD symptoms. What this study adds to the literature is that we confirm these findings for a lengthy follow-up period spanning from early childhood to adolescence, for both genders, using multiple assessments.

Keeping in mind that causal interpretations of results of longitudinal path analyses should be avoided and results should be interpreted with caution, the present findings do not suggest that ODD symptoms are a milder, earlier manifestation of CD. Neither do the results suggest that ODD symptoms may develop secondarily to CD symptoms, as none of the paths between ODD and CD symptom levels in the unidirectional and the cross-lagged models were significant. The results are in line with the study by Lahey and colleagues (9) that also used CBCL items to predict conduct problems from ADHD and ODD symptoms in a community-based sample. It should however be noted that the high temporal stability of both ODD and CD symptoms renders reciprocal associations between the two constructs less probable insofar as the greater the stability of a construct, the less the chance of finding other variables that potentially explain its variance [START_REF] Eccles | Studying the development of learning and task motivation[END_REF].

In contrast to the findings by Rowe et al [START_REF] Rowe | The relationship between DSM-IV oppositional defiant disorder and conduct disorder: Findings from the Great Smoky Mountains Study[END_REF], which are also based on a community-based sample, the temporal relations between ODD and CD symptoms did not differ by gender in this study. One possible reason for this is that whereas Rowe et al [START_REF] Rowe | The relationship between DSM-IV oppositional defiant disorder and conduct disorder: Findings from the Great Smoky Mountains Study[END_REF] based their results on analyses utilizing clinical cut-offs we examined the temporal relations between ODD and CD dimensionally in a sample that overall did not display very high symptom levels. Consequently, the findings of this study indicate that as long as negativistic and defiant behaviors in childhood are mild to moderate and do not reach clinical levels, boys and girls are equally likely to either outgrow these problems or not develop more serious conduct problems.

Future studies need to examine whether the above findings hold even after control for other comorbid psychopathology such as depression and anxiety that may have different relations with ODD and CD symptoms in boys and girls. Some previous studies suggest that children with ODD and ADHD are more likely to develop CD than children with ODD only [START_REF] Waschbush | A meta-analytic examination of comorbid hyperactive-impulsive-attention problems and conduct problems[END_REF]. Consequently, removal of ADHD symptoms from the measurement model should render the unidirectional model as better fitting than the correlational model. Nevertheless, results did not suggest such an effect although the concurrent associations between ADHD symptoms and both ODD and CD symptoms were significantly positive across all time points. Hence, it appears that in community-based samples the impact of ADHD symptoms on the temporal relations between ODD and CD is minimal. This study is not without limitations. First, like most previous longitudinal studies that examined the prospective relations between oppositional-, inattentive/hyperactive behaviors and later conduct problems the present analyses were not based on measures that were a perfect match to the DSM-IV symptom-criteria. A recent, large (N = 476), multi-ethnic, clinical study of children and adolescents confirmed the construct and discriminant validity of the DSM-oriented scales showing that the scales corresponded well (area under the curve values ≥ .71) with related clinical diagnoses (34). However, it should be noted that in this latter study, the DSM-oriented scales of the CBCL for ODD, CD, and ADHD did not show greater correspondence with corresponding clinical diagnoses than the CBCL empiricallyderived syndrome scales of Aggressive behavior, Rule-breaking behavior, and Attention problems.

Moreover, the present findings are quite consistent with the smaller set of studies that did use DSM-based measures in terms of the concurrent associations between ODD, CD, and ADHD (e.g., [START_REF] Lahey | Waxing and waning in concert: Dynamic comorbidity of conduct disorder with other disruptive and emotional problems over 7 years among clinic-referred boys[END_REF][START_REF] Biederman | The longterm longitudinal course of oppositional defiant disorder and conduct disorder in ADHD boys: findings from a controlled 10-year prospective longitudinal follow-up study[END_REF] suggesting that the findings of this study are relevant to DSM conceptualizations of these disorders.

Nevertheless, because different results may have been obtained had we used DSM-based measures, future longitudinal studies using such measures would be very useful. Second, because of the use of parent ratings, results may not generalize to studies using teacher or self-ratings of ODD and CD symptoms.

Finally, the reliability of the CD scale at the first two time points was relatively low. Inspection of the results of the CFAs we conducted showed that this was primarily caused by low loadings of two CD scale items namely, "Truancy" and "Threatens people." The results are not unexpected given that at the first two time points of the study participants were very young and therefore highly unlikely to attempt to be absent from school or have fully developed the language and cognitive skills in order to seriously threaten others.

One of the strengths of this study is that the multiple assessments allowed for the examination of competing models in terms of their ability to explain the dynamic, temporal linkage between ODD and CD symptoms. In conclusion, the findings suggest that the best predictor of subsequent CD symptoms is initial CD symptoms. However, ODD symptoms appear to carry unique information not captured by CD symptoms and given also their stability across childhood and early adolescence they deserve to receive attention beyond the context of being precursors to other disruptive behavior disorders. A clinical implication of the findings is that ODD and CD do not represent the same underlying pathology and possibly require different treatment.

34.

Ebesutani

  a precursor of CD. Perhaps the discrepant findings of the two above studies are due to the use of different ODD measures (i.e., DSM-III-R ODD diagnosis based on clinic interviews in the study by Burke et al; retrospective DSM-IV ODD diagnosis based on parent and teacher questionnaires in the study by Mannuzza et al). Third, as two literature reviews suggest

21 Figure 1 .Figure 2 .

 2112 Figure 1. Illustration of the three tested models Note: All three models include the same directional and correlation (solid) paths but, of the dotted paths, the cross-lagged model includes paths a, b, and c, the unidirectional model includes only paths b and c, and the correlational model includes only paths c.

  . The original sample of 2,600 children from 13 birth cohorts aged 4 to 16 was drawn from municipal registers that list all residents in the Dutch province of Zuid-Holland. A random sample of 100 children of each gender and age of Dutch nationality was drawn. Of the 2,447 parents reached, 2,076 (i.e., 84.8%) provided usable data. The sample in 1983 consisted of 1,106 boys and 1,060 girls. This sample was assessed biannually five times (until 1991;

	Time 5), once six years later (1997; Time 6) and then again in 2007 (Time 7). The present study is based on
	data from Time 1 to Time 5. As we aimed to examine the temporal relations between symptoms of ODD
	and CD from early childhood to adolescence, we selected an age-appropriate subsample of children who
	were aged 4-6 years at baseline. The number and age of the participants at each time point was: 485 (231
	boys) age 4 -6 years at Time 1, 373 (184 boys) age 6 -8 years at Time 2, 326 (167 boys) age 8 -10 years at
	Time 3, 311 (164 boys) age 10 -12 years at Time 4, and 305 (160 boys) age 12 -14 years at Time 5. We
	obtained written informed consent by all participants after the procedures were fully explained and the local
	ethical committee approved the data collection. The ethnic composition of the sample was 97% Dutch
	(remaining 3% came from Surinam, the Dutch Antilles, and Morocco). Participants who remained in the
	study at Time 5 did not differ significantly in any study variables from those who had dropped out (-0.16 ≤
	ts ≤ 1.82, ns).

  C, Bernstein A, Nakmura BJ, Chorpita BF, Higa-McMillan CK, & Weisz JR. (2010) Concurrent validity of the child behavior checklist DSM-oriented scales: Correspondence with DSM diagnoses and comparison to syndrome scales. J PSYCHOPATH BEHAV ASSESS 32:373-All correlations significant at p < .05 level or lower except for correlations marked with ª that were not significant.

	Lahey et al., 2009 Table 2 Table 3 15 ADHD T5	Community based; 6,994 .28 .18		4-7 .32	.42		8-13 .38	.26	Parent ratings (CBCL .29 .35 .36	Log linear regres
		children CBCL /4-18 items included in the ADHD, ODD, and CD symptom scales Correlation matrix and means (M) and standard deviations (SD) for observed variables items) .41 .46 .60 .69 .79 1.00	ODD and ADHD
											symptoms predic
		___________________________________________________________________________ ______________________________________________________________________________________ Table 3 continued	conduct problem
		384. ADHD (.70 ≤ α ≤ .75) _______________________________ ______________________________________________________________________________________ ODD (.70 ≤ α ≤ .74)
	Table 1 Mannuzza et al., Can"t concentrate _______________________________ Clinical sample; 207 1 2	3	6-12	4	5	18 Argues a lot 6	7	Parent and teacher 8 9	LRA predicting a
	2004			boys with DSM-II						ratings at baseline;	CD diagnosis
	Overview of prospective studies examining the temporal relations between ODD and CD hyperkinetic reaction Can"t sit still Disobedient at school diagnostic interviews 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9
	________________________________________________________________________________________________________ at follow-up Impulsive or acts without thinking Disobedient at home ______________________________________________________________________________________ 10 11 12 13 14 15 16
	______________________________________________________ Rowe et al., 2002 Community sample; 9-13 Talks too much _______________________________ ______________________________________________________________________________________ 4 annual Diagnostic interviews Stubborn	LRA predicting O
				1,420 children						assessments	CD at waves 2-4
	Reference Loud _______________________________ Sample 1 ODD T1 1.00		Baseline age		Follow-up age Hot temper	Diagnostic instrument	Data analyses wave 1
		2 ODD T2 16 SES T1		.54 -.13	1.00 -.13	(years) -.14 -.20	(years) -.15 -.13	-.15	-.11	-.12
	________________________________________________________________________________________________________
	Speltz et al., 1999 ______________________________________________________ Clinical sample of 79 boys 4-5.5 CD (.55 ≤ α ≤ .80) 3 ODD T3 .46 .57 1.00 -.19 -.18 -.16 -.24 -.20 -.22	6-7.5 1.00	Diagnostic interviews	Presence or abse
	August et al., 1999 Cruel to animals 4 ODD T4 M	Community-based; 308 with DSM-III-R ODD Runs away .44 .55 2.26 1.98	6-10 .59 1.84	1.00 2.10	11-15 2.07 1.06	Parent and teacher ratings (DSM-III-R diagnoses) 0.98 0.76 0.89	Presence or abse CD diagnosis at f
		Mean to others 5 ODD T5 0.83 2.95	children with disruptive Sets fires .39 .44 2.58 2.55 2.43	.51 2.27	.64 3.47	1.00	at baseline; diagnostic	ODD, or CD diag
	behavior problems Clinical sample of 151 Destroys other"s things Whittinger et al., Steals at home 6 CD T1 .48 .38 SD 1.96 1.84 1.89 6-13 .34	.35 1.89	11-18 .27 1.93 1.67 1.00	interview at follow-up Diagnostic interviews 1.62 1.23 1.62	follow-up (χ²) LRA predicting A
	Biederman et al., 2007		140 children with DSM-III-R children with DSM-IV ADHD	6-17			16-27 at	Diagnostic interviews (DSM-IV diagnoses)	LRA predicting p and CD diagnose
	2008	Lacks guilt 7 CD T2 1.69	ADHD diagnosis and 120 Steals outside home .36 .59 .35 .39 2.37 2.32 2.33 2.34 2.34	follow-up .35 .45 1.00 1.57	absence of CD di
		controls ______________________________________________________________________________________ Fighting Swears 8 CD T3 .29 .42 .50 .48 .32 .39 .50 1.00	follow-up
	Burke et al., 2005 Note: LRA = Logistic regression analysis; GEE = Generalized estimating equations; CBCL = Child Behavior Clinic-referred; 177 boys with 7-12 Annual Diagnostic interviews Lying, cheating Threatens people 9 CD T4 .21 .31 .33 .57 .49 .40 .47 .58 1.00 _______________________________	GEE predicting s
	Checklist		disruptive behavior disorders					assessments	in wave T + 1 fro
		Attacks people 10 CD T5 Note:		Truancy .20 .30		.31	.48		until age 18 .56 .29	.47	.47	.71
	Costello et al., 2003 Bad companions 1.00	Community-based; 1420 Vandalism		9-13			Annual	Diagnostic interviews	LRA predicting O
		children ___________________________________________________________________________ assessments (DSM-IV) 11 ADHD T1 .53 .37 .37 .34 .18 .40 .29 .26	.14	diagnosis at each
											until age 16	from previous tim
		.09ª	1.00
	Harvey et al., 12. ADHD T2	168 children with .36	.54		3 .36	.39		6 .22	.38	Diagnostic interviews .53 .40 .30	LRA predicting O
	2009	.28	.60	behavior problems at 1.00						diagnosis at follo
				baseline							from baseline sym
		13. ADHD T3		.28	.31		.47	.36		.20	.22	.30	.44	.22
		.15	.57	.68	1.00
	Lahey et al., 2002 14. ADHD T4	As in Burke et al (2005) above 7-12 .30 .25 .35	.48		Annual .29 .33	Diagnostic interviews .35 .43 .41	Modeled mean n
											assessments	(DSM-III, DSM-III-	ODD and CD sy
		.36	.50	.68	.75	1.00					over 7 years	R, & DSM-IV)	longitudinally in
											20

Acknowledgements: Support for the first author was provided by The Swedish Council for Working Life and Social Research.