S Baste/ Composites Science and Technology 61(2001)2285-2297 crack opening displacements of the transverse crack x2 system,U(Fig. 6)[12]. The distance between two crack a++≤, is related to the crack density. So, in an extensometer length L, the number n of cracks, 2a deep and 2b thick, is: with stiffness Cr and compliance Sr, which is embedded in an infinite homogeneous solid whose stiffness and n=L (3) compliance tensors are, respectively, Cand S.The material is loaded by uniform stress, o, or subjected to uniform strain, a, at infinity. Let the stress and strain The relationship between the inelastic strain and the fields in the inclusion be o and Er, respectively, so that crack opening displacement is then [15]: σr=CrEr,Er=S10r It is well known that the elastic field in the ellipsoidal inclusion is uniform [17] and can be evaluated as Thus, the inelastic strain is simply the density of transverse matrix microcracks multiplied by their aspect σr=B10,Er=AE The extension of the cracks is limited by the waviness Ar and Br are the crack localisation tensors of the r of the bundles and experimental observation of inelastic inclusion strains implies that crack opening displacement is not negligible. Therefore, it is necessary ider 3D Br=[+Q(S-S),A1=[-P(Cr-C)-.(8) defined cracks in order to evaluate the effective stiffness tensor of the damaged material [16]. The cracks are The solution of this problem requires the determina modelling by ellipsoidal voids. Their volume concentration tion of the tensor Q defined by: is defined through a unit cell; it represents the largest volume of material containing a single crack 0=C-CPC By using a homogenisation method that provides the elationship between the effective stiffness tensor andand the determination of the tensor P whose compo- the intensity of damage in the individual modes, it is nents are given by [18] possible to relate the micro- and macro-level damage measurement. The cracked material is substituted by an Pal- 4 Jo(a-a?+c202+bag>d2 abc Dik(o (10) equivalent homogeneous medium. Effects of damage are then described by the changes of the effective properties of the equivalent medium. where $2 is the surface of the unit sphere centred at the Cracks are consider as an ellipsoidal inclusion origin of(ol, (2, a3)space. The fourth order tensor D is defined by Dijk=o)@)gik with gik=[Cmnpa@n@g Jik (11) Turning now to the basic equations for composites, we note that in order for the concept of overall moduli Z2a02ab-=u Displacements, to be meaningful, it is necessary to consider macro- scopically uniform loading [19]. In such a case, the lal to the phase average stresses and strains are related to the o= bro and E=a e 2a/阝 Let c. denote the volume concentration of the rth ∑ Fig. 6. Inelastic strains; a macroscopic consequence of the micro. it follows[20] that the overall stifness C and compliancecrack opening displacements of the transverse crack system, U (Fig. 6) [12]. The distance between two cracks is related to the crack density. So, in an extensometer length L, the number n of cracks, 2a deep and 2b thick, is: n ¼ L  2a : ð3Þ The relationship between the inelastic strain and the crack opening displacement is then [15]: "in ¼ Lin L ¼ n2U L ¼ n2b L ¼ : ð4Þ Thus, the inelastic strain is simply the density of transverse matrix microcracks multiplied by their aspect ratio =b/c. The extension of the cracks is limited by the waviness of the bundles and experimental observation of inelastic strains implies that crack opening displacement is not negligible. Therefore, it is necessary to consider 3D￾defined cracks in order to evaluate the effective stiffness tensor of the damaged material [16]. The cracks are modelling by ellipsoidal voids. Their volume concentration is defined through a unit cell; it represents the largest volume of material containing a single crack. By using a homogenisation method that provides the relationship between the effective stiffness tensor and the intensity of damage in the individual modes, it is possible to relate the micro- and macro-level damage measurement. The cracked material is substituted by an equivalent homogeneous medium. Effects of damage are then described by the changes of the effective properties of the equivalent medium. Cracks are consider as an ellipsoidal inclusion: x2 1 a2 þ x2 2 c2 þ x2 3 b2 41; ð5Þ with stiffness Cr and compliance Sr, which is embedded in an infinite homogeneous solid whose stiffness and compliance tensors are, respectively, C and S. The material is loaded by uniform stress, ; or subjected to uniform strain, "; at infinity. Let the stress and strain fields in the inclusion be r and r, respectively, so that: r ¼ Cr"r; "r ¼ Srr: ð6Þ It is well known that the elastic field in the ellipsoidal inclusion is uniform [17] and can be evaluated as r ¼ Br ; "r ¼ Ar" ð7Þ Ar and Br are the crack localisation tensors of the r inclusion: Br ¼ ½ I þ Q Sð Þ r S 1 ; Ar ¼ ½ I P Cð Þ r C 1 : ð8Þ The solution of this problem requires the determina￾tion of the tensor Q defined by: Q ¼ C CPC ð9Þ and the determination of the tensor P whose compo￾nents are given by [18]: Pijkl ¼ abc 4 ð O Dijklð Þ !n a2!2 1 þ c2!2 2 þ b2!2 3
3=2 d ð10Þ where  is the surface of the unit sphere centred at the origin of (!1, !2, !3) space. The fourth order tensor D is defined by: Dijkl ¼ !l!jgik with gik ¼ Cmnpq!n!q  1 ik : ð11Þ Turning now to the basic equations for composites, we note that in order for the concept of overall moduli to be meaningful, it is necessary to consider macro￾scopically uniform loading [19]. In such a case, the applied stress is equal to the average stress, , and the phase average stresses and strains are related to the overall averages through r ¼ Br and "r ¼ Ar": ð12Þ Let cr denote the volume concentration of the rth phase. Since X r cr ¼ 1;  ¼ X r crr; " ¼ X r cr"r; ð13Þ it follows [20] that the overall stiffness C and compliance S are given by: Fig. 6. Inelastic strains; a macroscopic consequence of the micro￾scopic crack opening displacement. 2288 S. Baste / Composites Science and Technology 61 (2001) 2285–2297