Structural interfaces in linear elasticity. Part II: Effective properties and neutrality

Transcription

1 Journal of the Mechanics and Physics of Solids 55 (007) Structural interfaces in linear elasticity. Part II: Effective properties and neutrality. Bertoldi, D. Bigoni, W.J. Drugan 1 Dipartimento di Ingegneria Meccanica e Strutturale, Università di Trento, Via Mesiano 77, I Trento, Italy Received 9 February 006; received in revised form 13 June 006; accepted 15 June 006 Abstract The model of structural interfaces developed in Part I of this paper allows us to analytically attack and solve different problems of stress concentration and composites. In particular, (i) new formulae are given for effective properties of composite materials containing dilute suspensions of (randomly oriented) reinforced elliptical voids or inclusions; (ii) a new definition is proposed for inclusion neutrality (to account for the fact that the matrix is always overstressed, and thus non-neutral in a classical sense, at the contacts with the interfacial structure), which is shown to provide interesting stress optimality conditions. More generally, it is shown that the incorporation of an interfacial structure at the contact between two elastic solids exhibits properties that cannot be obtained using the more conventional approach of the zero-thickness, linear interface. For instance: contrary to the zero-thickness interface, both bulk and shear effective moduli can be optimized for a structural interface; effective properties higher that those possible with a perfect interface can be attained with a structural interface; and neutrality holds with a structural interface for a substantially broader range of parameters than for a zero-thickness interface. r 006 Elsevier Ltd. All rights reserved. eywords: Structural interfaces; Composite materials; Neutral inclusions; Reinforced cracks; Stress intensity factors Corresponding author. Tel.: ; fax: addresses: katia.bertoldi@ing.unitn.it (. Bertoldi), davide.bigoni@ing.unitn.it (D. Bigoni), drugan@engr.wisc.edu (W.J. Drugan). URL: 1 Permanent address: Department of Engineering Physics, 1500 Engineering Drive, University of Wisconsin- Madison, Madison, WI, 53706, USA /$ - see front matter r 006 Elsevier Ltd. All rights reserved. doi: /j.jmps

2 36. Bertoldi et al. / J. Mech. Phys. Solids 55 (007) Introduction The mechanical properties of the interphase, which separates inclusions from the matrix in composite materials, have been demonstrated to play an important role in the overall behavior of the composite. Often the interphase is modelled as a surface separating two materials, across which displacements may be discontinuous, but this discontinuity is (linearly or nonlinearly) related to tractions. A more realistic model which takes account of the actual interfacial microstructure is obtained through the concept of a structural interface, developed in Part I of this paper. The focus of the present Part II is to investigate the role of this microstructure on the mechanical properties of composite (fiber- and particulate-reinforced) materials, always under the assumptions of two-dimensional linear elasticity. These can be quantified through investigation of: (i) stiffness, and (ii) strength. In particular, i) Stiffness is quantified (in Section ) by finding the effective properties of a material containing a dilute suspension of randomly oriented elliptical inclusions connected to the matrix via a structural interface. It is shown that there are geometries of the interfacial structure yielding optimal properties in terms of shear and bulk stiffness. We determine for the first time the effective properties of a dilute suspension of randomly oriented fiber-reinforced elliptical voids, showing the strong effect of the fibers. In addition, we show that the effective properties tend to those of the matrix for a certain distribution of inclined fibers. In this sense, a reinforced void can be designed that leaves the stiffness of the matrix unchanged. ii) Strength is addressed (in Section 3) in the following way. A neutral inclusion, i.e. an inclusion which does not perturb the ambient stress/strain fields, is considered optimal from the point of view of strength, since it eliminates stress concentrations. It is shown that a circular inclusion coated by a continuous structural interface is neutral under both shear and equal biaxial tension, for a far broader material parameter range than for the linear interface analyzed by Ru (1998). In addition, the notion of weak neutrality, based on energetic arguments and more general than the concept of neutrality usually employed (e.g. Ru, 1998; Milton and Serkov, 001), is introduced. This is useful for analyzing discrete interfacial structures and reveals that interfacial structures can be tailored to produce: (a) weakly neutral elliptical inclusions coated by a discrete structural interface under remote shear but not under equal biaxial tension; (b) weakly neutral circular inclusions coated by a discrete structural interface under both shear and equal biaxial tension separately. In particular, to quote some among many contributions, Walpole (1978) showed that a thin coating on an inclusion strongly influences the stress fields; Benveniste (1985), Achenbach and Zhu (1989, 1990), Hashin (1990, 1991a, b) obtained expressions for the effective moduli; Levy (1996) introduced a nonlinear separation mechanism in the problem of effective transverse bulk response of a composite; Hashin (00) derived by a Taylor expansion method imperfect interface conditions which are equivalent to the effect of a thin elastic interface.

3 . The effective properties of a dilute composite containing inclusions with structural interfaces It is shown in this section that analytical expressions can be found for the effective elastic properties of a dilute suspension of randomly oriented reinforced elliptical voids and elliptical inclusions, connected to an elastic matrix by structural interfaces. The effective properties of the composite are shown to depend strongly on the geometry and stiffness of the interfacial structure, a finding allowing design of structural interfaces to obtain extreme properties of composites. We refer to the volume averaged stress r and strain ē r ¼ 1 Z r dv ¼ 1 Z xþ ds; e ¼ V V V qsðrn 1 Z e dv ¼ 1 Z ðu nþ V V V Sym ds, (1) qs (in which the suffix Sym denotes the symmetric part of a tensor) so that the effective elastic Lame constants l, m and two-dimensional bulk modulus are introduced as r ¼ l ðtr eþi þ m e ¼ ðtr eþi þ m dev e, () where dev and tr are the operators taking the deviatoric part and the trace of a tensor and the elastic constants appearing in Eq. () are related as ¼ m þ l ¼. Bertoldi et al. / J. Mech. Phys. Solids 55 (007) m k 1. (3) This section is organized as follows. First, the effective moduli of a dilute suspension of circular inclusions coated with a continuous double Warren structural interface (of the type analyzed in Part I, Section 4 of this paper see Fig. 1) are obtained as simple analytical expressions. Second, discrete interfacial structures are analyzed, reinforcing randomly oriented elliptical voids and inclusions. Fig. 1. Circular inclusion of radius R coated with a continuous double Warren structure of thickness d.

4 38. Bertoldi et al. / J. Mech. Phys. Solids 55 (007) Dilute suspension of circular inclusions connected to the matrix by a continuous structural interface Let us consider a dilute suspension of circular inclusions of radius R connected to an infinite matrix by a continuous double Warren truss structure of thickness d (Fig. 1). When a remote uniaxial stress s 1 11 is applied to the matrix, substitution of the complex potentials solution (given by Eq. (49) in Part I of this article) into Eqs. (1) and neglecting terms of Oðf Þ yields s xx ¼ s 1 11 þ F m B m ðr þ dþ s1 11 f, s yy ¼ F m þ B m ðr þ dþ s1 11 f ; s xy ¼ 0, xx ¼ ð1 þ kþ Þ s m þ yy ¼ ðkþ 3Þs m þ þ B mk þ F m ðr þ dþ m þ s1 11 f, B mk þ þ F m ðr þ dþ m þ s1 11 f ; xy ¼ 0, ð4þ where f ¼ðR þ dþ =r is the inclusion volume fraction, and the coefficients B m, F m are given by Eqs. (51) in Part I of this article. Introduction of (4) into Eq. () yields m m þ ¼ 1 B mðk þ þ 1Þ ðr þ dþ f ; þ ¼ 1 þ F mðk þ þ 1Þ ðr þ dþ ðk þ 1Þ f. (5) In the limit case of a zero-thickness linear interface (obtained by setting g ¼ c d and taking the limit as d! 0), the coefficients B m and F m are given in Eqs. (54) in Part I of this article, so that the effective properties of the composite are the same as those given by Thorpe and Jasiuk (199) and Bigoni et al. (1998). The effective moduli for an inclusion volume fraction of 0.1 are plotted in Figs. 4 as a function of the fiber inclination a. Matrix and inclusion are characterized by m =m þ ¼ 10 in Fig. and m =m þ ¼ 0:1 infig. 3, while both cases are considered in Fig. 4 (in all cases n ¼ n þ ¼ 1=3). Different values of the dimensionless fiber compliance parameter L c ¼ m þ =½ðk þ þ 1ÞkRŠ (see Eq. (65) in Part I of this article) have been considered and geometries defined by d=r ¼f0; 0:5; 0:5; 1g. Note that for the limiting case of a linear zero-thickness interface, when a ¼ 0 the tractions at the interface are given by s rr ¼ k1u r U; s ry ¼ 0, (6) while when a! p= they become s rr ¼ 0; s ry ¼ k1u y U. (7) In the case of a perfect interface, the averaged properties reduce to the well-known results (see, for instance, Willis, 198) m m þ ¼ 1 þ f ð1 þ kþ Þðm m þ Þ k þ m þ m þ ; þ ¼ 1 þ f ð1 þ kþ Þ½m ðk þ 1Þ m þ ðk 1ÞŠ ðk þ 1Þðm þ m þ ðk, 1ÞÞ (8)

5 . Bertoldi et al. / J. Mech. Phys. Solids 55 (007) Λ c =5 0.9 Λ c =5 μ μ δ / R= linear zero-thickness δ / R= linear zero-thickness μ μ δ / R= linear zero-thickness + Λ c =1 Λ 0.6 c =1 linear zero-thickness δ / R= Fig.. Effective shear modulus m (left) and D bulk modulus (right) (normalized by m þ and þ, respectively) of a dilute suspension (f ¼ 0:1) of circular elastic inclusions of radius R connected to the matrix by a continuous double Warren truss structure with thickness d and fibers inclination a. The inclusions are stiffer than the matrix (m =m þ ¼ 10) and n þ ¼ n ¼ 1 3, whereas the interfacial compliance is L c ¼ 5 (upper part) and L c ¼ 1 (lower part) Λ c =5 0.9 Λ c =5 μ μ δ / R= δ / R= linear zero-thickness Λ c =1 0.9 Λ c =1 δ / R=1.00 μ μ linear zero-thickness δ / R= linear zero-thickness Fig. 3. Effective shear modulus m (left) and D bulk modulus (right) (normalized by m þ and þ, respectively) of a dilute suspension (f ¼ 0:1) of circular elastic inclusions of radius R connected to the matrix by a continuous double Warren truss structure with thickness d and bars inclination a. The matrix is stiffer than the inclusions (m =m þ ¼ 0:1) and n þ ¼ n ¼ 1 3, whereas the interfacial compliance is L c ¼ 5 (upper part) and L c ¼ 1 (lower part).

6 40. Bertoldi et al. / J. Mech. Phys. Solids 55 (007) μ μ μ - / μ + =10 Λ c =0 0.5 rigid inclusions δ / R= μ - / μ + =10 Λ c =0 rigid inclusions δ / R= perfect interface perfect interface rigid inclusions 1.1 rigid inclusions μ μ μ - / μ + =0.1 1 μ - / μ + = Λ c =0 + Λ δ / R= c =0 δ / R= perfect interface perfect interface Fig. 4. Effective shear modulus m (left) and D bulk modulus (right) (normalized by m þ and þ, respectively) of a dilute suspension (f ¼ 0:1) of circular elastic inclusions of radius R connected to the matrix by a continuous double Warren truss structure with thickness d and bars inclination a. Both inclusions stiffer than the matrix (m =m þ ¼ 10, upper part) and inclusions less stiff than the matrix (m =m þ ¼ 0:1, lower part) are considered, whereas n þ ¼ n ¼ 1 3 and the bars are rigid (L c ¼ 0). Two perfect interface results are also plotted: one for the same m =m þ ratio as for the structural interface cases, and one for rigid inclusions. so that we get m=m þ ¼ 1:135, = þ ¼ 1:116 for m =m þ ¼ 10 and m=m þ ¼ 0:794, = þ ¼ 0:73 for m =m þ ¼ 0:1, values shown in Fig. 4, where bars having an infinite stiffness (i.e. L c ¼ 0) are considered. We conclude from Figs. 4 that While it is obvious that increasing (decreasing) the bars stiffness, the averaged bulk and shear moduli of the composite increase (decrease), it may be surprising to note that for inclusions less stiff than the matrix, the averaged moduli may be stiffer than the matrix moduli, if the interfacial structure is stiff enough (Fig. 4). This effect cannot be obtained with a linear, zero-thickness interface, which always decreases the stiffness of the composite material. As a function of the bars inclination a, the averaged bulk modulus has two stationary points, at a ¼ 0 and a ¼ p=. Moreover, a ¼ 0 corresponds to a maximum for if L c 4 dmþ ðr þ dþðk 1Þ m R. (9) ðk þ þ 1Þ For a zero-thickness linear interface, Eq. (9) yields the condition a ¼ 0 (corresponding to null tangential stiffness k y ¼ 0) for the maximum averaged bulk modulus, while the averaged shear modulus attains a maximum for a ¼ p=4 (corresponding to equal tangential and radial stiffnesses k r ¼ k y ¼ k). Therefore, for a zero-thickness linear interface it is not possible to maximize both the bulk and the shear moduli of the composite

7 . Bertoldi et al. / J. Mech. Phys. Solids 55 (007) material. It is clear from Figs. 3 4 that this is not true if the inclusions are connected to the matrix through a structural interface: in fact, a structural interface with bars characterized for instance by L c ¼ 1 and inclination a ¼ p= allows us to maximize both the bulk and the shear moduli of the composite material (Fig. 5)... Dilute suspension of randomly oriented elliptical voids and inclusions reinforced by discrete structures A dilute suspension of randomly oriented elliptical voids reinforced by an arbitrary distribution of bars or reinforced by a structural interface enclosing an elliptical inclusion, so that there are N junction regions on the boundary of each void, is now considered. The averaged stress and strain can be determined by using Eqs. (1), with reference to a circular contour of radius r, centered at the center of the elliptical void. To this purpose, we need the leading-order terms of the displacements and tractions, calculated on the large circle having rbr. These fields can be obtained by using the resultants P k and S k (in the radial and tangential directions, respectively), applied at the center of the kth junction region. We will determine now (employing the complex potentials given by Eq. (3) in Part I of this paper) the leading-order large-r terms of the displacement and stress fields, assuming that the void s boundary is loaded by N concentrated forces of tangential and normal components S k and P k, respectively. Fig. 5. Geometry of the interfacial structure permitting maximization of both the bulk and the shear moduli of the composite. Discrete (upper parts) and smeared (lower parts) structures.

8 4. Bertoldi et al. / J. Mech. Phys. Solids 55 (007) First we note that, since from the conformal mapping z ¼ R z þ m, (10) z we can obtain z as z ¼ z p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z 4mR (11) R and since we seek the leading-order (large r) terms, we expand Eq. (11) in z=r and find for z=r large (note that only the þ sign in Eq. (11) is physically correct) z ¼ r e ib ¼ z R þ O R ¼ reiy z R þ O R, (1) z so that we conclude that a circle of radius rbr is mapped into a circle of radius r ¼ r=r in the transformed domain such that y ¼ b. Therefore, we can express the leading-order terms in the far field displacements as u r ¼ r 8m U 1ðyÞþ R 8mr U ðyþþ R 4pmr U 3ðyÞ, u y ¼ r 8 m U 4ðyÞþ R 4m r U 5ðyÞþ R 4pmr U 6ðyÞ, ð13þ where U 1 ðyþ ¼4s 1 X 1 sin y þ s 1 aa½ðk 1Þþð3 aþ cos yš, a¼1 U ðyþ ¼4ð1 þ k þ m cos yþ sin ys 1 1 þ X a¼1 þð3 aþðmðcos 4y 3Þþcosyð1 þ kþþš, s 1 aa ½ð1 þ m Þ mð3 þ kþ cos y U 3 ðyþ ¼ XN X h F a k ð1 þ kþ cos y ^y þ k d p a þ cos ^y k d p a k¼1 a¼1 m cos ^y þ k þ d p i a, U 4 ðyþ ¼s 1 X 1 cos y s 1 aað3 aþ sin y; U a¼1 5ðyÞ ¼½ðk 1Þþm cos yšu 4 ðyþ, U 6 ðyþ ¼ XN X h F a k ðk 1Þ sin y ^y þ k d p a k¼1 a¼1 þ sin ^y k d p a m sin ^y þ k þ d p i a, ð14þ with d ij denoting the ronecker delta and F a k ¼fP k, S k g and ^y þ k ¼ b k þ y n k, ^y k ¼ b k y n k, where y n k denotes the angle between the inward normal at the void boundary and the x 1 (horizontal) axis at the node k.

9 . Bertoldi et al. / J. Mech. Phys. Solids 55 (007) From the far field displacement field, we obtain the leading-order terms in the far field stress s rr ¼ S 1 ðyþþ R r S ðyþþ R pr S 3ðyÞ, s ry ¼ S 4 ðyþþ R r S 5ðyÞþ R pr S 6ðyÞ, s yy ¼ S 7 ðyþþ R r S 8ðyÞþ R pr S 9ðyÞ, where ð15þ S 1 ðyþ ¼s 1 1 sin y þ 1 X a¼1 s 1 aa½1 þð3 aþcos yš, S ðyþ ¼4ðm cos y Þ sin ys 1 1 þ X a¼1 þ cos yðm cos y Þð3 aþš, s 1 aa½4m cos y 1 m S 3 ðyþ ¼ XN X h F a k 4 cos y ^y þ k d p a þ cos ^y k d p a k¼1 a¼1 m cos ^y þ k þ d p i a, S 4 ðyþ ¼s 1 1 cos y 1 X sin y s 1 aað3 aþ, a¼1 S 5 ðyþ ¼4½cos y m sin yšs 1 X 1 þ sin y a¼1 s 1 aa½m ð3 aþð1 þ m cos yþš, S 6 ðyþ ¼ XN X h F a k sin y ^y þ k d p a sin ^y k d p a k¼1 a¼1 þm sin ^y þ k þ d p i a, S 7 ðyþ ¼ s 1 1 sin y þ 1 X a¼1 s 1 aa½1 ð3 aþcos yš, S 8 ðyþ ¼ ms 1 X 1 sin 4y þ s 1 aa ½1 þ m mð3 aþð1 þ cos 4yÞŠ, a¼1 S 9 ðyþ ¼ XN X h F a k cos ^y k d p a m cos ^y þ k þ d p i a. ð16þ k¼1 a¼1

10 44. Bertoldi et al. / J. Mech. Phys. Solids 55 (007) Note that the resultant components P k and S k are functions of the geometry, the loading at infinity, and the elastic constants of the plane, elliptical inclusion and structural interface. In particular, compatibility at the elastic body/structure jth junction, where M bars converge, can be expressed as (see Eq. (9) in Part I of this paper, but now in terms of resultants, instead of quantities per unit length of the bar) X M ðp k þ is k Þe iyn k ¼ h¼1 k ðhkþ b ½ðuðx k Þ uðx h ÞÞ.e r ðhkþ Šer ðhkþ, (17) where uðx h Þ is the displacement at the hth junction central point x h, k ðhkþ b is the stiffness of the hk bar, and e r ðhkþ is the unit vector aligned parallel to the bar, e r ðhkþ ¼ x k x h jx k x h j. (18) Note that, with x h lying on the matrix, uðx h Þ is due to both the remote stress applied (u 1 ) and the tractions applied at the junction (u PS ) uðx h Þ¼u 1 ðx h Þþu PS ðx h Þ; (19) otherwise, if x h lies on the inclusion, uðx h Þ¼u PS ðx h Þ. (0) It is important now to note that u PS ðx h Þ is a linear function of the load components P k and S k, u PS ðx h Þ¼ XN j¼1 ðv hj P j þ W hj S j Þ, (1) where the coefficients are determined by employing the complex potentials given in Eqs. (4) and (3) in Part I of this article for the elliptical inclusion and matrix, respectively. Note that the compatibility Eq. (17) needs to be written at each node of the interfacial structure, so that, after separation of its real and complex parts, a linear system of N tot equations is obtained TfP 1 ; S 1 ;...; P Ntot ; S Ntot g T ¼ Ffu 1 1 ðx 1Þ; u 1 ðx 1Þ;...; u 1 1 ðx NÞ; u 1 ðx NÞg T, () N tot denoting the total number of nodes of the structure. Therefore, the traction resultant components P k and S k at the k-junction can be solved for from Eq. () as P k ¼ XN ðt 1 FÞ ðk 1Þðj 1Þ u 1 1 ðx jþþðt 1 FÞ ðk 1Þj u 1 ðx jþ, j¼1 S k ¼ XN ðt 1 FÞ kðj 1Þ u 1 1 ðx jþþðt 1 FÞ kj u 1 ðx jþ. j¼1 ð3þ Finally, to simplify the notation, we introduce the following matrices: A ¼ðT 1 FÞ ðk 1Þðj 1Þ ; B ¼ðT 1 FÞ ðk 1Þj ; C ¼ðT 1 FÞ kðj 1Þ ; D ¼ðT 1 FÞ kj : ð4þ

11 We consider now a remote uniaxial stress s 1 11 and employ Eq. (3) in Eqs. (13) and (15), which, substituted in Eqs. (1), allows us to obtain the averaged stress and strain in the circular region of radius r containing an elliptical reinforced void or inclusion inclined at an angle a with respect to the horizontal direction (truncated at first order in f) s 11 ¼ s 1 11 f s1 11 R XN. Bertoldi et al. / J. Mech. Phys. Solids 55 (007) þ m m cos a ab ½ðG 0 j A þ G 1 j B Þh ð0;0;1þ k þ f ð1 þ kþr s pabm þðg 0 j C þ G 1 j D Þh ð1;1;1þ k Š, s 1 ¼ fmr s 1 11 sin a ab þ f ð1 þ kþr s pabm þðg 0 j C þ G 1 j D Þ cosð^y þ k þ aþš, X N ½ðG 0 j A þ G 1 j B Þ sin ð^y þ k þ aþ s ¼ f s1 11 m R ab þ f ð1 þ kþr s pabm X N ½ðG 0 j A þ G 1 j B Þh ð0;;1þ þðg 0 j C þ G 1 j D Þh ð1; 1;1Þ k Š ð5þ k and 11 ¼ s 1 1 þ k 11 8m fr s 1 11 XN ½ðG 0 j A þ G 1 j B Þh ð0;0;kþ mð9 þ kþ cos a 4ð1 þ k þ m Þ k 16abm þðg 0 j C þ G 1 j D Þh ð1;1;kþ k Š, f ð1 þ kþr s pabm 1 ¼ fmr s 1 11ð3 þ kþ sin a 16abm f ð1 þ kþr s pabm þðg 0 j C þ G 1 j D Þ cosð^y þ k þ aþš, X N ½ðG 0 j A þ G 1 j B Þ sinð^y þ k þ aþ ¼ s 1 11 XN k 3 8m f 4ð1 k þ s1 11 R m Þþðk 3Þmcos a f ð1 þ kþr s abm 3pabm ½ðG 0 j A þ G 1 j B Þh ð0; ;kþ k þðg 0 j C þ G 1 j D Þh ð1; 1;kÞ k Š, ð6þ where f ¼ ab=r and G p j ¼½1 ð 1Þ p mš cos b j p p þ cos b j þ a þ p p, h ðp;g;qþ k ¼ cos ^y k p p m cos ^y þ k þ p p þ q cos ^y þ k þ a þ g p. ð7þ

12 46. Bertoldi et al. / J. Mech. Phys. Solids 55 (007) In order to analyze a dilute suspension of randomly oriented elliptical inclusions, an isotropic average, defined for an arbitrary function g of angle a as g is ¼ 1 Z p gðaþ da, (8) p is employed, yielding 0 s is 11 ¼ s1 11 f þ s1 m 11R ab þ f ð1 þ kþr s abpm þ B ðv ð1;1;0þ w ð1;0;1þ ÞþC ðv ð0;0;1þ X N ½A ðv ð0;0;0þ þ w ð1;1;1þ Þ þ w ð1;0;1þ ÞþD ðv ð1;1;1þ þ w ð1;1;1þ ÞŠ, s is 1 ¼ f ð1 þ kþr s abmp X N ½ðC B Þ cos ðb j ^y þ k Þ ða þ D Þ sin ðb j ^y þ k ÞŠ, and s is ¼ f s1 11 m R ab is 11 ¼ s1 11 þ B ðv ð1;1;0þ þ f ð1 þ kþr s abpm X N þ w ð1;0;1þ ÞþC ðv ð0;0;1þ 1 þ k 8m þ f 1 þ k þ s1 m 11R 4abm þ B ðv ð1;1;0þ is 1 ¼ k m sis 1, þ w ðk;1;0þ ÞþC ðv ð0;0;1þ ½A ðv ð0;0;0þ w ð1;1;1þ Þ w ð1;0;1þ ÞþD ðv ð1;1;1þ w ð1;1;1þ ÞŠ, ð9þ f ð1 þ kþr s abpm X N ½A ðv ð0;0;0þ w ðk;1;0þ ÞþD ðv ð1;1;1þ þ w ðk;0;0þ Þ þ w ðk;0;0þ ÞŠ, where is ¼ s1 11 v ðp;g;qþ k 3 8m þ f 1 k þ s1 m 11R 4abm þ B ðv ð1;1;0þ w ðk;1;0þ ÞþC ðv ð0;0;1þ f ð1 þ kþr s abpm X N ½A ðv ð0;0;0þ w ðk;0;0þ Þ þ w ðk;1;0þ ÞþD ðv ð1;1;1þ w ðk;0;0þ ÞŠ, ð30þ ¼ ½ð 1Þ p mš cos b j þ g p h ð 1Þ q cos ^y k þ q p m cos ^y þ k þ q p i, ¼ p cos b j ^y þ k þ g p q p. ð31þ w ðp;g;qþ We note that we have always found s is enforced by the isotropic averaging (8). 1 ¼ is 1 ¼ 0, a result consistent with the symmetry

13 . Bertoldi et al. / J. Mech. Phys. Solids 55 (007) We are now in a position to determine the effective moduli for the composite; employing Eqs. (), we obtain (considering only up through linear terms in f) ( m m ¼ 1 þ f ða þ bþ ð1 þ kþ 8mp þð1þkþ XN ½ðA þ D Þ cosðb 3abpm j ^y þ k Þ ðb C Þ sinðb j ^y þ k ÞŠ ), ( ¼ 1 þ f ð1 þ kþ 4ða þ b Þmp þðkþ1þ XN ½A ðb g ð0;0;0þ 8mabpðk 1Þ B ðabg ð1;0;0þ þd ða g ð1;1;0þ þ a g ð1;1;1þ abg ð1;0;1þ ÞŠ þ abg ð0;1;1þ Þ Þ C ðabg ð0;1;0þ b g ð0;0;1þ Þ ), ð3þ where g ðp;g;qþ ¼ cos b j þ q p cos b k þ g p cos y n k þ q p. (33) We note that the effective moduli given by Eq. (3), in the limit case of a void (i.e. in the absence of a structural interface), reduce to m m ða þ bþ ¼ 1 f ð1 þ kþ ; 4ab ¼ 1 f 1 þ k a þ b k 1 ab, (34) which coincide with those reported by Jasiuk et al. (1994) for a dilute composite with randomly oriented elliptical holes. As an application of the expressions for the effective properties (3), the effective shear and bulk moduli for an infinite elastic plane with a dilute suspension of randomly oriented cracks of length a reinforced with two fibers orthogonal to the fracture surface (having locations x and x from the central point of the crack) is considered (see the detail in Fig. 6, right). In this case, S k ¼ 0 and only u 1 contributes to the fiber elongation, so that A ¼ C ¼ D ¼ 0. For a crack density n, the volume concentration f is given by f ¼ npab, so that, taking the limit for b! 0, Eqs. (3) reduce to m m ¼ 1 ð1 þ kþa n 4 ¼ 1 ð1 þ kþa n ðk 1Þ p þ 1 þ k 8m p 1 þ k 4m X 4 X 4 B sin ðb j ^y þ k Þ!,! B sin b j sin b k sin y n k, ð35þ where, numbering the junctions as in the detail of Fig. 6 (right), B are B 11 ¼ B 41 ¼ B ¼ B 3 ¼ Vt b ½aA ðþ kð1 þ kþ 8pmŠ, B 1 ¼ B 31 ¼ B 1 ¼ B 4 ¼ VaA ð1þ kt b ð1 þ kþ, B j3 ¼ B j ; B j4 ¼ B j1, ð36þ

14 48. Bertoldi et al. / J. Mech. Phys. Solids 55 (007) μ μ Λ= x x x a a a Λ=0.0 Fig. 6. Effective shear modulus m (left) and D bulk modulus (right) (normalized by m and, respectively) for a dilute suspension of randomly oriented sharp cracks of length a reinforced by two symmetrical transverse fibers located at x and x, for different values of the fibers compliance parameter L. in which V ¼ 8kpm a k ð1 þ kþ ½ðA ð1þ Þ ða ðþ Þ Šþ16aA ðþ kð1 þ kþpm 64p m, A ðhþ ¼ X a¼1 " # ð 1Þ a 4b a sin b þ ðcos b ð 1Þ h cos b a Þ log 1 þð 1Þh cos ðb a þ bþ 1 þð 1Þ h, cos ðb a bþ ð37þ with b 1 ¼ b and b ¼ b þ. Note that in the absence of fibers, B 11 ¼ B 1 ¼ 0, Eqs. (35) reduce to the well-known formulae for a dilute suspension of randomly oriented cracks (achanov et al., 1994). The variation of averaged shear modulus m and D bulk modulus (normalized by m and, respectively) as a function of the fiber position is shown in Fig. 6, for t b =a ¼ , crack density n ¼ 0:001, and different values of the dimensionless compliance parameter L ¼ m þ =½ðk þ þ 1Þkt b Š (see Eq. (35) of Part I). First we note that as L increases, the averaged moduli tend to those corresponding to an unreinforced crack, m=m ¼ 0:79 and = ¼ 0:37. The effect of the fibers vanishes when they are close to the crack tip, whereas it is maximum for x=a 0:. We conclude that fibers close to the crack tip drastically reduce the stress singularity, but are almost ineffective for enhancing the overall elastic properties. As a final example, the effective shear and bulk moduli for an infinite elastic plane with a dilute suspension of randomly oriented elliptical (thin, a=b ¼ 0) voids reinforced by rigid fibers (L ¼ 0) is plotted in Fig. 7, for an inclusion volume fraction f ¼ 0:005. The results model the averaged stiffness coefficients of a cracked fiber-reinforced material, such as that reported in Fig. 1(B) of Part I of this article, a situation not previously investigated. The fibers are characterized by t b =a ¼ and two different distributions are considered: purely orthogonal to the major axis (marked in Fig. 7 as vertical ) and randomly distributed and oriented (marked as random in Fig. 7). The averaged properties for a dilute suspension of randomly oriented rigid inclusions are given by the well-known

15 . Bertoldi et al. / J. Mech. Phys. Solids 55 (007) rigid inclusion random 1 rigid inclusion vertical μ μ random vertical elliptical void N elliptical void N Fig. 7. Effective shear modulus m (left) and D bulk modulus (right) (normalized by m and, respectively) for a dilute distribution (f ¼ 0:005) of randomly oriented elliptical voids (a=b ¼ 0), reinforced by N rigid fibers (L ¼ 0). Two different fiber distributions are considered: purely vertical and randomly oriented and distributed. formulae m m þ ¼ 1 þ f m þ k þ ½ k þ ðm ÞŠ k þ ð1 m Þðk þ þ m ; Þ þ ¼ 1 þ f ðkþ m Þð1 þ k þ Þ k þ ð1 m, (38) Þ as reported by Thorpe and Sen (1985), so that for an inclusion characterized by a=b ¼ 0, m=m ¼ 1:04 and = ¼ 1:019. These values are not approached in the case of (highly dense) parallel vertical fibers, since in this case the reinforcement remains weak in the direction perpendicular to the fibers, but those values are approached in the case of randomly distributed fibers. Note that the oscillations in the graph are due to the fact that only one random distribution of fibers is considered for each N...1. Circular inclusions coated by a discrete structural interface: effective properties In the particular case of a circular inclusion of radius R connected to an infinite matrix having a circular hole of radius R þ d by a discrete structural interface, Eqs. (3) simplify to m ¼ 1 f ð1 þ kþ m 1 ( 1 þ k 8pm X N ½ðA þ D Þ cosðb j y k Þ ) ðb C Þ sinðb j y k ÞŠ, ð39þ " # f ð1 þ kþ ¼ 1 þ 1 þ 1 þ k X N ða cos b k 1 8mp j þ B sin b j Þ. As an example, the effective shear modulus m and bulk modulus (normalized by m þ and þ, respectively) are determined for a dilute suspension (f ¼ 0:1) of circular inclusions coated by a discrete structural interface with thickness d ¼ R. Two different interfacial

16 50. Bertoldi et al. / J. Mech. Phys. Solids 55 (007) Perfect inclusion 1. Perfect inclusion 1.1 μ μ 1 1 β 0.9 BM Double Warren γ=π/ Double Warren γ=π/8 circular void N 0.6 γ circular void N Fig. 8. Effective shear modulus m (left) and D bulk modulus (right) (normalized by m and, respectively) for a dilute distribution (f ¼ 0:1) of circular inclusions of radius R connected to an infinite matrix having a circular hole of radius R þ d (d=r ¼ 1) by a double Warren truss structure comprised of 4N rigid fibers (L ¼ 0). The matrix is stiffer than the inclusions (m =m þ ¼ 0:1) and n þ ¼ n ¼ 1 3. structures are considered: the discrete BM-model (from Bigoni and Movchan (00), see Part I of this article) and a double Warren truss structure with two different opening angles g ¼fp=3; p=8g (see detail in Fig. 8), corresponding to a fp=; p=4g. The elastic materials are characterized by m =m þ ¼ 10 and n ¼ n þ ¼ 1 3, while the bars of thickness t b=r ¼ are assumed to be rigid (L ¼ 0). The effective moduli are reported in Fig. 8 at increasing bar density N (4N for the double Warren structure). It is clear from the figure that the nonlocality introduced by the bars inclination increases the averaged properties of the composite. In the particular case of a dilute suspension of circular holes, the averaged properties are equal to m=m þ ¼ 0:733 and = þ ¼ 0:6, the values approached in the graphs when the numbers of bars approaches zero. On the other hand, for a dilute suspension of circular inclusions connected to the matrix by a perfect interface, the averaged properties are equal to m=m þ ¼ 1:14 and = þ ¼ 1:1, values approached in the graphs as the number of bars increases. In closure, we note that a dilute suspension of circular inclusions coated by a sufficiently dense structural interface is characterized by averaged properties higher than those corresponding to the perfect interface and that, increasing the bar density, the averaged properties tend to those predicted by the continuous model. 3. Optimal stress distribution: neutrality The possibility of tuning the properties of the interface, so that an optimal stress distribution is achieved, is investigated in this section. Optimality is intended here to correspond to a minimum stress concentration near the interface in the matrix, a condition which is fully satisfied when the inclusion is neutral. Previously, Ru (1998) analyzed the design of neutral elastic inclusions employing the zero-thickness linear interface, while Bigoni et al. (1998) investigated for the same structure a weaker condition of neutrality.

17 . Bertoldi et al. / J. Mech. Phys. Solids 55 (007) We show here that by employing a continuous structural interface, neutrality occurs for a range of material parameters broader than for a zero-thickness linear interface. A new concept of weak neutrality (coincident with the notions of neutrality employed by Ru, 1998 and Bigoni et al., 1998 only in particular cases) is introduced for inclusions coated by discrete structural interfaces, and it is shown that for a weakly neutral inclusion the perturbation introduced in the matrix is very localized, although non-null at every point of the matrix, as the notion of strong neutrality implies Strong- and weak-neutrality conditions for an inclusion connected to the matrix by a structural interface Let us consider a generic inclusion connected to an elastic matrix by a structural interface, as shown in Fig. 9, and subject to a remote uniform stress r 1. Denoting by a superscript p the perturbation induced by the inclusion, we have u ¼ u p þ u 1 ; e ¼ e p þ e 1 ; r ¼ r p þ r 1. (40) The strain energy in the elastic matrix contained within a contour G enclosing the inclusion, W m, can be expressed as W m ¼ 1 Z ðr 1 e 1 þ r p e p þ r 1 e p þ r p e 1 Þ, (41) V þ Γ x Γ _ x 1 Γ + Fig. 9. Elastic inclusion and structural interface.

18 5. Bertoldi et al. / J. Mech. Phys. Solids 55 (007) where V þ is the volume enclosed between G and the contour G þ, defining the boundary of the structural interface. On application of the divergence theorem, Eq. (41) becomes W m ¼ 1 Z ðt 1 u 1 þ t p u p þ t 1 u p þ t p u 1 Þ, (4) ~G where t ¼ rn denotes the traction vector and Z Z Z ¼ XNþ, (43) ~G G h¼1 x þ h where N þ is the number of the junction regions x þ h lying on Gþ (see Part I, Section ). The last term on the right-hand side of Eq. (43) is equal to the strain energy W i in the structural interface and inclusion (changed in sign). Therefore, the strain energy in the region enclosed by the contour G is the sum of the contributions of the matrix, structural interface and inclusion, viz. W ¼ W m þ W i ¼ 1 Z ðt 1 u 1 þ t p u p þ t 1 u p þ t p u 1 Þ, (44) G and it can be observed that W diverges as b!1. However, this divergence is due to the fact that the unperturbed solution corresponds to a homogeneously stressed infinite domain, so that its energy is necessarily infinite. Therefore, the quantity W 1 Z t 1 u 1 ¼ 1 Z ðt p u p þ t 1 u p þ t p u 1 Þ (45) G G remains finite. This is zero in the absence of any inclusion and can be viewed as the perturbation of the energy due to the presence of the inclusion. Therefore, we take as a definition of strong inclusion neutrality the condition that the energy perturbation (45) be zero for all the control volumes singled out by contour G, namely Z ðt p u p þ t 1 u p þ t p u 1 Þ¼0; 8G. (46) G This definition of neutrality is equivalent to that employed by Milton and Serkov (001) and Ru (1998). The problem with applying (46) to discrete structural interfaces is that the matrix will be always overstressed at the bar junctions, where a strong-neutrality condition will in general be violated. We prefer therefore to introduce a new, weak notion of neutrality. In particular, specializing G to a circle of radius d and taking the limit for d!1of Eq. (46) we obtain Z so that lim d!1 GðdÞ t p u p! 0, (47) we take as a definition of inclusion weak neutrality the condition that the energy perturbation (45) be zero for d!1, namely Z lim ðt 1 u p þ t p u 1 Þ¼0. (48) d!1 GðdÞ

19 . Bertoldi et al. / J. Mech. Phys. Solids 55 (007) That condition (48) is connected to neutrality is obvious since it is a weak form of (46). However, it is proven below that for a continuous structural interface (thus including as particular cases perfect and zero-thickness linear interfaces), connecting a circular inclusion to a matrix, condition (48) is equivalent to requiring the material s effective properties to be equal to the properties of the matrix material ( m=m ¼ = ¼ 1) for a dilute suspension of inclusions, a condition of neutrality also employed by Bigoni et al. (1998). 3.. Strong- and weak-neutrality for a circular inclusion connected to an infinite matrix by a continuous structural interface We start by considering a circular inclusion of radius R connected to an infinite matrix having a circular hole of radius R þ d by a continuous double Warren truss structure (see Fig. 1). Since when a uniform stress is applied at infinity, the stress and displacement fields are derived from Eqs. (49) of Part I of this article, choosing G as a circle of radius d centered at the inclusion center, Eq. (46) can be rewritten in matrix form as where r 1 Hr 1 ¼ 0; () det H ¼ 0, (49) r 1 ¼½s 1 11 ; s1 1 ; s1 ŠT, (50) and the coefficients of the symmetric matrix H are H 11 ¼ H 33 ¼ F mp 4d m þ ½d ðk þ 3Þ 4F m Šþ 1 4 H, H ¼ p d 6 m þ f6m m 1B mm m d þ B m d 4 ½d ð1 k þ ÞþB m ðk þ þ 3ÞŠg, H 13 ¼ H 11 1 H ; H 1 ¼ H 3 ¼ 0, ð51þ in which B m, F m and M m are given by Eqs. (51) in Part I of this article. The form of the neutrality condition given in Eq. (49) is instructive, since it reveals that if the eigenvalues of H are distinct, neutrality holds only for certain loading at infinity, corresponding to the eigenvector related to the vanishing eigenvalue. We are now in a position to compute the eigenvalues h i and the corresponding eigenvectors v i of the matrix H, which turn out to be h 1 ¼ pf m d m þ ½d ðk þ 3Þ 4F m Š, h ¼ h 3 ¼ p d 6 m þ f6m m 1B mm m d þ B m d 4 ½B m ðk þ þ 3Þþð1 k þ Þd Šg, v 1 ¼½1; 0; 1Š T ; v ¼ v 3 ¼½0; 1; 0Š T, ð5þ showing that an inclusion can be neutral for equibiaxial stress or pure shear. The condition of strong neutrality requires vanishing of h 1 or h for every d, whereas weak neutrality corresponds to the vanishing of h 1 or h, but computed at the limit d!1, so that

20 54. Bertoldi et al. / J. Mech. Phys. Solids 55 (007) Strong neutrality : Weak neutrality : F m ¼ 0 B m ¼ M m ¼ 0 F m ¼ 0 B m ¼ 0 for equibiaxial tension; for shear: for equibiaxial tension; for shear: (53) Therefore, there is no difference between strong and weak neutrality for equibiaxial remote stress, whereas the difference is in the requirement of M m vanishing for remote shear. Note that in the case of weak neutrality, for an infinite matrix characterized by k ¼ 3(k ¼ 1), the eigenvalue h 1 (h 3 ) is identically zero; for these cases higher-order terms should be considered in the energy. For a dilute suspension of circular inclusions connected to an infinite matrix by a continuous double Warren truss structure, the averaged properties are given by Eq. (5), so that it is clear from Eq. (53) that the condition of weak neutrality under equibiaxial tension is equivalent to = þ ¼ 1, whereas the condition of weak neutrality under shear is equivalent to m=m þ ¼ 1. 3 The following can be concluded from the conditions of strong and weak neutrality: It is possible to choose the interfacial properties to satisfy weak neutrality (coincident for circular inclusions with = þ ¼ m=m þ ¼ 1) for both remote equibiaxial stress and pure shear (i.e. F m ¼ B m ¼ 0, or, in other words, h 1 ¼ h ¼ 0). A circular inclusion connected to the matrix by a zero-thickness linear interface is weakly neutral for both remote shear and equibiaxial tension when the conditions found by Bigoni et al. (1998, their Eqs. (43) (44)) are met, which is possible only for inclusions stiffer than the matrix. For a circular inclusion coated by a continuous structural interface, the solution of the conditions B m ¼ F m ¼ 0 in the plane m =m þ versus d=r is reported in Fig. 10, for n þ ¼ n ¼ 1 3 and different bar inclinations a. Itis clear that employing a continuous structural interface, weak neutrality becomes possible also for m =m þ o1. In the case of strong neutrality, the stress at each point of the matrix is the same as in the absence of the inclusion. However, as in the case of the zero-thickness linear interface (Ru, 1998), differently from weak neutrality, it is not possible to choose the interfacial properties to satisfy strong neutrality for both remote equibiaxial stress and pure shear (i.e. F m ¼ B m ¼ M m ¼ 0); For remote equibiaxial traction, strong neutrality occurs when F m ¼ 0 (see Eq. (53)), which in terms of bar stiffness k becomes k ¼ m m þ ðr þ dþ ð1 k ÞRðR þ dþm þ cos a þðk þ 1Þm ðr cos a þ Rd þ d Þ. (54) 3 The coincidence between weak neutrality and the criterion = þ ¼ m=m þ ¼ 1 (for circular inclusions) is valid also as particular cases for a zero-thickness interface (in this case the coefficients B m and F m are given by Eqs. (54) in Part I of this article) and for a perfectly bonded interface.

21 . Bertoldi et al. / J. Mech. Phys. Solids 55 (007) μ - μ = δ R Fig. 10. Circular inclusion of radius R connected to an infinite matrix by a continuous double Warren truss structure of thickness d and bars inclination a. Combinations of parameters m =m þ and d=r for which weak neutrality is possible under both remote equibiaxial tension and shear (F m ¼ B m ¼ 0). Different bar inclinations a are considered μ - μ thickness linear interf =0 Neutrality possible for =0 Neutrality possible for =/π δ R Fig. 11. Circular inclusion of radius R connected to an infinite matrix by a continuous double Warren truss structure of thickness d and bar inclinations a ¼f0; p=g. Range of parameters m =m þ, and d=r for which inclusion strong neutrality becomes possible under remote equibiaxial stress. Admissibility corresponds to all points in both the gray zones for a ¼ p= and in the light gray zone for a ¼ 0. In the limit case of a linear zero-thickness interface [d! 0anda! arctanðcrþ, see Eq. (46) in Part I of this paper], Eq. (54) reduces to a condition found by Ru (1998) 4m m þ k r ¼ R½ðk þ 1Þm m þ ðk 1ÞŠ, (55) whereas k y remains unprescribed and neutrality is possible only for inclusions stiffer than the matrix, m =m þ X1. Assuming for simplicity k þ ¼ k, we report in Fig. 11 the regions in the plane m =m þ versus d=r where neutrality is possible (namely, when the

22 56. Bertoldi et al. / J. Mech. Phys. Solids 55 (007) compliance parameter k of the continuous structural interface is positive), for two bar inclinations a ¼ 0andp=. We note that all values of m =m þ lying above the curve marked a ¼ 0 correspond to strong neutrality under equibiaxial stress (we mean therefore points in the light gray zone). Strong neutrality under equibiaxial remote stress for a ¼ p= is possible at every point in the figure (thus in both the light and dark gray zones). Other values of a have been investigated and they always correspond to limiting curves lying below the limiting curve for a ¼ 0 shown in the figure. Therefore, for a circular inclusion coated by a continuous double Warren truss structure, differently from a linear zero-thickness interface, strong neutrality for equibiaxial tension is possible also for m =m þ o1. For remote shear stress, strong neutrality occurs when both B m and M m vanish, see Eq. (53). In particular, from the condition M m ¼ 0 we obtain a linear equation in k, which can be solved as function of m =m þ. Introduction of this computed value into the equation B m ¼ 0 yields a polynomial expression of second order for m =m þ. Assuming for simplicity k þ ¼ k, the values of the parameters m =m þ and d=r for which strong neutrality is possible for different values of bars inclination a are reported in Fig. 1. In the particular case of a linear zero-thickness interface, we obtain that neutrality is possible under shear for k ¼ m m þ Rðm m þ Þ ; a ¼ p 4, (56) so that k r ¼ k y ¼ k, a result coincident with that given by Ru (1998). Therefore, for a zero-thickness linear interface, the neutrality condition is possible only for inclusions stiffer than the matrix, m =m þ X1, whereas for a continuous structural interface this condition is not necessary if a4p=3. In addition, we note from Fig. 1 that under remote shear stress, strong neutrality is possible only for axp=4 (the case a ¼ p=4 is attained in the limit case of a zero-thickness linear interface). It is interesting to observe that for p=3oaop=, strong neutrality is possible on two separate branches: (i) for d=ro0: with m =m þ X1 and employing a continuous structural interface characterized by high stiffness k; (ii) for d=r40: also for inclusions less stiff than the matrix. μ-1.5 μ = δ 0. R μ μ = δ R Fig. 1. Circular inclusion of radius R connected to an infinite matrix by a continuous double Warren truss structure of thickness d and bars inclination a. Combinations of parameters m =m þ, and d=r for which strong neutrality is possible under remote shear stress, for different values of the bar inclination a. Admissibility corresponds to all points on the curves.

23 . Bertoldi et al. / J. Mech. Phys. Solids 55 (007) In closure, we emphasize that: the range of parameters for which neutrality occurs for a continuous structural interface is much broader and extends to values m =m þ o1 (inclusions less stiff than the matrix) than that obtained by employing the zero-thickness linear interface model (limited to inclusions stiffer than the matrix) Weak neutrality for an elliptical inclusion connected to an infinite matrix by a discrete structural interface Let us consider an elliptical inclusion connected to an infinite matrix by a structural interface, so that we can make recourse to Eqs. (13), (15) and (3). Substituting these equations into the weak-neutrality condition (48) and employing the representation (50) yields the condition of weak neutrality in the same form (49), but with H given now by H 11 ¼½þmð 7 þ 6mÞþkð þ m m ð1 þ kþ ÞŠp þ 4m fð3 kþ½ðm 3Þða ð0;0þ þ c ð0;0þ Þþð1 mþðb ð0;0þ þ d ð0;0þ ÞŠ þðkðm þ Þ 3mÞ½ðm 3Þða þð0;0þ c þð0;0þ Þþð1 mþðb þð0;0þ d þð0;0þ ÞŠg, H ¼ 16ðk 1Þp 4ðk 1Þ ½a þð1;1þ þ b þð1;1þ þ c þð1;1þ þ d þð1;1þ Š, m H 33 ¼ðþmð7þ6mÞ kðm þ m ð1 þ kþ ÞÞp þ 4m fð3 kþ½ðm þ 1Þða ð0;0þ þ c ð0;0þ Þ ð3þmþðb ð0;0þ þ d ð0;0þ ÞŠ þðkðm Þþ 3mÞ½ðm þ 1Þða þð0;0þ c þð0;0þ Þ ð3þmþðb þð0;0þ d þð0;0þ ÞŠg, H 1 ¼ ð1 þ kþ fð3 kþ½a ð0;1þ þ b ð0;1þ þ c ð0;1þ þ d ð0;1þ Š m þð1 kþðm 3Þða þð1;0þ þ c þð1;0þ Þþðk 1Þðm 1Þðd þð1;0þ þ b þð1;0þ Þ þðkð þ mþ 3mÞ½a þð0;1þ þ b þð0;1þ c þð0;1þ d þð0;1þ Šg, H 13 ¼ ½5 þ 3m kðm ð1 þ kþ þ 3ÞŠp þ 4m fð3 kþ½ðm 1Þða ð0;0þ þ c ð0;0þ Þ ð1þmþðb ð0;0þ þ d ð0;0þ ÞŠ þ½kð4 m þ m Þ 4þ3m 3m Šða þð0;0þ c þð0;0þ Þ ½kð4 þ m þ m Þ 4 3m 3m Šðb þð0;0þ d þð0;0þ Þg, H 3 ¼ H 1 þ ðk 1Þ ½c þð0;1þ c þð1;0þ b þð0;1þ þ b þð1;0þ þ d þð0;1þ m þ d þð1;0þ a þð0;1þ a þð1;0þ Š, ð57þ

24 58. Bertoldi et al. / J. Mech. Phys. Solids 55 (007) where a ði;gþ ¼ XN k;j¼1 b ði;gþ ¼ XN k;j¼1 c ði;gþ ¼ XN k;j¼1 d ði;gþ ¼ XN k;j¼1 A cos ^y k þ i p cos b j þ g p, B cos ^y k þ i p sin b j g p, C sin ^y k i p cos b j þ g p, D sin ^y k i p sin b j g p. ð58þ If the structural interface is symmetric with respect to both x 1 and x axes a ð0;1þ ¼ b ð0;1þ ¼ c ð0;1þ ¼ d ð0;1þ ¼ 0, a ð1;0þ ¼ b ð1;0þ ¼ c ð1;0þ ¼ d ð1;0þ ¼ 0, so that H 1 ¼ H 3 ¼ 0, (60) a condition which thoroughly simplifies the calculations. For the particular case of a structural interface symmetric with respect to both x 1 and x axes, we can compute the eigenvalues h i and the corresponding eigenvectors v i of the matrix H, which turn out to be h 1 ¼ H ; v 1 ¼½0; 1; 0Š T h ¼ H p ffiffi h 11þH 33 D ; v ¼ H p ffiffi i 11 H 33 D T; H 13 ; 0; 1 ð61þ h 3 ¼ H p ffiffi h 11þH 33 þ D ; v 3 ¼ H p ffiffi i 11 H 33 þ D T; H 13 ; 0; 1 with D ¼ 4H 13 þðh 11 H 33 Þ. Therefore assuming H 13 ðh 11 H 33 Þa0, the inclusion can be weakly neutral only when a uniform shear is applied at infinity and no neutrality under equibiaxial stress is found. Although we cannot exclude that either H 11 ¼ H 33 or H 13 ¼ 0 might be satisfied, we were unable to find a geometry for the interfacial structure to satisfy such a condition for an elliptical inclusion (in contrast to the circular case), so that we were unable to find weak neutrality under biaxial stress for elliptical inclusions or reinforced voids. As an example, an elliptical void with a=b ¼ 5 is considered, reinforced with 15 double Warren truss structural elements comprised of bars with thickness t b =a ¼ , as shown in Fig. 13. The infinite matrix is characterized by n ¼ 1 3 and loaded by a uniform remote shear stress s 1 1 ¼ m=100. The bars stiffness is selected to satisfy the weak neutrality condition given by Eq. (48) and corresponding to L ¼ 0:17. It is clear from Fig. 13, where the absolute value of the components of stress perturbation (normalized by the remote shear stress, i.e. js ij s 1 ij j=s 1 1 ) are shown, that, when the elliptical void is weakly neutral, the perturbation introduced is very localized near the bar junctions, whereas the perturbation ð59þ

25 . Bertoldi et al. / J. Mech. Phys. Solids 55 (007) Λ=0.17 Λ=0.50 σ 11 /σ 1 σ 11 /σ 1 Λ=0.17 Λ=0.50 σ 1 -σ 1 /σ 1 σ 1 -σ 1 /σ 1 Λ=0.17 Λ=0.50 σ /σ 1 σ /σ σ-σ σ 1 Fig. 13. Elliptical void reinforced by 15 double Warren truss structures and loaded by a remote shear stress: weak neutrality occurs for bars compliance L ¼ 0:17, so that in this case the stress fields are almost unperturbed (compare to the non-neutral case L ¼ 0:5). Maps of the absolute values of the perturbation in the stress components (normalized by the remote stress) are shown. spreads through the matrix when the bar stiffness is changed (see the other example with L ¼ 0:5) Weak neutrality for a circular inclusion reinforced by a discrete structural interface Let us consider a circular inclusion having radius R connected to an infinite matrix with a circular hole of radius R þ d by a structural interface. For this particular geometry m ¼ 0, y n k ¼ b k ¼ y k, ^y þ k ¼ y k and ^y k ¼ 0, so that a ð1;0þ ¼ a ð1;1þ ¼ 0; b ð1;0þ ¼ b ð1;1þ ¼ 0, c ð0;0þ ¼ c ð0;1þ ¼ 0; d ð0;0þ ¼ d ð0;1þ ¼ 0. ð6þ

26 60. Bertoldi et al. / J. Mech. Phys. Solids 55 (007) Therefore, coefficients (57) for the particular case of a structural interface symmetric with respect to both x 1 and x axes simplify to H 11 ¼ pð1 þ kþþ 1 þ k 4m fð3 kþðb ð0;0þ 3a ð0;0þ Þ þ ðk 1Þ½b þð0;0þ d þð0;0þ 3ða þð0;0þ c þð0;0þ ÞŠg, H ¼ 16pðk 1Þ 4ðk 1Þ ða þð1;1þ þ b þð1;1þ þ c þð1;1þ þ d þð1;1þ Þ, m H 33 ¼ pð1 þ kþþ 1 þ k 4m fð3 kþða ð0;0þ 3b ð0;0þ Þ þ ðk 1Þ½c þð0;0þ a þð0;0þ 3ðd þð0;0þ b þð0;0þ ÞŠg, H 13 ¼ pð5 3kÞþ 1 þ k 4m ½ðk 3Þða ð0;0þ þ b ð0;0þ Þ þ 4ðk 1Þða þð0;0þ b þð0;0þ c þð0;0þ þ d þð0;0þ ÞŠ, H 3 ¼ H 1 ¼ 0. (63) If the structure is symmetric about n distinct diametral lines at fixed angles p=n, we find that, in addition to Eqs. (58) we have a ð0;0þ ¼ b ð0;0þ, a þð0;0þ ¼ a þð1;1þ ¼ b þð0;0þ ¼ b þð1;1þ, d þð0;0þ ¼ d þð1;1þ ¼ c þð0;0þ ¼ c þð1;1þ, ð64þ so that Eqs. (63) reduce to H 11 ¼ H 33 ¼ ð1 þ kþp þ 1 þ k m ½ðk 3Þa ð0;0þ þ 4ð1 kþða þð0;0þ c þð0;0þ ÞŠ, H 13 ¼ðk 3Þ 4p þ k þ 1 m a ð0;0þ H 11, H 1 ¼ H 3 ¼ 0; H ¼ ðh 11 H 13 Þ. ð65þ For this particular case, the eigenvalues h i and the corresponding eigenvectors v i of the matrix H turn out to be h 1 ¼ H 11 þ H 13 ; v 1 ¼½1; 0; 1Š T ; ð66þ h ¼ h 3 ¼ H ; v ¼ v 3 ¼½0; 1; 0Š T ; showing that now weak neutrality is possible for shear and equibiaxial stress separately. It follows that for a circular inclusion connected to an infinite matrix by a (symmetric, discrete) structural interface, neutrality occurs: for remote equibiaxial tension when H 11 þ H 13 ¼ðk 3Þ 4p k þ 1 m a ð0;0þ ¼ 0 ¼) a ð0;0þ ¼ 4mp 1 þ k. (67)