Microsoft Word - numero_42_art_30.docx J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 280 Theoretical model of homogenized piezoelectric materials with small non-collinear periodic cracks Jean-Marie Nianga, Driss Marhabi Pôle de Recherche « Structures & Matériaux », Hautes Etudes d’Ingénieur – (HEI), 13 rue de Toul, 59046 Lille Cedex, France jean-marie.nianga@yncrea.fr ABSTRACT. An analytical model for the homogenization of a piezoelectric material with small periodic fissures is proposed on the basis of the method of asymptotic expansions for the elastic displacement, the electric scalar potential and the test functions. Starting from the variational formulation of the three-dimensional problem of linear piezoelectricity, we have at first obtained that concerning a cracked piezoelectric structure, before the implementation of homogenized equations for a piezoelectric structure with a periodic distribution of cracks. It then follows, the characterization of the homogenized law between the mechanical strain and the electric potential, on one hand, and the mechanical stress and the electric displacement, on the other hand. Contrary to the previous investigations, the focus of this paper is the development of a mathematical model taking the non-parallelism of cracks into account. KEYWORDS. Piezoelectric material; Asymptotic expansions; Homogenization; Variational formulation; Periodic cracks. Citation: Nianga, J.-M., Marhabi, D., Theoretical model of homogenized piezoelectric materials with small non- collinear periodic cracks, Frattura ed Integrità Strutturale, 42 (2017) 280-292. Received: 01.02.2017 Accepted: 04.07.2017 Published: 01.10.2017 Copyright: © 2017 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. INTRODUCTION he piezoelectric materials are used in an increasing way in technological applications [1-3]. Among the numerous problems which can arise, there is that concerning the global estimation of the homogenized characteristics of non-homogeneous materials, in particular those presenting a periodic distribution of singularities. Significant efforts had been made to the study of periodical cracks in linear piezoelectricity, through an extension to piezoelectric materials of the modeling of periodic cracks in elastic materials [4, 5]. Gao and al.[6] studied, in terms of the Parton assumption and Stroh formalism, the problem of a half-infinite crack in piezoelectric media with periodic crack; reducing it to Hilbert one and getting therefore the closed-form solutions in the media and inside the cracks. Wang and al. [7] provided a theoretical treatment of the dynamic interaction between cracks in a piezoelectric medium under anti-plane mechanical and in-plane electrical incident wave. They used Fourier transform to study the dynamic electromechanical behavior of a single crack, and solved the obtained singular integral equations by Chebyshev polynomials. The single crack solution was then implemented into a pseudo-incident wave method to account for the interaction between cracks. T J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 281 Somme years more-late, Han and al. [8] obtained the development of a mathematical model to predict the length scale for the spacing of transverse cracks forming in a piezoelectric material subjected to a coupled electro-mechanical external loading condition. In particular, they analyzed the interactions of a row of cracks periodically located in a piezoelectric material layer. Although, one of the remaining problems that need to be treated is that of a periodic array of non-collinear cracks. So, the present paper provides a theoretical model of homogenized piezoelectric materials with small non-collinear periodic cracks through an extension of previous works [9] and [10]. It is organized as follows: Section 2 describes the variational formulation for the three-dimensional problem of linear piezoelectricity. Section 3 develops a variational formulation for the problem of a fissured piezoelectric structure. In Section 4, are presented the homogenized problem of a piezoelectric material with small periodic cracks. Section 5 is then devoted to the formulation of the homogenized local problem in the homogenization period. The analysis of the relationship between the strain and the electric potential on one hand, and the stress and the electric field secondly, is presented in Section 6, just above the conclusion. VARIATIONAL FORMULATION FOR THE THREE-DIMENSIONAL PROBLEM OF LINEAR PIEZOELECTRICITY et  be an open connected domain of 3 with smooth boundary  made of two parts 1 and  2  in the mechanical sense, and of 3  and 4  in the electrical one. These parts of  represent portions of regular surfaces with smooth common boundary, respectively. Moreover,  may be divided into two parts by a smooth surface .S Figure 1: Representation of the open  . In the framework of linear piezoelectricity, the elastic and electric effects are coupled by the constitutive equations: k ij ijkl kij k j u a e E x      (1) k i ij j ikl l u D E e x      (2) where { }iu u is the elastic displacement, { }ij  is the symmetric stress tensor, { }iE E is the electric field vector, and { }iD D is the electric displacement vector, with (i, j, k, l) = (1, 2, 3). We now assume that the elastic coefficients at zero elastic field ijkla , the piezoelectric coefficients kije and the dielectric constants ij at vanishing strain satisfy the following symmetry and positivity properties: ; ;ijkl jikl klij kij kji ij jia a a e e      (3) 0, , 0 :ij ij ji ijkl kl ij ij jie e e a e e e e     (4) L J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 282 1 1, 0 :i ij j i i i           (5) where 1 and 0 are constants; a superimposed bar denoting the complex conjugate. But, under the quasi-electrostatic approximation [9], there exists an electric scalar potential  such that: i i E x     (6) Moreover, if 0 0 0, ,j jt u w and 0 are prescribed values per unit area, the mechanical boundary conditions can be written as: 0 1j ij in t on    (7) 0 2j ju u on   (8) and the electric ones are in the following forms: 0 3i in D w on    (9) 40 on    (10) The piezoelectric plate is supposed to be clamped by 2 ;  and n represents the unit outer normal to . If the body force and extrinsic bulk charge are assumed to be negligible, and D are divergence-free, i.e. 0 ij j in x     (11) 0i i D in x     (12) On the other hand, we have:   0i ij ij Sn on     (13)   0i i Su u on    (14)   0 Son     (15)   0i i i Sn D D on    (16) The variational problem (VP) corresponding to Eqs. (7) to (12) is obtained by introducing the following spaces:  1 02 2; ( ),i i iV u u H u u     (17) J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 283  14 4; ( ), 0V H        (18) Problem (VP): Find 2 4( , )u inV V  such that: 0 2 1 0 4 4 ( , ) ( , ) ( , ) ( , ) a u v b v t v ds v V c d u w ds V                          (19) where ( , ) ( ) ( )ijkl kl ija u v a e u e v dv   (20) ( , ) iijk k j v b v e dv x x        (21) ( , ) ij j i c dv x x            (22) ( , ) kikl l i u d u e dv x x          (23) Proposition1. Problem (VP) is equivalent to Eqs. (7) to (12). Proof. (19)1 is obtained by multiplying (11) par a test function vi and by integrating by parts; taking into account the boundary conditions (7) and (8). By analogy, we obtain (19)2, by multiplying (12) par a test function  and by integrating by parts; taking into account the boundary conditions (9) and (10). The coefficients ijkla are assumed to be continuous on S . For the existence and uniqueness of the solution of problem (VP), see [9]. VARIATIONAL FORMULATION FOR THE PROBLEM OF A FISSURED PIEZOELECTRIC STRUCTURE e now consider a piezoelectric structure containing a closed crack C, i.e. C C (24) where C is the closure of C, and where C is assumed to be smooth. Let us introduce the open subset ,C verifying: C C    (25) The local equations of linear piezoelectricity for a fissured piezoelectric structure can then be written as follows [10]: 0 ij C j in x     (26) 0i C i D in x     (27) W J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 284 0iu on  (28)   0i iu N on C (29) 0 on   (30)   0 on C  (31) , , , 11 2 ( ) ( ) ( )ijkl k l j kij k j ijkl k l j kij k j NNkl k l kNN k ia u n e E n a u n e E n a u e E N on C      (32) , 0NNkl k l kNN ka u e E on C  (33) 1 2 0i i i iD n D n on C   (34) where N and n represent the unit normal to C, outer to its side 1, see (Fig. 2), and the unit outer normal to the open domain ,C C    respectively. Figure 2: Representation of the fissured piezoelectric structure C . These relations express a compression on C according to (33), as well as the normality of the force which acts; with an opposition between the action and the reaction, according to (32). Consequently, the variational formulation (FVP) for the problem of such a fissured piezoelectric structure can be stated as follows: Problem (FVP): Find * *( , ) uu inV V   such that: * * ( , ) ( , ) 0 ( , ) ( , ) 0 ua u v u b v u v V c d u V                       (35) where  1( ); ( ); 0u i i C iV u u u H u      (36)   * ( ); ; 0u ui i i iV u u u V u N    (37)  1; ( ); 0CV H        (38) J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 285   * ; ; 0V V      (39) And where a, b, c, and d are bilinear forms on * * * * *, , ,u u uV V V V and V V    respectively. Proposition2. Problem (FVP) is equivalent to Eqs. (26) to (34). The proof is analogous to that of proposition 1, by taking into account Eqs. (29) and (31). HOMOGENIZED EQUATIONS-FORMAL EXPANSION e now consider a linear piezoelectric plate with a   periodic distribution of fissures, so that, the period Y of 3 ,R admits a smooth fissure C verifying: C Y   (40) Figure 3: Representation of the period Y with a smooth fissure C. The fissured material denoted by C is then defined as follows:  1 2 3( , , );C Cxx x x x y Y Y C         (41) And we assume that, there is no fissure intersecting the boundary  of the open . Introducing the following spaces:  1( ); ( ); 0u i i C iV u u u H u       (42)   * ( ); ; 0u ui i i iV u u u V u N     (43)  1; ( ); 0CV H        (44)   * ; ; 0V V      (45) The corresponding variational formulation ( FVP ) of such a piezoelectric problem in ,C is then defined as follows: W J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 286 Problem ( FVP ): Find * *( , ) uu inV V     such that: * * ( , ) ( , ) 0 ( , ) ( , ) 0 ua u v u b v u v V c d u V                             (46) In order to study the asymptotic behavior of the solution when  tends to zero, we use the classical following expansions, both for the unknown and the test functions: 0 1 2 2( ) ( ) ( , ) ( , ) ...u x u x u x y u x y      (47) 0 1 2 2( ) ( ) ( , ) ( , ) ...x x x y x y        (48) 0 1 2 2( ) ( ) ( , ) ( , ) ...v x v x v x y v x y      (49) 0 1 2 2( ) ( ) ( , ) ( , ) ...x x x y x y        (50) Introducing the following spaces:  1( ); ( );uYC i i CV v v v H Y Y periodic    (51)  ( ); ; 0u uYC i YCV v v v V v     (52)   * ( ); ; 0u uYC i YC i iV v v v V v N on C    (53)  * *( ); ; 0u uYC i YCV v v v V v     (53)  1; ( );YC CV H Y Y periodic     (54)  ; ; 0YC YCV V       (55)   * ; ; 0YC YCV V      (56)  * *; ; 0YC YCV V       (57) Comparing (47)-(50) with (46), we get the following relations: 0 0 1 1 0 00 0 1 1 11 0 0 1 10 0 ( ) ( ) ( ) ( ) ( ) ( ) j j j j j jk k k ijkl ijkl ijkl l i l i l i j jk i i i i ijkl ijk ijk l i k j k j v u v u v uu u u a dx a dx a dx x x x y y x v uu v u v u a dy e dx e y y x x x y                                                             0 0 1 11 1 0 1 1 3 0 ( ) ( ) 0; , ( ( ))i i i iijk ijk k j k j dx v u v u e dx e dy v v H y x y y                                                   (58) J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 287 0 0 1 1 0 00 0 1 0 01 1 0 0 1 11 ( ) ( ) ( ) ( ) ( ) ( ) ij ij ij j i j i j i k k ij ikl ikl j i l i i i dx dx dx x x x y y x u u dy e dx e dx y y x x xl y e                                                                                            1 10 0 1 1 0 1 1 0 ( ) ( ) 0; , ( )k kkl ikl l i l i u u dx e dy H y x y y                                                 (59) with  represents the operator average defined on any Y-periodic function f(y) by: 1 ( )f f y dyYY   (60) In the particular case where 1 1 1 1 ,v u and    we get: 0 00 1 0 00 1 0 1 1 3 0 ( ) ( ) 0; , ( ( )) j jk k i i ijkl ijk l l i k k j v uu u v u a dx e dx x y x x y x v v H                                         (61) 0 10 0 0 00 1 0 1 1 0 ( ) ( ) 0; , ( ) k k ij ikl j j i l l i u u dx e dx x y x x y x H                                               (62) Consequently, the corresponding homogenized equations in which there are no more fissures then follow: 0 0 1 0 1 0 0 ij j k k ij ijkl ijk l l k k x u u a e x y x y                                      (63) and 0 0 10 1 0 0 i i k k i ij ikl j l l D x u u D e xj y x y                                 (64) J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 288 HOMOGENIZED LOCAL PROBLEM IN THE PERIOD Y et us choose the fields 1 1v and  as: 1 1 1 * ( , ) ( , ) ( ) ( , ) ( ); 0 1; u u u u u u YC v x y u x y w y u x y D w V                  (65) 1 1 1 * ( , ) ( , ) ( ) ( , ) ( ); 0 1; YC x y x y w y x y D w V                           (66) where ( )D  represents the set of the infinitely derivable functions with compact support in  . When we take (64) and (65) into account, then the comparison of (58) and (59) with (60) and (61) respectively, gives the following equations, 10 1 10 1( ) ( ) 0 ( ); 0 1 u u j j u uk k i i ijkl ijk l l i k k j u u w uu u w u a dy e dy x y y x y y D                                                          (67) 0 11 10 1 ( ) ( ) 0 ( ); 0 1 k k ij ikl j j i l l i u uw w dy e dy x y y x y y D                                                               (68) Therefore, we locally obtain, respectively: 10 1 10 1 * ( ) ( ) 0; u u j j u uk k i i ijkl ijk YC l l i k k j w uu u w u a e w V x y y x y y                                           (69) 0 11 10 1 *( ) ( ) 0;k kij ikl YC j j i l l i u uw w e w V x y y x y y                                             (70) and the local homogenized problem (LHP) in Y, then follows: Problem (LHP): Find 1 1 * *( , ) u uYC YCu inV V   such that we obtain, for given 0 0( ) ( )u x and x : 10 1 10 1 * ( ) ( ) 0 u u j jk k i i ijkl ijkY Y l l i k k j u u YC w uu u w u a dy e dy x y y x y y w V                                               (71) and L J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 289 0 11 10 1 * ( ) ( ) 0k kij iklY Y j j i l l i YC u uw w dy e dy x y y x y y w V                                                   (72) All these results can then be summarized through the following proposition: Proposition3. Under the expansions (47) and (48) for the solution ( ( , ), ( , ))u x y x y  of Problem (46), the first term 0 0( ( ), ( ))u x x satisfies Eqs. (62)-(63) and appropriate boundary conditions. Furthermore, for given 0 0( ( ), ( )),u x x the field 1 1( ( , ), ( , ))u x y x y is the solution of the nonlinear problem (LHP; Eqs. 70-71), and ( 0ij , 0 iD ) is therefore, defined as functions of 0 0( ( )) ( ( )).x xgrad u x and grad x So, Eqs. (70)-(71) represent a nonlinear piezoelectric law. ANALYSIS OF THE (STRAIN, ELECTRIC POTENTIAL)-(STRESS, ELECTRIC DISPLACEMENT) LAW Remark1. Problem (LHP) can be written as in the following simplified form: Find 1 1 * *( , ) u uYC YCu inV V   such that we obtain, for given 0 0( ) ( )u x and x :     0 1 1 0 1 1 * ( ) ( ) ( ) ( ) ( ) ( ) 0 (.) (.) (.) (.) ; (.) ; (.) u ijkl klx kly jiyY u ijk kx ky ijyY k k klx kly kx l l l u u YC a h u h u h w u dy e h h h w u dy h h h x y x w V                             (73) and    0 1 1 0 1 1 * ( ) ( ) ( ) ( ) ( ) ( ) 0 (.) (.) ij jx jy iy ikl klx kly iyY Y iy k YC h h h w dy e h u h u h w dy h y w V                          (74) Remark2. Denoting 0 1 1 1 0 0( ); ( ); ; ; ;kl klx kl kly ij ij i iH h u h h u u u D D         (75) Problem (LHP) can then be formulated as follows: Find * *( , ) u uYC YCu inV V   such that we obtain, for given 6 3 kl kH and H R R :     * ( ) ( ) ( ) ( ) 0u uijkl kl kl ji ijk k k ijY Y u u YC a H h u h w u dy e H h h w u dy w V                (76) J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 290 and     * ( ) ( ) ( ) ( ) 0ij j j i ikl kh kl iY Y YC H h h w dy e H h u h w dy w V                     (77) Therefore, 0ij ij  and 0 i iD D can be written as follows:         ( ) ( ) ( ) ( ) ij ijkl kl kl ijk k k i ij j j ikl kl kl a H h u e H h D H h e H h u              (78) So, the homogenized (strain, electric potential)-(stress, electric displacement) law is characterized by the function defined by: ( , ) ( , )kl kl ij iH h D   (79) Nevertheless, for the study of (79), let us introduce the following functions, defined from 6 3R R towards R , by:       1 ( , ) ( ) ( ) 2 1 ( ) ( ) 2 sm s ijkl ij ij lm lmY ijk i i jk jkY W H H a H h u H h u dy Y e H h H h u dy Y          (80)       * 1( , ) ( ) ( ) 2 1 ( ) ( ) 2 sm s ij i i j jY ikl ik ik l lY W H H H h H h dy Y e H h u H h dy Y             (81) Moreover, the proposition that follows presents the main result of this analysis: Proposition 4. The functions defined above, through (80) and (81), are of class C1, positive; and D   satisfying the following relations: * 1 2 1 2 ij ij i i W H W D H              (82) Proof. As u and  are continuous functions of     1,2,3, 1,2,3ij i ii jH H and H H   , defined from 6 3( . )respR R towards ( . ),uYC YCV resp V  *W and W are then of class 0 .C Let us now introduce: * * ( ) ( ) ij ij ij i i i h h u H and h h u H         (83) J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 291 So, by virtue of (3)-(5), we can formulate the variation of W and W* as follows: * * * * * * * * * * * * * 1 1 1 2 2 1 1 2 2 1 1 1 2 2 1 2 ijkl ij kl ijkl ij kl ijk i jkY Y Y ijk i jk ijk i jkY Y ijkl ij kl ijkl ij kl ijk i ijY Y Y kl kl W a h h dy a h h dy e h h dy Y Y Y e h h dy e h h dy Y Y a h h dy a h H dy e h h dy Y Y Y H                                (84) * * * * * * * * * * * * * 1 1 1 2 2 1 1 2 2 1 1 1 2 2 2 ij i j ij i j ikl ik lY Y Y ikl ik l ikl ik lY Y ij i j ikl ik l j jY Y W h h dy h h dy e h h dy Y Y Y e h h dy e h h dy Y Y h h dy e h h dy D h Y Y                                 (85) Taking (76) and (77) into account, we get: 1 2 1 2 ij kl i i W H W D H              (86) CONCLUSION rom the variational formulation of the three-dimensional problem of Linear Piezoelectricity, we deduced that corresponding to a cracked piezoelectric structure. Considering afterward the case of a structure presenting a periodic distribution of cracks, we managed to build, on the homogenization period, the homogenized formulation of the corresponding problem, as a result of an asymptotic development of the solution. A non- linear law between the mechanical strain and the electric potential on one hand, and the mechanical stress and the electric displacement on the other hand, has been then established. REFERENCES [1] Dieulesaint, E., Royer, D., Ondes élastiques dans les solides. Application au traitement du signal, Paris, (1974). [2] Alshits, I., Darinskii, A.N., Lothe, J., On the existence of surface waves in half-anisotropic elastic media with piezoelectric and piezomagnetic properties, Wave. Motion., 16 (1992) 265 – 283. [3] Li, J.Y., Dunn, M.L., Micromechanics of magnetoelectroelastic composite materials: average fields and effective behavior, J. Intell. Mater. Syst. Struct., 7 (1998) 404 – 416. [4] Zhag, T.Y., Tong, P., Fracture mechanics for a mode III crack in a piezoelectric material, Int. J. Solids. Struct, 33 (1996) 343 – 359. [5] Suo, Z., Kuo, C.M., Barnett, D.M., Willis, J.R., Fracture mechanics for piezoelectric ceramics, J. Mech. Phys. Solids, 40 (1992) 739 – 765. [6] Gao, C.F., Wang, M.Z., Periodical cracks in piezoelectric media, Mech. Rech. Commun, 26 (1999) 427 – 432. F J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 292 [7] Wang, X.D., Meguid, S.A., Modelling and analysis of the dynamic behavior of piezoelectric materials containing interacting cracks, Mech. Materials, 32 (2000) 723 – 737. [8] Han, J.C., Wang, B.L., Electromechanical model of periodic cracks in piezoelectric materials, Mech. Materials, 37 (2005) 1180 – 1197. [9] Turbé, N., Maugin, G.A., On the linear piezoelectricity of composite materials, Math. Methods. Applied. Sciences, 14 (1991) 403 – 412. [10] Sanchez-Palencia, E., Non Homogeneous media and Vibration Theory, Lecture Notes in Physics., Springer, Berlin, (1980). << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. 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