Microsoft Word - 2289 M. K. Hussain et alii, Frattura ed Integrità Strutturale, 48 (2019) 599-610; DOI: 10.3221/IGF-ESIS.48.58 599 Focused on “Showcasing Structural Integrity Research in India” Evaluation of mixed mode (I/II) notch stress intensity factors of sharp V-notches using point substitution displacement technique Mirzaul Karim Hussain, K.S.R.K. Murthy Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati-781039, India mirzaul@iitg.ac.in, https://orcid.org/0000-0003-4300-360X ksrkm@iitg.ac.in, https://orcid.org/0000-0002-9112-8557 ABSTRACT. In this paper, determination of the accurate notch stress intensity factors (NSIFs) have been demonstrated using a recently proposed technique: The point substitution displacement technique (PSDT) for the sharp V- notched configurations. In this technique, certain optimal point(s) on the notch flanks are obtained where the displacements are found to be highly accurate. Using the PSDT, the NSIFs are determined from the finite element (FE) displacements at these optimal point(s). The NSIFs of two pure mode I and two mixed mode (I/II) examples have been determined and excellent agreement of the present results with the published results is observed. The PSDT is efficient, robust and easy to be implemented in the available FE code. KEYWORDS. V-notch; NSIF; Stress intensity; Finite element; Mixed mode Citation: Hussain, M. K., Murthy, K. S. R. K., Evaluation of mixed mode (I/II) notch stress intensity factors of sharp V-notches using point substitution displacement technique, Frattura ed Integrità Strutturale, 48 (2019) 599-610. Received: 30.11.2018 Accepted: 25.02.2019 Published: 01.04.2019 Copyright: © 2019 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 harp V-notches are frequently encountered in many engineering applications and structures. Due to the presence of these notches, a very high local stresses and stress gradient is observed close to the notch tip. The presence of notch drastically decreases the load-bearing capacity of the component and may cause damage to structures subjected to both static and variable loads. Therefore, a very accurate assessment of the strength of the singularity is desired near the notch tip. Williams [1] investigated plane V-notched configurations. In linear elastic fracture mechanics (LEFM), notch stress intensity factors (NSIFs) are widely used to characterize the singular stress field in the vicinity of the notch tip. The singularity stress field in the vicinity of the notch tip can be stated in the form of   1 ijKr f here K is the notch stress intensity factor, r is the radial distance from the notch tip,   1 is the order of stress singularity and  ijf are the angular stress components [1]. Due to the widespread use of the NSIFs in brittle [2-7] and fatigue [8, 9] fracture criteria, many efforts have been put forward to develop various post-processing techniques [10-29] to calculate NSIFs. Gross and Mendelson [10] and Carpenter [11] determined NSIFs of sharp V-notches using the boundary collocation method. Chen [12] proposed a body force method to evaluate the NSIFs for problems under tensile and in-plane bending loadings. Noda et al. [13] also used body force method to calculate the NSIFs. Ju and Chung [14] and Liu et al. [15] used S http://www.gruppofrattura.it/VA/48/2289.mp4 M. K. Hussain et alii, Frattura ed Integrità Strutturale, 48 (2019) 599-610; DOI: 10.3221/IGF-ESIS.48.58 600 the least-squares method for determining the NSIFs using the stresses obtained from finite element method (FEM). As these least-squares methods [14, 15] use the stress field, they require either very fine meshes at the notch tip or higher order Williams coefficients to attain the accurate NSIFs. Triefi et al. [16] developed a strain energy method to determine the mixed mode NSIFs of sharp V-notches. In another research, Lazzarin et al. [17] proposed a strain energy density technique for the rapid evaluation of NSIFs. Many researchers used fractral-like finite element method [18-20] and extended finite element method [21, 22] to calculate the NSIFs. Other researchers employed various path independent integrals [23-26] to estimate the NSIFs of sharp V-notches. Apart from these methods, Ayatollahi and Nejati [27] determined the NSIFs along with the higher order terms of Williams coefficients using an overdeterministic method utilizing FE displacements. Recently, Hussain and Murthy [28, 29] developed a collocation technique and a point substitute displacement technique (PSDT) to obtain the mixed mode (I/II) NSIFs using the FE displacements along the notch flank. In their method [29], certain optimal point(s) of the notch tip element along the notch flanks have been identified by minimizing the error in displacements, and interestingly, the displacements and its slope are found to be more accurate at those points. The PSDT utilizes the displacements on the notch flanks at these optimal point(s) to calculate NSIFs accurately. This method is efficient, simple and straightforward to be implemented in the available FE code and can be employed to find the NSIFs using manual calculations. It is worth noting here that, unlike the availability of well-known quarter point elements [30, 31] in the crack problems, no such popular singular elements are currently available for use in the sharp V-notched configurations. Consequently, any post-processing techniques for the determination of the NSIFs should be capable enough to determine the accurate NSIFs yet using the conventional elements at the notch tip and concurrently it should be simple enough to implement in the existing code. The main thrust of the present work is to demonstrate further the efficacy of the above technique, PSDT proposed by Hussain and Murthy [29] in terms accurate estimation of the NSIFs of the specimens having straight and curved boundaries under mode I and mixed mode (I/II) loading conditions. THEORETICAL BACKGROUND short description of the formulations for PSDT is presented in this section. However, for more details, one may refer to Ref. [29]. For a homogeneous elastic 2D problems containing sharp V-notch (in Fig. (1)), the displacement field at any nearby point  ,P r under any arbitrary in-plane loading can be given as [14]                                                           1 1 Re cos 2 cos 2 cos cos 2 2 Re cos 2 cos 2 sin sin 2 2 2 for I n II n I I I I I In n n n n n R n II II II II II IIn n n n n n R n n A u r u G B r u G n B (1)                                                         1 1 Re cos 2 cos 2 sin sin 2 2 Re cos 2 cos 2 cos cos 2 2 2 for I n II n I I I I In n n n n n n II II II II II IIn n n n n n R n n A v r G B r v G n B (2) where nA and nB are Williams’ coefficients corresponding to n -th term for mode I and mode II, respectively,  ,r denotes the polar coordinate components,  Re denotes real part of the variables, Kolosov constant  is equal to      3 1 for plane stress and 3 4 for plane strain conditions,   ( 180 2 ) is the notch angle (Fig. (1)),   2 1G E is the shear modulus,  and E are the Poisson’s ratio and Young’s modulus, respectively. In and  IIn are the mode I and mode II eigenvalues, respectively, and can be obtained by solving the following characteristic Eqn. (3). The constants displacements I Ru , II Ru and II Rv can be shown in Eqn. (4) [27]. A M. K. Hussain et alii, Frattura ed Integrità Strutturale, 48 (2019) 599-610; DOI: 10.3221/IGF-ESIS.48.58 601             sin 2 sin 2 0 sin 2 sin 2 0 I I n n II II n n (3)             0 2 0 2 1 1 1 ; sin ; ( cos ) 2 2 2 I II II R R Ru A u B r v B B r G G G (4) where 0A and 0B are Williams’ coefficients corresponding to the rigid body translation and 2B is Williams’ coefficient corresponding to the rigid body rotations. Figure 1: A notch geometry with a local coordinate system. Mode I and mode II NSIFs can be defined as [10]                                                  1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 lim 2 0 2 1 cos 2 cos 2 lim 2 0 2 1 cos 2 cos 2 I II I I I I I y r II II II II II xy r K r A K r B (5) Considering only the singular terms and constant displacement terms, the displacement field at any nearby point  ,P r under any arbitrary in-plane loading can be given as [14]                                                 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 2 1 cos 2 cos 2 cos cos 2 2 2 1 cos 2 cos 2 sin sin 2 sin 2 2 I II I I I I I II II II II II A u A r G G B r B r G G (6)                                                1 1 1 1 1 1 1 0 1 1 1 1 1 1 2 1 cos 2 cos 2 sin sin 2 2 2 1 cos 2 cos 2 cos cos 2 cos 2 2 I II I I I I I n II II II II II A v r B G G B r B r G G (7) where 1A and 1B are Williams coefficients for the singular terms for mode I and mode II, respectively. It can be shown that the notch opening displacement (NOD) can be written as [29]                                       1 1 1 1 1 1 1 1 1 1 cos 2 cos 2 sin 2 2 2 2 sin 2 I I I I I I I I I I A v v v v r A C r G (8) M. K. Hussain et alii, Frattura ed Integrità Strutturale, 48 (2019) 599-610; DOI: 10.3221/IGF-ESIS.48.58 602 Similarly, the notch sliding displacement (NSD) can be obtained as [29]                                             1 1 1 1 1 1 1 1 2 1 2 2 3 cos 2 cos 2 sin 2 sin 2 1 sin II II II II II II II II II II B u u u u r G B r B C r B C r G (9) where 1C , 2C and 3C are constants which depend upon eigenvalues,  and  . Thus, using Eqn. (8) the coefficient 1A can be calculated and 1B can be calculated from the NSD (Eqn. (9)) under all loading conditions (pure mode I, pure mode II and mixed mode (I/II)). Assuming the isoparametric quadratic quadrilateral elements are deployed at the notch tip, the FE displacement along notch flank nodes 1–2–4 (Fig. (2)) can be written with r being the distance from the notch tip as [29]     2FEv A r Br (10) where the constants A and B are constants and can be obtained from the FE displacements using the following equation                  12 2 2 2 2 4 4 4 FE FE A r r v B r r v (11) where 2 FEv and 4 FEv are FE displacements at nodes 2 and 4, respectively, and 2r and 4r are distances of nodes 2 and 4, respectively, from notch tip 1. The FE NOD can be expressed as   22 2FEv Ar Br (12) Figure 2: A notch flank finite elements around a notch tip. The residual  between the analytical NOD (Eqn. (8)) and FE NOD (Eqn. (12)) can be written as          1 22 2 1 12 2 2 IFEv v A C r Ar Br (13) M. K. Hussain et alii, Frattura ed Integrità Strutturale, 48 (2019) 599-610; DOI: 10.3221/IGF-ESIS.48.58 603 By minimizing the function  with respect to r and after some algebraic manipulations, r and 1A can be obtained as [29]            1 1 1 2 I op I B r r A (14)       1 1 2 1 1 1 op I I FE r rop op op op vAr Br A C r C r (15) Thus, by using the notch opening displacement  FEv at opr , 1A can be obtained using the Eqn. (15) and hence IK using the Eqn. (5). Taking logarithm on both sides of Eqn. (8), one can write  1 1 1ln ln ln Iv r A C (16) It can be seen that, Eqn. (16) is a straight line equation whose slope is equal to 1 I . It can be shown that the slope of the plot of     ln FEv versus  ln / Nr L (Eqn. (10)) becomes equal to 1 I at the optimal point optr (Refer [29] for more details). Here NL is the notch tip element length as shown in Fig. (2). Similar to the mode I problems, estimation of the opr or optr from u -displacement for mode II is not possible as   FEu (or   FEu ) contains the term IIRu (Eqn. (9)). Therefore, to calculate the mode II NSIFs, it is presumed that the mode II u - displacements are also accurate at the optimal points obtained for the mode I problems using Eq. (14). To calculate both the constants 1B and 2B , two nearby points  op opt Nr r L and    op op opt Nr r r L are considered [29]. Therefore, from the FE u -displacements at the two close points opr and opr , the Williams’ coefficient 1B can be obtained as [29]            1 1 1 1 1 2 op op II II FE FE r op r op op op op op u r u r B C r r r r (17) Again, by substituting FE NSD  op FE ru and  op FE ru at opr and opr , 1B can be obtained using the Eqn. (17) and hence IIK using the Eqn. (5). Finally, the NSIFs can be normalized as       ( ) 11 ( ) ( ) ( ) I II I II I IIF K a and  ( ) 1 ( ) ( ) I II I II I IIF K R F for the rectangular and circular plates, respectively. NUMERICAL EXAMPLES his section presents a numerical examination of the PSDT for the evaluation of the NSIFs under mode I and mixed mode (I/II) loading conditions. The FE analysis is carried out in ANSYS® [32]. Throughout the domain conventional eight noded quadratic elements are used and no singular elements are utilized at the notch tip. At the notch tip, they are collapsed to form a spider web pattern. Four problems viz. (a) Mode I example of a single edge notched plate under uniform tension (SENT), (b) mode I example of a single edge notched plate under in-plane bending (SENB), (c) mixed mode problem of an angled single edge notched plate under uniform tension (ASENT) and (d) mixed mode problem of a sharp V-notched Brazilian disc (SV-BD) are considered for the demonstration of the efficacy of the PSDT. The notch length to width ratio  0.5a w is considered for the single edged notch plates and the notch length to radius ratio  0.5a R is considered for the Brazilian disc. It is assumed that all the specimens are under plane stress condition. Consistent units are employed in all the examples. The thickness of all the specimens is considered as unity. T M. K. Hussain et alii, Frattura ed Integrità Strutturale, 48 (2019) 599-610; DOI: 10.3221/IGF-ESIS.48.58 604 Example 1: Mode I Problems The first set of examples considered are single edge notched plate under uniform tension (SENT) and single edge notched plate under in-plane bending (SENB) as pure mode I examples (as shown in Fig. (3)). The geometrical parameters considered for the SENT and SENB are as follows:  1.0w and  1.2h as shown in Fig. (3). The uniform tensile loading  (=1.0) and a bending loading 3 .2 /h w (=7.2) are considered for the SENT and SENB specimens, respectively (Fig. (3a) and 3(b)). The notch angles   0°, 15°, 30°, 45°, 60°, 75°, 90, 105° and 120° are considered for validating the present method. Young’s modulus  1E and Poisson’s ratio   0.25 are considered. One-half (shaded portion in Fig. (3a) and Fig. (3b)) of the geometries is modeled due to the symmetry with appropriate boundary conditions. The typical mesh with 211 number of elements (NE) and 702 number of nodes (NN) used for both the specimens is shown in Fig. (3c). The notch tip element to notch length ratio is considered as  0.025NL a for both the specimens. The optimum points  opt op Nr r L obtained using Eq. (14) are found to be centered around 0.79 for all the notch angles in the present examples (for a detailed description, one may refer to Ref. [29]). It has been observed that  FEv at the optimal points optr obtained using Eqn. (14) are very accurate. The mode I NSIFs calculated at the respective optimal points and are listed in Tab. (1) and Tab. (2) for the SENT and SENB specimens, respectively. Figure 3: (a) Single edge notched plate under uniform tension (SENT), (b) single edge notched plate under in-plane bending (SENB) and (c) typical FE mesh used for both SENT and SENB specimens (NE=211, NN=702).    IF Present Gross and Mendelson [10] Zhan and Hahn [33] Chen [12] 0 2.7856 2.8237 2.8301 – 15 2.8019 – – – 30 2.8279 2.8448 2.8408 2.8440 45 2.8775 – – – 60 2.9620 2.9716 2.9937 2.9700 75 3.1005 – – – 90 3.3191 3.3244 3.3716 3.3220 105 3.6532 – – – 120 4.1578 4.1458 4.0133 – Table 1: Normalized NSIF IF for the SENT specimen (  0.5a w ). M. K. Hussain et alii, Frattura ed Integrità Strutturale, 48 (2019) 599-610; DOI: 10.3221/IGF-ESIS.48.58 605    I F Present Chen [12] 0 1.4881 – 15 1.4922 – 30 1.5000 1.5010 45 1.5185 – 60 1.5537 1.5530 75 1.6153 – 90 1.7164 1.7140 105 1.8749 – 120 2.1185 – Table 2: Normalized NSIF IF for the SENB specimen (  0.5a w ). It can be observed that the NSIFs calculated from the present technique are in excellent agreement with the available results (% relative error is less than 1% with respect to Chen [12] results). Due to the implementation of the NOD, very accurate values of NSIFs can be attained with coarse meshes. Example 2: Mixed Mode Example of an Angled Single Edge Notched Plate under Uniform Tension (ASENT) The third example considered is a mixed mode example of an angled single edge notched plate under uniform tension (ASENT) to analyze the efficiency of the PSDT under mixed mode loading. The geometry and loading for ASENT are as follows:  1.0w ,  3.5h and   1.0 as shown in Fig. (4a). The notch inclination angles  =15°, 30° and 45° and notch angles   0°, 30°, 60°, and 90° are considered for determining the mixed mode NSIFs. A Poisson’s ratio   0.25 and Young’s modulus  1E are considered. A typical FE mesh for the ASENT is shown in Fig. (4b) for   30 and   30 with 531 number of elements and 1723 number of nodes. Fig. (4c) shows the enlarged portion near the notch tip with  0.025NL a . Figure 4: (a) Angled single edge notched plate under uniform tension (ASENT), (b) typical FE mesh used for the ASENT specimen (NE=531, NN=1723) and (c) mesh arrangement near the notch tip portion. M. K. Hussain et alii, Frattura ed Integrità Strutturale, 48 (2019) 599-610; DOI: 10.3221/IGF-ESIS.48.58 606 Similar to the pure mode I problem 1A can be obtained from  FEv at optr using Eqn. (14). 1B can be obtained from  FEu at two radii (  op opt Nr r L and    op op op Nr r r L ) using Eqn. (17).  opr can be considered as 0.002 for calculating the mode II NSIFs [29]. The normalized mixed mode (I/II) NSIFs obtained using the PSDT are listed in Tab. (3). Once again, the results obtained from the present method show a good agreement with the published results for various  and  . The results in Tab. (3) show that using PSDT by the mere substitution of FE NOD and NSD at the optimal points accurate values of mixed mode (I/II) NSIFs can be obtained.       I F IIF Present Chen [12] Present Chen [12] 15 0 2.7108 – 0.3970 – 30 2.7507 2.7670 0.5468 0.5410 60 2.8790 2.8870 0.7764 0.7660 90 3.2257 3.2300 1.1220 1.1160 30 0 2.0399 – 0.7122 – 30 2.5315 2.5460 1.0190 1.0100 60 2.6447 2.6530 1.4619 1.4280 90 2.9587 2.9640 2.0576 2.0950 45 0 2.1792 – 1.0078 – 30 2.2039 2.2170 1.3615 1.3520 60 2.2937 2.3040 1.9383 1.9110 Table 3: Normalized NSIFs IF and IIF for the ASENT specimen (  0.5a w ). Example 3: Mixed Mode Example of a Sharp V-notched Brazilian Disc (SV-BD) A sharp V-notched Brazilian Disc (SV-BD) as shown in Fig. (5) is considered as the fourth example for analyzing mixed mode (I/II) problem. The radius of the SV-BD is taken as  60R and a compressive loading 1.0F is applied as shown in Fig. (5). The notch length to radius ratio  0.5a R , notch inclination angles   0º, 10º, 20° and 30° and notch angles   30°, 60°, and 90° are considered for determining the mixed mode NSIFs for the SV-BD. A Poisson’s ratio   0.25 and Young’s modulus  1E are considered. The boundary conditions used for the FE analysis are shown in Fig. (5a). Fig. (5b) shows the typical mesh used for the SV-BD (NE=1833 and NN=5643). Fig. (5c) shows the enlarged portion near the notch tip. Figure 5: (a) Sharp V-notched Brazilian Disc (SV-BD), (b) typical FE mesh used for the SV-BD specimen (NE=1833, NN=5643) and (c) mesh arrangement near the notch tip portion. M. K. Hussain et alii, Frattura ed Integrità Strutturale, 48 (2019) 599-610; DOI: 10.3221/IGF-ESIS.48.58 607 The mode I and mode II NSIFs are also calculated in a similar way to that of the ASENT specimen. Tab. (4) lists the results of mode I and mode II NSIFs obtained using the PSDT. For comparison, the numerical and experimental results of Ayatollahi and Nejati [34] are also presented in Tab. (4) and it has been noticed that the present results are in good agreement with the published results. Thus, the results in all the examples indicate that the PSDT is simple and robust. The FE displacements at the optimal points are more accurate than the other points on the notch tip element. The PSDT can provide very good results with coarse mesh even with convention elements at the notch tip.       IF IIF Present Ayatollahi and Nejati [34] (FEM) Ayatollahi and Nejati [34] (Experimental) Present Ayatollahi and Nejati [34] (FEM) Ayatollahi and Nejati [34] (Experimental) 0 30 0.7213 0.7100 0.7400 0.0027 0.0000 0.0200 60 1.1530 1.1300 1.1700 0.0000 0.0000 0.0500 90 2.3108 2.2900 2.2900 0.0000 0.0000 0.0600 10 30 0.5727 0.5600 0.6000 0.7470 0.7200 0.7200 60 0.9599 0.9400 0.9900 1.2619 1.1700 1.1100 90 2.0187 2.0000 1.9500 2.1169 2.0200 2.1300 20 30 0.2148 0.2000 0.2200 1.2006 1.1800 1.1800 60 0.4940 0.4800 0.4800 1.9260 1.8600 1.8900 90 1.3013 1.2900 1.2700 3.0792 3.1200 3.0300 30 30 0.1980 – – 1.3447 – – 60 0.0555 -0.0100 0.0900 2.0788 2.0200 1.9200 90 0.4395 – – 3.2716 – – Table 4: Comparison of the normalized NSIFs IF and IIF for the SV-BD specimen ( a w 0.5). CONCLUSIONS n this paper, the efficacy of an FE based NSIF extraction technique: the point substitution displacement technique (PSDT) has been demonstrated by determining the NSIFs of various configurations under mode I and mixed mode (I/II) loading conditions. The NSIFs are determined from the FE notch flank displacements at the optimum point(s) in the notch tip elements. The efficacy of the present method is examined with two mode I and two mixed mode (I/II) problems. The results obtained using the present method is validated with the available results. The results show that the PSDT is simple, robust and easy to be implemented in the available FE code. Further, accurate values of mode I and II NSIFs can be obtained using coarse meshes. REFERENCES [1] Williams, M.L. (1952). Stress singularities resulting from various boundary conditions in angular corners of plates in extension, J. Appl. Mechs., 19, pp. 526–528. [2] Dunn, M.L. and Suwito, W. (1997). 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NOMENCLATURE a notch length ,A B parameters related to finite element displacements 0 0 2, ,A B B parameters related to rigid body motion  1, 2, 3,...nA n Williams coefficients for mode I  1, 3, 4,...nB n Williams coefficients for mode II E Young’s modulus ,I IIF F modes I and II normalized notch stress intensity factors F compressive point load on the sharp V-notched Brazilian disc G shear modulus h semi-height of the notched plate NL notch tip element length ,I IIK K modes I and II notch stress intensity factors R radius of the sharp V-notched Brazilian disc  residual  ,r polar coordinate components optr optimum point opr optimum radius ,u v notch field displacement w plate width  parameter related to notch angle  notch inclination angle  notch opening angle  Kolosov constant  1 1, I II modes I and II eigenvalues correspond to the singularity term  ,I IIn n modes I and II eigenvalues correspond to the n -th term  Poisson’s ratio  far field stress  ,u v notch opening and sliding displacements M. K. Hussain et alii, Frattura ed Integrità Strutturale, 48 (2019) 599-610; DOI: 10.3221/IGF-ESIS.48.58 610  ,FE FEv u finite element notch opening and sliding displacements ASENT angled single edge notched plate under uniform tension FE finite element FEM finite element method NE number of elements NN number of nodes NOD notch opening displacement NSD notch sliding displacement NSIF notch stress intensity factor PSDT point substitution displacement technique SENB single edge notched plate under in-plane bending SENT single edge notched plate under uniform tension SV-BD sharp V-notched Brazilian disc << /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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