Microsoft Word - numero_37_art_19 M. A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 37 (2016) 138-145; DOI: 10.3221/IGF-ESIS.37.19 138 Focussed on Multiaxial Fatigue and Fracture A multiaxial incremental fatigue damage formulation using nested damage surfaces Marco Antonio Meggiolaro, Jaime Tupiassú Pinho de Castro Pontifical Catholic University of Rio de Janeiro, PUC-Rio, R. Marquês de São Vicente 225, Rio de Janeiro, 22451-900, Brazil meggi@puc-rio.br jtcastro@puc-rio.br Hao Wu School of Aerospace Engineering and Applied Mechanics Tongji University, Siping Road 1239, 200092, Shanghai, P.R.China wuhao@tongji.edu.cn ABSTRACT. Multiaxial fatigue damage calculations under non-proportional variable amplitude loadings still remains a quite challenging task in practical applications, in part because most fatigue models require cycle identification and counting to single out individual load events before quantifying the damage induced by them. Moreover, to account for the non-proportionality of the load path of each event, semi-empirical methods are required to calculate path-equivalent ranges, e.g. using a convex enclosure or the MOI (Moment Of Inertia) method. In this work, a novel Incremental Fatigue Damage methodology is introduced to continuously account for the accumulation of multiaxial fatigue damage under service loads, without requiring rainflow counters or path-equivalent range estimators. The proposed approach is not based on questionable Continuum Damage Mechanics concepts or on the integration of elastoplastic work. Instead, fatigue damage itself is continuously integrated, based on damage parameters adopted by traditional fatigue models well tested in engineering practice. A framework of nested damage surfaces is introduced, allowing the calculation of fatigue damage even for general 6D multiaxial load histories. The proposed approach is validated by non-proportional tension- torsion experiments on tubular 316L stainless steel specimens. KEYWORDS. Multiaxial fatigue; Variable amplitude loads; Non-proportional multiaxial loads; Nested fatigue damage surfaces; Incremental damage calculation. INTRODUCTION ost fatigue crack initiation models need to properly identify load events before computing the damage induced by them. Hence their fatigue damage calculation routines need to include cycle counting algorithms like the well-known rainflow methodology for uniaxial loads. Cycle counting is necessary because traditional fatigue models are discrete in nature, since they only can accumulate damage after a load event (e.g. a half-cycle) is properly identified, detected e.g. from a load reversal or from a hysteresis loop that closes. However, the detection and counting of loading events can be a quite challenging task under multiaxial non-proportional (NP) histories. The existing multiaxial rainflow algorithms [1] are not trivial to apply. In fact, they are not even robust, since they can output very different half- M M. A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 37 (2016) 138-145; DOI: 10.3221/IGF-ESIS.37.19 139 cycles depending on the choice of the initial counting point of a periodic load history [2, 3]. Furthermore, multiaxial fatigue damage evaluation requires the semi-empirical calculation of path-equivalent stress or strain ranges from the rainflow-counted paths, increasing even more its computational burden [4]. On the other hand, a completely different fatigue calculation approach assumes damage as a continuous variable, whose increments can be computed as the loading proceeds. Most works based on this idea use Continuum Damage Mechanics concepts [5], which need to be supplemented by purely phenomenological damage evolution equations that are difficult to calibrate, to say the least. In fact, despite their academic appeal, such models remain controversial and have not found a wide acceptance in the fatigue design community. Other continuous damage approaches are based on an integration of elastoplastic work. However, the accumulated total work required to initiate a microcrack by fatigue certainly is not a material property. Moreover, the elastoplastic work still depends on the number of cycles, thus it is impossible to calculate without previous load cycle and/or reversal detection. Therefore, even if it could be assumed that fatigue damage can be quantified by this parameter, its calculation routine still would need to include a rainflow or other similar load event counter. Alternatively, instead of integrating dubious strain energy or energy-based damage parameters, a more reasonable path is to continuously quantify fatigue damage itself, using some well-proven model that can properly describe multiaxial fatigue damage in the material in question. The so-called Incremental Fatigue Damage (IFD) approach integrates the chosen parameter until reaching 1.0 or any other suitable critical-damage value using traditional accumulation concepts, as originally performed a long time ago for the uniaxial case by Wetzel under Topper's guidance in 1971 [6], and again by Chu in 2000 [7]. It is important to emphasize that in such calculations fatigue damage is continuously computed after each infinitesimal stress or strain increment, so its quantification does not require the prior identification of load cycles. In this work, the IFD approach is revisited and extended to general multiaxial fatigue problems, including no-proportional ones, based on a direct analogy with non-linear incremental plasticity concepts, however calculating damage instead of plastic strains at each load increment. THE INCREMENTAL FATIGUE DAMAGE APPROACH he Incremental Fatigue Damage approach was proposed for uniaxial load histories in Wetzel and Topper’s rheological model [6, 8]. It makes use of the derivative of the normal stress  with respect to damage D, called here generalized damage modulus D, thus D d dD D dD (1 D ) d        (1) Consider, for instance, a uniaxial constant amplitude loading history with stress amplitude a. During a loading half-cycle, the excursion of the stress  from a to a could be integrated according to Eq. (1) to find the associated fatigue damage D  1/2N, however without explicitly calculating the fatigue life N. Assuming the material is initially virgin, the damage D from the first half-cycle is initially zero in the initial valley when a and thus a)  0, and continuously grows toward D = 1/2N until  reaches the peak a, when a) = 2a. For simplicity, Wöhler’s stress-based fatigue damage model is adopted below (but strain-based models will be considered later). A simplified relation between the current stress state  and the continuous damage D from the half-cycle excursion a  a can then be obtained from Wöhler’s curve e.g. written in Basquin’s notation:   1/bb ba c a c a c( 2 N ) 2 [ ( )] 2 D D ( ) 2                   (2) The generalized damage modulus D during this half-cycle is thus such that   1/ba c a1 D dD d ( ) 2 [ b( )]           (3) from which the fatigue damage D  1/2N can be calculated using the integral T M. A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 37 (2016) 138-145; DOI: 10.3221/IGF-ESIS.37.19 140       aa aa 1/b 1/b 1/b a a a a c c c 1 1D d b( ) 2 2 2 N                          (4) If this conceptually simple procedure could be generalized to multiaxial NP variable amplitude loading (VAL) histories, integrating damage along a general multiaxial load path, then cycle identification, multiaxial rainflow counting, and stress (or strain) range calculations would not be required to obtain the fatigue damage D. However, this bold statement is easier said than done, since D depends not only on the current stress state ( in this uniaxial case), but also on the previous loading history (the value a from the last reversal), see Eq. (3). So, Incremental Fatigue Damage models need to allow D to vary as a function of the stress level and of the existing state of damage [9]. The history dependence of D, often neglected or overly simplified in the few IFD models proposed in the literature, is analogous to the load-order dependence of elastoplastic hysteresis loops. Chu [7] outlined the generalization of Wetzel’s rheological model to multiaxial loadings, indirectly detecting cycles using two simple rules. However, damage memory is not properly stored in that simple model for general NP VAL histories, where often no hysteresis loop actually closes and thus any virtual loop closure detection makes no sense. The main purpose of this work is to propose the improvements needed to properly extend the interesting IFD idea to general multiaxial loads. MULTIAXIAL INCREMENTAL FATIGUE DAMAGE APPROACH Stress-based Incremental Fatigue Damage Formulation n this work, instead of using rheological models, a direct analogy between IFD and incremental plasticity is adopted instead to store fatigue damage memory, using internal material variables. In incremental plasticity, a 5D deviatoric stress increment ds  can be used to calculate the associated 5D plastic strain increment plde  from the current generalized plastic modulus P, using a plastic flow rule [10-11]. In particular, it is well known that in the non-linear kinematic (NLK) incremental plasticity formulation, plastic memory is stored by the current arrangement among the hardening surfaces defined by their backstresses i  , from which the surface translation directions iv  are calculated (according to some translation rule) and combined with material coefficients pi to calculate the current plastic modulus P [10-11]. Therefore, no plastic straining occurs if the stress increment ds   happens inside the yield surface, whose radius should be equal or smaller than the cyclic yield strength SYc. The accumulated plastic strain p is then proportional to the integral of the scalar norm plde  of the deviatoric plastic strain increments. Let’s now rephrase the previous paragraph for the desired IFD model, based on the proposed direct analogy between plasticity and fatigue damage. In the IFD model presented here, a 5D deviatoric stress increment ds  can be used to calculate the associated 5D damage increment dD  from the current generalized damage modulus D, using a damage evolution rule. In the IFD formulation, damage memory is stored by the current arrangement among damage surfaces defined by their damage backstresses i  , from which the damage surface translation directions iv  are calculated (according to some translation rule) and combined with material coefficients di to calculate the current damage modulus D. No damage occurs if the deviatoric stress increment ds  happens inside the fatigue limit surface, whose radius should be equal or smaller than the fatigue limit of the material SL. The accumulated damage D is then equal to the integral of the scalar norm dD  of the 5D damage increments. The damage backstress vector   locates the center of the current fatigue limit surface, which can be decomposed as the sum of M damage backstresses  1  ,  2  , …, M  that describe the relative positions between centers of consecutive damage surfaces, as illustrated in Fig. 1 for a 2D case. Notice in Fig. 1 that each damage surface has a constant radius ri, while the radius differences between consecutive surfaces are ri  ri+1  ri. The fatigue limit and failure surfaces are defined, respectively, for i  1 and i  M  1, while the remaining i = 2, 3, …, M are the damage surfaces. The damage backstress lengths are always between i  0  , if consecutive centers coincide, and i ir    , if they are mutually tangent. I M. A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 37 (2016) 138-145; DOI: 10.3221/IGF-ESIS.37.19 141 1r fatigue limit surface '  1 '  2 '  3 '  s'  2r 3r 4r s1 s2 Figure 1: Fatigue limit, damage, and failure surfaces in a 2D deviatoric stress space for three moving nested surfaces, showing the damage backstress vector that defines the location of the fatigue limit surface center, and its three components that describe the relative positions between the centers of consecutive surfaces at each load event. The proposed multiaxial IFD model uses a 5D damage vector D   [D1 D2 D3 D4 D5]T that acts as an internal variable that stores the current multiaxial fatigue damage state (to account for the damage memory). The scalars D1 through D5 are signed damage quantities associated with each one of the directions of the 5D deviatoric stress vector s  , defined in [11]. In this way, the total accumulated damage D (which thus works for multiaxial fatigue problems analogously to the accumulated plastic strain p for multiaxial plasticity problems) is obtained from the length of the path described by the 5D damage vector D  , calculated in either continuous or discrete formulations from D dD dD D D        | | | |   (5) If a given stress state s  is on the fatigue limit surface with a normal unit vector n  , and if its infinitesimal increment ds  is in the outward direction, then Tds n 0     and a fatigue damage increment is obtained from a damage evolution rule (inspired on the analogous Prandtl-Reuss flow rule [10-11]): T MS NPdD D ds n n f f n               ( 1 ) ( ) ( ) ( , )      (6) M. A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 37 (2016) 138-145; DOI: 10.3221/IGF-ESIS.37.19 142 where MSf ( )  is a scalar mean stress function of the current 6D stress   to account for mean/maximum-stress effects, which can be defined e.g. from Goodman’s or Gerber’s am relations when applicable; and , NPf n ( )    is a NP function to account for the additional effects introduced by the non-proportionality of the load path. For materials that fail due to distributed damage in all directions, the mean stress function MSf ( )  could be based on the current hydrostatic stress h from   . On the other hand, for materials that fail due to a single dominant crack, like most metallic alloys (whose multiaxial fatigue damage parameters tend to be better described by the critical-plane approach), then MSf ( )  could be based on the normal stress  perpendicular to the considered candidate plane. Except for the failure surface (which never translates), during this damage process the fatigue limit and all damage surfaces suffer translations , if or , if i i i i i i i id d v dD r d r                     | | 0 | |     (7) where di are coefficients calibrated for each surface, and iv  are the damage surface translation directions adapted e.g. from the general translation rule from [11]. The current generalized damage modulus D is then obtained from the consistency condition, which guarantees that the current stress state is never outside the fatigue limit surface, taken from an analogy to the NLK hardening formulation for plasticity problems  M Ti iiD d v n        1   (8) allowing the calculation of the evolution of the damage vector D  using Eq. (6). The (scalar) accumulated damage D is then obtained from Eq. (5). This formulation can deal with any multiaxial stress history, proportional or NP, and eliminates the need to count cycles and find equivalent ranges, or even to define them. Indeed, for instance, Fig. 2 shows continuous IFD damage predictions for a material whose elastic Coffin-Manson’s parameters are c  772.5MPa and b  0.09, under the uniaxial loading history x = {0  300  300  300}MPa. Jiang-Sehitoglu’s translation rule was adopted with M  16 surfaces, calibrated between logarithmically spaced damage levels 108 and 0.01. x 10 -5 accumulated stress (MPa) a cc u m u la te d d a m a g e D theoretical (discrete) damage calculated continuous damage x 10 -5 0 200 400 600 800 1000 1200 1400 1600 0 1 2 3 4 5 6 7 -1.5 -1 -0.5 0 0.5 1 1.5 -300 -200 -100 0 100 200 300 signed damage D1 n o rm al s tr es s (M P a ) unloading loading Figure 2: Hysteresis loops relating applied stress and a signed damage state (left) and resulting accumulated damage (right) for a uniaxial constant amplitude loading history. Strain-based Incremental Fatigue Damage Formulation All the formulations and the example presented above assumed nominally linear elastic loading histories, whose damage can calculated from SN models such as Wöhler-Basquin’s and Goodman, but this is not a limitation for this methodology. M. A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 37 (2016) 138-145; DOI: 10.3221/IGF-ESIS.37.19 143 Indeed, the proposed IFD approach can be as well extended for elastoplastic loading histories, whose fatigue damage must be quantified by N models. However, instead of using fatigue limit and damage surfaces defined in stress spaces, strain spaces should be used in the continuous damage calculations in such cases. A generalized damage modulus D (instead of D) is thus defined, which for uniaxial loading histories becomes the derivative of the normal strain  with respect to damage D, thus D  d/dD. In the strain-based version of the proposed IF approach, a 5D deviatoric strain increment de  , defined in [10], is used to calculate the associated 5D damage increment dD  from the current D, using a suitable damage evolution rule. To do so, damage memory is stored by the current arrangement among damage surfaces defined by their damage backstrains i  , from which the damage surface translation directions iv  are calculated according to some translation rule and combined with material coefficients di to calculate the current D. The accumulated damage D is then equal to the integral of the scalar norm dD  of the damage increments. The same equations from the stress-based version can be used in the strain-based one, as long as the M damage surface backstrains  1  ,  2  , …, M  , radii ri, and radius differences ri  ri+1  ri between consecutive damage surfaces are all defined as strain (instead of stress) quantities. EXPERIMENTAL RESULTS he proposed IFD formulation is experimentally evaluated using complex 2D tension-torsion stress histories, applied on annealed tubular 316L stainless steel specimens in a multiaxial servo-hydraulic testing machine. The Coffin-Manson curve for this material is 0 277 0 5822 0 0119 2 N 0 758 2 N     . .. ( ) . ( ) , obtained from uniaxial N tests. The experiments consist of strain-controlled tension-torsion cycles applied to eight tubular specimens, each of them following one of the eight periodic x×xy/3 histories from Fig. 3. Tab. 1 compares the predicted and observed fatigue lives in number of blocks, where each block consists of a full load period. All predictions were performed using the strain- based version of the proposed incremental plasticity formulation, assuming for simplicity MSf 1( )   and , NPf n 1 ( )     in Eq. (6). Cross Circle Square Diamond Triangle 1 Triangle 2 Square/Cross Square/Circle/ Diamond Figure 3: Applied periodic x×xy/3 strain paths on eight tension-torsion tubular specimens, all of them with normal and effective shear amplitudes 0.6%. As shown in Tab. 1, albeit the proposed IFD method does not use any cycle detection or counting algorithm, all fatigue lives are predicted with relatively small errors, well within the usual scatter found in all fatigue life measurements. It also automatically applies Miner’s rule under VAL, as it can be seen in the loading path consisting of blocks of consecutive square and cross paths, since the predicted number of blocks 482 is such that 1/482  1/751  1/1314. Similarly, the T M. A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 37 (2016) 138-145; DOI: 10.3221/IGF-ESIS.37.19 144 predicted 327 blocks of consecutive square, circle and diamond paths is such that 1/327  1/751  1/996  1/1436. Miner’s rule was also confirmed within the observed experimental results, since e.g. in this latter case it would predict a life of 1/(1/772  1/837  1/976) = 285 blocks, almost the same value as the measured 288 blocks. It is important to note that all the predictions listed in Tab. 1 were based only on uniaxial Coffin-Manson data, without any posterior curve fitting procedure. Tension-Torsion path: predicted observed error Cross 1314 1535 14% Diamond 1436 976 47% Triangle 1 1135 842 35% Triangle 2 1180 840 40% Circle 996 837 19% Square 751 772 3% Square + Cross 482 342 41% Square + Circle + Diamond 327 288 14% Table 1: Predicted and observed lives, in number of blocks, for each applied path. CONCLUSIONS n this work, a continuous multiaxial Incremental Fatigue Damage formulation that does not needs cycle counting or path-equivalent estimations is proposed, based on a direct analogy with incremental plasticity models. Both proposed stress and strain-based approaches can be formulated using traditional stress, strain, or even energy-based SN and N damage models, such as Wöhler-Basquin, Coffin-Manson, Smith-Watson-Topper, or Fatemi-Socie, making it an attractive and practical tool for engineering use. In particular, the proposed IFD models do not require additional fitting parameters, or complex calibration routines, as opposed to equally continuous models that are based on traditional Continuum Damage Mechanics approaches. The results show that the proposed method is able to predict quite well multiaxial fatigue lives under complex tension-torsion histories, even though it does not require any cycle detection, multiaxial rainflow counting, or path-equivalent range computations. REFERENCES [1] Wang, C.H., Brown, M.W. Life prediction techniques for variable amplitude multiaxial fatigue - part 1: theories, J. Eng. Mater. Technology 118 (1996) 367-370. [2] Meggiolaro, M.A., Castro, J.T.P. An improved multiaxial rainflow algorithm for non-proportional stress or strain histories - part I: enclosing surface methods, Int. J. Fatigue 42 (2012), 217-226. doi:10.1016/j.ijfatigue.2011.10.014. [3] Meggiolaro, M.A., Castro, J.T.P. An improved multiaxial rainflow algorithm for non-proportional stress or strain histories - part II: the modified Wang-Brown method, Int. J. Fatigue 42 (2012) 194-206. doi:10.1016/j.ijfatigue. 2011.10.012. [4] Meggiolaro, M.A.; Castro, J.T.P.; Wu, H. Invariant-based and critical-plane rainflow approaches for fatigue life prediction under multiaxial variable amplitude loading, Procedia Engineering 101 (2015) 69-76. doi: 10.1016/ j.proeng.2015.02.010. [5] Kachanov, L.M. Introduction to Continuum Damage Mechanics, Springer (1986). [6] Wetzel, R.M. A Method of Fatigue Damage Analysis, Ph.D. Thesis, U. Waterloo, CA, (1971). I M. A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 37 (2016) 138-145; DOI: 10.3221/IGF-ESIS.37.19 145 [7] Chu, C.C. A new incremental fatigue method, in: ASTM STP 1389 (2000) 67-78. [8] Castro, J.T.P., Meggiolaro, M.A. Fatigue Design Techniques (in 3 volumes), CreateSpace, Scotts Valley, CA, USA (2016). [9] Kreiser, D., Jia, S.X., Han, J.J., Dhanasekar, M. A nonlinear damage accumulation model for shakedown failure, Int. J. Fatigue 29 (2007) 1523-1530. doi: 10.1016/j.ijfatigue.2006.10.023. [10] Meggiolaro, M.A., Castro, J.T.P., Wu, H. A general class of non-linear kinematic models to predict mean stress relaxation and multiaxial ratcheting in fatigue problems - Part I: Ilyushin spaces, Int. J. Fatigue 82 (2016) 158-166. doi: 10.1016/j.ijfatigue.2015.08.030. [11] Meggiolaro, M.A., Castro, J.T.P., Wu, H. A general class of non-linear kinematic models to predict mean stress relaxation and multiaxial ratcheting in fatigue problems - Part II: Generalized surface translation rule, Int. J. Fatigue 82 (2016) 167-178. doi: 10.1016/j.ijfatigue.2015.08.0310. << /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. Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.) /HUN /ITA /JPN /KOR /LTH /LVI /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.) /NOR /POL /PTB /RUM /RUS /SKY /SLV /SUO /SVE /TUR /UKR /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.) >> /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ << /AsReaderSpreads false /CropImagesToFrames true /ErrorControl /WarnAndContinue /FlattenerIgnoreSpreadOverrides false /IncludeGuidesGrids false /IncludeNonPrinting false /IncludeSlug false /Namespace [ (Adobe) (InDesign) (4.0) ] /OmitPlacedBitmaps false /OmitPlacedEPS false /OmitPlacedPDF false /SimulateOverprint /Legacy >> << /AddBleedMarks false /AddColorBars false /AddCropMarks false /AddPageInfo false /AddRegMarks false /ConvertColors /ConvertToCMYK /DestinationProfileName () /DestinationProfileSelector /DocumentCMYK /Downsample16BitImages true /FlattenerPreset << /PresetSelector /MediumResolution >> /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ] >> setdistillerparams << /HWResolution [2400 2400] /PageSize [612.000 792.000] >> setpagedevice