Ap1_02.vp 1 Introduction Stress intensity factors can be determined by ex- perimental, numerical or analytical methods. However, with complicated component and crack geometry or under complex loading only numerical procedures are applicable. Many programs have been designed recently to deal with fracture phenomena (e. g., Franc2d, Franc/FAM, Franc3d and AFGROW), which, in spite of great programming efforts, still show some deficiencies in range of functionality, operation comfort and reliability. Little evidence has yet been provided about the accuracy and suitability of such programs for solv- ing engineering problems [1, 2]. Present applications cover illustrative examples and simple problems [3, 4, 5]. Proven multi-purpose numerical programs such as MARC and ABAQUS usually possess no routines to find out stress intensity factors. To take advantage of the programs’ ex- tensive functions and high reliability a number of user subroutines for solving stress intensity factors need to be programmed. As regards the accuracy of stress intensity factors, only single programs have been compared with analytical solu- tions so far, and no comparison of the programs with each other has been presented. The objective of this study was therefore to confront the results of Franc2d, Franc3d and MARC with each other and for a better view also to verify their deviations from the analytical solutions. The analyses were conducted on simple models while also observing the in- fluence of mesh fineness, usability of available solvers and the overall performance of the programs. 2 Programs and models The tested programs were Franc2d Version 2.7 [6], Franc3d [7] Version 1.15, and MARC Version 2000 [8]. The first two come from Cornell University, New York, and as freeware they can be freely distributed. Franc2d (two-dimensional FRActure ANalysis Code) is based on the finite element method and enables analyses of two-dimensional problems with arbitrary component and crack geometries. Several methods are im- plemented for calculating stress intensity factors, from which the J-Integral Method [9] was chosen for the purposes of this study. Franc3d uses the boundary element method and was designed for solving three-dimensional fracture problems. Also here, arbitrary component and crack geometries can be analysed. Stress intensity factors are determined by the Displacement Correlation Method [10]. Both Franc2d and Franc3d possess further important functions for modeling various fracture phenomena, such as fatigue crack growth. Currently, a new version of Franc3d is being developed, which is based on the finite element method and offers a greater functional range. The finite element system MARC is suitable for analyses of general problems of engineering mechanics. To determine stress intensity factors the Displacement Cor- relation Method with a linear extrapolation from two nodes at each crack face [11, 12] has been implemented in the user procedures. As analytical solutions are known for only certain probe types, a single-edge cracked beam subjected to three-point bending and a flat plate with a semielliptical surface crack (Fig. 1) were chosen for the tests in this study. The analytical solution for the single-edge cracked beam is given by the equation K Fs BW a a W a W a W a W I � � � � � � � � � � � � � �3 2 199 1 215 393 27 2 . . . . � � � � � � � � � � � � � � � � � 2 3 21 2 1 a W a W (1) with the conditions s W a W � � �4 0 0 6, . .and The distribution of stress intensity factors along the crack front in a flat plate is expressed by the relation K c Y c a c W a b I n� � � � � � 1 � � � �, , , (2) with the conditions 0 0 5 0 10 0� � � � � � c a c W . , . and � � , where � is a complete elliptic integral of the second kind and Y is a geometric function. However, the function Y is not based on theoretical examinations but on experimental studies [13]. 62 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 41 No. 2/2001 Accuracy of Determining Stress Intensity Factors in Some Numerical Programs M. Vorel, E. Leidich At present, there are many programs for numerical analysis of cracks, in particular for determining stress intensity factors. Analyses of a single-edge cracked beam and flat plate with a semielliptical surface crack are presented in this study to examine the accuracy and applicability of the Franc2d and Franc3d programs. Further numerical computations of the MARC program and analytical solutions of stress intensity factors were included to compare the results with each other. For this purpose MARC was equipped with special user procedures. The influence of mesh fineness on the results was also investigated in all programs. The distributions of the stress intensity factors show good agreement in quality. The maximum deviations from the analytical solutions are 9.7 %. With greater numbers of elements programs Franc2d and Franc3d showed some instability, which currently reduces the usefulness and reliability of these promising tools for engineering applications. Keywords: fracture mechanics, stress intensity factors, numerical programs. The meshes of the analysed models were generated with the relevant preprocessors (Casca, OSM, Mentat) in three mesh densities at a time, in order to observe the influence of mesh fineness on stress intensity factors. Thus, from each examined probe type there were models with a coarse, medium fine and fine mesh in each program (Table 1). Although in general it is appropriate to take advantage of a model symmetry, here, with respect to future studies, complete models were created (Fig. 2). The basic elements were taken linear in Franc3d, and quadratic in Franc2d and MARC. The crack front (or crack tip in two-dimensional cases) formed in all programs collapsed quarter-point quadratic elements [14], the number of which varied in three-dimensional cases from 16 to 48 along the crack front. The rosette consisted of 6 to 8 collapsed ele- ments. The two dimensional models were considered as plain strain problems. The meshes of Mentat were generated by a newly introduced parametric modeling function. The load of the single-edge cracked beam models consisted of a sin- © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 63 Acta Polytechnica Vol. 41 No. 2/2001 y x z H = 40 H/2 = 20 b = 12.566 a = 4 c = 2 W = 20 = 0.3 E = 210000 MPa = 0.3 E = 210000 MPaW = 8 B = 4 H/2 = 18 a = 4 s/2 = 16 (a) (b) H/2 H/2 W s/2 F s/2 a B b b aa y c W t x ns H/2 H nn zt z Fig. 1: A single-edge cracked beam (a) and a flat plate with a semielliptical surface crack (b). Length measures in mm. Fig. 2: An example of the used Franc3d meshes; (a) single-edge cracked beam, (b) flat plate Elements Nodes coarse medium fine fine coarse medium fine fine Single-Edge Cracked Beam Franc2d/Casca 556 1636 2168 1645 4525 6395 Franc3d/OSM 936 2252 4020 902 2148 3876 MARC/Mentat 2760 8400 14080 13233 37939 62115 Centre Cracked Plate Franc3d/OSM 1206 2014 3084 1001 1918 2780 MARC/Mentat 7396 12880 17608 33143 55979 76341 Table 1: Numbers of elements and nodes of the analysed models gle force ( F = 500 N) with two-dimensional models, or of a uniformly distributed force ( f = 125 Nmm�1) in three-di- mensional models. The flat plate was loaded with normal stress (�n = 200 MPa) on the upper surface of the model. 3 Strategy and results A single-edge cracked beam was analysed in Franc2d, Franc3d and MARC. The flat plate was analysed as a sheer three-dimensional problem only in Franc3d and MARC. The analytical solutions were carried out for both probe types. After analysis there were three results (coarse, medium fine and fine mesh) for one probe type from each used program. In particular, these were three mesh qualities × three pro- grams from the single-edge cracked beam, and three mesh qualities × two programs from the flat plate. From each three results the optimum solutions were then chosen. The criteria for this were minimum deviation from the analytical solution on the one hand, and the lowest possible computational time on the other. In the end the programs’ optimum solutions were compared together with the analytical solutions in one diagram for each probe. The stress intensity factors were evaluated as a single value at a crack tip or as a course of values at a crack front. The Franc2d models were solved with an implicit direct solver, which required very low computational times (Fig. 3). There was no point in using more sophisticated solvers, as the problem was entirely linear. The Franc3d jobs were processed by the boundary element solver BES, which includes four different schemes. In this study the iterative scheme with out-of-core element storage proved best: the direct scheme could not be applied to larger models, as the programs crashed after the stiffness matrix assembly, and the other schemes turned out to run slightly more slowly. The MARC analyses were carried out with the iterative solver (in-core element storage, incomplete Choleski preconditioner). All computations were performed on an SGI Origin 2000 computer. The dependences of the stress intensity factors on the mesh fineness are displayed in Figs. 4 and 5. The optimum solutions of stress intensity factors are compared in the following diagrams, Fig. 6: • stress intensity factors in a single-edge cracked beam under single force or distributed force loading; three numeri- cal solutions and one analytical solution; F = 500 N or f = 125 Nmm�1 respectively; • stress intensity factors in a flat plate under normal stress loading; two numerical solutions and one analytical solu- tion; �n = 200 MPa. 4 Discussion With simple crack configurations, such as that of the sin- gle-edge cracked beam, mesh quality seems to have only a lit- tle influence on the values of the stress intensity factors (Fig. 4). The smallest differences can be observed for Franc2d. However, mesh quality appears to be significant in the case of more complicated crack forms, such as that of a semielliptical crack (Fig. 5). The stress intensity factors differ with MARC es- pecially at the crack edges; Franc3d shows consistent values from a certain mesh fineness. The overall comparison of stress intensity factors for a sin- gle-edge cracked beam was performed in the middle of the crack front, as these show more important evidence than the values at the crack ends. However, this was not the case for the semielliptical crack in the flat plate, where the stress intensity factors vary considerably along the whole crack front. The values of Franc3d are always somewhat higher than the analyt- ical solution (deviation up to 9.7 %, Fig. 6), whereas MARC delivers somewhat lower values (deviation up to 8.4 %). On the one hand, the deviations can be traced to the methods used (boundary element vs. finite element method) and ele- ments (linear vs. quadratic), but on the other hand, they may also result from the different methods of calculating the stress intensity factors. The deviations of the Franc2d values from the analytical solution are the lowest, which agrees with the Acta Polytechnica Vol. 41 No. 2/2001 64 28000 24000 20000 16000 12000 8000 4000 32000 0 Franc3dFranc2d MARC MARC 12000 10000 8000 6000 4000 2000 0 Franc3d C o m p u ta tio n al tim e [s ] 500 N 200 MPa coarse medium fine fine 5 25 40 24 98 65 0 90 0 63 00 coarse medium fine fine 28 61 37 49 12 60 36 60 33 19 12 66 3 3 1 9 2 0 1 0 9 6 0 Fig. 3: Computational time of the single-edge cracked beam (left) and flat plate (right) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 65 Acta Polytechnica Vol. 41 No. 2/2001 meshsolution analytical medium fine finecoarse 467.9 468.9 471.5470.7 Franc2d a) 560 520 480 440 400 600 500 N Franc3d b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 560 520 480 440 400 600 Coordinate along the crack front [1] analytical solution fine medium fine coarse Mesh quality: 500 N MARC c) 560 520 480 440 400 600 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Coordinate along the crack front [1] analytical solution fine medium fine coarse Mesh quality: 500 N Fig. 4: Influence of mesh fineness on stress intensity factors KI for a single-edge cracked beam using Franc2d (a), Franc3d (b) and MARC (c) Franc3d a) 460 420 380 340 500 300 0 20 40 60 80 100 120 140 160 180 Angle along the crack front [deg]y . analytical solution fine medium fine coarse Mesh quality: 200 MPa MARC b) 460 420 380 340 500 300 . 0 20 40 60 80 100 120 140 160 180 Angle along the crack front [deg]y analytical solution fine medium fine coarse Mesh quality: 200 MPa Fig. 5: Influence of mesh fineness on stress intensity factors KI at the flat plate using Franc3d (a) and MARC (b) supposition that domain integral methods (such as the J-Inte- gral Method) are more accurate than local methods (such as the Displacement Correlation Method) [15, 16]. The stress intensity factors determined by Franc3d at the crack edges in the flat plate do not correspond well to the other displayed solutions (Fig. 6). Stress intensity factors at the crack edges are in general dependent especially on the geometric configuration [13], but they should not acquire such a falling form. False values at crack edges are generally caused by some unsuitable treatment of the singularities which always exist at the ends of a crack front [17]. Here, this phenomenon as well as the sudden decline of the stress intensity factors at � = 90° may also be connected to the boundary element method that is used. Franc2d and Franc3d show some errors during both man- ual operation and computational processing. These emerge mostly with larger models (from about 4000 elements) and result in disfunction of some commands (e. g., manual rede- fining of elements). This is probably caused by deficient mem- ory management: during computational processing too high memory demand and falsely defined elements can lead to a program crash. 5 Conclusion In this study some analyses with Franc2d, Franc3d and MARC were conducted to compare the accuracy of deter- mining stress intensity factors and to examine the behavior of the programs. A single-edge cracked beam and a flat plate with a semielliptical surface crack were used as test models. Franc2d shows good accuracy, but it is applicable only to two-dimensional problems. Franc3d delivers acceptable values and appears to be a promising tool for engineer- ing applications. To this end, reliability and function range should be improved. MARC with special user procedures shows lower but certainly usable values. Although this multi- -purpose program shows high reliability, the programming effort to adapt it for solving fracture problems remains high. References [1] May, B.: Ein Beitrag zur praxisnahen Simulation der Aus- breitung von Ermüdungsrissen bei komplexer Beanspruchung. Düsseldorf, VDI Verlag, 1998 [2] FRANC3D. Documentation: Volume V, Validation/ Veri- fication, New York, Cornell University, 1998 [3] Schöllmann, M., Richard, H. A.: Franc/FAM – A software system for the prediction of crack propagation. Journal of Structural Engineering, 26, 1999, pp. 39–48 [4] Lewicki, D. G.: Crack Propagation Studies to Determine Benign of Catastrophic Failure Modes for Aerospace Thin-Rim Gears. NASA Technical Memorandum 107170, Cleve- land, NASA, 1996 [5] Lewicki, D. G., Sane, A. D., Drago, R. J., Wawrzynek, P. A.: Three-Dimensional Gear Crack Propagation Studies. NASA Technical Memorandum 208827, Cleveland, NASA, 1998 [6] Franc2d, A Two Dimensional Crack Propagation Simulator, User’s Guide, Version 2.7. New York, Cornell University, 1998 [7] Franc3d, A Three Dimensional Fracture Analysis Code, Con- cepts and User’s Guide, Version 1.14. New York, Cornell University, 1998 [8] MARC Volume A Manual: Theory and User Information, Version K7. Los Angeles, MARC Analysis Research Cor- poration, 1997 [9] Dodds, R. H. Jr., Vargas, P. M.: Numerical evaluation of domain and contour integrals for nonlinear fracture mecha- nics: formulation and implementation aspects. Illinois, Uni- versity of Illinois at Urbana-Champaign, 1988 [10] Shih, C. F., de Lorenzi, H. G., German, M. D.: Crack Extension Modelling with Singular Quadratic Isoparametric Elements. Int. J. Fracture, 12, 1977, pp. 647–651 [11] Mi, Y.: Three-Dimensional Analysis of Crack Growth. Southampton, Computational Mechanics Publications, 1996 66 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 41 No. 2/2001 560 520 480 440 400 600 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D = 6.3 %0.5 D = 0.4 %0.5 D = 4.5 %0.5 analytical solution Franc3d, coarse mesh Franc2d, medium fine mesh MARC, medium fine mesh a) 500 N 9D = 9.7 % D = 8.4 %90 D101= 12.2 % 460 420 380 340 500 300 0 20 40 60 80 100 120 140 160 180 analytical solution MARC, fine mesh Franc3d, medium fine mesh b) 200 MPa Angle along the crack front [deg]yCoordinate along the crack front [1] .. Fig. 6: Comparison of stress intensity factors KI for a single-edge cracked beam (a) and flat plate (b) 67 Acta Polytechnica Vol. 41 No. 2/2001 [12] Barsoum, R. S.: On the use of isoparametric finite elements in linear fracture mechanics. International Journal for Nu- merical Methods in Engineering, 10, 1976, pp. 25–37 [13] Sähn, S., Göldner, H.: Bruch- und Beurteilungskriterien in der Festigkeitslehre. Leipzig, Fachbuchverlag, 1989 [14] Bittnar, Z., Šejnoha, J.: Numerické metody mechaniky 2. Praha, Vydavatelství ČVUT, 1992 [15] Dhondt, G.: Mixed-Mode calculations with ABAQUS. In: “DVM Bericht 232: Festigkeits- und Bruchverhalten von Fügeverbindungen”, Berlin, DVM 2000, pp. 333–343 [16] Bažant, Z. P., Planas, J.: Fracture and size effect in concrete and other quasibrittle materials. Boca Raton, CRC Press, 1998 [17] Shivakumar, K. N., Raju, I. S.: Treatment of singularities in cracked bodies. International Journal of Fracture, 45, 1990, pp. 159–178 Dipl.-Ing. Michal Vorel phone: +490 371 531 4566 fax: +490 371 531 4560 e-mail: michal.vorel@mbv.tu-chemnitz.de Prof. Dr. Ing. Erhard Leidich Technical University Chemnitz Faculty of Mechanical Engineering and Process Technology Reichenhainerstraße 70 Chemnitz, D-09126, Germany << /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 false /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 (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /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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