Microsoft Word - numero_39_art_11 S. Seitl et alii, Frattura ed Integrità Strutturale, 39 (2017) 100-109; DOI: 10.3221/IGF-ESIS.39.11 100 Focussed on Modelling in Mechanics Numerical study and pilot evaluation of experimental data measured on specimen loaded by bending and wedge splitting forces S. Seitl Academy of Sciences of the Czech Republic, v. v. i., Institute of Physics of Materials, Brno, Czech Republic Brno University of Technology, Faculty of Civil Engineering, Institute of Structural Mechanics, Brno, Czech Republic seitl@ipm.cz, http://orcid.org/0000-0002-4953-4324 R. Diego Liedo Academy of Sciences of the Czech Republic, v. v. i., Institute of Physics of Materials, Brno, Czech Republic University of Oviedo, Department of Construction and Manufacturing Engineering, Campus de Viesques, Gijón, Spain uo212533@uniovi.es ABSTRACT. The fracture mechanical properties of silicate based materials are determined from various fracture mechanicals tests, e.g. three- or four- point bending test, wedge splitting test, modified compact tension test etc. For evaluation of the parameters, knowledge about the calibration and compliance functions is required. Therefore, in this paper, the compliance and calibration curves for a novel test geometry based on combination of the wedge splitting test and three-point bending test are introduced. These selected variants exhibit significantly various stress state conditions at the crack tip, or, more generally, in the whole specimen ligament. The calibration and compliance curves are compared and used for evaluation of the data from pilot experimental measurement. KEYWORDS. Numerical simulation; Stress intensity factor; T-stress; Concrete; Finite element method; Wedge splitting test; Three–point bending test. Citation: Seitl, S., Liedo, R. D., Numerical study and pilot evaluation of experimental data measured on specimen loaded by bending and wedge splitting forces, Frattura ed Integrità Strutturale, 39 (2017) 100-109. Received: 25.07.2016 Accepted: 22.09.2016 Published: 01.01.2017 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 or evaluation of fracture mechanical properties of materials like concrete, standardized methodology is not published yet. There is only recommendation for measurement of properties by RILEM [18]. In the literature, researchers used various specimen geometries for experimental measurement of fracture properties of concrete, F http://www.gruppofrattura.it/pdf/rivista/numero39/audio/11.mp3 S. Seitl et alii, Frattura ed Integrità Strutturale, 39 (2017) 100-109; DOI: 10.3221/IGF-ESIS.39.11 101 see e.g. wedge splitting test (WST) [4, 5, 7, 14, 15, 21, 23, 24, 31], three–point bend [10, 17], comparison between data from WST and three–point bend (3PB) tests are introduced in [12], modified compact tension (MCT) test [6] and another configurations can be found in handbooks e.g. [11, 27]. Note that from various geometry, the different values of fracture parameters can be obtained for the same material. Therefore the combined WST/3PBT geometry has been investigated, the variants are proposed in [29] Fig. 1 (Variant I represents the classical WST [7, 24] and is included in the study as a reference case), see in [29]. In all these cases the crack propagates from a notch provided on the top side of the specimen (in the groove for inserting of the WST loading fixtures. Finally, the variant IIIb differs from the variant III by the central notch provided also from the bottom surface.), provides a wide range of various stress distributions in the specimen ligament – from bending to tension – which is expected to result in the desired variety e.g. in the fracture process zone size and shape, fracture energy or fracture toughness, etc. In the paper, the numerical support (calibration and compliance curves) for evaluation of the experimentally obtain data is shown/introduced. The pilot numerical study of the selected shape of specimens by using Williams expansion was introduced in [22, 25, 28, 30]. The values of stress intensity factor (SIF), T-stress and crack opening displacement (COD, see Fig. 2) at the load line for load Psp = 1000 N = 1 kN are introduced. The changes of properties are compared and discussed. These changes could be obtained modifying the specimen length to width and the span to length ratios (and/or simultaneously the wedge angle). At the end of the contribution, examples of the evaluation of experimental data measured by using the studied combination [29] have been presented. variant I variant III variant II variant IIIb Figure 1: Considered variants for the combined bending/splitting configurations, taken from [29]. S. Seitl et alii, Frattura ed Integrità Strutturale, 39 (2017) 100-109; DOI: 10.3221/IGF-ESIS.39.11 102 Dimensions of specimens for all four geometry variants are summarized in Tab. 1 (dimensions common to all variants) and in Tab. 2, there are unique dimension of all studied variants (I, II, III and IIIb) with angles. Width Breadth Height Load position Groove depth W [mm] B [mm] H [mm] h [mm] dn [mm] 150 150 130 8 20 Effective width Groove width Load position Eccentricity Weff [mm] f [mm] i [mm] e [mm] 142 40 10 20 Table 1: Nominal variant dimensions and test geometry parameters, taken from [29]. Geometry variant Wedge angle Length Span Depth of top notch Depth of bottom notch Initial crack length Relative crack length Specimen set 2αw [º] L [mm] S [mm] c [mm] c1 [mm] a [mm] α = a/Weff [-] I, α1 30 150 0 13 - 25 0.18 I, α2 30 150 0 30 - 42 0.30 II, α1 15 300 270 15 - 27 0.19 II, α2 15 300 270 31 - 43 0.30 II, α3 15 300 270 54 - 66 0.46 III, α1 15, 30 600 540 13 - 25 0.18 III, α2 15, 30 600 540 35 - 47 0.33 III, α3 15, 30 600 540 54 - 66 0.46 IIIb, α1 15, 30 600 540 8 53 53 0.37 IIIb, α2 15, 30 600 540 9 81 81 0.57 Table 2: Nominal variant dimensions and test geometry parameters, taken from [29]. THEORETICAL BACKGROUND ccording to the two-parameter fracture mechanics approach which uses T-stress as a constraint parameter [1, 11, 13, 24, 34], the stress field around the crack tip of a two-dimensional crack embedded in an isotropic linear elastic body subjected to normal mode I loading conditions is given by the following expressions [33]:                                                                      2 3 cos 2 sin 2 cos 2 2 3 sin 2 sin1 2 cos 2 2 3 sin 2 sin1 2 cos 2 I xy I yy I xx          r K r K T r K (1) A S. Seitl et alii, Frattura ed Integrità Strutturale, 39 (2017) 100-109; DOI: 10.3221/IGF-ESIS.39.11 103 where r and θ are the polar coordinates and x and y are the Cartesian coordinates, both with their origins at the crack tip. KI is the stress intensity factor for mode I and T is the T-stress. Thus, in two-parameter based fracture mechanics, the stress field is expressed by means of the two parameters, the stress intensity factor KI and the T-stress (see e.g. [1, 11, 24]). The values of crack opening displacement at load line is measured in the axes of roller bearings through which the load of specimens is applied (see e.g. in [4, 14] and sketch of forces in Fig. 2). The applied load ratio between forces is following, [4, 14, 19, 24]: kPP vs 2 1  (2) where    wcw wc ctg k      1tan tan1 , (3) where αw is the angle of the slope of the wedge and µc refers to friction in the roller bearings. Figure 2: Detail of boundary conditions, see the half of specimen and the load application (Psp and Pv/2) with crack opening displacement (COD), position at load line. MODELING IN ANSYS he finite element software ANSYS [2] is used for numerical calculation of mentioned fracture parameters (K, T- stress and COD). Plots of variants of the test geometry selected for the experimental study are shown in Fig. 1. Note that geometries are symmetric for all considered specimen shapes (including boundary conditions); therefore, only one half of the problem was modelled like in [21, 22, 26]. The size of the smallest element in the crack tip is 5 × 10-5 mm. The specimen geometry IIIb could leads to crack closure, therefore the whole body of the specimen was modeled and the example of numerical model with applied boundary conditions is shown in Fig. 3. The crack length to depth ratio a/Weff varies from 0.1 to 0.9. The thickness B is taken as unity in the computations, conditions of plane strain was applied. T Psp COD Pv/2 2 S. Seitl et alii, Frattura ed Integrità Strutturale, 39 (2017) 100-109; DOI: 10.3221/IGF-ESIS.39.11 104 Figure 3: Example of numerical model, where boundary conditions are shown, with detail in the vicinity of the crack tip. The material input data for the concrete used in the numerical simulations were as follows: Ec = 33 000 MPa and νc = 0.2; and for the steel: Es = 210 000 MPa and νs = 0.3. For good comparison of numerically obtain results for all cases, the load was applied as splitting force Psp = 1 000 N = 1 kN. NUMERICAL RESULTS he numerically calculated values of KI and T-stress are given in Figs. 4 and 5, respectively. In the present paper, four cases of the specimens’ shapes/arrangements on the calibration curves were investigated, see Fig. 1 (I, II, III and IIIb) for wedge angle: 15, 20 25, 30. On the left side of Fig. 4, the wedge splitting test configuration for αw =15º is compared with bending/splitting combination as II and III and IIIb. All four studied cases show similar trend of the SIF results, when the configuration is changed the SIF value for the same load decrease. On the right side of Fig. 4, the IIIb configuration with various wedge angles 15º, 20º, 25º and 30º are shown. Up to a/Weff = 0.6 the values have a smooth character, however for the a/Weff reaches 0.6 till 0.9 the values change unpredictably, there is a dominant effect of the 3PB loading. Figure 4: Stress intensity factor (KI) as a function of the relative crack length, α/Weff, for the combined bending/splitting variants defined in Fig. 1, loaded by Psp = 1000 N, on the left side for wedge angle =15° and on the right side for variant IIIb for various wedge angles. On the left side of the Fig. 5, the values of the T-stress for the WST configuration are compared with combination of WST and 3PB as variants II, III and IIIb for angles 15º. As we suppose according to the reference [23] the function for WST vary from negative values for very short cracks to positive values for relative crack larger than 0.2. When the T S. Seitl et alii, Frattura ed Integrità Strutturale, 39 (2017) 100-109; DOI: 10.3221/IGF-ESIS.39.11 105 distance from the support is larger the values of the T-stress are always positive for II and III variants. For variants IIIb the values of T-stress are always negative. On the right side of the Fig. 5, the T-stress values for combinations of IIIb with various wedge angles 15º, 20º, 25º and 30º are shown. Figure 5: T-stress as function of the relative crack length, , for the combined bending/splitting variants defined in Fig. 1, loaded by Psp = 1000 N, on the left side for wedge angle =15° and on the right side for variant IIIb for various wedge angles. Figure 6: COD as a function of the relative crack length, α, for WST and combined bending/splitting variants defined in Fig. 1 (as II, III and IIIb), loaded by Psp = 1000 N, on the left side for wedge angle =15° and on the right side for variant IIIb for various wedge angle. The displacement, COD, in the line of the load is shown in Fig. 6. On the left side of the Fig. 6, the values of WST are compared with variants II and III for wedge angle 15°, where the splitting loading is dominant. It can be seen that for longer specimens the value of COD decreases. On the right side of the Fig. 6, it can be seen the COD for configuration IIIb and various wedge angles 15°, 20°, 25°, 30° when the 3PB load is dominant and the crack start from bottom side. S. Seitl et alii, Frattura ed Integrità Strutturale, 39 (2017) 100-109; DOI: 10.3221/IGF-ESIS.39.11 106 EVALUATION OF EXPERIMENTAL DATA AND DISCUSSION t should be mentioned that quasi-brittle fracture of concrete is akin to elastic-plastic fracture of metals. The ASTM standard on fracture toughness KIC [3] has clearly specified the conditions to avoid elastic-plastic or in study case quasi-brittle fracture, i.e. crack 10 times of characteristic crack etc. The simple methodology presented in this example is consistent with the ASTM standard for linear-elastic fracture, but can also cover the first part of the quasi- brittle diagram. The material and experimental procedure are described in [29]. For evaluation of data in our paper, knowledge about selected input data are needed, therefore the relative initial notch length a/Weff and the maximal value of splitting force Pspmax are shown in Tab. 3. Note that for I variants the values of SIF were evaluated according to [14] for wedge angle 30° and for all others the new calibration curves were used, see Fig. 4. Using the classical linear elastic fracture mechanics, the fracture toughness KIC can be worked out on the basis of the initial relative notch length. The following expressions are used: NI Nsp sp IC K BP P K 1000, 1000, max,  (4) Using the results (calibration curves for 1000N) presented in Fig. 4, relation between the maximum splitting force Eq. (4) for relative notch length a/Weff, we obtain results of fracture toughness of concrete, the results are shown in Tab. 4. It can be seen, that the values of fracture toughness are within the interval 0.2÷1.4 MPam1/2, see in [1, 9]. The values of fracture toughness have decreasing tendency when the relative notch length grow, this is in accordance with the results in [34], where for this effect explanation the changes of T-stress values is used. Variant, wedge angle a/Weff [-] Pspmax [kN] I, 30° 0.18 7.60 I, 30° 0.38 5.74 II, 15° 0.18 9.80 II, 15° 0.30 7.47 II, 15° 0.46 4.17 III, 15° 0.18 13.68 III, 15° 0.33 9.33 III, 15° 0.46 4.56 Table 3: Variants of evaluated experiments and corresponding wedge angle, and the relative notch length with maximal value of splitting force (both values are mean values from 3 up to 6 measurements, see detail in [29]. Variant, wedge angle a/Weff [-] KIC [MPa m1/2] I, 30° 0.18 0.60 I, 30° 0.38 0.62 II, 15° 0.18 0.68 II, 15° 0.30 0.68 II, 15° 0.46 0.59 III, 15° 0.18 0.67 III, 15° 0.33 0.65 III, 15° 0.46 0.47 Table 4: Variants of evaluated experiments and relative notch length with corresponding value of fracture toughness. I S. Seitl et alii, Frattura ed Integrità Strutturale, 39 (2017) 100-109; DOI: 10.3221/IGF-ESIS.39.11 107 CONCLUSIONS n this paper, the combinations of wedge splitting and three-point bending load applied on beam-shaped notched specimens are numerically analyzed. The numerically obtain data could be used for evaluation of experimentally obtain data as is shown in example. Based on the numerical results presented here, the following conclusions can be drawn:  The values of the stress intensity factor (KI) have the same trend in the whole range of the relative crack length  for specimen variants I, II and III, see Fig. 1.  The values of the T-stress increase with the distance of the two supports on the bottom side of the specimen, varies from negative to positive values with increasing relative crack length (a/Weff), for specimen variants I, II and III.  The values of COD increase in the whole range of the relative crack length (a/Weff) for all variants of the boundary conditions for specimen variants I, II and III.  The variant IIIb has a crack from the bottom part of the specimens, the crack growth is influenced by combination of the wedge splitting force which in turn leads to crack closure during the load of specimen, see in Fig. 6. 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DOI: 10.1016/j.crme.2014.03.001. << /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