This paper presents an attempt to find an optimal procedure for determining the height parameters of fracture surfaces. This is a useful task that may significantly increase the reliability of topographic analyses of solids. The paper focuses on seeking an optimum number of measuring sites to ensure sufficient reliability of the resulting height parameters determined by the confocal technique. The statistical tests show that the number may be close to 25 measuring sites

Ficker, T.

Acta Polytechnica

English

T. Ficker

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Acta Polytechnica Vol. 52 No. 2/2012An Optimal Procedure for Determining the Height Parameters ofFracture SurfacesT. FickerAbstractThis paper presents an attempt to find an optimal procedure for determining the height parameters of fracture surfaces.This is a useful task that may significantly increase the reliability of topographic analyses of solids. The paper focuseson seeking an optimum number of measuring sites to ensure sufficient reliability of the resulting height parametersdetermined by the confocal technique. The statistical tests show that the number may be close to 25 measuring sites.Keywords: profile analysis, 3D profile parameters, fracture surfaces, cement-based materials, confocal microscopy.1 IntroductionIn fracture mechanics there has been long-term inter-est in the topographic properties of fracture surfaces.It is believed that fracture surfaces bear valuable in-formation not only on the structure but also on themechanical properties of solids.Software analysis of surface topography requiresdigital replicas of the surfaces to be available. Amongseveral techniques available for this purpose, confo-cal microscopy is very convenient, since it producesthree-dimensional replicas without any contact withthe surface, i.e. without any mechanical damage tothe specimens. Confocal replicas are reconstructedfrom a series of horizontal sections, which are in factdigital two-dimensional microscopic images.As soon as the digital three-dimensional surfacerelief is formed, software analysis of the surface canbegin, and it can reveal various useful surface quan-tities, including the 3D-profile and 3D-roughness pa-rameters, which are often employed for classifyingsurface height irregularities [1–4]. These parametersare closely related to structural parameters such asporosity [5], or to mechanical properties, includingcompressive strength [6].The 3D-profile parameters Ha, Hq computed bymeans of the reconstructed surface profile f(x, y)within the plane rectangle L × MHa =1L · M∫∫(LM)|f(x, y)| dxdy (1)Hq =√1L · M∫∫(LM)[f(x, y)]2 dxdy (2)were tested previously [1, 5, 6], and were shown tobe reliable indicators of both the porosity and thecompressive strength of hydrated cement materials.However, the determination of parameters Ha, Hq isconnected with certain particularities that should beborne in mind when they are determined. The firstproblem is that we do not know the optimal numberof surface sites to be visited with the confocal mi-croscope in order to obtain reliable profile parametervalues. To resolve this problem, we performed a largeseries of measurements on a particular fracture sur-face and used statistical considerations to infer a suf-ficient number of site measurements to ensure preciseresults. This paper reports on the whole procedure.2 ExperimentThe fracture surface of hydrated cement paste (Fig.1) was tested. The test specimen was prepared fromordinary Portland cement CEM 42,5 I R-sc. Thewater-to-cement ratio was set to 0.5 and the periodof hydration was one year, i.e. the specimen was ofhigh maturity.Fig. 1: The surface of a fractured specimen of hydratedcement paste31Acta Polytechnica Vol. 52 No. 2/2012Fig. 2: Confocal relief of a fractured surfaceThe 3D profiles f(x, y) were created using anOlympus Lext 3100 confocal microscope. One ofthese profiles is shown in Figure 2. The profileswere formed by the software, which processed a se-ries of optical sections taken at various heights ofthe fracture surface. Approximately 200 image sec-tions were taken (magnification 20×) for each mea-sured surface site, starting from the very bottom ofthe surface depressions (valleys) and processing tothe very top of the surface protrusions (peaks) witha step of 1.28μm. The investigated area L × M =1 280μm× 1 280μm (1 024 pixels× 1 024 pixels) waschosen in 280 surface sites. These measuring sitesconsisted of seven groups, i.e. 5, 10, 15, 25, 50,75 and 100 measuring sites located randomly on thesurface. For each group of measurements, parame-ters Ha, Hq were computed and their averages weredetermined. In this way we obtained seven cou-ples of average values, whose statistical reliabilitiesincreased with increasing number of measurements.Naturally, the group of 100 measurements yieldedthe most reliable averages, and they were thereforeadopted as reference values H(100)a , H(100)q of highprecision close to exact values. The averages H(N)a ,H(N)q of the remaining groups N = 5, 10, 15, 50, and75 were then classified according to their percentagedeviation from the precise reference valuesPa(N) =|H(100)a − H(N)a |H(100)a× 100 (%) (3)Pq(N) =|H(100)q − H(N)q |H(100)q× 100 (%) (4)As the optimal number of measuring sites N , wechose a number which ensured that the percentagedeviation will be less than five, i.e. P (N) < 5%.3 Results and discussionIn Figure 3, there are two graphs Pa(N) and Pq(N)showing the behavior of percentage deviations fromthe reference average values H(100)a , H(100)q . Bothgraphs clearly indicate the optimum number of mea-suring sites to be close to N = 25. This numberof measuring sites ensures that the percentage de-viation is about 5%, which is a normal laboratorystatistical deviation. For larger N > 25, the result-ing profile parameters H(N)a , H(N)q would show stilllower statistical uncertainty but at the expense of anenormous measuring and computational effort. Onthe other hand, results with N < 25 rapidly increasetheir statistical uncertainty. For example, at N = 5the percentage deviation reaches a value of almost25%, which indicates rather high uncertainty and lowreliability.4 ConclusionThe statistical tests presented in this paper haveshown that the optimum number of measuring sur-face sites to determine 3D-profile parameters usingthe confocal microscopic technique is close to 25 sitesin order to ensure sufficient reliability of numerical re-sults. In practice, this means that the measurementsmay be performed within a matrix of 5 × 5 surfacepoints uniformly distributed over the tested surface.Fig. 3: Percentage deviations of average profile parameters from the reference value representing 100 site measurements32Acta Polytechnica Vol. 52 No. 2/2012AcknowledgementThis work was supported by the Ministry of Educa-tion, Youth and Sports of the Czech Republic underContract no. ME 09046 (Kontakt).References[1] Ficker, T., Martǐsek, D., Jennings, H. M.: Rough-ness of fracture surfaces and compressive strengthof hydrated cement pastes, Cem. Conr. Res. 40,2010, 947–955.[2] Qu, J., Stih, A. J.: Analytical surface roughnessparameters of a theoretical profile consisting of el-liptical arcs, Machining Science and Technology,7, 2003, 281–294.[3] Hong, E.-S, Lee, J.-S., Lee, I.-M.: Underestima-tion of roughness in rough rock joints, Int. J. Nu-mer. Anal. Meth. Geomech. 32, 2008, 1 385–1403.[4] Fardin, N., Stephansson, O., Jing, L.: The scaledependence of rock joint surface roughness, In-ternational Journal of Rock Mechanics & MiningSciences, 38, 2001, 659–669.[5] Ficker, T., Martǐsek, D., Jennings, H. M.: Rough-ness and porosity of hydrated cement pastes, ActaPolytechnica, 51, 2011, 7–20.[6] Ficker, T.: Fracture surfaces of porous materials,Acta Polytechnica, 51, 2011, 21–24.Prof. RNDr. Tomáš Ficker, DrSc.Phone: +420 541 147 661E-mail: ficker.t@fce.vutbr.czDepartment of PhysicsFaculty of Civil EngineeringBrno University of TechnologyVeveř́ı 95, 662 37 Brno, Czech Republic33

An Optimal Procedure for Determining the Height Parameters of Fracture Surfaces

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

Acta Polytechnica Vol. 52 No. 2/2012An Optimal Procedure for Determining the Height Parameters ofFracture SurfacesT. FickerAbstractThis paper presents an attempt to ﬁnd an optimal procedure for determining the height parameters of fracture surfaces.This is a useful task that may signiﬁcantly increase the reliability of topographic analyses of solids. The paper focuseson seeking an optimum number of measuring sites to ensure suﬃcient reliability of the resulting height parametersdetermined by the confocal technique. The statistical tests show that the number may be close to 25 measuring sites.Keywords: proﬁle analysis, 3D proﬁle parameters, fracture surfaces, cement-based materials, confocal microscopy.1 IntroductionIn fracture mechanics there has been long-term inter-est in the topographic properties of fracture surfaces.It is believed that fracture surfaces bear valuable in-formation not only on the structure but also on themechanical properties of solids.Software analysis of surface topography requiresdigital replicas of the surfaces to be available. Amongseveral techniques available for this purpose, confo-cal microscopy is very convenient, since it producesthree-dimensional replicas without any contact withthe surface, i.e. without any mechanical damage tothe specimens. Confocal replicas are reconstructedfrom a series of horizontal sections, which are in factdigital two-dimensional microscopic images.As soon as the digital three-dimensional surfacerelief is formed, software analysis of the surface canbegin, and it can reveal various useful surface quan-tities, including the 3D-proﬁle and 3D-roughness pa-rameters, which are often employed for classifyingsurface height irregularities [1–4]. These parametersare closely related to structural parameters such asporosity [5], or to mechanical properties, includingcompressive strength [6].The 3D-proﬁle parameters Ha, Hq computed bymeans of the reconstructed surface proﬁle f(x, y)within the plane rectangle L×MHa =1L ·M∫∫(LM)|f(x, y)| dxdy (1)Hq =√1L ·M∫∫(LM)[f(x, y)]2 dxdy (2)were tested previously [1, 5, 6], and were shown tobe reliable indicators of both the porosity and thecompressive strength of hydrated cement materials.However, the determination of parameters Ha, Hq isconnected with certain particularities that should beborne in mind when they are determined. The ﬁrstproblem is that we do not know the optimal numberof surface sites to be visited with the confocal mi-croscope in order to obtain reliable proﬁle parametervalues. To resolve this problem, we performed a largeseries of measurements on a particular fracture sur-face and used statistical considerations to infer a suf-ﬁcient number of site measurements to ensure preciseresults. This paper reports on the whole procedure.2 ExperimentThe fracture surface of hydrated cement paste (Fig.1) was tested. The test specimen was prepared fromordinary Portland cement CEM 42,5 I R-sc. Thewater-to-cement ratio was set to 0.5 and the periodof hydration was one year, i.e. the specimen was ofhigh maturity.Fig. 1: The surface of a fractured specimen of hydratedcement paste31Acta Polytechnica Vol. 52 No. 2/2012Fig. 2: Confocal relief of a fractured surfaceThe 3D proﬁles f(x, y) were created using anOlympus Lext 3100 confocal microscope. One ofthese proﬁles is shown in Figure 2. The proﬁleswere formed by the software, which processed a se-ries of optical sections taken at various heights ofthe fracture surface. Approximately 200 image sec-tions were taken (magniﬁcation 20×) for each mea-sured surface site, starting from the very bottom ofthe surface depressions (valleys) and processing tothe very top of the surface protrusions (peaks) witha step of 1.28μm. The investigated area L × M =1 280μm× 1 280μm (1 024 pixels× 1 024 pixels) waschosen in 280 surface sites. These measuring sitesconsisted of seven groups, i.e. 5, 10, 15, 25, 50,75 and 100 measuring sites located randomly on thesurface. For each group of measurements, parame-ters Ha, Hq were computed and their averages weredetermined. In this way we obtained seven cou-ples of average values, whose statistical reliabilitiesincreased with increasing number of measurements.Naturally, the group of 100 measurements yieldedthe most reliable averages, and they were thereforeadopted as reference values H(100)a , H(100)q of highprecision close to exact values. The averages H(N)a ,H(N)q of the remaining groups N = 5, 10, 15, 50, and75 were then classiﬁed according to their percentagedeviation from the precise reference valuesPa(N) =|H(100)a −H(N)a |H(100)a× 100 (%) (3)Pq(N) =|H(100)q −H(N)q |H(100)q× 100 (%) (4)As the optimal number of measuring sites N , wechose a number which ensured that the percentagedeviation will be less than ﬁve, i.e. P (N) < 5%.3 Results and discussionIn Figure 3, there are two graphs Pa(N) and Pq(N)showing the behavior of percentage deviations fromthe reference average values H(100)a , H(100)q . Bothgraphs clearly indicate the optimum number of mea-suring sites to be close to N = 25. This numberof measuring sites ensures that the percentage de-viation is about 5%, which is a normal laboratorystatistical deviation. For larger N > 25, the result-ing proﬁle parameters H(N)a , H(N)q would show stilllower statistical uncertainty but at the expense of anenormous measuring and computational eﬀort. Onthe other hand, results with N < 25 rapidly increasetheir statistical uncertainty. For example, at N = 5the percentage deviation reaches a value of almost25%, which indicates rather high uncertainty and lowreliability.4 ConclusionThe statistical tests presented in this paper haveshown that the optimum number of measuring sur-face sites to determine 3D-proﬁle parameters usingthe confocal microscopic technique is close to 25 sitesin order to ensure suﬃcient reliability of numerical re-sults. In practice, this means that the measurementsmay be performed within a matrix of 5 × 5 surfacepoints uniformly distributed over the tested surface.Fig. 3: Percentage deviations of average proﬁle parameters from the reference value representing 100 site measurements32Acta Polytechnica Vol. 52 No. 2/2012AcknowledgementThis work was supported by the Ministry of Educa-tion, Youth and Sports of the Czech Republic underContract no. ME 09046 (Kontakt).References[1] Ficker, T., Martiˇsek, D., Jennings, H. M.: Rough-ness of fracture surfaces and compressive strengthof hydrated cement pastes, Cem. Conr. Res. 40,2010, 947–955.[2] Qu, J., Stih, A. J.: Analytical surface roughnessparameters of a theoretical proﬁle consisting of el-liptical arcs, Machining Science and Technology,7, 2003, 281–294.[3] Hong, E.-S, Lee, J.-S., Lee, I.-M.: Underestima-tion of roughness in rough rock joints, Int. J. Nu-mer. Anal. Meth. Geomech. 32, 2008, 1 385–1403.[4] Fardin, N., Stephansson, O., Jing, L.: The scaledependence of rock joint surface roughness, In-ternational Journal of Rock Mechanics & MiningSciences, 38, 2001, 659–669.[5] Ficker, T., Martiˇsek, D., Jennings, H. M.: Rough-ness and porosity of hydrated cement pastes, ActaPolytechnica, 51, 2011, 7–20.[6] Ficker, T.: Fracture surfaces of porous materials,Acta Polytechnica, 51, 2011, 21–24.Prof. RNDr. Toma´sˇ Ficker, DrSc.Phone: +420 541 147 661E-mail: ﬁcker.t@fce.vutbr.czDepartment of PhysicsFaculty of Civil EngineeringBrno University of TechnologyVeveř´ı 95, 662 37 Brno, Czech Republic33

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An Optimal Procedure for Determining the Height Parameters of Fracture Surfaces

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