Detection of metal ﬁbres in cementitious composites based ...€¦ · of ﬁbres. For spherical...

Detection of metal fibres in cementitious composites based on signaland image processing approaches

JIŘÍ VALABrno University of Technology

Faculty of Civil EngineeringVeveřı́ 95, 602 00 BrnoCZECH [email protected]

LEONARD HOBSTBrno University of Technology

Faculty of Civil EngineeringVeveřı́ 95, 602 00 BrnoCZECH [email protected]

VLADISLAV KOZÁKAcademy of Sciences of the CRInstitute of Physics of Materials

Žižkova 22, 616 62 BrnoCZECH REPUBLIC

[email protected]

Abstract: Mechanical properties of cementitious composites, reinforced by metal fibres, are conditioned by the volumefraction and distribution of directions of fibres. However, their reliable non-destructive or low-invasive experimentalevaluation is a serious problem. The paper pays attention to four classes of such indirect methods. The first classrelies on the planar X-ray imaging, with the discrete fast Fourier transform applied to image processing. The secondone applies the magnetic approach, with certain electromagnetic alternative. The third one comes from the computedtomography, as an unique exact method for the detection of volume fraction without breaking the sample, with aninformation on (an)isotropy as a benefit. The last one is concentrated to the FEM modelling. Examples related toall sketched method from the experiments performed at the Brno University of Technology show the advantages andrestrictions of particular approaches.

Key–Words: Cementitious composites, non-destructive testing, signal and image processing, computational simula-tion.

1 Introduction

Advanced building structures make frequently use ofmaterials as silicate composites, reinforced by metalparticles (e. g. steel-fibre-reinforced concrete), prevent-ing the tension stresses and strains as sources of unde-sirable micro- an macro-cracking. Mechanical proper-ties of such composites are determined by the choice offibre properties and their volume fraction, location andorientation in the matrix, sensitive to the technologicalprocedures (as special compaction) and to the early-agetreatment – cf. [12], as well as by the bond / slip inter-face relations – cf. [3]. The employment of the destruc-tive approach relies usually on the separation of parti-cles, taken from the early-age matrix, alternatively ob-tained from the crushed part of the existing structure,in the laboratory; consequently the volume fraction ofparticles can be evaluated accurately, whereas any infor-mation related to the original orientation of particles ismissing. Moreover, such experiments with many struc-tures are not allowed by technical standards. This isa strong motivation for the employment of some re-liable non- or (at least) semi-destructive measurementmethods, applicable in situ, handling homogeneity andisotropy and detecting the volume fraction of fibres inthe material structure.

Regardless of the significant progress in this re-

search area in the last decade (for more historical re-marks and references see [8]), no inexpensive, robustand reliable method is available, thus all identificationapproaches rely on a) some indirect measurements andb) non-trivial numerical analysis, to handle a corre-sponding inverse problem – typically ill-posed, unsta-ble, etc., forcing artificial regularization. Since a) pro-duce quite other information than needed volume frac-tions and directional distributions of fibres, typicallydigital images in pixels or voxels, or electromagneticquantities detected on the specimen surface, some cal-ibration relations are needed, motivated by the phys-ical and geometrical similarity. Moreover, some rea-sonable algorithm for the evaluation of effective mate-rial properties, using the properties of matrix and parti-cles and the geometrical configuration, as input data, isneeded: from simple arguments from the mixture the-ory to complicated physical and mathematical homog-enization techniques (which will be specified later, inconnection with electromagnetic measurements).

In this paper we shall pay attention thanks to the re-search experience of the authors from BUT (Brno Uni-versity of Technology), namely to four representativeapproaches:

1. the planar X-ray imaging, with the discrete fastFourier transform applied to image processing,

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS Jiri Vala, Leonard Hobst, Vladislav Kozak

E-ISSN: 2224-3429 39 Volume 10, 2015

2. the magnetic approach, utilizing the Hall probeand advanced considerations on material homog-enization with certain electromagnetic alternative,

3. the computed tomography, as an unique exactmethod for the detection of volume fraction with-out breaking the sample, with an information on(an)isotropy as a benefit.

4. the finite element modelling, as a method for ex-act electromagnetic field modelling based on therandom fibre generation and identification.

2 First class of methods: analysis ofX-ray images

The radiographic approach, developed in [7], [22] fora rather large class of building materials, comes fromthe grey-scale planar images and some of their post-processing modifications, in particular:

1) the reduction of all fibres (whose length and thick-ness is known) to one-pixel thick black curves, fol-lowed by the simplified evaluation of their amountand orientation, by [7],

2) the application of the two-dimensional fast Fouriertransform by [10] and [15], avoiding most artifi-cial image changes, where the same as in 1) canbe identified with a special diffraction process:for the Cartesian coordinates x, y and the greylevel f(x, y), related to a square image containingN×N pixels (withN tending to∞ theoretically),with the associated image in the Fourier transformsF (u, v) in the Cartesian coordinate system, the di-rect and inverse Fourier transforms can be evalu-ated using the formulae

F (u, v) =N−1∑x=0

N−1∑y=0

f(x, y) exp(−2πi(ux+ vy)/N) ,

N2f(x, y) =N−1∑u=0

N−1∑v=0

F (x, y) exp(2πi(ux+ vy)/N)

moreover the power spectrum P (u, v) = |F (u, v)|2contain the useful information, needed for the deriva-tion of the histograms of fibre directions.

Figure 1 presents an example of such MATLAB-supported evaluation of fibre orientation in the fibreconcrete specimen; the utilized X-ray equipment isshown on Figure 2. In general, the radiographic analy-sis gets useful results related to preferential orientationsof fibres, although limited to data from planar images,even from several views to cubic specimens. The esti-mate of volume fraction of fibres is not very precise, atleast in the comparison with destructive tests.

Figure 1: Evaluation of fibre orientation from the X-rayimage (images from the top to bottom): original image,result of fibre localization, power spectrum P , resultingrose of fibre directions.


E-ISSN: 2224-3429 40 Volume 10, 2015

Figure 2: X-ray machine EcoRay HF 1040 with digi-tal recording to PC equipment (top photo). PeMaSo-01 depth probe for magnetic measurements (bottomphoto).

3 Second class of methods: numeri-cal treatment of magnetic measure-ments

Magnetic measurements like [24] and [5] rely on thedifferent values of relative permeability of fibres and amatrix, with possible alternative of electrical measure-ments and relative permittivity. The special experimen-tal configuration usually tries to force a (nearly) sta-tionary process, whose mathematical description workswith a differential operator close to the classical Laplaceone, to enable non-expensive software simulation. Fig-ure 3 shows the geometrical configuration of such pro-cess numerical simulation of such process in COMSOL:the magnetic field is generated by several permanentmagnets, located in the drilled hole (thus this methodcould be classified as low-invasive, not non-destructivecompletely), consequently the Hall effect based probefrom Figure 2 detects the magnetic field strength. Fig-ure 4 documents the numerical simulation of such ex-periment, applying the COMSOL supported planar fi-nite / infinite element technique: the influence of the ir-regularities caused by an artificial hall seems to be not

substantial. The comparative simulation, applying onlyselected functions of pde toolbox from MATLAB, leadsto the same conclusion.

The crucial problem is now to implement a cor-rect evaluation procedure for an effective relative per-meability (or permittivity) using the incomplete data onthe material microstructure and on relative permeabilityof fibres. For spherical particles the classical Maxwell-Garnett mixing formula is available; the generalizationof [6] comes from the so-called Brugemann approachand the repeated usage of similar ellipsoids as refer-ence volume elements, whereas [14] admits the pres-ence of multiple scattering, important for high volumefractions of fibres. No additional physical assumptionare needed, again for periodic spheres, in [13]: the aux-iliary problem, referring to the mathematical theory ofhomogenization of elliptic operators, can be then anal-ysed (including the existence and uniqueness of solu-tion, the convergence of sequences of approximate solu-tions, etc.) using the two-scale and similar convergencetheorems by [2]; the crucial (seemingly) explicit for-mula for the evaluation of an effective parameter value,comes from the method of oscillating test functions.

In [25] the difficulties with complex particle shapesare handled using the boundary integral approach,thanks to the knowledge of general solutions of theLaplace equation, with Heaviside characteristic func-tions of particles; [11] admits a priori anisotropic struc-tures. Some generalizations are available using the leastsquares and conjugate gradient approaches – cf. [21].Unfortunately, further substantial generalization of thisapproach (namely to non-periodic structure, avoidingall mixing tricks), lead to non-trivial (partially stillopen) problems of mathematical analysis, namely to theconvergence using probability measures by [20], thusvarious alternative statistical approaches, as that withSobol sensitivity indices and Monte Carlo simulationsby [9], have been developed.

The unique material characteristics included here isthe magnetic permeability µ [Vs/(Am)]; at least in thecase of silicate composites used in civil engineering µcan be set to 1 for the pure matrix, but no relevant con-stant is guaranteed by the producers of ferromagnetic fi-bres. In practice, the dimensionless relative permeabil-ity µr = µ/µ0, using the well-known magnetic constantµ0 = 4π · 10−7 Vs/(Am), is usually considered; simi-larly the relative permeabilities µc for the matrix and µsfor all fibres can be introduced. Fortunately, for a suf-ficiently slow volume fraction ξ of fibres (ξ ≤ 0.05 inreal experiments), following [6], under the assumptionof random orientation of fibres, we obtain an explicitmonotone and continuous dependence between µ and ξ


E-ISSN: 2224-3429 41 Volume 10, 2015

in the form

ξ = 1−M µs − µrµs − µc

(µcµs

)3L(1−2L)(2−3L),

where the factors

M =

(M1M2

)2(3L−1)2/((2−3L)(1+3L)),

L =ς

4ϑ3

(2ςϑ+ ln

ς − ϑς + ϑ

)are determined using the ratio ς of lengths of a majorand (both) minor axes of ellipsoidal particles (clearlyς > 1) for the simplifying notation ϑ =

√ς2 − 1 and

M1 = (1+3L)µc+(2−3L)µs,M2 = (1+3L)µr+(2−3L)µs. In particular, for a (theoretically) infinite lengthand zero diameter of particles we receive L = 1/3.Unfortunately, all attempts to generalize this result formore complicated distributions of fibre directions leadto unpleasant non-analytical integrals, with the duty oftheir non-trivial numerical evaluations.

Figure 5 documents the least squares based identi-fication of µr for 3 input data sets with assumed µr = 1for pure concrete and uncertain µs in all other cases,using the above sketched formulae for an isotropicmedium. the specimens (unlike the situation in situ)were prepared with exact volume fractions of fibres0.5 %, 1 % and 1.5 %. Other experiments with compa-rable results have been performed by the authors’ teamwith magnetic field induced by an electric coil. More-over, [4] presents a totally non-destructive equipment,applicable to the surface of a specimen (thus prefer-ring fibres close to such surface). All these result seemto give good estimates of volume fractions (whose im-provement using more advanced mathematical analysisis possible), but the differentiating between system andrandom errors in distributions of fibre directions is dif-ficult.

4 Third class of methods: computedtomography

A new approach to non-destructive analysis of struc-tures of cementitious composites, motivated by [16],[23] and [1], has been offered by the computed to-mography (X-ray CT), generating 3-dimensional im-ages from large series (slices) of 2-dimensional radio-graphic images taken around a single axis of rotation.The modern industrial tomograph, presented on Figure6, has been recently installed in the Central EuropeanInstitute CEITEC of BUT. Unlike most tomographs formedical applications, an analysed specimen is fixed onthe manipulation table of the tomograph, between the

Figure 3: Radially symmetric geometrical arrangementof the magnetic experiment (top) and computationalsimplification, including finite / infinite element mesh(bottom).

radiation source and the surface radiation detector, com-pound from a matrix of mini-detectors. During the rota-tion of the table the surface detector records successivechanges of X-ray radiation; consequently the special-ized computer software is needed to analyse the innerstructure of a specimen.

Several types of fibre concrete specimens have beentested using this equipment: Figure 7 shows the cubicspecimen, similar to that from Figure 1, and demon-strates the ability of the specialized software to recog-nize all fibres completely unlike all approximate esti-mates from separate planar images. Consequently vari-ous forms of histograms or graphical or 3-dimensionalroses of directions similar to 2-dimensional ones fromFigure 1 can be created. However, this is rather timeconsuming, expensive and not applicable to the fibreconcrete structures in situ. Nevertheless, this seems tobe a useful method to obtain a reliable reference basisfor all numerical simulation attempts with random po-


E-ISSN: 2224-3429 42 Volume 10, 2015

Figure 4: Results of COMSOL based on finite elementsimulation of stationary magnetic field strength.

Figure 5: Application of the least squares technique tothe identification of parameters ξ and µs from magneticmeasurements.


E-ISSN: 2224-3429 43 Volume 10, 2015

sitions and orientation of fibres.

Figure 6: Tomograph GE phoenix,v|tome|x L 240 (topphoto) and a cylindrical specimen fixed in its manipula-tor (bottom photo).

5 Fourth class of methods: FEMmodelling

The current trend of work in micromechaninics ad-dresses the industry requirements to decrease the de-pendence on experimental work, and complement itwith new numerical and/or analytical processes capa-ble of providing quickly and efficiently the same infor-mation. A highly attractive process to simulate the realbehaviour of composite is through finite element anal-ysis. For that, a representative volume element (RVE)of the materials needs to be defined and am equivalentrandom distribution of fibres generated.

The first issue concerning the use of RVE is its di-mension. The RVE cannot be too large as this wouldendanger the possibility to numerically to analyse it;however, it cannot be too small either as it could notbe representative of composite material, see [26]. Trias

Figure 7: Cubic fibre concrete specimen, edge length150 mm, required X-ray tube voltage 300 kV (imagesfrom the left to the right): axonometric view on its sur-face, axonometric view inside its structure, axonometricprojection of separated fibres in the cube specimen.

et al. [27] demonstrates that for long fibre composites avalue of 50x the fibre radius should be used.

The second issue involving the use of an RVE is thespatial arrangement of reinforcements which normallyis not periodic and is highly dependent upon manufac-turing process. [28] using homogenization theory con-cluded that distribution of reinforcements in the RVEdoes not affect to macroscopic response, but it signif-icantly affects the microscopic stress distribution andfollowing damage in the matrix.

Good review of some numerical methods for thefinite mesh generation can be found in [29] and [30].Digital image analysis provides a perfect replica of thereal composite, but can be extremely time and resourceconsuming as it requires specific software and hard-ware. To generate a random distribution of fibres iscoupled with a statistical analysis and verified by setsof experiments.


E-ISSN: 2224-3429 44 Volume 10, 2015

6 Conclusion

This paper should be understood as the comparativestudy to the most promising non-destructive approachesto macroscopic identification of content and random lo-cation of fibres in the structure of cementitious com-posites. However, all approaches have strong restric-tions: serious obstacles to get some reasonable estimateof volume fraction of fibres, as the most requested pa-rameter, in the first case, expensive and fastidious ex-perimental setting in the third case, interpretable as themore sophisticated upgrade of the first one, both tech-nical and computational difficulties in the second case.

For the successful computational detection of vol-ume fraction and preferential orientation of fibres, mak-ing use of their ferromagnetic properties, both underlaboratory and in situ conditions, the crucial point ofall considerations is the development of a homogeniza-tion procedure, specific to the analysed class of mate-rials, including its formal verification and its validityrange. This leads to non-trivial problems of both physi-cal and mathematical analysis, whose validation seemsto be available thanks to the progress in the image pro-cessing techniques.

Acknowledgements: The financial support of theFAST-S-14-2490 research project at BUT is acknowl-edged.

References:

[1] N. Baddour, Generalized Fourier diffraction the-orem for tomography. Proceedings of the 6-thWSEAS Inter. Confer. on Simulation, Modellingand Optimization in Lisbon (Portugal), 2006,pp. 411–416.

[2] D. Cioranescu and P. Donato, An Introduction toHomogenization. Oxford University Press, 1999.

[3] V. M. C. F. Cunha, J. A. O. Barros and J. M. Sena-Cruz, An integrated approach for modelling thetensile behaviour of steel fibre reinforced self-compacting concrete. Cement and Concrete Re-search 41, 2011, pp. 64–76.

[4] M. Faifer, L. Ferrara, R. Ottoboni and S. Toscani,Low frequency electrical and magnetic methodsfor non-destructive analysis of fiber dispersionin fiber reinforced cementitious composites: anoverview. Sensors 13, 2013, pp. 1300–1318.

[5] M. Faifer, R. Ottoboni, S. Toscani and L. Ferrara,Nondestructive testing of steel-fiber-reinforcedconcrete using a magnetic approach. IEEE Trans.on Instrumentation and Measurement 60, 2011,pp. 1709–1711.

[6] S. Giordano, Effective medium theory for dielec-tric ellipsoids. Journ. of Electrostatics 58, 2003,pp. 59–76.

[7] L. Hobst, O. Anton, J. Vodička and J. Ščučka,Homogeneity detection of fibre-concrete struc-tures by using radiographic technique. In:Nondestructive Testing of Mater. and Struct.,Springer 2013, pp. 323–328.

[8] L. Hobst and P. Bı́lek, Various control methods de-veloped for fibre concrete structures. Recent ad-vances in integrity, reliability and failure – 4-thInter. Confer., Funchal, 2013, pp. 721-730.

[9] Z. Kala, Geometrically non-linear finite elementreliability analysis of steel plane frames with ini-tial imperfections. Journ. of Civil Engineering andManagement 18, 2012, pp. 81–90.

[10] S. Kärkkäinen and E. B. Vedel Jensen, Estima-tion of fibre orientation from digital images. Im-age Anal. and Stereology 20, 2001, pp. 199–202.

[11] M. Y. Koledintseva, R.E. DuBroff andR.W. Schwartz, Maxwell-Garnett rule for di-electric mixtures with statistically distributedorientations of inclusions. Progress In Electro-magnetics Research 99, 2009, pp. 131–148.

[12] A. Krasnikovs, V. Zaharevskis, O. Kononova,V. Lusi, A. Galushchak and E. Zaleskis, Fiber con-crete properties control by fibers motion – investi-gation in fresh concrete during casting. IndustrialEngineering – 8th Inter. DAAAM Baltic Confer. inTallin, 2012, Part V: Mater. Eng., #10, pp. 6.

[13] G. Kristensson, Homogenization of spherical in-clusions. Progress in Electromagnetic Research42, 2003, pp. 1–25.

[14] P. Mallet, C. A. Guérin and A. Sentenac, Maxwell-Garnett mixing rule in the presence of multiplescattering: derivation and accuracy. Phys. ReviewB 72, 2005, 14205/1–9.

[15] N. E. Mastorakis and N. S. Swamy, Spectraltransformations for two-dimensional filters viaFFT. IEEE Trans. on Circuits and systems – I:Fundamental Theory and Applications 49, 2002,pp. 827–831.

[16] P. J. M. Monteiro, C. Y. Pichot and K. Belkebir,Computer tomography of reinforced concrete. In:Mater. Sci. of Concrete, Chapter 12, American Ce-ramics Society, 1998.

[17] G. Nguetseng and N. Svanstedt, σ-convergence.Banach Journ. of Mathematical Analysis 5, 2011,pp. 101–135.

[18] M. Pieper and P. Klein, Application of simpleperiodic homogenization techniques to non-linearheat conduction problems in non-periodic porousmedia. Heat and Mass Transfer 48, 2012, pp. 291–300.


E-ISSN: 2224-3429 45 Volume 10, 2015

[19] M. Ya. Sushko, Effective permittivity of mixturesof anisotropic particles. Journ. of Physics D: Ap-plied Physics 42, 2009, 155410: 9 pp.

[20] N. Svanstedt, Multiscale stochastic homogeniza-tion of convection-diffusion equations. Applica-tions of Math. 53, 2008, pp. 143–155.

[21] J. Vala, Least-squares based technique for identi-fication of thermal characteristics of building ma-terials. Inter. Journ. of Math. and Comp. in Simu-lation 5, 2011, pp. 126–132.

[22] J. Vala, L. Hobst, V. Kozák, Non-destructive de-tection of metal fibres in cementitious composites.Proceedings of the WSEAS Inter. Confer. on Mate-rials, Advances in Engineering Mech. and Mater.,Santorini Greece, 2014, pp. 125–128.

[23] G. Weidemann, R. Stadie, J. Goebbels andB. Hillemeier, Computer tomography study of fi-bre reinforced autoclaved aerated concrete. Mater.Testing 50, 2008, pp. 278–285.

[24] H.-J. Wichmann, H. Budelmann and A. Holst, De-termination of steel fiber dosage and steel fiberorientation in concrete. In: Nondestructive Testingof Mater. and Struct., Springer 2013, pp. 239–245.

[25] K. W. Whites and F. Wu, Effects of particle shapeon the effective permittivity of composite materi-als with measurements for lattices of cubes. IEEETrans. on Microwave Theory and Tech. 50, 2002,pp. 1723–1729.

[26] L. Mishnaevsky Jr. and S. Schmauder, ContinuumMesomechanical finite element modelling in ma-terials development a state-of-the-art review. Ap-plied Mech. Rev., 54, 1, 2001, pp. 49–69.

[27] D. Trias, J. Costa, A. Turon and J. Hurtado, Deter-mination of the critical size of a statistical repre-sentative volume element (SRVE) for carbon re-inforced polymers. Acta Mater. 54, 13, 20006,pp. 3471–3484.

[28] S. Schmauder and L. Mishnaevsky Jr., Microme-chanics and nanosimulation of metals and com-posites. Springer, 2008, pp. 430.

[29] D. Raabe, Computational materials science: Thesimulation of materials microstructures and prop-erties. Wiley, 1998.

[30] A. R. Melro, P. P. Camanho and S. T. Pinho, Gen-eration of random distribution of fibres in longfibre reinforced composites. Composites Scienceand Technol. 68, 2008, pp. 2092–2102.


E-ISSN: 2224-3429 46 Volume 10, 2015

Detection of metal ﬁbres in cementitious composites based ...€¦ · of ﬁbres. For spherical...

Documents

Transcript of Detection of metal ﬁbres in cementitious composites based ...€¦ · of ﬁbres. For spherical...