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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
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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.
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS Jiri
Vala, Leonard Hobst, Vladislav Kozak
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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
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS Jiri
Vala, Leonard Hobst, Vladislav Kozak
E-ISSN: 2224-3429 41 Volume 10, 2015
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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-
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS Jiri
Vala, Leonard Hobst, Vladislav Kozak
E-ISSN: 2224-3429 42 Volume 10, 2015
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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.
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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.
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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.
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS Jiri
Vala, Leonard Hobst, Vladislav Kozak
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.
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Vala, Leonard Hobst, Vladislav Kozak
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