A MICROWAVE PROBE FOR THE NON-DESTRUCTIVE DETERMINATION OF THE STEEL FIBER CONTENT IN CONCRETE SLABS

Transcription

1 A MICROWAVE PROBE FOR THE NON-DESTRUCTIVE DETERMINATION OF THE STEEL FIBER CONTENT IN CONCRETE SLABS A. Franchois 2, L. Taerwe 1 and S. Van Damme 2 1 Dept. of Structural Engineering, Ghent University, Belgium. 2 Dept. of Information Technology (INTEC-IMEC), Ghent University, Belgium. Abstract A non-destructive measuring technique to determine the fiber content in hardened steel fiber reinforced concrete is presented. The technique is based on an accurate open-ended coaxial probe reflectometry method for measuring the effective permittivity of the fiber reinforced concrete and on a mixing rule from the classical homogenization theory for deriving the fiber content from the measured permittivity. A microwave probe has been constructed. Experimental results for reinforced concrete slabs with various fiber contents confirm the potential interest of this technique. 1. Introduction At present, the dosage and uniformity of the fiber distribution in steel fiber reinforced concrete (SFRC) is mainly checked in a destructive way [1]. In this paper a nondestructive measurement technique is presented to determine the fiber content in hardened SFRC. The technique is based on the use of radio frequency (RF) waves in the range 100 MHz 800 MHz, which exhibit satisfactory penetration in dry concrete and the reflective properties of which are sensitive to the presence of the highly conductive steel fibers. Such a wave is guided onto a SFRC slab by means of an open-ended coaxial probe, which is placed on the surface of the slab and which also guides back the reflected wave into a reflection coefficient measurement system. The reflection coefficient depends on the permittivity or dielectric constant of the SFRC, which in turn is related to the fiber content. The development of open-ended coaxial probe methods for applications in non-destructive evaluation has been widely reported in the literature, in particular to measure the permittivity of liquids, biological materials [2],[3] and homogeneous solids [4]. In this paper such a method is applied to measure the effective permittivity of SFRC, which is an inhomogeneous mixture of concrete and fibers, if observed at the length scale of the fibers. Since the volume fraction of the fibers in SFRC usually is below 1%, a classical homogenization approach can be used to relate it 249

2 to the effective permittivity of the mixture [5]. A coaxial probe has been constructed and the measurement technique has been tested in a laboratory environment on a number of slabs with various fiber contents. In Section 2 a homogenization approach for SFRC is discussed. The open-ended coaxial probe technique is described in Section 3. Experimental results are presented in Section Effective dielectric properties of SFRC The interaction between an electromagnetic wave and matter is a complex phenomenon. At the atomic scale the applied field displaces electrically charged particles and electrical currents from their equilibrium states, creating electric and magnetic dipole moments. The resulting electric and magnetic fields in the matter show strong fluctuations over very short distances. The exact knowledge of the microscopic charge, current, dipole and field distributions is not of interest for applications at a macroscopic scale. Therefore, a spatial averaging is applied, yielding macroscopic distributions, which satisfy the wellknown Maxwell equations. For example, the numerous individual electric dipole moments generated in the molecules of a dielectric material by an applied electric field E a (r, t), with r a position vector and t time, are replaced by an electric polarization density P(r, t), i.e. the electric dipole moment per unit of volume. Note that vectors are in boldface. In this paper a harmonic time dependence e jt, with = 2f the angular frequency, is adopted, hence all field quantities are of the form A(r, t) = Re[A(r) e jt ], where A is a complex vector and Re stands for the real part. As usual, the time dependence is omitted, and all equations are expressed in terms of the complex vectors. The relation between the resulting (macroscopic) field E in the dielectric and P is given by P = ( 0 ) E, (1) where (r) in Farad/m is the (complex) permittivity of the material and 0 = F/m is the permittivity of vacuum. The permittivity is a measure for the polarizability of a material. The polarizability of vacuum is zero. It is usual to introduce the dimensionless relative permittivity r (r) as = r 0. (2) Values of r as a function of the frequency are tabulated in the literature for many materials [6], including concrete [7]. Parameter models are reported for certain substances, such as water [8]. SFRC is a heterogeneous mixture of the mortar matrix, coarse aggregates, air inclusions and steel fibers. When the largest particle dimension is much smaller than the 250

3 ε r,eff Fig. 1: The effective relative permittivity of SFRC as a function of the fiber volume fraction for l/d = wavelength of the interrogating wave, the mixture can be regarded as an equivalent homogeneous material with an effective (relative) permittivity eff ( r, eff ). Homogenization is in fact similar to the spatial averaging procedure as described in the beginning of this section, but now from a lower to a higher macroscopic level. In this paper SFRC is considered as a two-phase mixture consisting of a homogeneous concrete host medium with permittivity h in which the fibers are embedded. Since the amount of fibers typically ranges between 20 kg/m 3 and 80 kg/m 3, which corresponds to a volume fraction between and 0.01, the mixture is sparse, hence a mixing rule from the classical homogenization approach can be applied [5]. Such rules predict the effective permittivity of the mixture as a function of the permittivities, shapes and volume fractions of the constituents, under the quasi-static assumption that the wavelength be much larger than the particle dimensions. For the homogenization of SFRC the fibers are approximated by prolate spheroids, which are needle like ellipsoids of revolution, with semi-axes a 1 >> a 2 = a 3. The length 2a 1 and volume of the (prolate) spheroid are chosen equal to the length l and volume of the actual fiber. As a consequence, the aspect ratio A s = a 1 /a 2 of the spheroid usually is smaller than the aspect ratio l/d of the actual fiber, where d is the diameter, but the volume fractions of the spheroids and of the fibers are identical. The following 'Maxwell-Garnett' formula for randomly oriented conducting spheroids in a homogeneous host medium can be derived V f 1 31 V V [%] f 1 1 1, N N N eff h (3) f 251

4 where V f is the fiber volume fraction and where N 1 << N 2 = N 3 are the depolarization factors of the spheroid. These are geometrical factors, which only depend on the particle shape, in this case on A s [5]. It follows that for given permittivity of the concrete host and aspect ratio of the spheroids, the resulting effective permittivity only depends on the fiber volume fraction. Fig. 1 shows the effective relative permittivity as a function of the fiber volume fraction in the range 0 V f 0.01 for l/d = 54.5, computed with (3). 3. The open-ended coaxial probe measurement method An open-ended coaxial probe is a section of coaxial waveguide in which waves can propagate in the axial direction in both senses. The guide consists of two concentric conductors with a dielectric filling material in between (Fig. 2). The actual probe in this paper is 130 mm long, has an inner conductor with radius a = 5.9 mm, an outer conductor with radius b = 20 mm, a Teflon dielectric filling material and a flange extending 60 mm from the outer conductor (Fig. 3). In this paper it is assumed that only the Transverse ElectroMagnetic (TEM) mode can propagate in the guide, which means that the electric and magnetic field vectors are directed perpendicularly (transverse) to the axial direction. Adopting cylindrical coordinates with z along the probe axis and = (, ) a position vector in the transverse plane, the general solution for the (complex) electric field vector in the guide is the superposition of a wave which propagates in the positive z-direction, with a (complex) coefficient A +, and a wave which propagates in the negative z-direction, with a (complex) coefficient A, E(, z) = A + E() e jkz + A E() e +jkz, (4) where k = /v is the propagation constant in the guide, v is the velocity of light in the guide and E() is the z-independent transverse electric field pattern of the TEM-mode b 2a Flange ffl r;ef f Fig. 2: A schematic representation of the probe. 252

5 Fig. 3: The actual probe. The values of the coefficients A + and A depend on the boundary conditions at both ends of the waveguide, thus on the source at the one end and the material at the other end. The flanged open-end of the guide is positioned on the surface of the slab with unknown relative permittivity r,eff. The other end is connected to an automated network analyzer (ANA), which is a system to measure reflection coefficients. The ANA generates an incident RF wave, which propagates down to the surface of the slab and penetrates into the concrete. The reflected wave propagates back up to the ANA. The reflection coefficient is defined as the ratio of the complex amplitudes of the reflected and incident waves, hence, with the z-axis pointing downward, as K A e (5) A j2kz z. The phase of K(z) varies with the position z along the waveguide. On the one hand K(0) = A /A + at the probe-end (z = 0) is related to the normalized aperture admittance Y( r,eff, f ) in a simple manner. On the other hand the exact relation between Y and r,eff is quite complex and the inversion of Y for r,eff cannot be done explicitly. Therefore, the closed form rational function approximation of [9] is applied here Y r, eff, N p ja r, eff 2 n1p 1 np j a, (6) M P Q 1 m1q 0 mq q ja r, eff 2 n m 253

6 where np and mq are model coefficients. These model coefficients are reported in [9] for the class of 50 Teflon filled coaxial probes with N = M = 4 and P = Q = 8. They are valid in the frequency range 81 MHz < f < 1.5 GHz for the probe of Fig. 3. The selection of the measurement frequency is subject to the constraints imposed by the homogenization and by the sensitivity properties of the probe [10]. The inversion of (6) leads to an 8 th degree polynomial in the square root of r,eff, which can be solved efficiently for the appropriate root. In a real measurement setup, imperfections in the cables, connectors and in the probe itself cause unwanted reflections, which lead to errors in the measured reflection coefficient K m. Thus, it is necessary to perform a calibration in order to determine a set of correction coefficients [2]. The calibration consists of measuring the reflection coefficients for the probe subsequently terminated with 3 well-known calibration standards: air, a metal plate and a Teflon slab. Then the calibrated reflection coefficient K(0) at the probe-end is obtained from the measured value K m after applying the appropriate corrections. 4. Experimental results The experimental setup is shown in Fig. 4. It consists of the open-ended coaxial probe, an Agilent 8714ET network analyzer and a PC. Measurements were performed for three concrete slabs with dimensions 600 mm 600 mm 100 mm: slab S00 without fibers, slab S20 with 20 kg/m 3 fibers (V f = ) and slab S40 with 40 kg/m 3 fibers (V f = ). The fibers are straight wires with l = 30 mm, d = 0.55 mm, hence l/d = 54.5 and A s = 44.5, and with hooks at the ends. Measurement of the permittivity of slab S00 yielded r,h = 7. This value is also used as the permittivity of concrete in the SFRC slabs. Next, a series of measurements was performed on both SFRC slabs by positioning the center of the probe along a square grid of measurement points with spacing 20 mm. Significant local fluctuations in the permittivity between neighboring measurement points were observed. This is probably due to the fact that the dimensions of the interrogation area of one measurement, formed by the annular probe aperture between the inner and outer conductor (less than 40 mm 40 mm) are not very large relative to the fiber length. Therefore, a spatial averaging is carried out by replacing the permittivity measurement in each gridpoint with the average over 3 3 gridpoints, i.e. the original point and its neighbors, such that the interrogation area is increased to approximately 80 mm 80 mm. The result is shown in Fig. 5 for an 8 8 grid on slab S20 at 600 MHz. The gray scale represents the fiber content in kg/m 3 obtained by inversion of (3). It can be seen that the fiber distribution still shows local variations. This may be due to an imperfect mixing of the fibers. The average value over all the grid points is lower than the expected value of 20 kg/m 3. Future work will concentrate on further improving the measurement accuracy and the inversion rule. 254

7 Fig. 4: The experimental setup. kg/m Fig. 5: The averaged measured fiber content for a SFRC slab with 20 kg/m 3 fibers. 255

8 5. Conclusion A non-destructive measurement technique for the determination of the steel fiber content in SFRC was presented. The technique is based on the use of an open-ended coaxial probe and a classical homogenization approach. The experimental results demonstrate that the proposed technique is of potential interest for quantifying the fiber content. This easy-to-use technique is also promising from a practical point of view. References 1. Taerwe, L., van Gysel, A., de Schutter, G., Vyncke, J. and Schaerlaekens, S., 'Quantification of variations in the steel fibre content of fresh and hardened concrete', Proceedings of the Third RILEM Workshop 'High Performance Fiber Reinforced Cement Composites (HPFRCC 3)', Ed. H.W. Reinhardt and A.E. Naaman, Rilem Proceedings PRO 6, Mainz (Germany), May 1999, Burdette, E.C., Cain, F.L. and Seals, J., 'In vivo probe measurement technique for determining dielectric properties at VHF through microwave frequencies', IEEE Trans. Microwave Theory Tech. 28 (1980) Franchois, A., Pineiro, Y. and Lang, R.H., 'Microwave Permittivity Measurements of Two Conifers', IEEE Trans. Geoscience Remote Sens. 36 (1998) Ulaby, F.T., Bengal, T.H., Dobson, M.C., East, J.R., Garvin, J.B. and Evans, D.L., 'Microwave dielectric properties of dry rocks', IEEE Trans. Geoscience Remote Sens. 28 (1990) Sihvola, A., 'Electromagnetic Mixing Formulas and Applications' (Inst. Elect. Eng., London, U.K., 1999, Electromagnetic Waves Series). 6. von Hippel, A.R., Ed., 'Dielectric Materials and Applications' (Artech House, London, U.K., 1954). 7. Ghodgaonkar, D.K., Majid, W.M.B.W.A. and Majid, R.B.A., 'Accurate measurement of electromagnetic properties of concrete for non-destructive evaluation at microwave frequencies', Proceedings Int. Conf. on Concrete Durability and Repair Technology, Eds. R.K. Dhir and M.J. McCarthy, Dundee, 8-10 September 1999, Klein, L.A. and Swift, C.T., 'An improved model for the dielectric constant of sea water at microwave frequencies', IEEE Trans. Antennas Propagat., 25 (1977), Anderson, J.M., Sibbald, C.L. and Stuchly, S.S., 'Dielectric measurements using a rational function model', IEEE Trans. Microwave Theory Tech. 42 (1994) Van Damme, S., Franchois, A., De Zutter, D. and Taerwe, L., 'Non-destructive determination of the steel fiber content in concrete slabs with an open-ended coaxial probe', submitted for IEEE Trans. Geoscience Remote Sens. 256