Classical elastic wave features like pulse velocity and attenuation have been used for decades for concrete condition characterization. Relatively recently the effect of frequency has been studied showing no doubt over the dispersive behavior of the material. Despite the experimental evidence, there is no unified theory to model the material and explain this phase velocity change at frequencies below 200 kHz. Herein, the Mindlin's strain gradient elastic theory including the additional micro-stiffness and micro-inertia parameters is considered as an alternative of multiple scattering theory. Experimental results are produced from material with dictated microstructure using a specific diameter of glass beads in cement paste. Results show that Mindlin's theory provides conclusions on the microstructure of the material and is suitable for describing the observed dispersion in different length scales (from millimeters in the case of mortar to several centimeters in the case of concrete).

## 1. Introduction

Well known traditionally used wave features like pulse velocity and attenuation have proven very useful for the characterization of concrete materials and structures. However, with the increasing need for better performance, more detailed assessment, and detection of smaller defects in this multi-scale material, researchers gradually started from the 1990s to investigate the possibility of exploiting the effect of the applied frequency.^{1} It has been shown that when a longitudinal wave travels through concrete it exhibits significant dispersion which is indicated by the rapid change of phase velocity at low frequencies where the material microstructure is smaller than the wavelength of the incident wave. The phenomenon of dispersion is more pronounced in concrete compared to cement paste and mortar due to the larger dimension of the aggregate inclusions while it is influenced by the water and the sand content.^{2}

Moreover, experiments on concrete with simulated damage in the form of light inclusions showed a strong dependence of longitudinal and Rayleigh wave velocity on the frequency which actually was qualitatively close to the theoretical prediction from multiple scattering theory considering the problem of scattering on cavities.^{3,4} Scattering models, accounting for the interaction between inclusion size and wavelength, have also shown good results in case of fresh concrete considering again cavities in liquid matrix, a phenomenon that has been defined as “bubble resonance.”^{5} However, scattering models exhibit limitations on the scatterers content unless special procedures are applied.^{6} In addition, for the problem of stiff scatterers in a less stiff matrix (like aggregates in cement paste), the theoretical dispersion curves cannot closely follow the increasing trend of experimentally measured phase velocity^{7} with the exception of Molero *et al.*^{8} In the latter, good agreement is noticed for frequencies up to 5 MHz but the lower frequency band of more practical interest (below 500 kHz) is not treated in detail while additionally the cementitious specimens contain very small size of scatterers (<1 mm). Furthermore, the theoretical scattering curves and the experimental ones are far from the value predicted by well-known composite mixture models like Christensen's.^{9} Diffuse ultrasound has yielded correlations with the degree of damage^{10} but cannot easily relate to the characteristic microstructure size. Finally, Biot's theory^{11} has long been introduced but it mainly concerns fluid saturated porous media being therefore suitable to treat problems of hydrating concrete without evidence of adequacy to explain dispersive phenomena in hardened concrete.

Since different theories prove insufficient to globally describe the phenomenon of wave dispersion in concrete, the simple strain gradient elastic theory of Mindlin has alternatively been proposed in literature. According to it, the microstructure is taken into account with the introduction of two internal length scale material parameters, namely, the micro-stiffness (*g*) and micro-inertia (*h*). In a recent study^{12} it was shown that the theoretical dispersion curve is suitable to closely follow the experimentally measured increasing trend of phase velocity in hardened concrete up to 200 kHz, while at the same time it worked equally well for the case of fresh mortar. In the second case the inverse trend is exhibited with decreasing phase velocity for elevating frequencies. The average inclusion size of hardened concrete in that work was approximately 20 mm which is characteristic of the concrete material.

The aim of the current paper is to examine the validity of Mindlin's theory in different length scales of microstructure and characterize its efficiency to model the cementitious media of any composition. Therefore, the aggregates in this study are replaced by glass beads with controlled inclusion size in order to eliminate the variance in the microstructure size and the calculated values of the micro-stiffness and micro-inertia coefficients are related to the known microstructural scale.

## 2. Wave dispersion equation of Mindlin and its micro-structural coefficients

In his celebrated paper of 1964, Mindlin presented a generalized continuum theory for describing the behavior of materials with microstructure.^{13} The theory, in its simplest form, introduces only two intrinsic lengths, one for stiffness (notated with *g*) and one for inertia (notated with *h*), which give a measure of the area of microstructure that influences macrostructure and that is responsible for the dispersive behavior of longitudinal waves propagating in nonhomogeneous materials. The simplified form is reached by considering (a) the potential and kinetic energy density as a function of strains and the gradient of strains, the velocities and the gradient of velocities, respectively, and (b) longitudinal wave propagation in one direction. Assuming then harmonic excitation of the aforementioned microstructured medium leads to the following dispersion equation of longitudinal waves in the model of Mindlin:

In Eq. (1)*V _{p}* is the calculated phase velocity (in m/s),

*ω*is the angular frequency (in rad/s),

*c*is the wave velocity for frequency tending to zero (in m/s), while

*g*and

*h*represent the microstructural coefficients (length scale, in m). It should be mentioned that the original version of Mindlin's theory does not include damping although this can be added by an additional damping factor as in classical elasticity. For more information on the simplifications, the original wave equation and the derivation of the final wave dispersion equation the interested reader is referred to Mindlin

^{13}and Iliopoulos

*et al.*

^{12}

## 3. Experimental details and phase velocity calculation

In the current study, ultrasonic through transmission experiments are performed on hardened concrete samples where sand and coarse aggregates are substituted by spherical glass beads of two different sizes (namely, 8 mm and 600–800 *μ*m) and three different volumetric contents (nominally 9%, 16%, and 23%) in order to better control the microstructure. The matrix is composed of cement and water with the water over cement ratio (w/c) being equal to 0.5 by mass. For the experiments two ultrasonic broadband longitudinal wave contact transducers with central frequency 1 MHz are used while the thickness of the specimens varies from 25 mm for the samples composed of small sized spheres (600–800 *μ*m) to 50 mm for the samples containing 8 mm glass spheres. Vaseline grease is used as the coupling material.

The dispersion curve (or the phase velocity of each frequency component) is calculated considering a wideband excitation and the use of the aforementioned sensors similarly to Sachse and Pao.^{14} Applying first the fast Fourier transformation, the “phase” of the excited and the received wave is calculated and unwrapped. The difference of the unwrapped phases is then considered for each frequency component of the Fourier spectrum leading finally to the calculation of the phase velocity for that frequency. The description of the whole procedure is beyond the scope of this manuscript. The interested reader can be directed to the original paper of Sachse and Pao.^{14} It is mentioned that only the initial 1.5 cycle of the waveform is used while the rest is zero-padded. This is to avoid any reflection or other types of waves arriving later than the longitudinal, as well as the possible ringing behavior of the sensor.^{2} The received time domain signals are sampled with 20 MHz frequency and are composed of 1024 samples.

## 4. Results and discussion

Indicative experimental dispersion curves for concrete with different inclusion size and different volumetric content are shown in Figs. 1(a) and 1(b), respectively, with black dots. The *y* axis of the curves represents the phase velocity (in m/s) while the *x* axis refers to frequency values expressed both in rad/s [angular frequency–see Eq. (1)] and in kHz for easier interpretation and practical reasons. As shown in Fig. 1(a), the larger the size of inclusion (8 mm instead of 600 *μ*m) the higher the values of phase velocity and the more steep their increase even for the same inclusion volume fraction (16%). Similarly to Fig. 1(a), the effect of volumetric content of small sized glass spheres (600–800 *μ*m) is shown in Fig. 1(b). As the content increases, the curve is clearly translated to higher levels while the “bending” of the curve is slightly shifted to lower frequencies.

At a next step, the wave dispersion equation of Mindlin [Eq. (1)] is applied on the experimental data (continuous red line) providing the parameters *g*, *h*, and *c* based on the non-linear least squares fitting principle. In the cases described herein and all cases examined in total, Mindlin's dispersion relation is able to give a very close match to the experimental results as indicated by the high values of the *R*^{2} coefficient of determination (values higher than 0.98). An important observation comes from the resulting values of the microstructural coefficients which are also presented in Fig. 1. First the micro-stiffness (*g*) obtains values higher than micro-inertia (*h*) by a factor of 2–3. This is consistent with the physical model of the system as the inclusions are harder than the solid matrix. In the case of inclusions in liquid matrix, micro-inertia obtains higher values than micro-stiffness as demonstrated in Iliopoulos *et al.*^{12} In addition, the values of the microstructural coefficients show a firm relation to the inclusion size and second to the inclusion content. In Fig. 2(a) one can see the correlation of the average micro-stiffness *g* value vs the inclusion diameter. This average comes from five different specimens with an excellent match between experimental and theoretical curves, showing a goodness of fit *R*^{2} value higher than 0.98. Indicatively, for the case of the small microstructure of 600 *μ*m, the *g* values range between 2.4 and 3.3 mm averaging at 2.8 mm. While for the small size of inclusion the average value of *g* is 2.8 mm, for inclusions of 8 mm it becomes 4.5 mm. The curve is completed with the micro-stiffness calculated for concrete with average inclusion size of 23 mm that was obtained from a previous study.^{12} The aggregate properties in Ref. 12 are similar to the inclusion properties in this study. More specifically, the aggregates in the first case have a mass density and Young's modulus equal to 2650 kg/m^{3} and 70 GPa, respectively, while in the second case the mass density is 2460 kg/m^{3} and the modulus of elasticity equal to 80 GPa. It is obvious that as the actual characteristic size of the material's microstructure increases from the order of hundreds of *μ*m to mm to cm, the micro-stiffness exhibits an also very clear increase following the same order of magnitude. At the same time the micro-inertia, *h*, shows a similar behavior [see Fig. 2(b)] climbing from 1 to 2.4 mm and finally to 16 mm for the case of concrete. Unlike Ref. 12 where a small change of the w/c ratio causes only a small change of the microstructural coefficients, the change of the inclusion diameter in this study is sufficient to change even the order of magnitude of them.

A smaller but also noteworthy connection exists between these coefficients and the inclusion volumetric content. The decreasing trend implies on the one hand a weakening of the strain gradients connected to a higher degree of mixture homogenization as the inclusions content becomes higher and dominates the matrix [Fig. 3(a)]. On the other hand as the inclusion content increases, the neighborhood of a representative inclusion particle becomes more and more stiff, rendering any micro-movement around the particle's equilibrium position more difficult and decreasing thus the micro-inertia coefficient [Fig. 3(b)]. The results of Figs. 2 and 3 are indicatively fitted without suggesting that these are the unique possible fits for each case. The importance lies on the monotonic correlation that is shown at least up to the range of 25 mm, between the micro-structural coefficients and the actual inclusion size or volume content.

The exact relation between the coefficients *g* and *h* and the microstructure cannot be accurately determined. However, it is obvious that these parameters follow the increase of the physical size of the inclusions, which in sequence means bigger representative volume element, or longer interparticle distances or larger size of other characteristic microstructural descriptors. Based on the good matching between Mindlin's theory and experiments the suitability of the first is again highlighted while it is equally interesting one to notice the realistic magnitude of the microstructural coefficients which is of the order of the inclusions.

As a general comment, the difference in phase velocity curves for different inclusion sizes does not decrease when moving to lower frequencies, see Fig. 1(a). However, it should be mentioned that even if the problem is seen from the linear elastic scattering point of view, the two cases do not fall both into the “Rayleigh” or long wavelength regime. Taking the dimensionless parameter, *a = πD/λ*, where *D* is the inclusion diameter and *λ* is the wavelength, the domain of Rayleigh scattering is considered for $a\u226a1$. For the inclusion of 8 mm spheres, for the lowest frequency tested (78 kHz) and for the longitudinal wave velocity of 2752 m/s, the wavelength *λ* is equal to 35.3 mm. This provides an “*a*” value of 0.71 which is close to 1. On the other hand, for the case of the small sized spheres (0.6 mm) and the same frequency, the velocity is 1676 m/s while the wavelength is 21.5 mm corresponding to an “*a*” value of 0.088. Therefore, while for the small sphere the Rayleigh regime can be supported (0.088 $\u226a$ 1), this cannot be said for the larger inclusions. In addition, when microstructural phenomena dominate, it is reasonable to expect that the behavior departs away from the linear elastic case (*g* = *h* = 0). In this case the microstructural coefficients are of the order of mm to cm which is quite large. In literature *g*, *h*, as low as 0.1 mm,^{15} resulted in dispersion curves much different than the linear elastic solution.

## 5. Conclusions

In this study, the dispersive nature of hardened cementitious specimens with different size and volumetric content of inclusions is modeled by the wave dispersion equation of Mindlin providing the values of its microstructural coefficients. So far, those coefficients were property-free mathematical only values that are not explicitly correlated with the microstructural geometry or mix design parameters. Herein, a physical meaning is assigned to those microstructural coefficients by linking them with the size of the inclusion and the inclusion volume fraction. This study shows that the microstructure model and the corresponding wave equation proposed by Mindlin is suitable to describe the experimental dispersion of hardened concrete, while the derived microstructural coefficients follow the actual size of the microstructure dictated by the inclusions. Apart from the scientific interest, this approach opens the way for microstructure characterization (e.g., size and volume fraction of inclusions or distributed cracking) based on the experimental dispersion curve having also strong practical interest.

## Acknowledgments

The corresponding author gratefully acknowledges FWO Research Foundation-Flanders for the financial support of this study (scholarship and travel grant for two-month research stay at CBM, Munich). The authors also wish to acknowledge the reviewers' contribution to the improvement of the discussion of the paper.