4. Experimental analysis

### 4.1 Experimental set up

Experiments are conducted generating ultrasonic signals through a waveform generator. Ultrasonic signals defined as monochromatic waves of amplitude Ainp and frequency w<sup>0</sup> are used to excite the sample under test not far from one of its compressional resonance mode (the fundamental in general)

$$u(t) = A\_{imp} \cos\left(w\_0 t\right) \tag{15}$$

The emitter transducer is glued to the sample using a linear coupling (phenyl salicylate, for instance). A second (identical) transducer is used to detect the response of the material under test, and it is connected to a digital oscilloscope for data acquisition. Signals are recorded in a short time window once stationary conditions are reached. In order to excite longitudinal modes, the transducers are put, in general, on opposite faces of the samples.

The experimental procedure starts by detecting the output signal at a very low excitation level Ainp. The latter is chosen the lowest possible in order to not have any change in the mechanical properties of the material under test. To verify this, we should have a good signal-to-noise ratio to generate output signals emerging from the noise level. In most of the presented experiments, the lowest amplitude was chosen as Ainp <sup>0</sup> ¼ 5mV:Under these conditions, the sample under test behaves almost linearly, and the recorded low amplitude response will be termed as "linear signal" v0ð Þt . The linear signal measurement, performed without amplification in general, is followed by N acquisitions repeated increasing the amplitude of excitation up to a maximum level. The recorded N signals við Þt , ið Þ ¼ 1; ::; N are recorded, each corresponding to an excitation amplitude Ainp <sup>i</sup> .

#### 4.2 The scaling subtraction method (SSM)

Under a dynamic excitation, the presence of nonlinearity can be detected through the validity of the superposition principle, which represents a requirement for a system to be linear. By considering a linear function F, if v tðÞ¼ A v0ð Þt is the input function, we have

$$\mathfrak{u}\_A = F(A\nu\_0) = AF(\nu\_0) = Au\_0 \tag{16}$$

F is the transfer function, u denotes the output signal, and A is the amplification factor. Here, u0 is the response at the excitation amplitude v0.

Consider an elastic wave propagating in a microdamaged medium. In such a case, one might expect that if the propagation excites the nonlinearity of the system, it will consequently break the superposition property. Therefore, if the exciting wave is generated at amplitude A0, small enough so that nonlinearity of the medium is negligible, the system will behave linearly and its response is u0ð Þt . At a larger excitation amplitude A, the response u tð Þ of the same system is no longer equal to

$$
\mu\_{ref}(t) = \frac{A}{A\_0} \,\,\mu\_0(t) \tag{17}
$$

in log-log scale, the above equation reduces to a straight line 20 log<sup>10</sup> θ ¼ a<sup>0</sup>

Time Domain Analysis of Elastic Nonlinearity in Concrete Using Continuous Waves

Consolidated granular media and in particular concrete exhibit a strong nonlinear hysteretic elastic behavior when excited by ultrasonic wave perturbations

damaged [29]. A significant enhancement of the nonlinear response can be created by one of the numerous damages that might occur within concrete structures via quasi-static loading [30, 31], thermal stresses [8, 32, 33], carbonation [34], corrosion [35], and salt expansion [36]. Here, we show how different damage types affect the nonlinear observations derived from the scaling subtraction method, thus suggesting SSM to be a suitable method to monitor damage evolution in time. We also wish to highlight how nonlinear indicators defined using the SSM approach (particularly the slope b) allow to discriminate between different types of

One of the major effects that create damage in concrete is the application of mechanical loads in the presence of discontinuities. Indeed, discontinuity surfaces are very often the place from where damage may begin its progression. The effects could be the increase in crack density [37] and/or the crack openings [38, 39], depending on the nature of discontinuity such as existing cracks [40] or weak layers

In [25], one specimen with an internal discontinuity surface was produced by piling up two concrete cubes (measuring 10 cm on each side). The two pieces were joined using a thin layer of cement paste. Concrete cubes were produced using a concrete mix with CEM II A-L 42.5 R cement, ordinary aggregates (max.

size = 16 mm) and a water to cement ratio equal to 0.74, with no admixtures. Their age at the date of testing was approximately 6 months. The evaluation of the mechanical characteristics of the concrete was performed using mono-axial static

tudinal wave speed in the cube was measured to be V ¼ 3850m=s and the density of

As explained above, the evolution of damage as a function of the applied load can be followed using different nonlinear indicators. However, in these experiments, it is important to note that the frequency analysis (using FFT for instance) was not efficient, since the nonlinear indicators related to the possible generated frequencies (higher order harmonics) were below the noise level. Therefore, the application of the traditional nonlinear elastic wave spectroscopy is not expected to be efficient in detecting the presence of nonlinearity during these experiments.

. The longi-

compression tests that resulted in a compressive strength of 24 N/mm<sup>2</sup>

the cubes <sup>ρ</sup> <sup>¼</sup> <sup>2330</sup> kg=m3. For this experiment, one emitter (T1) and three receivers (R2, R3, and R4) were used, as schematized in Figure 3. Two receivers (R2 and R3) performed direct transmission measurements; whereas, the third (R4) was used in indirect transmission mode. It is important to note that the direct path from the emitter to R2 and R4 crosses the discontinuity of the concrete, while the path to receiver R3 does not (See Figure 3). Only the results obtained from receiver

[23–28]. The nonlinear behavior is strongly enhanced when the concrete is

5. Application to nonlinear characterization of concrete

5.1 Load effects on discontinuities in concrete

R4 will be shown here for the sake of conciseness.

As an alternative, the nonlinear indicator could be defined as the value of θ at a

þ b20 log<sup>10</sup> x:

damage.

[41–43].

147

fixed value of x (excitation level).

DOI: http://dx.doi.org/10.5772/intechopen.82621

which would be the response of that system if it remains linear even at large amplitudes. Therefore, the difference between the two responses can be taken as an indicator of nonlinearity. The nonlinear scaled subtracted signal w tð Þ, termed SSM signal, is introduced as (see Figure 2)

$$
\omega(t) = \mathfrak{u}(t) - \mathfrak{u}\_{\rm ref}(t) \tag{18}
$$

This time domain analysis of elastic nonlinearity, called scaling subtraction method (SSM), has proved to be sensitive to damage detection and easy to set [16].

From the quantitative point of view, if exciting signals are in the form of monochromatic continuous waves and measurements are taken in standing wave conditions, the SSM signal w(t) is also a continuous wave. Thus, a parameter could be introduced either as the maximum or the "energy" of the nonlinear signal w(t) as

$$\theta = \max(w(t))\tag{19}$$

$$\theta'=\mathbf{1}/T\int\_{0}^{T}w^2(t)dt\tag{20}$$

where T is the wave period. The parameter could then be shown as a function of excitation amplitude in order to highlight nonlinearity.

From the point of view of damage monitoring, for a given sample state, a quantitative nonlinear indicator must be defined. To this purpose, we observe that in materials exhibiting hysteresis, a power law holds in the form

$$
\theta = \mathfrak{ax}^b \tag{21}
$$

where x is the maximum of the output amplitude. Thus, experimental data could be fitted to derive the coefficient a and the parameter b, normally called slope, since

Figure 2. Basic principle of SSM analysis.

Time Domain Analysis of Elastic Nonlinearity in Concrete Using Continuous Waves DOI: http://dx.doi.org/10.5772/intechopen.82621

in log-log scale, the above equation reduces to a straight line 20 log<sup>10</sup> θ ¼ a<sup>0</sup> þ b20 log<sup>10</sup> x:

As an alternative, the nonlinear indicator could be defined as the value of θ at a fixed value of x (excitation level).
