3. Nonclassical nonlinear wave propagation in solids

The above discussed theory revealed to be nonsufficient to describe nonlinear behavior of elastic waves in diverse solids (concrete, cracked metals or composites, rocks, etc.). Indeed, these materials revealed that they belong to the class of nonlinear mesoscopic elastic materials (NMEM), where several experimental observations are in contradiction with the classical Landau theory expectations. In quasi-static stress-strain experiments (performed on sandstone, for instance), the dependence of stress on strain was nonlinear, hysteretic and showed the presence of memory effect [16]. Such evolution cannot be predicted in the framework of classical nonlinear Landau theory. Therefore, we need to introduce an additional term into the definition of stress by taking into account new parameters such as the stress dependence on the sign of the strain rate.

In addition to the quasi-static behavior described above, other observations suggest that nonlinearity in NMEM materials should have a different origin from that of atomistic nonlinearity and should therefore be related to the material structure. Most of these experiments are showing an anomalous dynamic behavior in NMEM materials, that is, when a time dependent perturbation is applied. In the different work in the literature, there is an agreement about the fact that most undamaged materials such as intact aluminum, Plexiglas, and monocrystalline solids show only a very small Time Domain Analysis of Elastic Nonlinearity in Concrete Using Continuous Waves DOI: http://dx.doi.org/10.5772/intechopen.82621

nonlinear response related to strain at the atomic scale. In this range, their behavior is well described by the Landau theory. Evidence of such nonlinearity is only manifested at moderately high strain levels. Indeed, when these materials are microdamaged, their behavior at strain amplitudes ε>10�<sup>7</sup> is more similar to that of Berea sandstone. Here, we should note that classical nonlinearity remains present, and as strain grows larger, its effect is hidden by stronger effects due to the presence of hysteresis. The following equation was therefore written as:

$$\frac{\partial^2 u}{\partial t^2} = \mathbf{C}\_0^2 \frac{\partial^2 u}{\partial \mathbf{x}^2} \left( \mathbf{1} + \beta \left. \frac{\partial u}{\partial \mathbf{x}} + \delta \left( \frac{\partial u}{\partial \mathbf{x}} \right)^2 + \dots \right) + H \left[ \varepsilon, \text{sign} \left( \frac{\partial \varepsilon}{\partial t} \right) \right] \tag{14}$$

H is a function describing hysteretic nonlinearity, and <sup>∂</sup><sup>ε</sup> <sup>∂</sup><sup>t</sup> is the strain rate. Note that this function depends on the strain rate and on the strain history as well. However, we should point out that an analytic expression of the H function is still missing the reason for which only few discrete models have been proposed to reproduce and give some understanding of the experimental observations related to nonlinear mesoscopic elastic materials [17–19].

As it was discussed for the classical nonlinearity, the dynamic nonlinear response of NMEM may manifest itself in a variety of ways. Many indicators can therefore be defined to link the detected strain amplitude of the driving frequency to resonance shift, harmonics amplitude, break of the superposition principle, etc. In that case, additional indicators not existing for classical nonlinear materials can be introduced, in the sense that the observed effects on mesoscopic materials might be very different depending on the excitation duration (seconds, minutes, etc.). Therefore, two categories of experiments can be defined: fast dynamics, when the experiment lasts one or few periods of the perturbation, and slow dynamics when the response of the system is tracked on much longer time scale to observe conditioning and relaxation.

#### 3.1 Fast dynamics

where CL is the wave speed in the linear medium. Eq. (9) can be rewritten as

� �

<sup>∂</sup>x2 <sup>1</sup> <sup>þ</sup> <sup>2</sup><sup>β</sup> <sup>∂</sup><sup>u</sup>

The velocity then becomes strain dependent and could be affected by any change in the strain amplitude of the propagating wave as a consequence of the change in the

As a consequence, in the case of a nondispersive medium, the same dependence could be observed and measured for the resonance frequency, wr, the latter being

> <sup>r</sup> � <sup>w</sup><sup>2</sup> L w2 l

The strain dependence of velocity is resulting in a shift of the resonance frequency when strain (or stress) amplitude increases. Therefore, the relation between

∝2

<sup>&</sup>gt;∝<sup>β</sup> <sup>&</sup>lt;ϵð Þ<sup>t</sup> <sup>&</sup>gt; <sup>þ</sup> <sup>δ</sup><ϵ<sup>2</sup>

<sup>2</sup> δ ϵ<sup>2</sup> max .

When the excitation is considered as a sinusoidal function, the frequency shift

The above discussed theory revealed to be nonsufficient to describe nonlinear behavior of elastic waves in diverse solids (concrete, cracked metals or composites, rocks, etc.). Indeed, these materials revealed that they belong to the class of nonlinear mesoscopic elastic materials (NMEM), where several experimental observations are in contradiction with the classical Landau theory expectations. In quasi-static stress-strain experiments (performed on sandstone, for instance), the dependence of stress on strain was nonlinear, hysteretic and showed the presence of memory effect [16]. Such evolution cannot be predicted in the framework of classical nonlinear Landau theory. Therefore, we need to introduce an additional term into the definition of stress by taking into account new parameters such as the stress

In addition to the quasi-static behavior described above, other observations suggest that nonlinearity in NMEM materials should have a different origin from that of atomistic nonlinearity and should therefore be related to the material structure. Most of these experiments are showing an anomalous dynamic behavior in NMEM materials, that is, when a time dependent perturbation is applied. In the different work in the literature, there is an agreement about the fact that most undamaged materials such as intact aluminum, Plexiglas, and monocrystalline solids show only a very small

wL <sup>∝</sup> <sup>1</sup>

wr � wL wL

C2 � <sup>C</sup><sup>2</sup> L C2 L

<sup>∝</sup> <sup>w</sup><sup>2</sup>

the resonance frequency shift and the strain amplitude could be written as

could be reasonably dependent on the higher order expansion term, where

∂x

� � is the wave speed. Note that, when higher nonlinear

<sup>¼</sup> C2 <sup>∂</sup><sup>2</sup> u

<sup>∂</sup>x2 (10)

ð Þt > (13)

<sup>∂</sup><sup>x</sup> <sup>þ</sup> <sup>3</sup><sup>δ</sup> <sup>∂</sup><sup>u</sup> ∂x

.

(12)

� �<sup>2</sup> h i

<sup>L</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup><sup>β</sup> <sup>∂</sup><sup>u</sup>

<sup>¼</sup> βϵ <sup>þ</sup> δϵ<sup>2</sup> (11)

∂2 u <sup>∂</sup>t2 <sup>¼</sup> C2 L ∂2 u

parameters are considered, the wave speed becomes C2 <sup>¼</sup> <sup>C</sup><sup>2</sup>

<sup>C</sup><sup>2</sup> � <sup>C</sup><sup>2</sup> L C2 L

wr � wL wL

max, which makes wr�wL

3. Nonclassical nonlinear wave propagation in solids

<

<sup>L</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup><sup>β</sup> <sup>∂</sup><sup>u</sup> ∂x

where C<sup>2</sup> <sup>¼</sup> <sup>C</sup><sup>2</sup>

Acoustics of Materials

elastic modulus.

<sup>δ</sup><ϵ<sup>2</sup>ð Þ<sup>t</sup> <sup>&</sup>gt; <sup>¼</sup> <sup>1</sup>

142

<sup>2</sup> δ ϵ<sup>2</sup>

dependence on the sign of the strain rate.

proportional to the velocity

During a dynamic experiment, fast dynamic effects appear rapidly (the very early pico or nanoseconds are sufficient to observe the amplitude dependence) and could be observed using standing or transient waves.

Harmonic generation consists in exciting a sample with a source function and analyzing signals detected by receivers in the frequency domain by determining harmonic amplitudes via a Fourier analysis. Most experiments were conducted using compressional waves; however, some bending [20] and torsional wave measurements have been conducted as well [21].

The plot of the second and third harmonic amplitudes as a function of the strain amplitude of the fundamental remains a power law <sup>y</sup> <sup>¼</sup> ax<sup>b</sup>, as for classical nonlinearity. However, the calculation of the slope b of the same curve plotted in logarithmic scale provides a new quantitative information (i.e., in contradiction with what theoretically expected), certainly linked to the nature of the nonlinearity, where the exponent revealed to be the same for the second and third harmonics. In addition, for the same dynamic strain, the amplitude of the third harmonic (i.e., odd harmonics) is larger than the one corresponding to the second harmonic (i.e., even harmonics).

#### 3.2 Nonlinear resonance frequency shift

Nonlinear effects can also be determined through the amplitude dependence of the resonance frequency through the technique named nonlinear resonant ultrasound spectroscopy (NRUS). The amplification provided by resonance makes

NRUS one of the most sensitive ways to observe nonlinear behavior, even at small dynamic strains (<sup>ε</sup> � <sup>10</sup>�8Þ. In general, we can excite resonances by sweeping upward and downward around a given resonance frequency, and frequency sweeps are repeated at successively increasing amplitude over the same frequency interval. The frequency shift resulting from the different strain amplitudes helps learning about the nature of nonlinearity. Note that the dependence of the normalized frequency shift on the strain is a power law y <sup>¼</sup> axb, where the exponent measured in different experiments revealed to be (b ¼ 1) again different from the prediction of the classical Landau theory.

3.2.1.2 Relaxation

strain level (equivalent to 10�<sup>6</sup> or 10�<sup>5</sup>

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

4. Experimental analysis

4.1 Experimental set up

chosen as Ainp

input function, we have

145

amplitude, materials state, strain duration, etc.) [11, 22].

compressional resonance mode (the fundamental in general)

recorded, each corresponding to an excitation amplitude Ainp

4.2 The scaling subtraction method (SSM)

put, in general, on opposite faces of the samples.

Relaxation starts right after the full conditioning (see Figure 1). In practice, a first frequency sweep around a given resonance mode is performed at a small strain amplitude to verify that the material is relaxed (resonance frequency and damping remain unchanged in time). Then, the same excitation is applied at a very large

Time Domain Analysis of Elastic Nonlinearity in Concrete Using Continuous Waves

sweeps are repeated at the lowest excitation amplitude (linear excitation �10�<sup>8</sup>

order to probe relaxation around the excited resonance mode. Relaxation takes relatively large time and changes as the logarithm of time before the final recovery over minutes, hours, or days depending on the conditioning characteristics (strain

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

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

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

<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

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

) for few minutes. Afterward, successive

u tðÞ¼ Ainp cos wð Þ <sup>0</sup>t (15)

<sup>i</sup> .

uA ¼ F Av ð Þ¼ <sup>0</sup> AF vð Þ¼ <sup>0</sup> Au<sup>0</sup> (16)

) in
