2. Classical nonlinear theory

#### 2.1 Stress-strain relations

In the approximation of small deformations, the free energy of an elastic system can be expanded as a power series with respect to the strain tensor. The general expression for the free energy of an isotropic body in the third approximation can be reduced by the convention of Einstein notation to

$$F = f\_{\;\;0} + \mathcal{C}\_{ijkl}\;\epsilon\_{ij}\epsilon\_{kl} + \mathcal{C}\_{ijklmn}\;\epsilon\_{ij}\epsilon\_{kl}\epsilon\_{mn} \tag{1}$$

where f0 is the constant, and Cijkl and Cijklmn are, respectively, the second and third order elastic tensors. For small strain amplitudes, the cubic term in the elastic energy can be neglected and the free energy reduces to

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

$$F = f\_{\ o} + C\_{ijkl} \, e\_{ij} e\_{kl} \tag{2}$$

In the specific case of isotropic materials, the expression could be further reduced as

$$F = f\_{\text{ o }} + \mu \left| \epsilon\_{\vec{\eta}} \right|^2 + \frac{1}{2} \left\lambda \left| \epsilon\_{\vec{\eta}i} \right|^2 \tag{3}$$

where μ and λ are the Lamé coefficients.

elastic waves is no longer valid. In such case, wave propagation (or even in standing wave conditions) is accompanied by the generation of amplitude dependent higher harmonics and sidebands [6, 10]. In addition, in resonance experiments, the increase in the dynamic perturbation creates a decrease in the elastic modulus of the propagating medium. This effect, which might be local or global, is observed through a decrease in the resonance frequency showing thus a softening in the elastic properties with an increase in damping around the excited resonance mode [11]. For relatively large excitation amplitudes, in the reversible regime, experiments show the presence of conditioning which means that the softening of elastic properties persists even when the excitation is switched off. In this case, at a given excitation amplitude, it takes seconds to minutes to stop the softening process and leads the medium into a new "equilibrium" state. The conditioning stops when the

excitation is switched off. In this case, the medium needs minutes to days (depending on its state) to go back to its initial elastic state. This process is called

So far in the literature, the aforementioned nonlinear effects have all been grouped into the same class in contrast with the classical nonlinearity well described by the Landau theory. However, a link has been postulated between the macroscopic nonlinear response and the microstructure of the different media [14]. Neutron scattering measurements have confirmed that the nonlinear behavior is localized in small regions close to discontinuities [15]. The observed regions might have different properties which makes the physical mechanisms described by the constitutive equation different. This makes the development of research around the existing relationship between the microscopic features and the different macroscopic observations interesting for basic research and for microcracks diagnosis in complex media.

Based on the above definitions, concrete as a consolidated granular medium is complex. Indeed, it exhibits a complicated nonlinear elastic behavior including hysteresis, harmonics generation, loss of reciprocity, etc. The complex structure of concrete can be seriously affected when damage is present. The latter, which can be of different origins, leads to important changes in the quasi-static (for advanced damage stages) and dynamic (for early damage stages) responses of concrete. Steel corrosion in reinforced concrete elements, mechanical stresses, thermal stresses, chemical attack by expansive agents, etc. have all negative effects on the concrete load carrying capacity, since they all lead to an increase in crack density and propagation. In this chapter, we propose to study the efficiency of a time domain analysis of elastic nonlinearity using continuous waves propagating in progressively

In the approximation of small deformations, the free energy of an elastic system can be expanded as a power series with respect to the strain tensor. The general expression for the free energy of an isotropic body in the third approximation can

where f0 is the constant, and Cijkl and Cijklmn are, respectively, the second and third order elastic tensors. For small strain amplitudes, the cubic term in the elastic

F ¼ f <sup>0</sup> þ Cijkl ϵijϵkl þ Cijklmn ϵijϵklϵmn (1)

relaxation and evolves in log-time [11–13].

Acoustics of Materials

damaged concrete samples.

2.1 Stress-strain relations

140

2. Classical nonlinear theory

be reduced by the convention of Einstein notation to

energy can be neglected and the free energy reduces to

At larger but still infinitesimal strains, the contribution of the third-order elastic term in the free energy expansion can no more be neglected. In such a case, the general expression for an isotropic medium becomes

$$F = f\_{\;\;0} + \mu \left. c\_{\dot{\text{ij}}} \right. + \frac{1}{2} \left. \lambda \left. c\_{\dot{\text{ii}}} \right. \right. + \frac{1}{3} \left. \mathbf{A} \left. c\_{\dot{\text{ij}}} \right. c\_{\dot{\text{ik}}} \right. + \mathbf{B} \left. c\_{\dot{\text{ij}}} \right. \right. \\ \left. \left. \mathbf{C} \left. c\_{\dot{\text{ii}}} \right. \right. \tag{4}$$

where A, B, and C are the components of the third order elastic tensor for isotropic bodies. Consequently, the components of the stress tensor, obtained deriving the free energy with respect to strain are

$$
\sigma\_{\text{i}\natural} = \frac{\partial F}{\partial \epsilon\_{\text{i}\natural}} = 2\mu \,\epsilon\_{\text{i}\natural} + \frac{1}{2} \,\lambda \,\epsilon\_{\text{ii}\parallel} \,\delta\_{\text{i}\parallel} + \frac{1}{3} \,\text{A} \,\epsilon\_{\text{ik}} \epsilon\_{\text{kl}} + 2 \,\text{B} \,\epsilon\_{\text{i}\parallel} \,\delta\_{\text{i}\parallel} + \text{C} \,\epsilon\_{\text{ii}} \,^2 \delta\_{\text{i}\parallel} \tag{5}
$$

Since strains are still infinitesimal, the corresponding elastic constants of the medium are

$$\mathbf{K}\_{\rm ijkl} = \frac{\partial \sigma\_{\rm ij}}{\partial \epsilon\_{\rm kl}} \tag{6}$$

In the 1-D case, these equations reduce to the compressional stress σ

$$\boldsymbol{\sigma} = \left(2\boldsymbol{\mu} + \lambda\right)\boldsymbol{\epsilon} + \left(\frac{1}{3}\mathbf{A} + \mathbf{3B} + \mathbf{C}\right)\boldsymbol{\epsilon}^2\tag{7}$$

The elastic modulus becomes

$$\mathbf{K} = \mathbf{K}\_0 (\mathbf{1} + \beta \mathbf{e}) \tag{8}$$

K0 is the Young modulus, and <sup>β</sup> <sup>¼</sup> <sup>1</sup> <sup>3</sup>Aþ3BþC <sup>2</sup>μþ<sup>λ</sup> is the quadratic nonlinear coefficient for longitudinal waves in isotropic media. Note that higher order terms of the elastic modulus can be obtained if we develop the free energy to the fourth order. In such a case, the modulus K would be K <sup>¼</sup> K0 <sup>1</sup> <sup>þ</sup> βϵ <sup>þ</sup> δϵ ð Þ <sup>2</sup> <sup>þ</sup> … , where <sup>δ</sup> <sup>¼</sup> 3lþ2m <sup>2</sup>μþ2<sup>λ</sup> is the cubic nonlinear coefficient and l, m are the 3rd order elastic constants (called Murnaghan constants).

#### 2.2 Wave equation

In the 1-D case, the equation of motion corresponding to a longitudinal plane wave propagating in a quadratic nonlinear medium can be written as

$$\frac{\partial^2 \mathbf{u}}{\partial \mathbf{x}^2} - \frac{1}{\mathbf{C}\_{\mathbf{L}}^2} \frac{\partial^2 \mathbf{u}}{\partial \mathbf{t}^2} = -2\mathfrak{d} \frac{\partial^2 \mathbf{u}}{\partial \mathbf{x}^2} \frac{\partial \mathbf{u}}{\partial \mathbf{x}} \tag{9}$$

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

$$\frac{\partial^2 \mathbf{u}}{\partial \mathbf{t}^2} = \mathbf{C}\_{\mathbf{L}}^2 \frac{\partial^2 \mathbf{u}}{\partial \mathbf{x}^2} \left[ \mathbf{1} + 2 \mathfrak{P} \, \frac{\partial \mathbf{u}}{\partial \mathbf{x}} \right] = \mathbf{C}^2 \frac{\partial^2 \mathbf{u}}{\partial \mathbf{x}^2} \tag{10}$$

nonlinear response related to strain at the atomic scale. In this range, their behavior is

þ …

<sup>þ</sup> <sup>H</sup> <sup>ε</sup>;sign <sup>∂</sup><sup>ε</sup>

∂t � � � � (14)

<sup>∂</sup><sup>t</sup> is the strain rate. Note

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

Time Domain Analysis of Elastic Nonlinearity in Concrete Using Continuous Waves

<sup>þ</sup> <sup>δ</sup> <sup>∂</sup><sup>u</sup> ∂x � �<sup>2</sup>

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

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.

During a dynamic experiment, fast dynamic effects appear rapidly (the very early pico or nanoseconds are sufficient to observe the amplitude dependence) and

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 mea-

The plot of the second and third harmonic amplitudes as a function of the strain

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

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).

!

of hysteresis. The following equation was therefore written as:

∂x

H is a function describing hysteretic nonlinearity, and <sup>∂</sup><sup>ε</sup>

nonlinear mesoscopic elastic materials [17–19].

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

could be observed using standing or transient waves.

surements have been conducted as well [21].

3.2 Nonlinear resonance frequency shift

∂2 u <sup>∂</sup> <sup>t</sup><sup>2</sup> <sup>¼</sup> <sup>C</sup><sup>2</sup> 0 ∂2 u <sup>∂</sup> <sup>x</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>β</sup> <sup>∂</sup><sup>u</sup>

3.1 Fast dynamics

143

where C<sup>2</sup> <sup>¼</sup> <sup>C</sup><sup>2</sup> <sup>L</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup><sup>β</sup> <sup>∂</sup><sup>u</sup> ∂x � � is the wave speed. Note that, when higher nonlinear parameters are considered, the wave speed becomes C2 <sup>¼</sup> <sup>C</sup><sup>2</sup> <sup>L</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup><sup>β</sup> <sup>∂</sup><sup>u</sup> <sup>∂</sup><sup>x</sup> <sup>þ</sup> <sup>3</sup><sup>δ</sup> <sup>∂</sup><sup>u</sup> ∂x � �<sup>2</sup> h i. 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 elastic modulus.

$$\frac{\mathbf{C}^2 - \mathbf{C}\_{\mathbf{L}}^2}{\mathbf{C}\_{\mathbf{L}}^2} = \beta \epsilon + \delta \mathbf{e}^2 \tag{11}$$

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 proportional to the velocity

$$\frac{\mathbf{C}^2 - \mathbf{C}\_{\mathrm{L}}^2}{\mathbf{C}\_{\mathrm{L}}^2} \propto \frac{w\_r^2 - w\_L^2}{w\_l^2} \propto 2 \frac{w\_r - w\_L}{w\_L} \tag{12}$$

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

$$<\frac{w\_r - w\_L}{w\_L} > \infty \left| <\epsilon(t) > +\delta < \epsilon^2(t) > \right. \tag{13}$$

When the excitation is considered as a sinusoidal function, the frequency shift could be reasonably dependent on the higher order expansion term, where <sup>δ</sup><ϵ<sup>2</sup>ð Þ<sup>t</sup> <sup>&</sup>gt; <sup>¼</sup> <sup>1</sup> <sup>2</sup> δ ϵ<sup>2</sup> max, which makes wr�wL wL <sup>∝</sup> <sup>1</sup> <sup>2</sup> δ ϵ<sup>2</sup> max .
