1. Introduction

Numerous experimental observations have shown that microinhomogeneous media manifest a "new" nonlinear elastic behavior different from the one explained by the classical theory of nonlinear elasticity proposed by Landau and applicable to homogeneous and crystalline media. The "new" properties, described in the framework of the nonlinear mesoscopic elasticity (NME) formalism, include most types of rocks, concrete, bones, damaged metals, and composites, which manifest the same macroscopic observations despite the existing differences in their microstructures and chemical constituents [1–8]. Indeed, all these materials share the characteristic of being complex with contacts, or microdefects (i.e., cracks), grain boundaries, dislocations, etc.

Observations on these materials concern quasi-static and dynamic acoustic experiments. In quasi-static tests, the stress-strain relationship is governed by an unusual multivalued function where the observed hysteretic loop contains small inner loops showing the presence of a memory effect [9]. In dynamic experiments with a propagating acoustic wave, the proportionality between input and output

elastic waves is no longer valid.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 relaxation and evolves in log-time [11–13].

F ¼ f <sup>0</sup> þ Cijkl ϵijϵkl (2)

2 ϵii þ 1 3 C ϵii

A ϵikϵkl þ 2 B ϵij δij þ C ϵii

<sup>ϵ</sup><sup>2</sup> (7)

<sup>2</sup>μþ<sup>λ</sup> is the quadratic nonlinear coefficient

K ¼ K0ð Þ 1 þ βϵ (8)

<sup>2</sup> (3)

<sup>3</sup> (4)

δij (5)

<sup>2</sup>μþ2<sup>λ</sup> is the

<sup>∂</sup><sup>x</sup> (9)

(6)

2

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

2 þ 1 <sup>2</sup> λ ϵii

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

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

> 1 3

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

Kijkl <sup>¼</sup> <sup>∂</sup>σij ∂ϵkl

> 1 3

<sup>3</sup>Aþ3BþC

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

In the 1-D case, the equation of motion corresponding to a longitudinal plane

∂2 u ∂x2

∂u

∂2 u <sup>∂</sup>t2 ¼ �2<sup>β</sup>

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

wave propagating in a quadratic nonlinear medium can be written as

∂2 u <sup>∂</sup>x2 � <sup>1</sup> C2 L

cubic nonlinear coefficient and l, m are the 3rd order elastic constants (called

A þ 3B þ C

λ ϵii δij þ

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

σ ¼ ð Þ 2μ þ λ ϵ þ

A ϵij ϵikϵkl þ B ϵij

F ¼ f <sup>0</sup> þ μ ϵij

Time Domain Analysis of Elastic Nonlinearity in Concrete Using Continuous Waves

where μ and λ are the Lamé coefficients.

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

F ¼ f <sup>0</sup> þ μ ϵij

<sup>σ</sup>ij <sup>¼</sup> <sup>∂</sup><sup>F</sup> ∂ϵij

The elastic modulus becomes

Murnaghan constants).

2.2 Wave equation

141

K0 is the Young modulus, and <sup>β</sup> <sup>¼</sup> <sup>1</sup>

medium are

general expression for an isotropic medium becomes

2 þ 1 <sup>2</sup> λ ϵii 2 þ 1 3

deriving the free energy with respect to strain are

¼ 2μ ϵij þ

1 2

reduced as

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 damaged concrete samples.
