3.2.1.1 Conditioning

Conditioning or softening of the material takes place at dynamic strains corresponding to �10�<sup>6</sup> . Note that full conditioning could be obtained quite rapidly (few seconds to minutes), which seems to be long enough to allow neglecting its effects in fast dynamic experiments (much shorter time-scale). However, experiments show that most of conditioning occurs mostly instantaneously, which makes the coupling between fast dynamics and conditioning unavoidable, in the sense that the same wave propagating in a dynamic experiment is causing non-negligible self-conditioning. The elastic modulus decreases continuously during the dynamic excitation of NMEM until the material reaches a new equilibrium state where no more change takes place (see Figure 1). The amount of conditioning depends on the excitation time and amplitude [11, 22].

#### Figure 1.

Scheme of the experimental set-up (left); frequency (or elastic modulus) conditioning and relaxation evolution as a function of time (right).

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

#### 3.2.1.2 Relaxation

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

Slow dynamics is by far the most typical characteristic of the nonclassical NMEM. It refers to the logarithm dependence recovery of the elastic modulus to the original initial value (i.e., at rest) after being excited and therefore softened by a large amplitude strain. The log-time evolution of the elastic modulus resembles to a creeplike behavior observed in quasi-static experiments. However, it is important to note that slow dynamics is more likely a new creep behavior due to the fact that modulus is not following the symmetry of the strain. Furthermore, contrary to creep experiments, slow dynamics is a reversible and repeatable behavior, and observations on rocks and some damaged metals are performed at dynamic strain levels two or three orders of magnitude below those of a typical creep experiment. Slow dynamics

includes two different scale mechanisms: conditioning and relaxation.

Conditioning or softening of the material takes place at dynamic strains

(few seconds to minutes), which seems to be long enough to allow neglecting its effects in fast dynamic experiments (much shorter time-scale). However, experiments show that most of conditioning occurs mostly instantaneously, which makes the coupling between fast dynamics and conditioning unavoidable, in the sense that the same wave propagating in a dynamic experiment is causing non-negligible self-conditioning. The elastic modulus decreases continuously during the dynamic excitation of NMEM until the material reaches a new equilibrium state where no more change takes place (see Figure 1). The amount of conditioning depends on the

Scheme of the experimental set-up (left); frequency (or elastic modulus) conditioning and relaxation evolution

. Note that full conditioning could be obtained quite rapidly

of the classical Landau theory.

3.2.1 Slow dynamic effects

Acoustics of Materials

3.2.1.1 Conditioning

Figure 1.

144

as a function of time (right).

corresponding to �10�<sup>6</sup>

excitation time and amplitude [11, 22].

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 strain level (equivalent to 10�<sup>6</sup> or 10�<sup>5</sup> ) for few minutes. Afterward, successive sweeps are repeated at the lowest excitation amplitude (linear excitation �10�<sup>8</sup> ) in 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 amplitude, materials state, strain duration, etc.) [11, 22].
