^{1,a)}, A. S. Gliozzi

^{1}, C. L. E. Bruno

^{2}and P. Antonaci

^{2}

### Abstract

Concrete, particularly if damaged, exhibits a peculiar nonlinear elastic behavior, which is mainly due to the coupling between nonequilibrium and nonlinear features, the two of which are intrinsically connected. More specifically, the formulation of a constitutive equation able to properly predict the dynamic behavior of damaged concrete is made difficult by the concomitant presence of two mechanisms: The modification of the microstructure of the medium and the transition to a new elastic state caused by a finite amplitude excitation (conditioning). Memory of that new state is kept when the excitation is removed, before relaxation back to the original elastic state takes place. Indeed, besides accounting for linear and nonlinear parameters, a realistic constitutive equation to be used in reliable prediction models should take into account nonequilibrium effects. Specific parameters, sensitive to finite amplitude excitations, should be introduced to provide information about conditioning effects. In this paper, experimental results indicating that nonlinearity of damaged concrete is memory-dependent will be presented and the implications of such findings in the development of physical models, with relevant outcomes for the characterization of hysteretical features, will be discussed.

I. INTRODUCTION

II. EXPERIMENTAL ANALYSIS

A. Experimental setup

B. Data analysis

III. RESULTS

A. Effects of conditioning on the material parameters

B. Effects of conditioning on nonlinear measurements

C. Nonlinearity and amplitude of conditioning

IV. DISCUSSION

V. CONCLUSIONS

### Key Topics

- Elasticity
- 15.0
- Adhesion
- 6.0
- Constitutive relations
- 6.0
- Materials properties
- 5.0
- Wave attenuation
- 5.0

## Figures

(Color online) Waveforms in a nonlinearity experiment: (a) Actual output signal related to a generic high excitation amplitude (dashed line) and its respective linear reference signal (solid line); (b) zoom of the waveforms reported in (a) over the time window used for the analysis; (c) SSM signal equal to the difference between the waveforms in (b).

(Color online) Waveforms in a nonlinearity experiment: (a) Actual output signal related to a generic high excitation amplitude (dashed line) and its respective linear reference signal (solid line); (b) zoom of the waveforms reported in (a) over the time window used for the analysis; (c) SSM signal equal to the difference between the waveforms in (b).

(Color online) Experiment E1. (a) Conditioning indicator [circles, see Eq. (1)] and maximum of the output signal (squares) vs the excitation number *n* for repeated bursts at constant amplitude of 12 V. (b) Wave velocity (circles) and maximum of the output signal (squares) vs the excitation number *n* recorded from a sequence of pulses at 1 V, each sent between two consecutive bursts at 12 V.

(Color online) Experiment E1. (a) Conditioning indicator [circles, see Eq. (1)] and maximum of the output signal (squares) vs the excitation number *n* for repeated bursts at constant amplitude of 12 V. (b) Wave velocity (circles) and maximum of the output signal (squares) vs the excitation number *n* recorded from a sequence of pulses at 1 V, each sent between two consecutive bursts at 12 V.

(Color online) Experiment E2. Nonlinearity indicator [see Eq. (2)] as a function of the output intensity *x* for the relaxed sample (diamonds) and after two levels of conditioning at different amplitudes (circles and squares, respectively, referring to measurements after conditioning at 10 and 15 V).

(Color online) Experiment E2. Nonlinearity indicator [see Eq. (2)] as a function of the output intensity *x* for the relaxed sample (diamonds) and after two levels of conditioning at different amplitudes (circles and squares, respectively, referring to measurements after conditioning at 10 and 15 V).

(Color online) Experiment E3. Repetitions of the measurement of the nonlinearity indicator [see Eq. (2)], without conditioning. The first measurement, circles in the plot, differs from the other repeated measurements due to nonequilibrium effects.

(Color online) Experiment E3. Repetitions of the measurement of the nonlinearity indicator [see Eq. (2)], without conditioning. The first measurement, circles in the plot, differs from the other repeated measurements due to nonequilibrium effects.

(Color online) Experiment E4. Comparison of the nonlinearity indicators [see Eq. (2)] for experiments conducted at different rates. In the fast experiment (circles) is calculated using single burst excitations with increasing amplitude. In the slow experiment (squares), trains of ten bursts at each of the increasing amplitudes are sent in the sample and, for each amplitude, the signals to be processed are the response of the sample to the last burst of the ten composing each train.

(Color online) Experiment E4. Comparison of the nonlinearity indicators [see Eq. (2)] for experiments conducted at different rates. In the fast experiment (circles) is calculated using single burst excitations with increasing amplitude. In the slow experiment (squares), trains of ten bursts at each of the increasing amplitudes are sent in the sample and, for each amplitude, the signals to be processed are the response of the sample to the last burst of the ten composing each train.

(Color online) Experiment E5. (a) Nonlinearity indicator [see Eq. (2)] as a function of the output signal intensity *x* for different levels of conditioning; (b) coefficient of the power law fitting function [Eq. (4)] as a function of the conditioning amplitude.

(Color online) Experiment E5. (a) Nonlinearity indicator [see Eq. (2)] as a function of the output signal intensity *x* for different levels of conditioning; (b) coefficient of the power law fitting function [Eq. (4)] as a function of the conditioning amplitude.

(Color online) Experiment E5. (a) Conditioning indicator [see Eq. (1)] measured during the conditioning process performed before the nonlinearity measurement discussed in Fig. 6(a); (b) evolution of time parameter (, circles) and asymptotic value (, squares) obtained fitting the curves of Fig. 6(a) with the function reported in Eq. (5).

(Color online) Experiment E5. (a) Conditioning indicator [see Eq. (1)] measured during the conditioning process performed before the nonlinearity measurement discussed in Fig. 6(a); (b) evolution of time parameter (, circles) and asymptotic value (, squares) obtained fitting the curves of Fig. 6(a) with the function reported in Eq. (5).

(Color online) Correlation between the fitting parameter *a* of the nonlinearity indicator [taken from Fig. 6(b)] and the fitting parameters of the temporal evolution of the conditioning indicator [taken from Fig. 7(b)]: Circles for ; squares for .

(Color online) Correlation between the fitting parameter *a* of the nonlinearity indicator [taken from Fig. 6(b)] and the fitting parameters of the temporal evolution of the conditioning indicator [taken from Fig. 7(b)]: Circles for ; squares for .

(Color online) Experiment E6. (a) Nonlinearity indicator [see Eq. (2)] as a function of the output signal intensity *x* for different durations of the conditioning process expressed as the number of conditioning bursts. (b) Coefficient *a* as a function of the number of bursts used to condition the sample.

(Color online) Experiment E6. (a) Nonlinearity indicator [see Eq. (2)] as a function of the output signal intensity *x* for different durations of the conditioning process expressed as the number of conditioning bursts. (b) Coefficient *a* as a function of the number of bursts used to condition the sample.

(Color online) Experiment E7. (a) Nonlinearity indicator [see Eq. (2)] as a function of the output signal intensity *x* for the sample in the relaxed state (triangles) and immediately after full conditioning at 15 V (circles), denoting an increase in the nonlinearity of the sample. Repeating the measurement (without further conditioning), the curves slowly relax (rightward triangles, stars, diamonds, and squares) toward the initial curve corresponding to the relaxed state. (b) Nonlinearity parameter *a* [obtained fitting the plots of (a) with the power law expressed by Eq. (4)] as a function of time, after conditioning the sample at 15 V. The dashed line represents the value of the coefficient *a* measured on the relaxed sample.

(Color online) Experiment E7. (a) Nonlinearity indicator [see Eq. (2)] as a function of the output signal intensity *x* for the sample in the relaxed state (triangles) and immediately after full conditioning at 15 V (circles), denoting an increase in the nonlinearity of the sample. Repeating the measurement (without further conditioning), the curves slowly relax (rightward triangles, stars, diamonds, and squares) toward the initial curve corresponding to the relaxed state. (b) Nonlinearity parameter *a* [obtained fitting the plots of (a) with the power law expressed by Eq. (4)] as a function of time, after conditioning the sample at 15 V. The dashed line represents the value of the coefficient *a* measured on the relaxed sample.

(Color online) Simulation results. (a) Nonlinearity indicator [see Eq. (2)] vs output signal intensity *x* [see Eq. (3)] for the relaxed sample (stars) and after increasing levels of conditioning [squares, circles, and triangles, to be compared with the experimental data of Fig. 6(a)]; (b) conditioning indicator [see Eq. (1)] vs the number of conditioning bursts *n* for increasing levels of conditioning [compare with the experimental data of Fig. 7(a)]. There is a qualitative agreement with the experimental data.

(Color online) Simulation results. (a) Nonlinearity indicator [see Eq. (2)] vs output signal intensity *x* [see Eq. (3)] for the relaxed sample (stars) and after increasing levels of conditioning [squares, circles, and triangles, to be compared with the experimental data of Fig. 6(a)]; (b) conditioning indicator [see Eq. (1)] vs the number of conditioning bursts *n* for increasing levels of conditioning [compare with the experimental data of Fig. 7(a)]. There is a qualitative agreement with the experimental data.

(Color online) Power law fitting curves with free exponents (solid lines) and fixed exponents (dashed lines) for three experimental data sets selected among those reported in Fig. 6(a).

(Color online) Power law fitting curves with free exponents (solid lines) and fixed exponents (dashed lines) for three experimental data sets selected among those reported in Fig. 6(a).

## Tables

Description of the experiments discussed in the paper.^{a}

Description of the experiments discussed in the paper.^{a}

Parameter values relative to the numerical simulation results shown in Figs.11(a) and 11(b).

Parameter values relative to the numerical simulation results shown in Figs.11(a) and 11(b).

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