**1. Introduction**

The safety and durability of reinforced concrete (RC) structures during their service lifetime are mainly dependent on reinforcement corrosion, which can lead to concrete cracking [1–3]. Such corrosion-induced cracking will harm the internal concrete structure, resulting in a decline in the service life of the concrete [4,5]. Hence, developing a convenient and precise evaluation method for the corrosion of RC structures has become an urgent problem to be solved [6,7].

Acoustic emission (AE) refers to the phenomenon of elastic waves released during material fracture [8–10]. The AE method can capture the generation process of microcracks inside the concrete structure dynamically and in real time [11,12]. It can also detect the internal damage of a concrete structure and provide early warning. Thus, the AE rate process theory has been widely used to quantitatively investigate the internal damage evolution of concrete materials.

Thus far, many efforts have been made to investigate the characteristics of AE parameters during the loading process of concrete [13–15]. Ohtsu et al. [16] proposed the AE rate theory to evaluate the compressive strength of actual concrete structures and the process of steel corrosion with the moment tensor theory [17]. Suzuki et al. [18] took samples from an existing bridge structure and performed AE monitoring in the uniaxial compression process. They noted that the variation of damage was consistent with that of the compressive strength. Suzuki et al. [19] also studied the uniaxial compression process of concrete samples, which were taken from a canal wall, by applying the AE monitoring method. Based on the AE rate process theory and the damage mechanism, they put forward the implementation steps to determine the relative damage and further evaluate the damage

**Citation:** Chen, Y.; Zhu, S.; Ye, S.; Ling, Y.; Wu, D.; Zhang, G.; Du, N.; Jin, X.; Fu, C. Acoustic Emission Study on the Damage Evolution of a Corroded Reinforced Concrete Column under Axial Loads. *Crystals* **2021**, *11*, 67. https://doi.org/ 10.3390/cryst11010067

Received: 30 December 2020 Accepted: 12 January 2021 Published: 15 January 2021

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to the samples. Zhou et al. [20] nondestructively obtained the elastic modulus and initial material damage by combining the AE rate process theory and the basic damage mechanics theory. In order to achieve a real-time quantitative assessment of steel strand damage, Deng et al. [21] assumed the number of AE events and stress levels of steel strands in a mathematical model and established the AE probability density function and damage evolution model for steel strands, enabling the real-time monitoring of the tensile damage of steel strands by AE technology [22].

This research conducted axial load experiments of RC columns with 0%, 5%, 10% and 15% corrosion rates. The corrosion rate and AE parameters were introduced to the AE rate process theory as the damage characteristic parameters of reinforced concrete. Then, a modified damage evolution model of a corroded RC column was proposed. This model provides real-time monitoring of the damage of corroded RC and quantitatively evaluates the damage degree from corrosion.

### **2. Damage Model**

The AE phenomenon can reflect the damage and deterioration of a concrete structure under loading [23]. In the literature [24], the AE rate process theory was introduced into the mathematical model for concrete under uniaxial compression. When the load level increases from *V* to *V + dV*, the probability density function *f*(*V*) of the AE event can be expressed as follows:

$$f(V)dV = dN/N\tag{1}$$

where *N* is the total number of AE events with the load level *V* increasing from the initial state.

In Ohtsu's model [16], *f*(*V*) can be used to represent the AE rate and is approximately expressed by hyperbolic functions as follows:

$$f(V) = a/V + b \tag{2}$$

where *a* and *b* are experimental parameters. Substituting Equation (2) into Equation (1) will give the relationship between the load level *V* and the total number of AE events *N* as follows:

$$N = cV^d \exp(bV) \tag{3}$$

where *a*, *b* and *c* are AE test parameters, and a reflects the number of microcracks in the material. When *a* > 0, a higher AE rate can appear at lower stress levels, which indicates multiple cracks. When *a* < 0, a low AE rate appears at lower stress levels, which corresponds to no cracks or a small number of cracks.

In Dai-Labuz's model [25], the relationship between *V* and *N* can be expressed as follows:

$$V = aN + c\ln(1 + qN)\tag{4}$$

where *a*, *c* and *q* are related parameters for the AE.

After substituting Equation (4) into Equation (2), the probability density function *f*(*V*) can be expressed as follows:

$$f(V) = \frac{1}{N\_0} (\frac{1 + qN}{a + cq + aqN}) \tag{5}$$

After integrating the probability density function over a range of load level *V*, the damage degree of the specimen under this load level can be obtained:

$$D = \int\_0^V f(V)dV\tag{6}$$

During the loading process of reinforced concrete, a large number of cracks will appear at the internal structure due to steel corrosion. Meanwhile, the corrosion degree of reinforcement will have an influence on the number of cracks on the structure during the entire loading process. However, limited research has been conducted on the damage evolution of corroded reinforced concrete [26], and there is not enough convincing evidence to show the diversity in the structural damage evolution of RC under different corrosion degrees [27]. Thus, it is necessary to fill these gaps. In this research, on the basis of Dai-Labuz's model [25], the load level and accumulative AE hit number in the loading process can be expressed as follows:

$$N = s(V) \tag{7}$$

where *s(V)* is determined by experiments.

After taking the derivative of Equation (7) and substituting it into Equation (2), the following equation can be obtained:

$$f(V) = \frac{1}{N\_0} \frac{ds(V)}{dV} \tag{8}$$

where *N<sup>0</sup>* is the accumulative AE hit number when the load level reaches the ultimate load.

Finally, according to Equation (6), the damage degree of RC at load level *V* can be obtained as follows:

$$D = \frac{1}{N\_0} \int\_0^V ds(V) \tag{9}$$

### **3. Experimental Program**
