**2. Theory**

According to the Joule effect and the magnetization theory of ferromagnetic material, there is a functional relationship between the stress of rebar and the change in magnetic permeability [30,31]. In Equation (1), *μ* is the permeability of rebar, *μ*<sup>0</sup> is the vacuum permeability, *λ*<sup>s</sup> is the axial deformation constant, *M*<sup>s</sup> is the saturation magnetization, *K*<sup>u</sup> is the uniaxial magnetic anisotropy constant, *H*<sup>R</sup> is the excitation magnetic field, and *θ*<sup>0</sup> is the angle between the magnetic field and the easy magnetization axis [32].

$$\sigma = E \frac{3\lambda\_{\text{sf}}M\_{\text{S}}}{2K\_{\text{u}}} (\mu - \mu\_0) H\_{\text{R}} \sin^2 \theta\_0 \cos \theta\_0 \tag{1}$$

A magnetic resonance sensor [28] was used to monitor the working stress of the rebar. The sensor's two coils are the excitation and induction coils. The coil is wound on the PVC skeleton, as shown in Figure 1. The equivalent circuit diagram [28] of the magnetic resonance sensor is shown in Figure 1. *L*<sup>T</sup> and *L*<sup>R</sup> are the inductance of the excitation coil and induction coil, respectively. *C*<sup>T</sup> and *C*<sup>R</sup> are the excitation and induction coil's compensation capacitors, respectively. *u*CT and *u*CR are the voltage of the compensation capacitor of the excitation coil and the induction coil. *R*<sup>T</sup> and *R*<sup>R</sup> are the internal resistance of the excitation coil and the induction coil, respectively. The voltage source is AC power, and the input voltage is *u*in. The millivoltmeter is regarded as a load connected in series with an induction coil, and its equivalent resistance is *R*L.

**Figure 1.** The magnetic resonance sensor and equivalent circuit diagram schematic.

According to Kirchhoff's voltage law [33], the self-impedance of the excitation coil and the induction coil is *Z*<sup>T</sup> and *Z*R, respectively, as shown in Equations (2) and (3). The loop current *I*<sup>R</sup> of the induction coil is shown in Equation (4), where *j* is the imaginary part of the complex number, *U*in is the effective value of *u*in, *ω* is the angular frequency of *u*in, and *M* is the mutual inductance between the excitation coil and the induction coil. According to the coupled mode equation of LC coupled circuit [34], the relationship between coupling coefficient *κ* and mutual inductance *M* can be expressed as Equation (5); *ω*<sup>0</sup> is the resonant frequency.

$$Z\_{\rm T} = R\_{\rm T} + j\omega L\_{\rm T} + \frac{1}{j\omega C\_{\rm T}} \tag{2}$$

$$Z\_{\rm R} = R\_{\rm R} + R\_{\rm L} + j\omega L\_{\rm R} + \frac{1}{j\omega C\_{\rm R}} \tag{3}$$

$$\dot{I}\_{\rm R} = \frac{-j\omega M \dot{U}\_{\rm in}}{Z\_{\rm T} Z\_{\rm R} + \left(\omega M\right)^2} \tag{4}$$

$$M = \frac{2\kappa\sqrt{L\_{\text{R}}L\_{\text{T}}}}{\omega\_0} \tag{5}$$

A rebar with a cross-sectional area of *A*iron is placed in a magnetic resonance sensor. *A*air is the cross-sectional area of the nonmagnetic material between the coil and the rebar. The voltage source provides alternating current for the excitation coil. Under the action of alternating current, the excitation coil generates an excitation magnetic field [35,36]. The excitation coil and the induction coil are resonantly coupled. An excitation magnetic field of the magnetized rebar is generated in the induction coil. The magnetic field is expressed as *H*R, which has a functional relationship with the coupling coefficient *κ*, as shown in Equation (6). *N*<sup>R</sup> is the number of turns of the induction coil. *l*<sup>R</sup> is the effective magnetic circuit length of the induction coil. According to electromagnetic induction law, the induced voltage of the induction coil can be obtained by the magnetic flux in the area around the coil [37], as shown in Equation (7), where *Φ* is the magnetic flux around the area of the induction coil, and *t* is the time.

$$H\_{\rm R} = \frac{N\_{\rm R} \dot{l}\_{\rm R}}{l\_{\rm R}} = \frac{-j\omega N\_{\rm R} 2\kappa \sqrt{L\_1 L\_2} l I\_{\rm in}}{\left[ Z\_{\rm T} Z\_{\rm R} + \left( \frac{2\omega \kappa \sqrt{L\_1 L\_2}}{\omega\_0} \right)^2 \right] l\_{\rm R} \omega\_0} \tag{6}$$

$$
\mu\_{\rm CR} = N\_{\rm R} \frac{d\Phi}{dt} = N\_{\rm R} \frac{d(\mu H\_{\rm R} A\_{\rm iron} + \mu\_0 H\_{\rm R} A\_{\rm air})}{dt} \tag{7}
$$

Combined with the electric power calculation formula, the excitation coil's input power *P*in and the millivoltmeter's output power *P*<sup>o</sup> as the load can be calculated, respectively. The results are shown in Equations (8) and (9). The transmission efficiency can be obtained as shown in Equation (10). *X*<sup>T</sup> = *ωL*T−1/*ωC*T, *X*<sup>T</sup> = *ωL*R−1/*ωC*R.

$$P\_{\rm in} = \frac{\boldsymbol{\mathcal{U}\_{\rm in}}}{R\_{\rm R}} = \frac{\left\{\boldsymbol{R}\_{\rm T} \left[\left(\boldsymbol{R}\_{\rm R} + \boldsymbol{R}\_{\rm L}\right)^{2} + \boldsymbol{X}\_{\rm R}\right] + \omega^{2}\boldsymbol{M}^{2} \left(\boldsymbol{R}\_{\rm R} + \boldsymbol{R}\_{\rm L}\right)\right\} \boldsymbol{U}\_{\rm in}\boldsymbol{\mathcal{Z}}}{\left[\left.\boldsymbol{R}\_{\rm T} \left(\boldsymbol{R}\_{\rm R} + \boldsymbol{R}\_{\rm L}\right) - \boldsymbol{X}\_{\rm T}\boldsymbol{X}\_{\rm R} + \omega^{2}\boldsymbol{M}^{2}\right] + \left[\left.\boldsymbol{R}\_{\rm T}\boldsymbol{X}\_{\rm R} + \left(\boldsymbol{R}\_{\rm R} + \boldsymbol{R}\_{\rm L}\right)\boldsymbol{X}\_{\rm T}\right]^{2}}\right.\tag{8}$$

$$P\_0 = I\_{\mathbb{R}} \,^2 R\_L = \frac{\omega^2 M^2 R\_{\mathbb{L}} \, ^2 \mathrm{L} \, ^2 \mathrm{L}}{\left[ R\_{\mathbb{T}} (R\_{\mathbb{R}} + R\_{\mathbb{L}}) - X\_{\mathbb{T}} X\_{\mathbb{R}} + \omega^2 M^2 \right] + \left[ R\_{\mathbb{T}} X\_{\mathbb{R}} + (R\_{\mathbb{R}} + R\_{\mathbb{L}}) X\_{\mathbb{T}} \right]^2} \tag{9}$$

 $\eta \quad = \frac{P\_{\text{in}}}{P\_{\text{in}}} \times 100\%$ 
$$= \frac{\omega^2 M^2 R\_{\text{L}}}{R\_{\text{T}} \left[ \left( R\_{\text{R}} + R\_{\text{L}} \right)^2 + \chi\_{\text{R}}^2 \right] + \omega^2 M^2 \left( R\_{\text{R}} + R\_{\text{L}} \right)} \times 100\% \text{ } \tag{10}$$

When the induction coil resonates, *X*<sup>R</sup> = 0, the transmission efficiency reaches the maximum, and the measured induced voltage is the highest. In the working stress monitoring

experiment, the rebar is used as the core of the induction coil. The change of permeability of rebar caused by working stress also causes the induction coil's inductance change. After the inductance changes, the resonant frequency of the induction coil changes, as shown in Equation (11). When the resonant frequency of the induction coil deviates from the initial resonant frequency, the coil coupling coefficient *κ* and the sensor induced voltage are significantly reduced, thereby improving the sensitivity of the rebar working stress monitoring.

$$\omega\_0 = \frac{1}{2\pi\sqrt{L\_\text{R}C\_\text{R}}} = \frac{1}{2\pi\sqrt{C\_\text{R}}} \frac{1}{\sqrt{(\mu A\_{\text{iron}} + \mu\_0 A\_{\text{air}})\frac{N\_\text{R}^2}{I\_\text{R}}}} \tag{11}$$

The above relationship is solved simultaneously to explore the internal relationship among stress, magnetism, and electricity. The change in stress will lead to the change of permeability of the rebar. The relationship between induced voltage and permeability can be simplified from Equations (7)–(12), where *f(u*CR*)* is the function of induced voltage *u*CR representing permeability *μ*. The induced voltage *u*CR is related to the coupling coefficient *κ*. For a specific rebar and sensor, the relationship between the sensor's induced voltage and the rebar's working stress can be expressed as Equation (13); *g(u*CR*)* is the function of the induced voltage *u*CR representing the working stress *σ*.

$$\mu = \frac{\int \mu\_{\rm CR} dt - N\_{\rm R} H\_{\rm R} \mu\_{0} A\_{\rm air}}{N\_{\rm R} H\_{\rm R} A\_{\rm iron}} = f(\mu\_{\rm CR}) \tag{12}$$

$$
\sigma = h[f(\mu\_{\text{CR}}) - \mu\_0] = \lg(\mu\_{\text{CR}}) \tag{13}
$$

Through the above derivation, it can be found that the rebar working stress is related to the sensor's induced voltage. Therefore, the induced voltage of the magnetic resonance sensor can be used to evaluate the working stress of the rebar. To verify the feasibility of the magnetic resonance monitoring method (REME) for rebar working stress monitoring, rebar working stress monitoring experiments were carried out.
