## 4 Multi-Physics Simulation Analyses of GMM-FBG Current Sensor
### Multi-Physics Simulation Methodology  
The magnetostrictive performances of the GMM rod under the multiple physical loads of magnetic field, thermal field and stress field can be simulated by establishing the constitutive model of nonlinear coupling solid mechanics, magnetic field and heat transfer based on thermodynamic theory. A nonlinear-coupling constitutive model that can comprehensively  
Figure 1: The overview (**left panel**) and the local scaled (**right panel**) schematics of the GMM-ferrite assembly component in the GMM-FBG current sensor model. describe GMM physical responses synergistically under mechanical, magnetic and thermal loads is utilized for predicting the magnetostrictive and magnetization curves under strain field at different temperatures [19]. A GMM rod is generally applied by external magnetic field or stress along the axial direction in practical applications, which means the magnetostrictive strain and magnetization generated under a mechanical-magnetic-thermal multi-field are along the axial direction of the GMM rod. Therefore, it is supposed that GMM is a nonlinear isotropic material with the stress/strain, magnetic field and magnetization being correlated to each other by the magnitude of mechanical tensor components and the magnetic vector, and the GMM anisotropic compensation temperature Tr (easy axis conversion temperature) should be used to construct the multi-physics coupling constitutive model [20; 21]. The GMM rod of the GMM-FBG current sensor designed in this paper is freely stretching due to the air gap at the top of the hemispherical surface. The magnetic field introduced by ferrite into GMM is mainly along the cylindrical axis of the GMM rod. Therefore, no pre-stress arises in the GMM rod during current detection, and the stress/strain originates from magnetostriction and thermal expansion, in which the temperature effect on the thermal expansion and magnetostriction should be adequately considered. Complying with the definitions of magnetostriction modulus \(E_{\rm m}=\partial\sigma/\partial\lambda\), relative permeability \(\mu_{\rm r}=(1+\partial M/\partial H)\) and piezomagnetic coefficient \(d_{\rm m}=\partial\lambda(\sigma\),\(H)/\partial H\) (\(\lambda\), \(\sigma\), \(M\) and \(H\) denote the magnetostrictive coefficient, strain, magnetization intensity and magnetic field strength, respectively), the constitutive model used for the GMM rod in multi-physics finite element simulations is simplified to the thermo-magnetostriction constitutive relations as follows:  
\[E_{\rm m}=\left(\frac{1}{E_{\rm s}}+\frac{2\lambda_{\rm s}}{3\sigma_{\rm s}} \Bigg{(}1-\frac{M^{2}}{\left[M_{\rm s}(T-T_{\rm r})\right]^{2}}\Bigg{)}+\frac {8k[\lambda_{\rm s}+\beta(T-T_{\rm r})]^{2}M^{2}}{\left[\mu_{0}\sigma[M_{\rm s }(T-T_{\rm r})]^{3}\sec^{2}(\pi M/2M_{\rm s}(T-T_{\rm r}))\right]}\Bigg{)}^{-1} \tag{5}\]  
\[\mu_{\rm r}=1+\left\{\frac{\pi}{2 kM_{\rm s}(T-T_{\rm r})}\sec^{2}\left(\frac{ \pi M}{2M_{\rm s}(T-T_{\rm r})}\right)\right\}^{-1} \tag{6}\]  
\[d_{\rm m}=\frac{4 kM[\beta(T-T_{\rm r})+\lambda_{\rm s}]}{\pi[M_{\rm s}(T-T_{\rm r })]\sec^{2}\left(\frac{\pi M}{2M_{\rm s}(T-T_{\rm r})}\right)} \tag{7}\]  
\[M=\frac{2M_{\rm s}(T-T\,r){\rm arctan}(\,H)}{\pi},\quad M_{\rm s}(T-T_{\rm r}) =M_{\rm s}\left(\frac{T_{\rm r}-T}{T_{\rm c}-T_{\rm r}}\right)^{X},\quad k= \frac{\pi\chi_{\rm m}}{2M_{\rm s}(T-T_{\rm r})} \tag{8}\]  
where the characteristic parameters of material attributes are defined and symbolized with specific values as listed in Table 1. The magnetostrictive response curves of ferrite-GMM assembly for current detection at different temperatures are simulated in a multi-field coupling model of solid mechanics/magneto-statics/thermal expansion with the expressions (5)-(8) for magnetostrictive material characteristics being input into the material attributes of ferrite-GMM as implemented by COMSOL Multiphysics software(COMSOL Inc., Stockholm, Sweden)