## 2 Materials and Methods
### Hysteresis and Sensitivity Mechanisms
#### 2.1.1 Mechanisms of Hysteresis
The alteration in resistance resulting from the displacement of the ith polymer chain is as follows:  
\[R_{i}=\frac{R_{\text{bb}}}{R_{\text{ia}}}=\left(\frac{LN_{0}s}{L_{0}Ns_{0}} \right)exp[\gamma(s-s_{0})] \tag{6}\]  
\(R_{\text{ib}}\) represents the resistance value after loading (\(\Omega\)), \(R_{\text{ia}}\) denotes the resistance value before loading (\(\Omega\)), \(L_{0}\) signifies the number of particles forming a single conducting path in the initial state, \(N_{0}\) represents the number of conducting paths in the initial state, and \(s_{0}\) indicates the distance between neighbouring particles in the initial state (m). The separation between the surfaces of two adjacent particles subjected to an external force can be articulated as follows:  
\[s=s_{0}\big{(}1+\dot{\varepsilon}t\big{)} \tag{7}\]  
Substituting Equation (8) into Equation (7) produces the associated coefficient of variation for resistance \(R_{i}\). \[\left\{ \begin{array}{c}R_{i}=R_{i0}(1+\varepsilon)exp(k_{i}\varepsilon)\\ R_{i0}=\frac{LN_{0}}{L_{0}N}\\ k_{i}=\tau s_{0}\end{array} \right. \tag{8}\]  
We simplify the model by treating \(R_{i0}\) as a constant, resulting in the following equation:  
\[\frac{\text{d}R_{i}}{\text{d}\varepsilon}=R_{i0}(1+k_{i}+k_{i}\varepsilon) \text{exp}(k_{i}\varepsilon) \tag{9}\]  
[PERSON]'s experimental studies indicate that stress correlates with the logarithm of the applied strain rate across a broad spectrum of strain rates [44]. We employ a logarithmic function of the strain rate to represent the relative equilibrium resistance, which escalates with strain as indicated in Equation (9). Consequently, the equivalent resistance \(R_{\infty}\) is computed as follows:  
\[R_{\infty}=R_{\infty}^{0}+R_{\infty 0}\varepsilon\exp(k_{\infty 0}\varepsilon)(1+a\text{ln}(1+b \frac{\int_{0}^{\varepsilon}\dot{\varepsilon}de^{\prime}}{\varepsilon})) \tag{10}\]  
\(\frac{\int_{0}^{\varepsilon}\dot{\varepsilon}de^{\prime}}{\varepsilon}\) represents the mean strain rate. Inserting Equations (9) and (10) into Equation (4) yields the relative resistance variation as a function of strain, namely, the resistive hysteresis model:\[(\frac{R}{R_{0}})(t)=R_{\infty}^{0}+R_{\infty 0}\varepsilon(t)exp(k_{ \infty 0}\varepsilon(t))(1+ahn(1+b\frac{\int_{0}^{t}\dot{a}dt^{\prime}}{\varepsilon(t )})) \tag{11}\] \[+\sum_{i=1}^{n}\int_{0}^{t}R_{i0}(1+k_{i}+k_{i}\varepsilon(t^{ \prime}))exp(k_{i}\varepsilon(t^{\prime}))\dot{\varepsilon}(t^{\prime})exp(- \frac{t-t^{\prime}}{\tau_{i}})dt^{\prime}\]  
The hysteresis of the composite sensing material, as indicated by Equation (11), is associated with the mechanical viscoelasticity of the rubber matrix and the adhesion between the filler and the matrix. Hysteresis is influenced by the material type, content, and fabrication process, with the filler content being the primary factor affecting the mechanical viscoelasticity of the composite material when the filler composition is fixed. Additionally, the preparation temperature significantly impacts the adhesion of the filler to the matrix. This paper examines the influence of four elements on the hysteresis of composite sensing materials: the amount of Fe\({}_{3}\)O\({}_{4}\), CNT, MLG, and the material MT.