## 2 Materials and Methods  
### Hysteresis and Sensitivity Mechanisms  
#### 2.1.1 Mechanisms of Hysteresis  
Hysteresis defines the variability of the output signal during the application and release of pressure. At the same time, the resistive viscoelasticity of conductive polymer composites is influenced not only by the mechanical viscoelasticity of the matrix but also by the conductive network established through the modulation of nanoparticles in reaction to external forces [16]. This study characterises the resistive hysteresis of sensing materials by integrating the viscoelastic model with electron tunnelling theory. The multi-branch model [41] (one-dimensional generalised viscoelastic model or Prony series) expresses total stress by the equation derived from the Boltzmann superposition principle:  
\[\sigma(t)=\sigma_{\infty}+\int_{0}^{t_{s}}\sum_{i=1}^{n}\dot{e}\sigma_{i} \text{exp}\left(-\frac{t-t^{\prime}}{\tau_{i}}\right)\text{d}t^{\prime} \tag{1}\]  
where \(\sigma(t)\) represents the stress at time t (Pa), \(\sigma_{\infty}\) denotes the equilibrium stress (Pa), \(\sigma_{i}\) signifies the ith corresponding coefficient, and \(\dot{e}\) indicates the constant elongation, with \(\tau_{i}\) being the relaxation period of the ith branch, \(t_{s}\) denotes the time at which the loading phase concludes, \(t^{\prime}\) represents a specific time point in the past used for calculating the stress magnitude at the current moment, and \(n\) signifies the number of non-equilibrium branches. This is achieved by substituting the pressure variable in Equation (1) with the resistance value:  
\[\left(\frac{R}{R_{0}}\right)(t)=R_{\infty}+\sum_{i=1}^{n}\int_{0}^{t}\frac{ \text{d}R_{i}}{\text{d}t^{\prime}}\exp(-\frac{t-t^{\prime}}{\tau_{i}})\text{ d}t^{\prime} \tag{2}\]  
In this context, \(R\) and \(R_{0}\) denote the resistance and initial resistance of the sensing material at time t, respectively (\(\Omega\)); \(R_{\infty}\) represents the relative equivalent resistance (\(\Omega\)); and \(R_{i}\) is the ith coefficient that characterises the impact of the load on the resistance. Equation (2) may also be expressed as follows:  
\[\left(\frac{R}{R_{0}}\right)(t)=R_{\infty}+\sum_{i=1}^{n}\int_{0}^{t}\frac{ \text{d}R_{i}}{\text{d}\varepsilon(t^{\prime})}\frac{\text{d}\varepsilon(t^{ \prime})}{\text{d}t^{\prime}}\exp(-\frac{t-t^{\prime}}{\tau_{i}})\text{d}t^{\prime} \tag{3}\]  
where \(\varepsilon(t^{\prime})\) represents the strain and may be any continuous function. Consequently, the relative resistance change \(\left(\frac{R}{R_{0}}\right)(t)\) can be articulated by the following equation:  
\[\left(\frac{R}{R_{0}}\right)(t)=R_{\infty}+\sum_{i=1}^{n}\int_{0}^{t}\frac{ \text{d}R_{i}}{\text{d}\varepsilon}\dot{e}\exp\left(-\frac{t-t^{\prime}}{\tau _{i}}\right)\text{d}t^{\prime} \tag{4}\]The theory of electron tunnelling is employed to ascertain the relevant coefficients \(R_{i}\). It is sourced from the literature [42; 43]:  
\[\left\{ \begin{array}{c}R=\left(\frac{L}{N}\right)\left(\frac{8.8\mu\hbar s}{32 \tau e^{2}}\right)\text{exp}(\gamma s)\\ \gamma=\frac{4\pi(2m\omega)^{0.5}}{\hbar}\end{array} \right. \tag{5}\]  
\(L\) represents the number of particles constituting a single conducting channel, \(N\) denotes the number of conducting channels, \(h\) signifies [PERSON]'s constant, and \(s\) indicates the distance between the surfaces of two adjacent particles (m). \(a^{2}\) represents the effective cross-section for tunnelling (m\({}^{2}\)), \(e\) denotes the electron charge, \(m\) signifies the electron mass (kg), \(\phi\) indicates the potential barrier height between adjacent particles, and \(\gamma\) denotes the tunnelling decay constant in the tunnelling effect. The alteration in electrical conductivity results from the relaxing of various segments of the polymer chains.