## 2 Materials and Methods
### Materials  
The material used for the construction of the flexible system of the fiber-constrained flexible actuator designed in this study is Ecoflex00-30 silicone from Smooth-On, whose material performance parameters are shown in Table 2.  
The constraint fiber was Kevlar, a material with high strength, wear resistance and extremely high tear resistance. Its parameters are as follows: strength is 3.6 Gpa, elongation modulus is 131 Gpa, elongation at break is 2.8% and axial elongation coefficient is \(-2\times 10^{-6}/\)K.  
The strain limiting layer was made of glass fiber fabric, which has good permeability and good compatibility with silicone, preventing the flexible actuator chamber from separating from the silicone when it expands and deforms.  
ABAQUS has excellent simulation capabilities for non-linear finite element analysis [8]. The finite element analysis in this paper was done in the ABAQUS/Standard module. For hyper-elastic materials such as silica, description methods could usually be divided into two main categories: phenomenological models, such as the Ogden model [9] and the Yeoh model [10], and intrinsic models based on thermodynamic statistics, such as the Arruda-Boyce model [11]. The latter was not suitable for modelling flexible actuators, which required more complex models in order to capture the material properties in multiple experimental tests [12]. In this paper, the Ogden model was used to describe the material properties of Ecoflex00-30. Its strain energy could be expressed as Equation (2):  
\[U=\sum_{i=1}^{N}\frac{2\mu_{i}}{\alpha_{i}^{2}}\Big{(}\lambda_{1}^{\alpha_{1} }+\lambda_{2}^{\alpha_{2}}+\lambda_{3}^{\alpha_{3}}-3\Big{)}+\sum_{i=1}^{N} \frac{1}{D}(J-1)^{2i} \tag{2}\]  
where \(\mu_{i}\) and \(\alpha_{i}\) are material constants, N is the number of terms in the function; \(\lambda_{i}\) is the principal elongation, \(D_{i}\) is the volume deformation coefficient and \(J\) is the ratio of deformed volume to undeformed volume.  
Considering silica gel as an incompressible material in this study, the principal elongations \(\lambda_{1}\), \(\lambda_{2}\) and \(\lambda_{3}\) satisfy Equation (3):  
\[\lambda_{1}\lambda_{2}\lambda_{3}=1 \tag{3}\]  
This material property was investigated by [PERSON] et al., whose 3 rd order Ogden model coefficients [10] are shown in Table 3.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Ecoflex00-30** \\ \hline Forming stiffness & 00-30 HA \\ density & 1.07 g/cm\({}^{3}\) \\ Young’s modulus & 68.9 Kpa \\ Poisson’s ratio & 0.499 \\ Tensile strength & 1378 Kpa \\ Fracture strain & 900\% \\ Forming shrinkage & 3000 cps \\ \hline \hline \end{tabular}
\end{table}
Table 2: Ecoflex00-30 material parameters.  
In this study, Kevlar was described using a linear elastic material model with a Young's modulus of 31,067 MPa, a Poisson's ratio of 0.36, a circular cross-sectional shape and a radius r set at 0.08 mm.  
The strain limiting layer was a glass fiber layer, and was treated as an elastomer. The strain limiting layer was described using a shell elastic material model, with a Young's modulus of 73,100 Mpa and a Poisson's ratio of 0.22.