The necessity of the inclusion of the second invariant of the left Cauchy-Green deformation tensor $\mathbf{B}$, namely $I_2$, in the strain energy function $W$ of rubber-like materials is analysed. Universal relationships that underline such necessity are revisited, and experimental data are examined to establish the trends in the variation of $\partial W/\partial I_2$. Corroborated by the established experimental trends, we consider (meso)structural arguments to devise a plausible approach for incorporation of $I_2$ into the $W$ function. On the basis of the molecular theory of rubbers and considering the entanglements as a topological tube constraint, our analysis confers that a first approximation of $W\left(I_1,I_2\right)$ is of the form $W\left(I_1,I_2\right)=f\left(I_1\right)+g\left(I_2\right)$. The $f\left(I_1\right)$ contribution may be that of any classical generalised neo-Hookean model, and the functional form of $g\left(I_2\right)$ is directly deduced from the tube model of entangled molecules. An additional logarithmic functional form of $I_2$ is also devised based on the rational approximation of the response function ${\beta }_{-1}$. The ensuing additive-type $W\left(I_1,I_2\right)$ models are then compared with experimental datasets. While this additive consideration may not be sufficient to account for all aspects of the mechanics of rubber-like materials, the fitting results demonstrate an eminent improvement in the predictions of the additive-type models compared with generalised neo-Hookean models having the same number of constitutive parameters. These analyses underline the central role of $I_2$ in modelling the finite deformation of rubber-like materials.

包含左Cauchy-Green变形张量的第二个不变量的必要性 $\mathbf{乙}$，即 ${\mathrm{一世}}_{\mathrm{2个}}$，在应变能函数中 $\mathrm{w\; ^}$分析了类似橡胶的材料。再次强调了强调这种必要性的普遍关系，并检查了实验数据以建立这种变化的趋势。$\partial \mathrm{w\; ^}/\partial {\mathrm{一世}}_{\mathrm{2个}}$。在既定的实验趋势的支持下，我们考虑了（中观）结构论证，以设计出一种可行的方法来纳入${\mathrm{一世}}_{\mathrm{2个}}$ 进入 $\mathrm{w\; ^}$功能。在橡胶分子理论的基础上，并将缠结视为拓扑管约束，我们的分析认为第一近似为$\mathrm{w\; ^}（{\mathrm{一世}}_{\mathrm{1个}}，{\mathrm{一世}}_{\mathrm{2个}}）$ 的形式 $\mathrm{w\; ^}（{\mathrm{一世}}_{\mathrm{1个}}，{\mathrm{一世}}_{\mathrm{2个}}）=F（{\mathrm{一世}}_{\mathrm{1个}}）+G（{\mathrm{一世}}_{\mathrm{2个}}）$。这$F（{\mathrm{一世}}_{\mathrm{1个}}）$ 贡献可以是任何经典的广义新霍克模型的贡献，以及 $G（{\mathrm{一世}}_{\mathrm{2个}}）$是直接从纠缠分子的管模型推导出来的。的另一种对数功能形式${\mathrm{一世}}_{\mathrm{2个}}$ 也基于响应函数的有理逼近而设计 ${\beta }_{-\mathrm{1个}}$。随后的添加剂类型$\mathrm{w\; ^}（{\mathrm{一世}}_{\mathrm{1个}}，{\mathrm{一世}}_{\mathrm{2个}}）$然后将模型与实验数据集进行比较。尽管这种加法考虑因素可能不足以说明类橡胶材料力学的所有方面，但拟合结果表明，与相同数量的广义新胡克模型相比，加法类型模型的预测有了显着改善。本构参数。这些分析强调了${\mathrm{一世}}_{\mathrm{2个}}$ 在模拟橡胶状材料的有限变形中。