考虑缠结效应的超弹性本构模型

经典熵弹性模型, 如 Neo-Hookean模型和Arruda-Boyce八链模型, 被广泛应用于预测橡胶等软材料的超弹性力学行为. 然而, 大量实验结果也显示仅采用一套模型参数, 这类模型不能同时准确地描述橡胶在多种加载模式下的应力响应. 为了克服上述模型的不足, 本文在熵弹性的模型基础上引入缠结约束效应. 微观上, 采用Langevin统计模型来表征熵弹性变形自由能, 通过管模型(tube model)引入缠结约束自由能, 并基于仿射假设, 建立微观变形与宏观变形之间的映射关系. 在宏观上, 所建立的超弹性模型的Helmholtz自由能同时包含熵弹性和缠结约束两部分, 其中熵弹性自由能与经典的Arruda-Boyce八链模型一致, 依赖于柯西-格林应变张量的第一不变量, 而缠结约束自由能依赖于柯西-格林应变张量的第二不变量. 与文献中的实验结果对比发现, 该三参数模型能准确地预测实验中所测得的橡胶材料在单轴拉伸、纯剪切和等双轴拉伸变形条件下的应力响应, 也能较好地描述不同预拉伸比条件下双轴拉伸实验结果. 最后, 本文比较了所建立的基于应变不变量的缠结约束模型与文献中相关的缠结约束模型在多种加载模式下自由能的异同. 总的来说, 本文所建立的本构理论能准确模拟橡胶等软材料的大变形力学行为, 对其工程应用有促进作用. 

HYPERELASTIC MODEL WITH ENTANGLEMENT EFFECT

Classic hyperelastic models, such as the Neo-Hookean model and Arruda-Boyce eight-chain model, have been widely adopted to describe the mechanical response of rubbers. However, the experimental data has shown that using a single set of parameters these models have difficulty in accurately predicting the measured stress-strain relationship of rubbers under various loading modes. For example, the Arruda-Boyce model fails to describe the stress response in biaxial loading conditions of Treloar's classic experiments. To address this limitation, this work develops a hyperelastic theory incorporation the entanglement effect. At the microscale, the Langevin statistical model is adopted for the entropic part and the tube model is used for the entanglement part. The affine assumption is used to construct the micro-macro mapping. Macroscopically, the Helmholtz free energy of the model consists of both an entropic part and an entanglement part. The entropic part has the same form as the eight-chain model, depending on the first invariant of the Cauchy-Green deformation tensor. In contrast, the entanglement part is a function of the second invariant of the Cauchy-Green deformation tensor. Compared with the eight-chain model and Neo-Hookean model, the developed model with three parameters provides a greatly improved prediction on the experimentally measured stress response of rubbers in uniaxial, pure shear and equibiaxial loading conditions, as well as that of biaxial tension tests with different pre-stretch ratios. The model shows superior prediction ability compared with the classic models, such as the Neo-Hookean model, the eight-chain model, the Yeoh model and the generalized Rivlin model. Finally, the work also compares the free energy density of entanglement part developed in this work and those of the related models in the literature. The constitutive theory developed in this work can accurately predict the large deformation behaviors of rubbers and other related soft materials, which can potentially benefit their engineering applications.