标题： 关于聚合物网络的统计物理和热力学：一种用于熵弹性的小变形理论 显示英文标题

标题： On the statistical physics and thermodynamics of polymer networks: a Eulerian theory for entropic elasticity

摘要： 本研究提出了一种欧拉理论，以阐明聚合物网络中的分子运动学及其与连续体变形的联系，其基础是基本的统计物理和热力学。 三个关键创新被引入：1. 通过一个全局热力学平衡条件来描述网络行为，该条件最大化所有链段的可访问微观状态数，而不是直接处理传统方法中常用的单链模型。 然后在欧拉框架中提出一个变分问题，以在几何波动约束下确定这种平衡状态。 其解重新获得了经典的单链模型，并揭示了链运动学对连续体变形的依赖性。 2. 发现链的拉伸和取向概率可通过欧拉对数应变和空间方向明确指定。 该产生的超弹性模型仅包含两个物理参数，优于具有相同参数数量的现有模型。 它进一步为先前表现出优越预测能力的模型提供了物理解释：在中等变形下，该模型等价于Biot链模型，而在小应变极限下则收敛到经典的Hencky应变能。 3. 一种新的双轴不稳定性作为链取向的相变出现。 在足够大的变形下，链越来越多地沿主要拉伸方向排列，导致其他方向上的密度减少。 因此，非主要拉伸方向上的应力会降低，因为链密度的损失超过了链力的增加。 对于等双轴张力，由于在没有任何扰动的情况下实现两个主拉伸的完美相等在实践中是不可能的，因此会引发不稳定性。 显示英文摘要

摘要： This study presents a Eulerian theory to elucidate the molecular kinematics in polymer networks and their connection to continuum deformation, grounded in fundamental statistical physics and thermodynamics. Three key innovations are incorporated: 1. The network behavior is described through a global thermodynamic equilibrium condition that maximizes the number of accessible microstates for all segments, instead of directly dealing with the well-established single-chain models commonly adopted in traditional approaches. A variational problem is then posed in the Eulerian framework to identify this equilibrium state under geometric fluctuation constraints. Its solution recaptures the classical single-chain model and reveals the dependence of chain kinematics upon continuum deformation. 2. The chain stretch and orientation probability are found to be explicitly specified through the Eulerian logarithmic strain and spatial direction. The resulting hyperelastic model, with only two physical parameters, outperforms the extant models with same number of parameters. It further provides a physical justification for prior models exhibiting superior predictive capabilities: the model becomes equivalent to the Biot-chain model at moderate deformations, while converging to the classical Hencky strain energy in the small strain limit. 3.A novel biaxial instability emerges as a phase transition in chain orientation. At sufficiently large deformation, chains increasingly align with the primary stretched direction, depleting their density in other directions. Consequently, the stresses in non-primary stretched directions would decrease as the loss in chain density outweighs the gain in chain force. For equal biaxial tension, instability is therefore triggered because perfect equality of the two principal stretches without any perturbation is practically unachievable.