Constitutive modeling of dielectric elastomers has been of long standing interest in mechanics. Over the last two decades rigorous constitutive models have been developed that couple the electrical response of these polymers with large deformations characteristic of soft solids. A drawback of these models is that unlike classic models of rubber elasticity they do not consider the coupled electromechanical response of single polymer chains which must be treated using statistical mechanics. The objective of this paper is to compute the stretch and polarization of single polymer chains subject to a fixed force and fixed electric field using statistical mechanics. We assume that the dipoles induced by the applied electric field at each link do not interact with each other and compute the partition function using standard techniques. We then calculate the stretch and polarization by taking appropriate derivatives of the partition function and obtain analytical results in various limits. We also perform Markov chain Monte Carlo simulations using the Metropolis and umbrella sampling methods, as well as develop a new sampling method which improves convergence by exploiting a symmetry inherent in dielectric polymer chains. The analytical expressions are shown to agree with the Monte Carlo results over a range of forces and electric fields. Our results complement recent work on the statistical mechanics of electro-responsive chains which obtains analytical expressions in a different ensemble.

介电弹性体的本构建模长期以来一直是力学领域的兴趣所在。在过去的二十年中，已经开发出严格的本构模型，将这些聚合物的电响应与软固体的大变形特征结合起来。这些模型的一个缺点是，与橡胶弹性的经典模型不同，它们没有考虑必须使用统计力学处理的单个聚合物链的耦合机电响应。本文的目的是使用统计力学计算受固定力和固定电场影响的单个聚合物链的拉伸和极化。我们假设在每条链路上施加的电场感应的偶极子不会彼此相互作用，并使用标准技术计算配分函数。然后，我们通过对分配函数取适当的导数来计算拉伸和极化，并在各种限制下获得分析结果。我们还使用 Metropolis 和伞状采样方法执行马尔可夫链蒙特卡罗模拟，并开发了一种新的采样方法，该方法通过利用介电聚合物链中固有的对称性来提高收敛性。分析表达式与蒙特卡罗结果在一定范围的力和电场上一致。我们的结果补充了最近在电响应链的统计力学方面的工作，这些工作在不同的集合中获得了分析表达式。我们还使用 Metropolis 和伞状采样方法执行马尔可夫链蒙特卡罗模拟，并开发了一种新的采样方法，该方法通过利用介电聚合物链中固有的对称性来提高收敛性。分析表达式与蒙特卡罗结果在一定范围的力和电场上一致。我们的结果补充了最近在电响应链的统计力学方面的工作，这些工作在不同的集合中获得了分析表达式。我们还使用 Metropolis 和伞状采样方法执行马尔可夫链蒙特卡罗模拟，并开发了一种新的采样方法，该方法通过利用介电聚合物链中固有的对称性来提高收敛性。分析表达式与蒙特卡罗结果在一定范围的力和电场上一致。我们的结果补充了最近在电响应链的统计力学方面的工作，这些工作在不同的集合中获得了分析表达式。