Multiscale Modeling of Strain-Induced Crystallization in Polymers

本项目将受应变诱导结晶（SIC）影响的聚合物视为由具有不同程度网络规则的区域组成的非均质介质。这样的概念允许描绘成核和晶体区域的生长，以及有效材料参数的变化取决于施加的应变水平。该模型在热力学上是一致的。它是基于自由亥姆霍兹能和耗散的假设。它们主要包括由于变形和结晶而产生的体积项和表面项。外部变量为变形量和温度，内部变量为非弹性变形量和网络规则度。根据最大耗散原理推导了它们的演化方程。为了模拟热效应，考虑了潜热和温度变化的影响。所解释的框架有几个优点。首先，它适合回答哪个过程主要影响SIC的关键问题：新晶体区域的成核还是已经存在的晶体区域的生长。其次，所提出的模型对于标准多尺度有限元概念的直接实现是理想的。这种数值均匀化方法符合有限应变理论，适用于尺度特征长度比趋于零的情况。这两个特征都是有效建模碳化硅的必要条件。该项目还包括对该过程的随机方面的研究，其中引入了可观察变量的分布函数来表示相关数量的期望值。考虑控制体积的有效能量，导出了必要的演化方程。这里的主要目标是研究成核和评估具有不同网络规律的区域的平均大小。所列出的任务的解决方案将有可能实现最终的项目目标：SIC的高级模拟，这可以大大有助于更有效地设计和使用聚合物。这是因为SIC必须被理解为一种已经成功应用于某些橡胶材料的增强材料。所提出的概念具有普遍性，可以作为涉及内部微观结构演变的类似过程建模的基础。

The present project treats a polymer affected by the strain induced crystallization (SIC) as a heterogeneous medium consisting of regions with the different degree of network regularity. Such a concept allows depicting the nucleation and the growth of crystalline regions as well as the change of effective material parameters depending on the level of the strain applied. The model proposed is thermodynamically consistent. It is based on the assumptions for the free Helmholtz energy and dissipation. Both of them primarily include bulk- and surface terms due to the deformation and crystallization. The external variables are deformations and temperature, whereas the inelastic deformations and degree of the network regularity are internal variables. Their evolution equations are derived according to the principle of maximum of dissipation. The influences of latent heat and of temperature change are implemented in order to simulate thermal effects. The explained framework is advantageous for several reasons. First, it is suitable to answer the crucial question of which process predominantly influences SIC: the nucleation of new crystalline regions or the growth of already existing ones. Secondly, the proposed model is ideal for a direct implementation within the standard multiscale finite element concept. This numerical homogenization procedure is compatible with the theory of finite strains and is applicable for modeling the cases where the ratio of characteristic lengths of scales tends to zero. Both of these features are necessary for the effective modeling of SIC. The project also includes a study of stochastic aspects of the process, where a distribution function for the observable variables is introduced to express the expectation value of relevant quantities. The necessary evolution equation is derived by considering the effective energy of a control volume. The main goals here are to study nucleation and to evaluate the average size of the regions with different regularities of the network. The solution of the tasks itemized will make it possible to achieve the final project goal: the advanced simulations of SIC which can significantly contribute to the more efficient designing and usage of polymers. This is especially motivated by the fact that SIC has to be understood as a kind of reinforcement already successfully applied for some rubber materials. The proposed concepts are of general nature and can be taken as a basis for the modeling of similar processes involving the evolution of the internal microstructure.