An isogeometric finite element formulation of electromechanically coupled beams with cross-sectional deformations and frictional contact for dielectric elastomer smart textiles

基于纺织品的电容式压力传感器已被开发用于许多应用，包括可穿戴电子产品、人造皮肤和结构健康监测。为了开发一种创新的传感器设计或机构，需要一种高效、准确的仿真方法。该项目旨在开发一种新的梁的有限元公式，用于在单个纤维水平上对机电耦合纺织结构进行直接数值模拟。每根纤维都是通过一个梁的公式来建模的，与砖单元的公式相比，它需要的自由度要小得多。我们考虑在径向上有多层材料的圆形截面：导体被介电弹性体包围。我们设想的电容式压力传感器的工作可以简单地解释为在光纤交叉点产生的等效电容器。导体的电荷是相反的，因此在它们之间产生电场，使中间的弹性体层中的电荷极化。弹性体受到外部负载的压缩，电容因交叉导体之间的距离和接触面积的变化而变化。因此，在电容式传感器的模拟中，将截面应变纳入新的梁公式中是很重要的，这与传统方法相反，传统方法通常假设刚性截面。我们采用无约束董事的Cosserat梁公式，其中董事的拉伸和旋转有效地表示截面应变。我们进一步丰富了截面应变作为不相容模式，使用增强假设应变（EAS）方法。机电耦合可能发生在几个部分，包括本构定律、导体之间的静电力和接触条件。我们采用自由能函数的本构律，它由三部分组成；机械，电气和极化。新的梁公式不假设任何零应力条件，并表示三维应力、应变和电场，从而可以直接应用三维本构定律。在光束对光束接触公式中，我们采用高斯点对面接触算法。使用NURBS（非均匀有理b样条）基函数的等几何分析给出了沿梁的纵向和周向的高阶连续性，从而产生了最近点投影的鲁棒局部牛顿-拉夫森过程。在电接触中，我们还研究了极化弹性体和带电导体之间的静电力的影响。该研究项目旨在有助于更深入地了解智能纺织品的工作原理。

Textile-based capacitive pressure sensors have been developed for many applications including wearable electronics, artificial skin, and structural health monitoring. For developing an innovative design or mechanism of the sensors, an efficient and accurate simulation method is necessary. This project aims at developing a new finite element formulation of beams for a direct numerical simulation of electromechanically coupled textile structures at the level of individual fibers. Each fiber is modeled by a beam formulation, which requires much less degrees-of-freedom, compared with the brick element formulation. We consider a circular cross-section having multiple layers of material in the radial direction: a conductor surrounded by a dielectric elastomer. The operation of our envisioned capacitive pressure sensors can be simply explained by an equivalent capacitor generated at the crossing point of the fibers. The conductors are charged oppositely, so that an electrical field is generated between them, which polarizes the charges in the elastomer layer in between. The elastomer is compressed by an external load, and the capacitance changes due to the change of the distance between crossing conductors, and the contact area. Therefore, in the simulation of the capacitive sensors, it is significant to incorporate the cross-sectional strains in the new beam formulation, which is in contrast to the conventional methods, which typically assume rigid cross-sections. We employ a Cosserat beam formulation with unconstrained directors, where the stretching and rotation of directors represent the cross-sectional strains efficiently. We further enrich the cross-sectional strains as incompatible modes, using an enhanced assumed strain (EAS) method. The electromechanical coupling may occur in several parts including constitutive laws, electrostatic force between conductors, and contact conditions. We employ a constitutive law from a free energy function, which consists of three parts; mechanical, electrical, and polarization. The new beam formulation does not assume any zero stress conditions, and represent 3-D stress, strain, and electrical fields, which enables a straightforward application of 3-D constitutive laws. In the beam-to-beam contact formulation, we employ a Gauss point-to-surface contact algorithm. An isogeometric analysis using NURBS (Non-uniform rational B-spline) basis functions gives a higher order continuity along the beam’s longitudinal and circumferential directions, which yields a robust local Newton-Raphson process for the closest point projection. In the electrical contact, we also investigate the effect of the electrostatic force between polarized elastomers, and charged conductors. The research project is intended to contribute to a deeper understanding of how smart textiles work.