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，需要一种高效而准确的仿真方法。该项目旨在开发一种新的梁的有限元公式，用于在单个纤维的水平上直接数值模拟机电耦合的纺织结构。每根纤维都用梁公式来建模，与砖单元公式相比，梁公式需要的自由度要少得多。我们考虑在径向具有多层材料的圆形横截面：由介电弹性体包围的导体。我们设想的电容式压力传感器的工作原理可以简单地用在光纤交叉点产生的等效电容来解释。导体的电荷相反，因此在它们之间产生一个电场，使介于两者之间的弹性体层中的电荷极化。弹性体在外加载荷的作用下受到压缩，其电容随相交导线间距和接触面积的变化而变化。因此，在电容式传感器的模拟中，将横截面应变纳入新的梁公式中具有重要意义，这与通常假定刚性横截面的传统方法不同。我们采用具有无约束指向矢的Cosserat梁公式，其中指向矢的伸缩和旋转有效地表示了横截面应变。我们进一步丰富了横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，包括本构关系、导体间的静电力和接触条件。我们采用自由能函数的本构定律，它由三部分组成：机械部分、电气部分和极化部分。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，这使得三维本构定律的直接应用成为可能。在束-束接触公式中，我们采用了高斯点面接触算法。利用NURBS(非均匀有理B-Spline)基函数进行等几何分析，可以得到沿梁的纵向和周向的高阶连续性，从而为最近点投影产生稳健的局部牛顿-拉夫森过程。在电接触方面，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。该研究项目旨在帮助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.