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

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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.