Nonlinear mechanics and parametric sensitivity of a displacement-dependent soft porous PDMS-based rectangular capacitive sensor: instability analysis

Abstract In many micro-capacitive pressure sensors, electrostatic forces are weak at larger gaps but strengthen considerably as the gap narrows. Achieving smooth movement and displacement necessitates a low initial mechanical stiffness in the sensor structure. This study introduces a porous polydimethylsiloxane (PDMS) elastomeric filler, which exhibits displacement-dependent porosity, leading to an increase in stiffness with displacement, a characteristic distinct from traditional air-gap designs where stiffness remains constant. This paper investigates the static instability of a micro-capacitive rectangular plate (MCRP), a crucial component in capacitive pressure sensors. The MCRP is supported by a porous PDMS filler and subjected to external pressure and electrostatic bias actuation. ​A novel aspect of this research involves realistically modeling the PDMS filler’s behavior by establishing a power-law relationship between its Young’s modulus and porosity, which is essential for accurately predicting the sensor’s response.​The MCRP is influenced by three primary forces: a downward measurement pressure, an upward mechanical pressure from the deflected PDMS filler, and a downward electrostatic pressure from the bias voltage. Utilizing nonlinear strain effects, the Airy stress function, and the Hamiltonian principle, we derive the governing coupled partial differential equations (PDEs) for static deflection, incorporating new terms that describe the PDMS filler’s behavior. These equations are then simplified using Galerkin’s method into a set of nonlinear algebraic equations and solved numerically. The sensitivity of the designed capacitive pressure sensor exhibits a remarkable increase when the porosity power index is below its critical value of $$\:{n}_{cr}\cong\:1.18$$ .​ Moreover, an increased power index value impacts sensitivity in two distinct ways. Firstly, it results in a reduction in the overall sensitivity of the sensor. Secondly, and conversely, it contributed to an expansion of the measurable pressure range. The finding can be beneficial for a variety of target applications as wearable electronics, health monitoring, robotics and human-machine interaction.