Slim Elastic Structures with Transversal Isotropic Material Properties under Finite Deformations

Compliant mechanisms offer numerous advantages compared to their rigid counterparts such as a reduction in part number, the corresponding reduction in assembly time and the avoidance of coulomb friction and the resulting wear. In small scale applications i.e. microelectromechanical systems classical joints cannot be manufactured in a reasonable cost range and compliant solutions usually apply.

The existing applications and simulation strategies have in common that the initial and mostly also the deformed configuration is planar and the mechanisms only undergo small deformations.

The presented work sketches the idea of modelling compliant mechanisms using beam-like structures which can be subjected to finite deformations. These beam elements are described on velocity level in convected coordinates. The rate description transforms the nonlinear initial boundary value problem (IBVP) into a linear boundary value problem (BVP) which is embedded in a nonlinear initial value problem (IVP). In the case of one dimensional (beam) elements the BVP can be solved accurately using a Runge Kutta (RK) integration scheme. For a sufficiently smooth behavior of the integrated function a 4th order RK integration shows accurate results [

1

]. In dynamic simulations the smooth behavior is usually not achieved for the deformations and for the kinetic variables as well. Here the use of lower order integration schemes leads to more reasonable results.

The description of the constitutive behavior of the beam material follows the above mentioned rate description by using the objective Truesdell stress rate. The corresponding material tensor which connects the Truesdell stress rate with the deformation gradient is determined by differentiating an appropriate strain energy function with respect to the left Cauchy-Green deformation tensor. Hyperelastic isotropic and transversal-isotropic materials [

2

] are used for the compliant members.

The use of transversal-isotropic material leads to a coupling between the bending and the torsional deformation which allows i.e. the generation of complex spatial movements with initially planar geometry and fully planar loads.