Non-linear vibration analysis of visco-elastically damped composite structures by multilevel finite elements and asymptotic numerical method

Abstract

Composite materials and structures are inherently in-homogeneous across multiple scales. Multi-scale modelling offers opportunities to apprehend the coupling of material behaviour and characteristics from the micro- to meso- and macro-scales. In this paper, a multi-scale finite element method (FE2) is proposed to compute the modal properties of visco-elastic heterogeneous composite materials in terms of damping frequencies and modal loss factors. In the proposed FE2-based vibration analysis, two finite elements (FE) calculations are carried out in a nested manner, one at the macro-scale and the other at the micro-scale. Unlike conventional analysis, the developed analysis does not require homogenized constitutive properties because these are derived from the micro-scale FE simulations at the representative volume element (RVE) level. The non-linearity at the micro-scale is accounted by using a frequency dependent Young’s modulus. The Asymptotic Numerical Method (ANM) and its automatic differentiation is used to solve the non-linear numerical problem. ANM consists of solving an analytical non-linear problem with a path-following (or continuation) method associated with a high-order perturbation technique. Compared with existing methods, the originality of the proposed approach lies in its ability to account for the frequency dependence of Young’s modulus in visco-elastic microstructure. Using the automatic differentiation makes the proposed approach enough flexible and generic to deal with damped and undamped vibration analyses of composite materials structures.

Keywords

• Vibrations
• Damping
• Multiscale finite element method (FE2)
• Asymptotic Numerical Method
• Nonlinear homogenization
• Automatic differentiation