Growth-induced instabilities for transversely isotropic hyperelastic materials

TL;DR

This study investigates how fiber stiffness influences growth-induced buckling and post-buckling behaviors in transversely isotropic bilayer structures, revealing the dominant energy contributions and stability thresholds through numerical analysis.

Contribution

It introduces a mixed variational formulation for analyzing growth-induced instabilities considering fiber stiffness and limits, with detailed numerical insights into energy contributions and bifurcation points.

Findings

Both wavelength and critical growth decrease with increasing fiber stiffness for the first instability.

Secondary buckling is minimally affected by fiber stiffness, occurring perpendicular to the fiber direction.

Energy analysis shows isotropic and anisotropic contributions dominate at different buckling stages.

Abstract

This work focuses on planar growth-induced instabilities in three-dimensional bilayer structures, i.e., thick stiff film on a compliant substrate. Growth-induced instabilities are examined for a different range of fiber stiffness with a five-field Hu-Washizu type mixed variational formulation. The quasi-incompressible and quasiinextensible limits of transversely isotropic materials were considered. A numerical example was solved by implementing the T2P0F0 element on an automated differential equation solver platform, FEniCS. It was shown that both the wavelength and critical growth parameter g decrease by increasing the fiber stiffness for the first instability, which is obtained along the stiff fiber direction. The effect of the fiber stiffness is minor on the secondary buckling, which was observed perpendicular to the fiber direction. For a range of fiber stiffnesses, bifurcation points of instabilities were also determined by monitoring displacements and energies. The energy contributions of layerswith different ranges of fiber stiffnesses were examined. It is concluded that the energy release mechanism at the initiation of the primary buckling is mainly due to isotropic and anisotropic contributions of the stiff filmlayer. For high fiber stiffnesses, the effect of the anisotropic energy on the first buckling becomes more dominant over other types. However, in the secondary instability, the isotropic energy of the film layer becomes the dominant one. Numerical outcomes of this study will help to understand the fiber stiffness effect on the buckling and post-buckling behavior of bilayer systems.