Ruga-formation instabilities of a graded stiffness boundary layer in a neo-Hookean solid

Abstract

We present an analysis of ruga-formation instabilities arising in a graded stiffness boundary layer of a neo-Hookean half space, caused by lateral plane-strain compression. In this study, we represent the boundary layer by a stiffness distribution exponentially decaying from a surface value Q 0 to a bulk value Q B with a decay length of 1/ a . Then, the normalized perturbation wavenumber, k ¯ = k / a , and the compressive strain, ε , control formation of a wrinkle pattern and its evolution towards crease or fold patterns for every stiffness ratio η = Q B / Q 0 . Our first-order instability analysis reveals that the boundary layer exhibits self-selectivity of the critical wavenumber for nearly the entire range of 0< η <1, except for the slab ( η =0) and homogeneous half-space ( η =1) limits. Our second-order analysis supplemented by finite-element analysis further uncovers various instability-order-dependent bifurcations, from stable wrinkling of the first order to creasing of the infinite-order cascade instability, which construct diverse ruga phases in the three-dimensional parameter space of ( ε , k ¯ , η ) . Competition among film-buckling, local film-crease and global substrate-crease modes of energy release produces diverse ruga-phase domains. Our analysis also reveals the subcritical crease states of the homogeneous half space. Our results are, then, compared with the behaviour of equivalent bilayer systems for thin-film applications.