Direct numerical simulations of three-dimensional surface instability patterns in thin film-compliant substrate structures

Abstract

A comprehensive numerical study of three-dimensional surface instability patterns is presented. The formation of wrinkles is a consequence of deformation instability when a thin film, bonded to a compliant substrate, is subject to in-plane compressive loading. We apply a recently developed computational approach to directly simulate complex surface wrinkling from pre-instability to post-instability in a straightforward manner, covering the entire biaxial loading spectrum from pure uniaxial to pure equi-biaxial compression. The simulations use embedded imperfections with perturbed material properties at the film-substrate interface. This approach not only triggers the first bifurcation mode but also activates subsequent post-buckling states, thus capable of predicting the temporal evolution of wrinkle patterns in one simulation run. The state of biaxiality is found to influence the surface pattern significantly, and each bifurcation mode can be traced back to certain abrupt changes in the overall load–displacement response. Our systematic study reveals how the loading condition dictates the formation of various instability modes including one-dimensional (1D) sinusoidal wrinkles, herringbone, labyrinth, and checkerboard.

Surface Wrinkling versus Global Buckling Instabilities in Thin Film‐Substrate Systems under Biaxial Loading: Direct 3D Numerical Simulations

“…This solution was based on superposition of two perpendicular 1D modes, and is apparently inconsistent with the predictions from other works. [14,15,17,18] From our previous simulation studies, [36,37] the numerically predicted wavelength for the square-checkerboard mode is in line with Equation (2).…”

“…In this “embedded imperfections” approach, [ 34 ] one or more regular elements at the interface of the film‐substrate structure are endowed with a perturbed elastic property to help activate bifurcation modes without the need of any multi‐step procedure. The present work is an extension of our earlier studies, [ 35,36 ] now for the first time involving full three‐dimensions and post‐instability modes under various biaxial loading and geometric conditions. A wide variety of possibilities can thus be explored.…”

“…The versatility of this approach in reliably generating deformation instabilities in the film-substrate system has been studied in detail in our previous studies. [36,37] The imposed boundary conditions are shown in Figure 1b,c. The roller boundary condition is imposed on the faces z = 0 and x = 0, and a corner point at the bottom of the substrate is fully fixed as shown in Figure 1b.…”

“…The amplitude of the surface wrinkles, A , is yet another important wrinkling parameter. The wrinkling amplitude can be written in a general form [ 14,36 ] as $$\begin{equation}A = {{\Psi}}{\left( {\frac{e}{{{e_{{\rm{cr}}}}}} - 1} \right)^{1/2}}\end{equation}$$where Ψ = t f for the sinusoidal wrinkles and Ψ =$${t_{\rm{f}}} \cdot \sqrt {8/[( {3 - {\nu _{\rm{f}}}} )( {1 + {\nu _{\rm{f}}}} )]} $$ for the square‐checkerboard pattern, [ 14,17 ] and $$e/{e_{{\rm{cr}}}}$$ is the applied compressive strain normalized by the critical value at the onset of primary instability mode.…”

Thermal conductivity of crack-containing media: A numerical study

“…Note that the same type of unit-cell approach has been broadly employed in micromechanical numerical modeling of composite materials and structures. [47][48][49] The present study extends it to periodic cracks, as cracks may be viewed as a "second phase" in the matrix material. The simulations presented in Results were in two dimensions (2D), so it corresponds to a plate-like specimen containing through-thickness cracks.…”