Session: 12-09-01: Instabilities in Solids and Structures

Paper Number: 150241

150241 - Upper Bound Analysis of the Second Bifurcation of Herringbone Patterns 

   Surface instability is one of the classical but challengeable research problems in solid mechanics because most scientists recognize the importance of a basic understanding of evolution and morphogenesis of living systems, such as cortical folding in mammalian brains. Because brains consist of two layers of gray and white matter, the most elementary model is a compressed film bonded to a compliant substrate. The emerging surface patterns gradually evolve as the compressive stress in the film increases beyond a critical buckling stress, i.e., as the overstress increases. A small degree of overstress induces hexagonal and square dimple patterns, whereas a large degree of overstress causes the occurrence of herringbone patterns.

   In this study, we perform upper bound analysis [1–3] of the second bifurcation of herringbone patterns [4,5], which appear on compressed films bonded to compliant substrates. This approach is inspired by the peculiarity of the herringbone patterns in reciprocal space. A trinomial expression consists of three rectangular checkerboard modes, which lead to the inclusion of three geometric constants that characterize the width, breath, and inclination angle to express the variety of the herringbone patterns. The first term is the rectangular checkerboard mode with the critical wavelength at the first bifurcation, whereas the second and third terms are found to be critical not at the first bifurcation but at the second bifurcation on the first bifurcated path [5]. These two terms have the constraint imposed by the wavenumbers in reciprocal space. The constraint explicitly generates the negative interaction energy between specific terms in the Airy stress function, which contributes to the energy reduction in the pattern evolution. The energy minimization analysis [1–3] also elucidates that the first bifurcation can smoothly induce the transformation from a quasi-stripe pattern to the herringbone pattern.

 

Keywords: Thin films, Bilayers, Upper bound analysis, Energy minimization, Buckling, Wrinkling, Bifurcation

 

[1] Cai, S., Breid, D., Crosby, A.J., Suo, Z., Hutchinson, J.W., 2011. Periodic patterns and energy states of buckled films on compliant substrates. J. Mech. Phys. Solids 59, 1094–1114.

[2] Abu-Salih, S., 2017. Analytical study of electromechanical buckling of a micro spherical elastic film on a compliant substrate part II: Postbuckling analysis. Int. J. Solids Struct. 110-111, 251–264.

[3] Zhao, Y., Jie, Z., Zhang, Y., Jiang, C., Cao, Y., 2024. Negative Gaussian curvature regulated pattern evolution on curved bilayer system. Int. J. Mech. Sci. 267, 108969.

[4] Miyoshi, H., Matsubara, S., Okumura, D., 2021. Bifurcation and deformation during the evolution of periodic patterns on a gel film bonded to a soft substrate. J. Mech. Phys. Solids 148, 104272.

[5] Kikuchi, S., Matsubara, S., Nagashima, S., Okumura, D., 2022. Diversity of the bifurcations and deformations on films bonded to soft substrates: robustness of the herringbone pattern and its cognate patterns. J. Mech. Phys. Solids 159, 104757.

Presenting Author: Dai Okumura Nagoya University

Presenting Author Biography: Dai Okumura, Dr.Eng.
Professor
Department of Mechanical Systems Engineering
Nagoya University
Chikusa-ku, Nagoya 464-8603, Japan

Authors:

Dai Okumura Nagoya University
Seishiro Matsubara Nagoya University
So Nagashima Nagoya University
Hiro Tanaka University of Hyogo