Research Papers

Multimodal Surface Instabilities in Curved Film–Substrate Structures

[+] Author and Article Information
Ruike Zhao

Soft Active Materials Laboratory,
Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139

Xuanhe Zhao

Soft Active Materials Laboratory,
Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139;
Department of Civil and Environmental
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: zhaox@mit.edu

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 5, 2017; final manuscript received May 29, 2017; published online June 13, 2017. Editor: Yonggang Huang.

J. Appl. Mech 84(8), 081001 (Jun 13, 2017) (13 pages) Paper No: JAM-17-1237; doi: 10.1115/1.4036940 History: Received May 05, 2017; Revised May 29, 2017

Structures of thin films bonded on thick substrates are abundant in biological systems and engineering applications. Mismatch strains due to expansion of the films or shrinkage of the substrates can induce various modes of surface instabilities such as wrinkling, creasing, period doubling, folding, ridging, and delamination. In many cases, the film–substrate structures are not flat but curved. While it is known that the surface instabilities can be controlled by film–substrate mechanical properties, adhesion and mismatch strain, effects of the structures’ curvature on multiple modes of instabilities have not been well understood. In this paper, we provide a systematic study on the formation of multimodal surface instabilities on film–substrate tubular structures with different curvatures through combined theoretical analysis and numerical simulation. We first introduce a method to quantitatively categorize various instability patterns by analyzing their wave frequencies using fast Fourier transform (FFT). We show that the curved film–substrate structures delay the critical mismatch strain for wrinkling when the system modulus ratio between the film and substrate is relatively large, compared with flat ones with otherwise the same properties. In addition, concave structures promote creasing and folding, and suppress ridging. On the contrary, convex structures promote ridging and suppress creasing and folding. A set of phase diagrams are calculated to guide future design and analysis of multimodal surface instabilities in curved structures.

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Grahic Jump Location
Fig. 1

Surface instabilities on curved film–substrate structures in nature and engineering applications. (a) Instabilities on flat structure (from left to right: crease, ridge, double, and wrinkle); (b) multimodal instabilities on concave structures (from left to right: cross section of colon, ductus deferens, muscular artery, and bronchus); and (c) multimodal instabilities on convex structures (from left to right: cross section of bitter melon, cactus, tree stump, and cereus).

Grahic Jump Location
Fig. 2

Schematics for plane-strain deformation of concave film–substrate structure (a)–(c) and convex film–substrate structure (d)–(f). (a) and (d) represent the reference states; (b) and (e) denote the homogeneous current states; and (c) and (f) illustrate the patterned current states.

Grahic Jump Location
Fig. 3

Instability patterns on curved film–substrate structure and their FFT characterization. (a) and (b) Patternless state and instability patterns of wrinkle, crease, fold, double, and ridge on concave and convex film–substrate structures, respectively. (c) FFT characterization of the instability patterns.

Grahic Jump Location
Fig. 5

Phase diagrams for mismatch-strain induced instability patterns on curved film–substrate structure with different normalized curvatures. (a)ρ=−0.1, (b)ρ=−0.04,(c)ρ=0, (d)ρ=0.04, and (e)ρ=0.1. Blue solid curve denotes the theoretical buckling condition. Black solid curves represent simulated phase boundaries. Colored regions represent different phases: purple for flat, green for winkle, orange for crease, yellow for fold, red for double, blue for ridge, and dark blue for disordering. The instability morphologies of the highlighted points (A–O) are shown in Fig. 6 (see color figure online).

Grahic Jump Location
Fig. 4

Curvature effects on the onset of wrinkling for the curved film–substrate systems. (a) Critical mismatch strain for systems with different normalized curvatures. The solid curves from left to right are for ρ = 0, 0.02, 0.04, and 0.1. The dotted curves from left to right are for ρ = −0.02, −0.04, and −0.1 (color version online). (b) Changing of critical mode number nc with film–substrate modulus ratio for structures with different curvatures.

Grahic Jump Location
Fig. 6

The instability morphologies of the highlighted points (A–O) on the phase diagrams in Fig. 5. The contour plots show the maximum in-plane nominal strain for each case. The five rows from top to bottom in each column represent five different curvatures ρ=−0.1,−0.04, 0, 0.04, and 0.1. Column (a) includes (A–E), which have the mismatch strain and modulus ratio εM=0.5,μf/μs=30. Column (b) includes (F–J), which have the mismatch strain and modulus ratio εM=0.6,μf/μs=30. Column (b) includes (K–O), which have the mismatch strain and modulus ratio εM=0.6,μf/μs=300.



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