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Research Papers

The Primary Bilayer Ruga-Phase Diagram II: Irreversibility in Ruga Evolution

[+] Author and Article Information
R. Zhao, M. Diab, K.-S. Kim

School of Engineering,
Brown University,
Providence, RI 02912

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 7, 2016; final manuscript received May 28, 2016; published online June 27, 2016. Editor: Yonggang Huang.

J. Appl. Mech 83(9), 091004 (Jun 27, 2016) (9 pages) Paper No: JAM-16-1229; doi: 10.1115/1.4033722 History: Received May 07, 2016; Revised May 28, 2016

When an elastic thin-film/substrate bilayer is cyclically compressed with a large plane-strain stroke, various surface morphologies develop either reversibly or irreversibly with cyclic hysteresis. Here, we examine the cyclic morphology evolution with extensive finite-element analyses and present a generic irreversibility map on the primary bilayer Ruga-phase diagram (PB-RPD). The term “PB” refers to a system of a film on a substrate, both of which are incompressible neo-Hookean, while the term “Ruga-phase” refers to the classification of corrugated surface morphologies. Our generic map reveals two configurational irreversibility types of Ruga-phases during a loading and unloading cycle. One, localization irreversibility, is caused by unstable crease localization and the other, modal irreversibility, by unstable mode transitions of wrinkle-Ruga configurations. While the instability of crease localization depends mainly on smoothness of the creasing surface or interface, the instability of Ruga-mode transition is sensitive to film/substrate stiffness ratio, film/substrate strain mismatch (εps), and material viscosity of the bilayer. For small strain mismatches (εps ≲ 0.5), PB Ruga structures are ordered; otherwise, for large strain mismatches, the Ruga structures can evolve to ridge configurations. For evolution of ordered Ruga phases, the configurational irreversibility leads to shake-down or divergence of cyclic hysteresis. Underlying mechanisms of the cyclic hysteresis are found to be the unstable Ruga-phase transitions of mode-period multiplications in the loading cycle, followed by either mode “locking” or primary-period “switching” in the unloading cycle. In addition, we found that the primary-period switching is promoted by the strain mismatch and material viscosity. These results indicate that various Ruga configurations can be excited, and thus, diverse Ruga-phases can coexist, under cyclic loading. Our irreversibility map will be useful in controlling reversibility as well as uniformity of Ruga configurations in many practical applications.

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Figures

Grahic Jump Location
Fig. 1

(a) The primary bilayer RPD: k¯ is the scaled stiffness ratio and ε is the compressive strain–(I) flat phase; (II) SM wrinkle phase; (III) global crease localization; (IV) DM wrinkle phase; (V) QM wrinkle phase; (VI) fold phase; (VII) global fold localization; (IIc), (IVc), and (Vc) three setback-crease phases; and (Vcf) crease-fold phase; A is the subcritical crease strain limit; B is the Biot critical strain of creasing; DIR is the M-period doubling limit; and QIR is the M-period quadrupling limit. The five-colored regions represent irreversible Ruga-phases. (b) Illustration of P-period and M-period in wrinkling (b1), doubling (b2), and quadrupling (b3).

Grahic Jump Location
Fig. 2

Cooperative creasing irreversibility for k¯=0.59 ( R=15): (a) ΔΩ is the crease-tip depth difference normalized by critical onset-wrinkling wavelength l; h is the film thickness; εW=0.08, εD=0.19, and εC=0.22 are the critical strains for wrinkling, doubling, and creasing modal bifurcations; ε1=0.24, ε2=0.35, and ε3=0.36 are the three particular strains chosen for comparison of loading and unloading. (b) Contour plots of loading and unloading FEM results for ε1, ε2, and ε3.

Grahic Jump Location
Fig. 3

Reversibility of PB without strain mismatch: ΔΩ is the amplitude difference normalized by critical onset-wrinkling wavelength, and ε is the compressive strain. (a) Bilayer with modulus ratio R = 90 (k¯=0.32): The traces coincide during the loading–unloading cycle indicating reversibility of the Ruga modes. (b) Bilayer with modulus ratio R = 3000 (k¯=0.10): Both DM2 and QM2 exhibit hysteretic behavior during unloading with mode locking; the FEM contour plots show the Ruga-phases under four specific compressive strains ε1, ε2, ε3, and ε4 during loading and unloading.

Grahic Jump Location
Fig. 4

P-period switching irreversibility of PB with strain mismatch (R = 104 and prestretch ratio λps = 1.4) during loading/unloading cycle: ΔΩ is the amplitude difference normalized by critical onset-wrinkling wavelength l; ε is the compressive strain; ΔΩs and ΔΩl are the normalized amplitude difference between the shallow wrinkle valley and two neighboring deep valleys; solid curves represent loading (DM1 → QM1 → F); dashed curves represent unloading; and P is the critical point, where shallow wrinkle valley becomes flat. (a1) Unloading from M-period doubling (DM1 → SM2). (a2) Normalized strain energy (W/W(o)) variation along a loading/unloading process near the mode-doubling point. (b1) Unloading from M-period quadrupling (QM1 → DM2 → SM2): G+ and R+ are the unloading starting points of two quadrupling bifurcations, G and R are the points of excited-mode DM2, and black star represents the configuration of excited-mode SM2. (b2) Normalized strain energy (W/Wo) variation along a loading/unloading process near the mode-quadrupling point. (c) FEM contour plots of Ruga evolution during a loading–unloading cycle: (c1) and (c3) represent the primary modes through transition (SM1 → DM1 → QM1) and (c4) represents excited-mode DM2 during unloading by snap-buckling from QM1.

Grahic Jump Location
Fig. 5

Vertical surface velocity of P-period switching: Upper curves of (a), (b), and (d)–(f) represent the velocity profile of a selected range marked by dashed line box in (c), and lower curves correspond to the surface profiles (Ω normalized by critical onset-wrinkling wavelength l). (a) Unloading from QM1 to current DM1 with small viscosity. (b) Loading to current DM1: (d)–(f) unloading from QM1 to SM2 with viscoelastic loss tangent of 2×10−4.

Grahic Jump Location
Fig. 6

Schematics of surface localization and Ruga modes of a neo-Hookean bilayer system subject to a loading–unloading cycle: (a) Three types of global Ruga localization—crease, fold, and ridge. (b) Mode transitions during loading–unloading cycle: Solid and dashed arrows represent loading and unloading processes, bottom row represents Ruga evolution pathway for mode locking irreversibility [(I) flat phase, (II) primary single-mode wrinkle phase SM1, (IV) primary double-mode wrinkle phase DM1, (V) primary quadruple-mode wrinkle phase QM1], {(SMi), (DMi), and (QMi), i = 2,4} represents the excited-modes observed in P-period switching during cyclic loading, and dashed boxes represent the predicted excited-modes.

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