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Shaoting
Lin‡
^{a},
Tal
Cohen‡
^{bc},
Teng
Zhang‡
^{ad},
Hyunwoo
Yuk
^{a},
Rohan
Abeyaratne
^{a} and
Xuanhe
Zhao
*^{ac}
^{a}Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. E-mail: zhaox@mit.edu
^{b}School of Engineering and Applied Science, Harvard University, Cambridge, MA 02138, USA
^{c}Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
^{d}Department of Mechanical Engineering, Syracuse University, Syracuse, NY 13244, USA

Received
21st July 2016
, Accepted 4th October 2016

First published on 4th October 2016

Soft elastic layers with top and bottom surfaces adhered to rigid bodies are abundant in biological organisms and engineering applications. As the rigid bodies are pulled apart, the stressed layer can exhibit various modes of mechanical instabilities. In cases where the layer's thickness is much smaller than its length and width, the dominant modes that have been studied are the cavitation, interfacial and fingering instabilities. Here we report a new mode of instability which emerges if the thickness of the constrained elastic layer is comparable to or smaller than its width. In this case, the middle portion along the layer's thickness elongates nearly uniformly while the constrained fringe portions of the layer deform nonuniformly. When the applied stretch reaches a critical value, the exposed free surfaces of the fringe portions begin to undulate periodically without debonding from the rigid bodies, giving the fringe instability. We use experiments, theory and numerical simulations to quantitatively explain the fringe instability and derive scaling laws for its critical stress, critical strain and wavelength. We show that in a force controlled setting the elastic fingering instability is associated with a snap-through buckling that does not exist for the fringe instability. The discovery of the fringe instability will not only advance the understanding of mechanical instabilities in soft materials but also have implications for biological and engineered adhesives and joints.

Here we show that a constrained soft elastic layer with a comparable thickness and width can indeed undergo mechanical instability, which forms on its exposed surfaces but is localized at the constrained fringes of the layer (Fig. 1b). When subjected to tension, the middle portion of the layer elongates nearly uniformly but the constrained fringe portions of the layer deform nonuniformly. As the applied stretch reaches a critical value, the exposed surfaces of the fringe portions begin to undulate periodically without debonding from the rigid bodies, giving the fringe instability (Fig. 1b).

While both the fingering^{17,18} and fringe instabilities occur on the elastic layers' exposed surfaces, the two modes of instabilities are dramatically different. To quantitatively understand the fringe instability and its differences from fingering instability, we combine experiments, theory and numerical simulations to show that: (i) the deformed layer's meniscus, prior to fringe instability, is not parabolic as is the meniscus before fingering instability (Fig. 1c and d). (ii) In load-controlled elongations, the reported fingering instability is associated with a snap-through buckling, which manifests as peaks on the stress–strain curves for relatively thin samples.§ Such snap-through does not exist in relatively thick specimens, in which the fringe instability dominates. (iii) The critical applied stretch for the onset of fringe instability increases with the decrease of the layer's width–thickness ratio to a finite value of 3.9, which is associated with a constant nominal stress level of 3.8 times of the layer's shear modulus. (iv) The wavelength of the fringe instability scales with the elastic layer's width, but the wavelength of the fingering instability scales with the layer's thickness. The discovery of the fringe instability and quantitative comparisons between the fringe and fingering instabilities will advance the current understanding of mechanical instabilities in soft materials and biological adhesives capable of large deformation. Moreover, the fundamental differences in the mechanical response of constrained elastic layers that differ only by their dimensions are expected to be useful in the design and engineering of advanced adhesives and joints,^{6,24} as well as various sealants, insulators and bearings.

Fig. 2 Schematic illustrations of the experimental setup for the observation of the elastic instabilities in constrained soft elastic layers. |

Instead, as the soft elastic layer (W/H = 2) is stretched to a critical point λ_{c} ≈ 3.2, the exposed surface of the fringe portions becomes unstable – beginning to undulate periodically while the middle portion of the layer maintains uniform elongation (Fig. 3c and Movies S1, S2, ESI†). If the applied stretch further increases, the undulation in the fringe portions increases in magnitude while maintaining a constant wavelength. The layer maintains adhering on the rigid bodies throughout the process of deformation and fringe instability (Fig. S2, ESI†). Once the applied stretch is relaxed, the elastic layer restores its undeformed state. Evidently, the fringe instability is qualitatively different from the fingering instability that occurs in relatively thin elastic layers (e.g., W/H = 8 in Fig. S5 and Movie S3, ESI†). Fig. 3d further shows that the numerical simulation can quantitatively predict the experimental observations of fringe instability. The simulation also confirms that the middle portion of the layer deforms almost uniformly while the fringe portions undergo the instability.

We make a single assumption on the deformation of the layer, that is: any horizontal plane in the layer at the undeformed state remains planar upon deformation^{30} (see the simulation results in Fig. 3a for validation of the assumption). Based on the above assumption and the incompressibility of the elastic layer, we can express the deformation gradient of the layer as (see detailed derivation in the ESI†)

(1) |

The elastic layer is taken as a neo-Hookean material with strain energy density function . By minimizing the elastic energy of the layer, the meniscus shape function is found to obey a first-order differential equation

(2) |

(3) |

Solving eqn (2) and (3) yields the meniscus shape function λ_{X}, deformation gradient F and elastic energy density Ψ of the layer as a function of the applied stretch λ. The total elastic energy of the layer per unit length in the Y direction, E, can then be calculated by integrating Ψ over the volume of the layer. We further define the averaged nominal stress applied on the layer as the applied force divided by the undeformed horizontal cross-section area, which can be calculated as

(4) |

Accordingly we can derive relations between the applied stretch λ, the applied nominal stress S and the meniscus shape λ_{X} for layers with a wide range of width–thickness ratios. In comparison with both experiments and simulations, it is found that the present analytical solution provides accurate predictions of the meniscus shape even for exceedingly high applied stretches and across the entire regime of specimen dimensions considered in this study. For example we show in Fig. 3b, the meniscus shape of a layer with W/H = 2 at applied stretch of λ = 2 given by the theory, experiments and numerical simulations. It can be seen that the theory can accurately predict the non-parabolic shape of a relatively thick layer (i.e., W/H = 2) under high stretches.

On the other hand, for relatively thick layers (e.g., W/H = 2.5, 2, 1.5, 1 and 0.5 in Fig. 4), the curves of S vs. λ obtained from experiments, simulations and theory are all monotonic; and the fringe instability is observed in these samples. Strikingly, from both experimental and simulation results, we find that the critical nominal stress for fringe instability in layers with decreasing W/H ratios approaches an approximately constant value of S_{c} ≈ 3.8μ (Fig. 4b and 5b). Returning to the analytical results in Fig. 4c, and according to the above argument, we may thus obtain an approximate theoretical stability limit by assuming that the fringe instability sets in at the same constant, level of stress from the transition point (where the fingering instability peak vanishes) and to lower width–thickness ratios, as shown by the continuation of the dashed line therein. In addition, different from the subcritical fingering instability,^{17} the fringe instability forms gradually with negligible hysteresis on the pattern amplitude vs. applied stretch curves obtained from loading and unloading of the sample (Fig. S9, ESI†).

To identify the critical width–thickness ratio (W/H)_{c} for the transition between the fingering and the fringe instabilities, we performed a series of experiments and simulations in an intermediate range of width–thickness ratios (e.g., W/H = 3.8, 4.0, 4.2, 4.3, 4.4, 4.5, 5.1). As shown in Fig. 4a, the non-monotonic behavior of the nominal stress-stretch curves disappears as W/H decreases to 4.4 in the experiments. In simulations, the critical width–thickness ratio is identified as (W/H)_{c} = 4. This slight difference between the experiment and simulation is possibly due to the deviation of the layer's mechanical properties from the neo-Hookean model. In this study, we take the simulation result (W/H)_{c} = 4 as the critical width–thickness ratio for the transition between the fingering and fringe instabilities.

We denote the thickness of the middle and fringe portions at the undeformed state as H_{m} and H_{f}, respectively. Based on the minimum layer thickness for the fringe instability, i.e., (W/H)_{c} = 4, we can further obtain H_{f} = W/4 and H_{m} = H − H_{f}. At the critical point for fringe instability, the stretch in the fringe portion λ_{f} is independent of W/H and can be obtained from the simulation results for W/H = 4 as λ_{f} = 1.8. The critical stretch in the middle portion λ_{m} is dictated by the nominal stress-stretch relation in plane-strain tension, i.e. S_{c}/μ = λ_{m} − λ_{m}^{−3} = 3.8, which gives λ_{m} = 3.9. Therefore, the asymptotic solution of the critical stretch for the onset of the fringe instability can be expressed as

(5) |

In Fig. 5c, we summarize the critical stretch levels for both the fringe and fingering instabilities obtained from experiments, simulations and theory. It can be seen that the above linear relation and the simulations can consistently predict the critical stretches for both types of instabilities. The analytical solution further provides a tight upper bound for the critical stretches in the entire range.

While it is known that the wavelength of the fingering instability l_{finger} scales with the elastic layer's thickness H (Fig. 5d), the wavelength of the fringe instability l_{fringe} does not follow such scaling since the thickness of the middle portions does not affect the fringe instability wavelength. Instead, the relevant length scale for the fringe instability wavelength is the fringe portion's thickness, which scales with the layer's width. Therefore, the wavelength for fringe instability scales with the elastic layer's width, instead of its thickness. This dependence has been validated by both the experimental and simulation results in Fig. 5d. In addition, by fitting to the experimental and simulation results, we can further obtain the pre-factors for the scales, i.e., l_{fringe} ≈ 0.45W.

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## Footnotes |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6sm01672c |

‡ These authors contribute equally to this paper. |

§ For brevity, throughout the text we refer to layers of thickness that is much smaller than the in-plane dimensions as ‘thin layers’, while ‘thick layers’ are considered to have a thickness of the order of the in-plane dimensions or larger. |

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