An algorithmic approach to multi-layer wrinkling
Introduction
Though wrinkling in bi-layer systems is a well understood phenomenon [1], [2], [3], [4], [5], [6], [7], the study of wrinkling in multi-layer systems is incomplete. Multi-layer systems, as illustrated in Fig. 1, do not necessarily have a single layer clearly defined as the film or a single layer clearly defined as the substrate. Instead, there are multiple layers some of which may contribute to the axial and bending stiffness of the film and some of which may contribute to the stiffness of the substrate [8], [9], [10]. Understanding wrinkling in multi-layer systems is important for interpreting the wrinkling behavior of some biological systems, such as the gut [11], [12], [13], the skin [14], [15], [16], or the lungs [17], [18]. The patterns which emerge from these biological multi-layer systems can inspire engineering design, such as engineering surfaces which utilize wrinkling to access multiple length scales and increase surface area [19], [20], [21]. In buckling based metrology, a better understanding of wrinkling in multi-layer films will enhance the study of novel multi-layer systems [22], [23], [24], [25]. In addition, in engineering stretchable electronics, a deeper understanding of multi-layer wrinkling can be used to design systems in which wrinkling is used to prevent high levels of strain in stiff or brittle layers [26], [27]. In this paper, we use epidermal electronics [28], [29], [30], [31], [32] as an example to motivate the study of multi-layer wrinkling. Epidermal electronics are a strong motivation because, as seen in Fig. 2, they connect to multi-layer wrinkling in two ways. First the electronic devices themselves are often multi-layer systems [28], [33]. Second, the skin is a multi-layer structure thought to have as many as six mechanically distinct layers which influence its wrinkling behavior [34], [35]. We present a multi-layer model suitable for capturing the wrinkling behavior of the device–skin system.
With regard to previous research specific to multi-layer wrinkling [8], we primarily build on four previous works to construct our approach to multi-layer wrinkling. First, Stafford et al. [37] proposed a method for combining thin, experimentally observed, surface layers of finite thickness with the film layer by treating the film as a composite beam. Second, Jia et al. [9] provided an analytical solution verified by numerical results to account for an intermediate layer that will combine with either the film or the substrate depending on the geometric and material properties of the system. Third, Huang et al. [38] demonstrated that after a finite substrate depth is reached, substrate depth no longer influences wrinkling behavior. Finally, in our recent work [10] we proposed an analytical solution verified by numerical results to account for wrinkling in multi-layer systems where the additional interfacial layers are assumed to be thinner than the film.
Here, we extend the solution presented in Lejeune et al. [10], which is restricted to bi-layer systems with additional thin interfacial layers, to systems with multiple layers that are not restricted in thickness or material properties. Then, we apply our extended multi-layer wrinkling model to a device–skin system. The rest of the paper is organized as follows. In Section 2, we clearly explain the methodology for performing this extension with an analytical solution verified by numerical results. In Section 3, we show an example of our multi-layer wrinkling model applied to epidermal electronics. Finally, we conclude our paper in Section 4.
Section snippets
Multi-layer model
The equations for critical strain , critical wavelength , and critical wave number that are sufficient to describe wrinkling initiation in a bi-layer system are quite simple. Our intention is to preserve this simplicity by extending the bi-layer solution to multiple layers following an intuitive procedure. We begin with the equation relating axial strain to wave number used to derive for bi-layer wrinkling [1], [39], where and refer to the plane
Application to epidermal electronics
As illustrated in Fig. 2, images and studies of the human skin indicate that it has a multi-layer structure [45]. Here, we present a model of the skin which has five mechanically distinct layers [35]. Due to the variability in parameters reported in the literature [46], [47], [48], [49], [50], both due to differences in experimental set up and variation across subjects and locations on the body [51], the parameters chosen for our model are approximations. Similar to the abrupt mode switch
Conclusion
In this paper, we outline a procedure for computing the analytical solution for wrinkling instability initiation in multi-layer systems. We begin by presenting our previously developed solution for multi-layer wrinkling in systems with a limited number of very thin interfacial layers. Then, we extend this solution to multi-layer systems with an unlimited number of layers and unconstrained layer thickness. Performing this extension has utility in many areas, ranging from stretchable electronics
Acknowledgment
Financial support for this research was provided by the National Science Foundation through CAREER Award CMMI-1553638 and the National Science Foundation Graduate Research Fellowship under Grant No. DGE-114747.
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