On the stability of a rod adhering to a rigid surface: Shear-induced stable adhesion and the instability of peeling
Introduction
The emergence of soft, miniaturized, and biologically inspired systems has led to particular interest in the dry adhesion of elastic rods. Relevant studies have examined the role of rod adhesion in microelectronic switches (Adams and McGreur, 2010), MEMS stiction (de Boer and Michalske, 1999), soft lithography stamp printing (Hui et al., 2002), muscle crossbridges (Stewart et al., 1987), nanotubes (Glassmaker and Hui, 2004), and gecko-inspired microfiber array adhesives (Majidi, 2009). These works are also related to extensive body of work on peeling problems (see Burridge and Keller, 1978; Plaut et al., 2001a, Plaut et al., 2001b; Podio-Guidugli, 2005, and references therein).
In the works on dry adhesion of rods, the conditions for static equilibrium are commonly derived from the stationarity of an energy functional that is composed of elastic bending energy and the work of adhesion. The purpose of the present paper is to develop stability and instability criteria for these equilibrium configurations. Of particular interest are the stability properties of shear-induced adhesion and a peeling process. These two examples are relevant to microfiber adhesive arrays. The stability of one of these examples can be exploited to promote adhesion while the instability can be used to deactivate the adhesion.
A prototypical problem featured in rod theory-based models for systems with adhesion is shown in Fig. 1. A rod of length is subject to a terminal load P and has a portion of length bonded to a surface. The equilibrium conditions for the rod dictate that the length depends on the components of P, the geometric and material properties of the rod, and the adhesion energy . Dating to the seminal work by Kendall (1971), variational principles can be used to show that the equilibrium configuration of the rod is such that the potential energy is extremized with respect to changes in . Some authors, e.g., de Boer and Michalske (1999) and Mastrangelo and Hsu (1993), also examine the second derivative and postulate minimization of as a function of as a stability criterion.
We concentrate our attention on Euler's theory of the elastica as this is the predominant rod theory featured in the literature on dry adhesion of rods. The criteria are developed by exploiting classical investigations of the second variation by Legendre and Jacobi.1 Our work is intimately related to the extension of these classical works to develop stability criteria for tree-like structures composed of elastic rods which was recently presented in O'Reilly and Peters (2012). A further consequence of our treatment is the ability to relate the criteria to nonlinear treatments of buckling instabilities in rods (which can be found in works by Born, 1906, Jin and Bao, 2008, Maddocks, 1984, Manning, 2009, O'Reilly and Peters, 2011, O'Reilly and Peters, 2012 and many others).
An outline of this paper is as follows: In Section 2, the relevant background material on adhesion and the elastic rod theory are introduced, and in Section 3 it is used to formulate a variational principle for the problem of an elastica which is in contact with a rigid surface. Particular attention is also placed on compatibility conditions which must be satisfied at the edge of the contact region. Section 4 is devoted to discussions on stability criteria. In particular, two stability criteria are developed in 4.3 The condition N1, 4.4 The condition S1, respectively. The first criterion is a necessary condition for stability, which we denote by N1. As in traditional treatments of buckling instabilities, the criterion N1 features the search for a bounded solution to a Riccati equation but with two subtle, yet important, differences. First, the domain of integration for the Riccati equation depends on the loading, adhesive strength, and geometric and material properties of the rod. Second, the solution to the Riccati equation must also satisfy a terminal inequality in order for N1 to hold. Our work in this respect can be viewed as an extension of the aforementioned nonlinear treatments of buckling instabilities to cases where adhesion is present. Section 4.5 of the paper is devoted to a discussion of a stability criterion where as a function of is examined.
For some of the problems of interest in this paper, we were fortunate to be able to establish a sufficient condition for stability, which we label S1, by slightly modifying a result of Gelfand and Fomin (1963). The criteria N1 and S1 are illustrated in Section 5 by applying them to three examples that have received significant attention in the literature in the past three decades. We also illustrate how the criteria N1 and S1 compare to stability treatments where the potential energy is expressed as a function of the adhesion length and the system parameters. The paper closes with a discussion of possible extensions to the criteria and open issues. The paper contains three appendices which are devoted to a proof of the criterion N1 and the establishment of explicit solutions for the deformed shape of the rod using elliptic functions.2
Section snippets
Background on adhesion energy and the elastica
The elastica is the simplest nonlinear theory of a deformable rod. Here we are interested in using this theory to model the adhesion of a rod-like body to a surface. We assemble all the background in this section that is required to formulate variational principles for the problems of interest and to examine the stability of the resulting equilibrium configurations.
Formulation of the problem of an elastica in contact with a rigid surface
In the applications considered in this paper we assume that the assigned force and the assigned moment . It follows that the local balance laws (6) simplify dramatically to the statements that the force and moment are piecewise constant throughout segments of the elastica.
For a rod which is in contact with a rigid substrate, the total potential energy of the rod will be composed of the potential energy of terminal forces and moments, the integral of the strain energy per unit
Conditions for stability from the second variation
Necessary conditions for the functional to be minimized includes the vanishing of the first variation and the non-negativity of the second variation. We now explore these conditions and use them to establish a necessary condition, which we label N1, for stability. For some cases, we are able to establish a sufficient condition, which is referred to as S1, for stability. This section of the paper concludes with a discussion of an alternative criterion for stability which is based on computing
Examples
The stability criteria N1 and S1 and the criterion (34) featuring are illustrated using three loading conditions that commonly arise in engineered systems. All of these examples represent special cases of the general loading condition illustrated in Fig. 1. In the first example, and so we can appeal to the necessary condition N1 to conclude instability and the sufficiency condition S1 to draw conclusions about stability. However, for the remaining examples, and so we are
Concluding remarks
Adhesion and buckling instabilities govern elastic deformation and contact between an elastica and a rigid halfspace. We have simultaneously addressed both instability modes with a comprehensive analysis that uniquely combines stationary principles and the calculus of variations with classical transformations by Legendre and Jacobi. In particular, we showed that stability requires the existence of solutions to a Riccati equation which satisfy an inequality. This inequality is intimately related
Acknowledgments
The work of OMOR was partially supported by the National Science Foundation of the United States under Grant no. CMMI 0726675. He is also grateful to Daniel M. Peters for several helpful discussions on stability criteria. The authors are grateful to Professor George G. Adams (Northeastern University) and the anonymous reviewers for their helpful comments on earlier drafts of the paper.
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