Bio-inspired mechanics of reversible adhesion: Orientation-dependent adhesion strength for non-slipping adhesive contact with transversely isotropic elastic materials

Geckos and many insects have evolved elastically anisotropic adhesive tissues with hierarchical structures that allow these animals not only to adhere robustly to rough surfaces but also to detach easily upon movement. In order to improve our understanding of the role of elastic anisotropy in reversible adhesion, here we extend the classical JKR model of adhesive contact mechanics to anisotropic materials. In particular, we consider the plane strain problem of a rigid cylinder in non-slipping adhesive contact with a transversely isotropic elastic half space with the axis of symmetry oriented at an angle inclined to the surface. The cylinder is then subjected to an arbitrarily oriented pulling force. The critical force and contact width at pull-off are calculated as a function of the pulling angle. The analysis shows that elastic anisotropy leads to an orientation-dependent adhesion strength which can vary strongly with the direction of pulling. This study may suggest possible mechanisms by which reversible adhesion devices can be designed for engineering applications.     

Orientation-dependent adhesion strength of a rigid cylinder in non-slipping contact with a transversely isotropic half-space

Recently, Chen and Gao [Chen, S., Gao, H., 2007. Bio-inspired mechanics of reversible adhesion: orientation-dependent adhesion strength for non-slipping adhesive contact with transversely isotropic elastic materials. J. Mech. Phys. solids 55, 1001–1015] studied the problem of a rigid cylinder in non-slipping adhesive contact with a transversely isotropic solid subjected to an inclined pulling force. An implicit assumption made in their study was that the contact region remains symmetric with respect to the center of the cylinder. This assumption is, however, not self-consistent because the resulting energy release rates at two contact edges, which are supposed to be identical, actually differ from each other. Here we revisit the original problem of Chen and Gao and derive the correct solution by removing this problematic assumption. The corrected solution provides a proper insight into the concept of orientation-dependent adhesion strength in anisotropic elastic solids.     

Non-slipping adhesive contact between mismatched elastic spheres: A model of adhesion mediated deformation sensor

A generalized JKR model is established for non-slipping adhesive contact between two dissimilar elastic spheres subjected to a pair of pulling forces and a mismatch strain. We discuss the full elastic solution to the problem as well as the so-called non-oscillatory solution in which tension and shear tractions along the contact interface is decoupled from each other. The model indicates that the mismatch strain has significant effect on the contact area and the pull-off process. Under a finite pulling force, a pair of adhering spheres is predicted to break apart spontaneously at a critical mismatch strain. This study suggests an adhesion mediated deformation sensing mechanism by which cells and molecules can detect mechanical signals in the environment via adhesive interactions.     

Non-slipping adhesive contact between mismatched elastic cylinders

We have recently proposed a generalized JKR model for non-slipping adhesive contact between two elastic spheres subjected to a pair of pulling forces and a mismatch strain (Chen, S., Gao, H., 2006c. Non-slipping adhesive contact between mismatched elastic spheres: a model of adhesion mediated deformation sensor. J. Mech. Phys. Solids 54, 1548–1567). Here we extend this model to adhesion between two mismatched elastic cylinders. The attention is focused on how the mismatch strain affects the contact area and the pull-off force. It is found that there exists a critical mismatch strain at which the contact spontaneously dissociates. The analysis suggests possible mechanisms by which mechanical deformation can affect binding between cells and molecules in biology.     

Mechanics of adhesive contact at the nanoscale: The effect of surface stress

At small length scales, the adhesion and surface effect are of great significance, both of which play important roles in the contact between two elastic solids. In this study, the classical Johnson–Kendall–Roberts (JKR) adhesive contact theory is generalized to the nanoscale at which the surface effect is considered. The influence of the surface stress on the JKR adhesive contact is investigated by employing the non-classical Boussinesq fundamental solutions. It is found that, compared with the classical theory, the pull-off force increases while the critical contact radius decreases as a result of the surface effect. Numerical results show that a relative error of 10% can be introduced in the pull-off force when the indenter radius is less than 20 nm. A detailed theoretical analysis of this interesting phenomenon is presented based on dimensional analysis, and two scaling laws for the adhesive contact at the nanoscale are constructed. These two new scaling laws reveal that the pull-off force is relevant to the elastic properties of the bulk materials, which is different from the classical adhesive contact theory. The present work is promising for the engineering applications in micro-electro-mechanical systems (MEMS) and nano-intelligent devices.     

Inclined circular flat punch on a transversely isotropic piezoelectric half-space

Summary Utilizing the general solution of transversely isotropic piezoelectricity, the paper analyzes the problem of an inclined rigid circular flat punch indenting a transversely isotropic piezoelectric half-space. The potential theory method is employed and generalized to take into account the effect of the electric field in piezoelectric materials. Assuming that the punch is maintained at a constant electric potential, exact expressions for the elastoelectric field are derived in terms of elementary functions. It is noted that the solution corresponding to a flat circular punch centrally loaded by a concentrated force can be obtained as a special case. Received 15 December 1998; accepted for publication 9 March 1999     

Exact analysis of the orientation-adjusted adhesive full stick contact of layered structures with the asymmetric bipolar coordinates

The adhesion failure has become one dominant factor in determining the reliability and service life of miniaturized devices subject to loadings with arbitrary orientations. This article establishes an adhesive full stick contact model between an elastic half-space and a rigid cylinder loaded in any direction. Using the Papkovich-Neuber functions, the Fourier integral transform, and the asymmetric bipolar coordinates, the exact solution is obtained. Unlike the Johnson-Kendall-Roberts (JKR) model, the present adhesive contact model takes into account the effects of the load direction as well as the coupling of the normal and tangential contact stresses. Besides, it considers the full stick contact which has large values of the friction coefficient between contacting surfaces, contrary to the frictionless contact supposed in the JKR model. The result shows that suitable angles can be found, which makes the contact surfaces difficult to be peeled off or easy to be pressed into.

     

Mechanics of axisymmetric wavy surface adhesion: JKR–DMT transition solution

Recent work on the mechanics of detachment of a rigid sphere from an elastic axisymmetric wavy surface in the presence of JKR adhesion has shown that the presence of small-amplitude waviness introduces instabilities into the detachment process which dissipate mechanical energy. These instabilities result in interface toughening and strengthening; both the external work and peak force required for separation of a wavy interface are higher than those for a flat interface. In this paper, we summarize the key dimensionless parameters governing axisymmetric wavy surface adhesion in the JKR regime. We then proceed to derive a solution for the JKR–DMT adhesion transition for the axisymmetric wavy surface contact problem using a Maugis–Dugdale cohesive zone formulation. The phenomenon of interface toughening and strengthening due to the presence of surface waviness is seen to be restricted primarily to the JKR adhesion regime.     

DYNAMIC CONTACT STIFFNESS OF VIBRATING RIGID SPHERE CONTACTING SEMI-INFINITE TRANSVERSELY ISOTROPIC VISCOELASTIC SOLID

Dynamic contact stiffness at the interface between a vibrating rigid sphere and a semi-infinite transversely isotropic viscoelastic solid is investigated. An oscillating force superimposed onto a static compressive force in the vertical direction excites the vibration of a rigid sphere, which causes variable contact radius and contact pressure distribution in the contact region. The assumption of a sufficiently small oscillating force yields a dynamic contact-pressure distribution of a constant contact radius, which gives dynamic contact stiffness at the interface between the rigid sphere and the semi-infinite solid. Numerical calculations show the influence of vibration frequency of the sphere, and elastic constants of the transversely isotropic solid on dynamic contact stiffness, which benefits quantitative evaluation of elastic constants and orientation of single hexagonal grains by resonance-frequency shifts of the oscillator in resonance ultrasound microscopy.     

Detachment of a rigid solid from an elastic wavy surface: Theory

The mechanics of detachment between a wavy elastic half space and a rigid solid is considered. Solutions for the axisymmetric problem of a rigid sphere and the plane strain problem of a rigid cylinder detaching from a wavy surface are developed. The interacting solids are taken to be in complete contact over a finite area initially. It is shown that the surface waviness makes the detachment process unstable, with the interface separating in alternating stable and unstable segments. Each unstable segment dissipates mechanical energy, leading to an increase in the total work of separation compared to that of a flat surface. Further, waviness causes the maximum separation force or the pull-off force to increase during detachment, resulting in an apparent toughening of the interface. This mechanism provides an alternative explanation to the experimental observations in the literature that roughness can sometimes lead to increase in pull-off force. It also illustrates the role of roughness in the attachment capability of several insect feet possessing soft pads. The basic solution presented here can be used to analyze the detachment of surfaces with multiple scale roughness as well. The solution also suggests strategies to improve reversible adhesion of a soft material by designing optimal surface topographies.     

Complex variable formulation for non-slipping plane strain contact of two elastic solids in the presence of interface mismatch eigenstrain

In this paper, the problems of non-slipping contact, non-slipping adhesive contact, and non-slipping adhesive contact with a stretched substrate are sequentially studied under the plane strain theory. The main results are obtained as follows:(i) The explicit solutions for a kind of singular integrals frequently encountered in contact mechanics (and fracture mechanics) are derived, which enables a comprehensive analysis of non-slipping contacts. (ii) The non-slipping contact problems are formulated in terms of the Kolosov–Muskhelishvili complex potential formulae and their exact solutions are obtained in closed or explicit forms. The relative tangential displacement within a non-slipping contact is found in a compact form. (iii) The spatial derivative of this relative displacement will be referred to in this study as the interface mismatch eigenstrain. Taking into account the interface mismatch eigenstrain, a new non-slipping adhesive contact model is proposed and its solution is obtained. It is shown that the pull-off force and the half-width of the non-slipping adhesive contact are smaller than the corresponding solutions of the JKR model (Johnson et al., 1971). The maximum difference can reach 9% for pull-off force and 17% for pull-off width, respectively. In contrast, the new model may be more accurate in modeling the non-slipping adhesion. (iv) The non-slipping adhesions with a stretch strain (S-strain) imposed to one of contact counterparts are re-examined and the analytical solutions are obtained. The accurate analysis shows that under small values of the S-strain both the natural adhesive contact half-width and the pull-off force may be augmented, but for the larger S-strain values they are always reduced. It is also found that Dundurs’ parameter β may exert a considerable effect on the solution of the pull-off problem under the S-strain.These solutions may be used to study contacts at macro-, micro-, and nano-scales.     

Exact Solutions for Linearly Electroelastic Plate-Like Bodies

A three-dimensional, explicit and exact solution is derived for a transversely isotropic, linearly electroelastic body in the form of a right cylinder of arbitrary cross section, being simply supported and connected to ground over its lateral boundary, and subject to an arbitrary distribution of force and charge over its end faces. When electric phenomena are ignored, this solution reduces to the solution given in [9] for linearly elastic plate-like bodies.     

A generalized JKR-model for two-dimensional adhesive contact of transversely isotropic piezoelectric half-space

The aim of the present paper is to investigate the adhesive behavior between a transversely isotropic piezoelectric half-space and a cylinder punch subjected to combined mechanical and electric loads under plane-strain condition. The effect of adhesion is described by using a generalized JKR-model which can account for the non-slip condition in the contact regions. Analytical function theory is employed to find the solution of the resulting singular integral equations. Our analysis shows that the adhesive contact behavior for different types of piezoelectric materials may be quite different. The results obtained in this paper may be helpful to understand the contact mechanics of piezoelectric materials at micro-scale.     

A circular elastic cylinder under its own weight

An exact analysis of deformation and stress field in a finite circular elastic cylinder under its own weight is presented, with emphasis on the end effect. The problem is formulated on the basis of the state space formalism for axisymmetric deformation of a transversely isotropic body. Upon delineating the Hamiltonian characteristics of the formulation, a rigorous solution which satisfies the end conditions is determined by using eigenfunction expansion. The results show that the end effect is significant but confined to a local region near the base where the displacement and stress distributions are remarkably different from those according to the simplified solution that gives a uniaxial stress state. It is more pronounced in the cylinder with the bottom plane being perfectly bonded than in smooth contact with a rigid base.     

Exact Lévy-Type Solutions for Plate Bending Exist for Transversely Isotropic but Not for General Monoclinic Materials

An exact three-dimensional Lévy-type solution for the bending of an elastic slab is one in which there are edge loads only and the unknown displacement and stresses have very simple polynomial dependence on the thickness coordinate. Lévy obtained such a solution in 1877 for a linearly elastic, isotropic, plate-like body. The most general material that allows bending and stretching to be uncoupled is monoclinic (13 elastic constants). However, it is shown that only transversely isotropic materials (5 elastic constants) admit exact solutions having polynomial dependence in the thickness direction. Such solutions are listed explicitly.     

轴对称横观各向同性热弹性圆柱的分解

为了得到精确的应力场、位移场、温度场,将扭转圆轴的精化理论研究方法推广到轴对称横观各向同性热弹性圆柱。利用Bessel函数以及轴对称横观各向同性热弹性圆柱的通解,给出了轴对称横观各向同性热弹性圆柱的分解定理。根据柱面齐次边界条件获得了精确的精化方程,精化方程可以分解为一阶方程、超越方程、温度方程,从而将横观各向同性热弹性圆柱的轴对称问题分解为轴向拉压问题、超越问题、热-应力耦合问题。超越部分对应端部自平衡情况,可以清晰地了解到端部应力分布对内部应力场的影响,热-应力耦合部分对应无外加应力场时圆柱内部因温度变化引起的热应力。     

Non-slipping adhesive contact of a rigid cylinder on an elastic power-law graded half-space

In this paper, adhesive contact of a rigid cylinder on an elastic power-law graded half-space is studied analytically with the theory of weakly singular integral equation and orthogonal polynomial method. Emphasis is placed on the coupling effect between tangential and normal directions which was often neglected in previous works. Our analysis shows that the coupling effect tends to reduce the contact area in the compressive regime. The effect of bending moment on the adhesion behavior is also examined. Like a pull-off force, there also exists a critical bending moment at which the cylinder can be bended apart from the substrate. However, unlike pull-off force, the critical bending moment is insensitive to the gradient exponent of the graded material.     

Mechanics of adhesive contact on a power-law graded elastic half-space

We consider adhesive contact between a rigid sphere of radius R and a graded elastic half-space with Young's modulus varying with depth according to a power law E=E₀(z/c₀)^k (0<k<1) while Poisson's ratio ν remaining a constant. Closed-form analytical solutions are established for the critical force, the critical radius of contact area and the critical interfacial stress at pull-off. We highlight that the pull-off force has a simple solution of P_cr=−(k+3)πRΔγ/2 where Δγ is the work of adhesion and make further discussions with respect to three interesting limits: the classical JKR solution when k=0, the Gibson solid when k→1 and ν=0.5, and the strength limit in which the interfacial stress reaches the theoretical strength of adhesion at pull-off.     

Stress Singularity in an Elastic Cylinder of Cylindrically Anisotropic Materials

The issue of stress singularity in an elastic cylinder of cylindrically anisotropic materials is examined in the context of generalized plane strain and generalized torsion. With a viewpoint that the singularity may be attributed to a conflicting definition of anisotropy at r=0, we study the problem through a compound cylinder in which the outer cylinder is cylindrically anisotropic and the core is transversely isotropic. By letting the radius of the core go to zero, the cylinder becomes one with the central axis showing no conflict in the radial and tangential directions. Closed-form solutions are derived for the cylinder under pressure, extension, torsion, rotation and a uniform temperature change. It is found that the stress is bounded everywhere, and singularity does not occur if the anisotropy at r=0 is defined appropriately. This revised version was published online in July 2006 with corrections to the Cover Date.