## Abstract

The plane strain problem of an elastic cylinder in adhesive contact with a stretched substrate is studied via a generalized JKR model taking into account the transmission of both tangential and normal tractions across the contact interface. The width of the contact region is determined from the Griffith energy balance near the contact edge. In the absence of external loading, the tangential traction is found to have a negligible effect on the contact size. As an external stress is applied to stretch the substrate, the contact solution exhibits three distinct regimes characterized by two threshold strains: (i) the size of the contact region is hardly affected by the applied loading when the substrate strain is below the first threshold level; (ii) the contact size decreases quickly with stretch as the substrate strain increases to between the two threshold levels; (iii) the contact size approaches zero when the substrate strain exceeds the second threshold level. Interestingly, these results share a number of common features with the experimentally observed cell reorientation on a cyclically stretched substrate. An approximate solution is presented in an appendix to represent the numerical results in closed form.

## 1. Introduction

Studies of adhesive contact between elastic bodies have received significant attention during the past decade (Muller *et al*. 1980; Carpick *et al*. 1996; Baney & Hui 1997; Greenwood 1997; Johnson & Greenwood 1997; Barthel 1998; Greenwood & Johnson 1998; Kim *et al*. 1998; Robbe-Valloire & Barquins 1998; Shull 2002; Morrow *et al*. 2003; Schwarz 2003). A general conclusion is that adhesion between solid surfaces tends to increase the size of the contact region above the prediction of the classical Hertz theory. The most widely known models of adhesive contact were developed by Johnson *et al*. (1971), Derjaguin *et al*. (1975) and Maugis (1992). In the Johnson–Kendall–Roberts (JKR) model, an equilibrium contact area is established via Griffith energy balance between elastic energy and surface energy, which results in compressive stress in the central region of contact and crack-like singular tensile stress near the edge of contact; the contact area remains finite until a critical pull-off force is reached. In the Derjaguin–Muller–Toporov (DMT) model, molecular forces outside the Hertz contact area are considered, but these forces are assumed not to change the contact profile of the Hertz solution. Maugis (1992) developed a unified model linking the JKR and DMT models by extending the Dugdale model (Dugdale 1960) of a plastic crack to the case of adhesive contact between two elastic spheres.

The theories of adhesive contact have also been used to understand biological adhesion mechanisms such as the hierarchical adhesion structures on the foot of Gecko (e.g. Gao & Yao 2004; Glassmaker *et al*. 2004; Hui *et al*. 2004; Gao *et al*. 2005*b*). The JKR model has also been applied to cell adhesion (Chu *et al*. 2005). Adhesion among different types of cells and between cells and substrates is of interest to many fields of biology including embryonic development, cancer metastasis, cellular transport, endo- and exocytosis, tissue and cellular engineering (Alberts *et al*. 1994; Bao & Suresh 2003; Gao *et al*. 2005*a*). Mechanical signals are found to play an important role in cell adhesion. For example, experiments over the last two decades (Buck 1980; Dartsch & Hammerle 1986; Shirinsky *et al*. 1989; Kanda & Matsuda 1993; Neidlinger-Wilke *et al*. 1994; Wang *et al*. 1995; Kemkemer *et al*. 1999; Wang 2000; Neidlinger-Wilke *et al*. 2001; Moretti *et al*. 2004) have shown that cells cultured on a cyclically stretched substrate tend to reorient themselves away from the stretch direction. An important fact first noted by Dartsch & Hammerle (1986) is that cells do not respond to small stretch amplitudes (less than 2%), suggesting that there exists a threshold stretch amplitude to initiate cell reorientation. Above this threshold, an increasing number of cells begin to respond to substrate deformation by reorienting themselves away from the stretch direction. The larger the stretch amplitude, the more cells reorient. Neidlinger-Wilke *et al*. (1994) reported that almost all cells joined the reorientation process once the stretch amplitude exceeds a second threshold level around 5–6%. These experimental observations indicate that the response of cells to substrate stretch can be divided into three regimes with two threshold stretch amplitudes: (i) the cells show no significant response to substrate deformation when the stretch amplitude lies below the first threshold level; (ii) the cells begin to actively reorient themselves away from the stretch direction when the substrate is stretched to between the two threshold amplitudes; (iii) nearly all cells join the reorientation process when the stretch amplitude exceeds the second threshold level. Wang *et al*. (1995) and Wang (2000) have shown that, despite the complex underlying biological responses, the final aligning angle of cells can be calculated based on the principle of minimum strain energy. Here, we show that models based on contact mechanics may also be useful for understanding the behaviours of cells on stretched substrates.

Mechanical properties of cells are highly viscoelastic due to the organization of actin cytoskeleton (Howard 2001). There is a growing body of literature (Maugis & Barquins 1978; Hui *et al*. 1998; Lin *et al*. 1999; Barthel & Haiat 2002; Haiat *et al*. 2003) on viscoelastic Hertzian and adhesive contact problems. In general, contact between viscoelastic bodies is more complicated than the corresponding elastic problems. Viscoelastic contact solutions are often history dependent, in which case there is no unique relation between the contact size and the applied load. In some cases, cells may respond to mechanical forces in a strongly linear manner (Yang & Saif 2005), although nonlinear responses are generally expected. To avoid excessive complications, in this first study, we will limit our attention to the linear elastic problem of a circular cylinder in adhesive contact with a stretched substrate. While this model is far from being realistic with respect to the viscoelastic properties of cells, it provides a limiting solution of a viscoelastic body in response to cyclical stretching at a sufficiently high frequency. More sophisticated modelling of the mechanical behaviours of cells with more realistic constitutive laws (viscoelastic, nonlinear, etc.) and geometrical configurations (more realistic cell shapes and substrate geometries) will be left to future work.

In most of the existing models on contact mechanics, tangential tractions inside the contact region are either neglected or assumed to be independent of the normal traction (Johnson 1985, 1997). Kendall (1975) investigated the effects of shrinkage stress on a brittle interfacial failure of a bonded laminate. Experiments (Savkoor & Briggs 1977) showed that an applied tangential force can reduce the area of contact between elastic solids. To model cells adhering to substrates via focal adhesion (Burridge *et al*. 1988), we develop generalized JKR models in which the contact interface is treated as a well-bonded region with no slippage. In this case, the stress field near the contact edge has an oscillatory feature similar to that near an interfacial crack between two dissimilar materials.

A number of two-dimensional plane strain problems of adhesive contact involving elastic cylinders have been studied in the past. Roberts & Thomas (1975) conducted experiments on glass cylinders contacting rubber slabs. Barquins (1988) derived solutions of contacting cylinders using the method of Greenwood & Johnson (1981). Chaudhury *et al*. (1996) studied a cylinder in contact with a rubber, with the edge of contact modelled as a mode I crack. In contrast to these previous studies, our work is focused on cases in which shear traction becomes so important that interfacial fracture mechanics must be used to provide a more accurate description of the stress field near the contact edge.

## 2. Model

The plane strain problem under consideration is shown in figure 1, where an elastic cylinder adheres to a semi-infinite elastic substrate via intermolecular forces between the solid surfaces. Following the JKR model (Johnson *et al*. 1971), the equilibrium contact area can be determined from the Griffith energy balance between the elastic energy and surface energy. A uniaxial strain is then imposed to stretch the substrate to a given strain level. During the stretch of the substrate, the contact region is assumed to be perfectly bonded except that the edge of contact shifts according to the changing balance between elastic energy and surface energy. As in almost all contact mechanics theories (Johnson 1985), the contact width is assumed to be small compared to the radius of the cylinder such that the deformation of the cylinder can be approximated by that of an elastic half-space. The objective is to calculate the equilibrium width of the contact area as a function of the substrate strain.

A Cartesian reference coordinate system (*x*,*y*) is placed at the centre of the contact region with half-width *a* (figure 1). The normal and tangential tractions along the surface of the cylinder inside the contact area will be denoted as *P*(*x*) and *Q*(*x*), respectively. The edges of the contact region resemble two opposing interfacial cracks under plane strain deformation. It is known (Erdogan 1965; Rice 1965; Westmann 1965) that the stress field near the tip of an interfacial crack is oscillatory with energy release rate(2.1)where *κ* is an oscillation index to be defined in equation (3.26), *K* is a complex-valued stress intensity factor in equation (3.32) and *E** is an effective Young's modulus in equation (3.5). The Griffith energy balance between elastic and surface energies can be written as(2.2)where Δ*γ* is the work of adhesion; *γ*_{1}, *γ*_{2} are the intrinsic surface energies of the two solids and *γ*_{12} is the surface energy of the contact interface.

## 3. General solution

According to the model, the continuity of displacements across the contact interface can be expressed as(3.1)where denotes the displacement in the *x* (*y*) direction of material *i* (*i*=1, 2) along the interface, *R* is the radius of the cylinder and *δ* is the displacement of the centre of the cylinder during with contact formation.

Differentiating equation (3.1) with respect to *x* yields(3.2)The surface displacements of an elastic half-space can be related to the surface tractions *Q*(*x*) and *P*(*x*) via Green's functions of an elastic half-space. When this is done, equation (3.2) becomes(3.3)where *E*_{1}, *ν*_{1}, *E*_{2}, *ν*_{2} denote Young's modulus and Poisson's ratio of the two solids, respectively, as shown in figure 1.

Equation (3.3) can be further simplified as(3.4)where *E** is an effective modulus defined by(3.5)and *β* is Dundurs' parameter (Dundurs 1969),(3.6)It is convenient, following a similar analysis of interfacial crack problem by Yu & Yang (1995), to rewrite equation (3.4) in a matrix form:(3.7)where(3.8)To solve equation (3.7), we define(3.9)where and . Equation (3.9) leads to the following relations:(3.10)and(3.11)where superscripts ‘+’ and ‘−’ stand for the limits of *F*_{k}(*z*) as *y*→0^{+} and *y*→0^{−}, respectively. With these relations, equation (3.7) can be written as(3.12)where(3.13)Before solving equation (3.12), we first consider the eigenvalue problem(3.14)Inserting (3.8) and (3.13) into (3.14) yields two eigenvalues(3.15)and two normalized eigenvectors(3.16)where(3.17)Introducing the following auxiliary functions,(3.18)and multiplying equation (3.12) by leads to the following decoupled equation:(3.19)where(3.20)The decoupled matrix equation (3.19) consists of two inhomogeneous Hilbert equations,(3.21)The solutions to the above can be obtained following the standard procedure (Carrier *et al*. 1983):(3.22)(3.23)(3.24)(3.25)where(3.26)*κ* is an oscillation index and *r* is the stress singularity near the contact edge. *c*_{1} and *c*_{2} are two constants to be determined by the loading boundary conditions.

According to the first equation of (3.8), (3.10) and (3.18), the tangential and normal tractions of the cylinder inside the contact region (|*x*|≤*a*) can be obtained from the following relations:(3.27)Substituting (3.22)–(3.25) into (3.27) yields the general solution(3.28)where(3.29)The condition that both *P*(*x*) and *Q*(*x*) should be real requires .

With no applied load on the cylinder, the equilibrium condition yields(3.30)In fact, one can easily show that both *c*_{1} and *c*_{2} are real constants. Numerical calculations suggest that *c*_{1} and *c*_{2} are nearly zero. While we have not been able to devise a rigorous proof that they are indeed zero, we can show that they are indeed zero to at least third order in Dundurs' constant *β*, which can be treated as a small parameter in its admissible range −1/4<*β*<1/4. Therefore,(3.31)We introduced a complex-valued stress intensity factor near *x*=*a* as(3.32)which by equation (3.31) has the solution(3.33)The equilibrium width 2*a* of the contact area is to be determined from the Griffith energy balance. Combining equations (2.1), (2.2) and (3.33) yields an implicit equation,(3.34)which relates the contact size *a* to the substrate strain *ϵ*. Note that Δ*γ* is the work of adhesion defined in equation (2.2).

## 4. The homogeneous case

The case when the cylinder and substrate have identical elastic properties,(4.1)is of special interest due to its simplicity. In this case, the stress field near the contact edge has the conventional square root singularity with(4.2)The shear and normal stress distributions inside the contact region can be obtained from equation (3.28) as(4.3)(4.4)The stress intensity factors near the contact edge are(4.5)

(4.6)The mode I solution of equation (4.6) is consistent with that given in Chaudhury *et al*. (1996) and the mode II solution of equation (4.5) is consistent with a solution for a similar external crack problem given in Tada *et al*. (2000). Inserting equations (4.5) and (4.6) into the Griffith energy balance(4.7)yields the following equation:(4.8)to determine *a* as a function of *ϵ*. When *ϵ*=0, the solution is(4.9)Using *a*_{0} to normalize equation (4.8) leads to(4.10)where(4.11)The cubic equation (4.10) has an explicit solution,(4.12)For *ϵ*→0, the solution behaves as(4.13)and for *ϵ*→∞, the contact half-width asymptotically approaches zero as(4.14)

## 5. Discussion of the general solution

### (a) The two-dimensional JKR model

In the special case when tangential tractions inside the contact region is neglected and *ϵ*=0, our model reduces to the two-dimensional JKR model (Barquins 1988; Chaudhury *et al*. 1996). In this case, the normal stress distribution along the interface is determined from(5.1)In the absence of an applied load, the solution to (5.1) is(5.2)which exhibits the usual crack-like singularity near the contact edge *x*=±*a*; *E** is the effective modulus defined by equation (3.5). Inserting the stress intensity factor(5.3)into the Griffith energy balance *G*=Δ*γ* yields the contact half-width of the two-dimensional JKR model,(5.4)

### (b) Generalized two-dimensional JKR model with no interfacial slippage

We have assumed on a generalized JKR model such that slippage is not allowed at the contact interface. In the absence of substrate stretch (*ϵ*=0), the general solution discussed in §3 indicates that the normal and shear tractions inside the contact region are given by(5.5)where(5.6)*r* being given by (3.26). Inserting the corresponding stress intensity factor(5.7)into Griffith energy balance in (3.34) yields the contact half-width(5.8)which can be compared to the two-dimensional JKR solution in (5.4). In particular, the ratio(5.9)only depends on Dundurs' parameter *β*. Numerical evaluation of (5.9), as plotted in figure 2, indicates that the ratio *a*_{0}/*a*_{JRK} is close to 1 for the entire admissible range of Dundurs' parameter −1/4<*β*<1/4, with maximum difference reaching only about 2%. This result indicates that, in the absence of external loading, the coupling between normal and shear tractions in the contact region is practically negligible. For all practical purposes, we shall take(5.10)

### (c) Effect of substrate stretch on contact width

In the case of substantial substrate stretch, the coupling between the shear and normal tractions in the contact area becomes increasingly important. In this case, the contact half-width *a* is related to the substrate strain *ϵ* according to equation (3.34) which is numerically solved and plotted in figure 3. Similar to the solution (4.10) and (4.11) in the homogeneous case, we normalize the contact half-width *a* by *a*_{0} of (5.8) and introduce a non-dimensional parameter,(5.11)

Figure 3*a* plots the relationship between the normalized contact half-width *a*/*a*_{0} and the substrate stretch *ϵ* for different values of *λ*. The result indicates that the behaviour of *a*/*a*_{0} can be characterized by three distinct regimes with two threshold strain levels: (i) the contact width is hardly influenced by the applied loading when the substrate strain is below the first threshold; (ii) as the substrate strain increases to between the two threshold values, the contact width begins to decrease significantly in response to the applied loading; (iii) the adhesion fails with almost no contact possible when the substrate strain exceeds the second threshold.

The homogeneous solution in equation (4.10) has suggested that the normalized contact half-width *a*/*a*_{0} depends on the substrate strain *ϵ* only through the parameter combination *λϵ*. Interestingly, this is found to be true also for the general bi-material case. Figure 3*b* plots the relation between *a*/*a*_{0} and *λϵ* for bi-materials along with the closed-form solution (4.12) for the homogeneous case. The results show that equation (4.12), which is strictly valid only in the homogeneous case, is found to match the numerical solution for bi-materials very well. In other words, it seems that the properties of the bi-material may strongly affect the parameters *a*_{0} and *λ*, but they do not alter the relationship between *a*/*a*_{0} and *λϵ*.

Figure 4 shows three contour plots of the relationship between the non-dimensional parameter *λ* and the substrate strain *ϵ* for three different ratios of *a*/*a*_{0}=0.1, 0.5 and 0.9, corresponding to *λϵ*=1.56, 0.65, 0.27, respectively. These plots show the regime in which the contact width becomes sensitive to substrate stretch. For a broad range of *λ*=*R*/*a*_{0}, the sensitive regime of substrate strain is on the order of a few percent.

Although the above results are derived for the adhesive contact of an elastic cylinder with a stretched substrate, the behaviours of the contact area show several features which appear to be qualitatively similar to that of cells cultured on a cyclically stretched substrate. Interestingly, experiments on cell reorientation in response to cyclic substrate stretch also show three characteristic regimes with two threshold stretch amplitudes (Dartsch & Hammerle 1986; Neidlinger-Wilke *et al*. 1994; Wang *et al*. 1995; Wang 2000). It was found that cells do not respond to stretch amplitudes smaller than 1–2% (Dartsch & Hammerle 1986). Once this first threshold is reached, an increasing number of cells begin to response to substrate stretch by reorientating themselves away from the stretch direction. The cell reorientation leads to decreased contact width in the direction of stretch and increased contact width in the transverse direction. As the stretch amplitude increases beyond a second threshold level around 5–6%, almost all cells reorient away from the stretch direction (Neidlinger-Wilke *et al*. 1994). These features and the associated strain levels appear to be in good agreement with our analysis. For the third regime in our analysis, the contact radius becomes quite small and approaching zero with increasing stretching. However, the contact area does not vanish and does not imply full delamination of cells in this regime. An interpretation of this result is that for large stretching strains cells must fully reorient to minimize the contact area.

Why does an elastic contact mechanics model produce results in qualitative agreement with the mechanical behaviour of cells which is expected to be strongly viscoelastic? A possible explanation is that the elastic model provides a limiting solution of a viscoelastic body in response to cyclical load at a sufficiently high frequency. In addition, the mechanical response of cells to mechanical forces may be strongly linear (Yang & Saif 2005). The present study shows that it is indeed promising to use mechanics models to help explain biological behaviours in response to mechanical forces. More sophisticated model with more realistic mechanical properties and geometrical configurations will be left to future work.

## Acknowledgments

Support of this work has been provided by the Max Planck Society, NSFC (nos. 10202023, 10272103) and Key Project from the CAS (grant no. KJCX2-SW-L2). The authors would like to express their gratitude to Dr Ralf Kemkemer of the Max Planck Institute for Metals Research for reading the manuscript and for providing helpful references on experiments of cells on cyclically stretched substrates.

## Appendix A Closed-form approximate solution for adhesive contact between a two-dimensional elastic cylinder and a stretched substrate

As discussed in §5*c*, the general solution for adhesive contact between a two-dimensional elastic cylinder and a stretched substrate can be approximated by the following simple closed-form solution which can be used for practical purposes,(A1)where(A2)

(A3)

(A4)

- Received October 30, 2004.
- Accepted July 14, 2005.

- © 2005 The Royal Society