## Abstract

Fibrillar structures are common features on the feet of many animals, such as geckos, spiders and flies. Theoretical analyses often use periodical array to simulate the assembly, and each fibril is assumed to be of equal load sharing (ELS). On the other hand, studies on a single fibril show that the adhesive interface is flaw insensitive when the size of the fibril is not larger than a critical one. In this paper, the Dugdale–Barenblatt model has been used to study the conditions of ELS and how to enhance adhesion by tuning the geometrical parameters in fibrillar structures. Different configurations in an array of fibres are considered, such as line array, square and hexagonal patterns. It is found that in order to satisfy flaw-insensitivity and ELS conditions, the number of fibrils and the pull-off force of the fibrillar interface depend significantly on the fibre separation, the interface interacting energy, the effective range of cohesive interaction and the radius of fibrils. Proper tuning of the geometrical parameters will enhance the pull-off force of the fibrillar structures. This study may suggest possible methods to design strong adhesion devices for engineering applications.

## 1. Introduction

The attachment systems in jumping spiders and geckos are believed to rely entirely on van der Waals forces between the finely structured feet and substrates (Autumn *et al.* 2000, 2002; Russell 2002; Kesel *et al.* 2003). Of particular significance are the many fibrils on the feet of these animals. Fibrillar structure is an important feature of these adhesion systems.

Owing to the reliance on van der Waals forces, the ability of geckos to adhere to substrates is primarily controlled by mechanics rather than surface chemistry. Hence, the geometry and material properties of the structure must play a pivotal role in enhancing the adhesion (Gorb & Scherge 2000; Niederegger *et al.* 2002; Geim *et al.* 2003; Gao *et al.* 2005; Huber *et al.* 2005; Spolenak *et al.* 2005; Yao & Gao 2006; Chen & Gao 2007). Various mechanical models have been developed to model specific hairy attachment systems (e.g. Hui *et al.* 2002; Persson 2003; Glassmaker *et al.* 2004; Hui *et al.* 2004) within the framework of adhesive contact mechanics (e.g. Johnson *et al.* 1971; Derjaguin *et al.* 1975; Roberts & Thomas 1975; Muller *et al.* 1980; Greenwood & Johnson 1981; Barquins 1988; Maugis 1992; Carpick *et al.* 1996; Chaudhury *et al.* 1996; Baney & Hui 1997; Johnson & Greenwood 1997; Barthel 1998; Greenwood & Johnson 1998; Kim *et al.* 1998; Robbe-Valloire & Barquins 1998; Morrow *et al.* 2003; Schwarz 2003; Chen & Wang 2006). In particular, the Johnson–Kendall–Roberts (JKR) model (Johnson *et al.* 1971) of contact mechanics has been used to show that splitting of a single contact into multiple smaller contacts always results in enhanced adhesion strength (Autumn *et al.* 2002; Arzt *et al.* 2003), and developed to explain the other adhesion problems in biology (Chen & Gao 2006*a*–,*c*). One of the puzzling predictions of the JKR-type model is that the spatula structure of geckos can be split ad infinitum to support arbitrarily large body weights. This is clearly impossible as the adhesion strength cannot exceed the theoretical strength of van der Waals interaction (Gao *et al.* 2005; Tang *et al.* 2005). In order to explain why the characteristic size of the fibrillar ultrastructure of bioattachment systems falls in a narrow range between a few hundred nanometres and a few micrometres and the optimal conditions under which the theoretical pull-off force can be achieved, Gao & Yao (2004) found that the robust design of shape-insensitive optimal adhesion becomes possible only when the diameter of the fibre is reduced to length scales of the order of 100 nm and the optimal adhesion could be achieved by a combination of size reduction and shape optimization. The smaller the size, the less important the shape. In order to simulate the fibrillar structure, Glassmaker *et al.* (2004) and Hui *et al.* (2004) studied the concept of equal load sharing (ELS) for a perfect interface containing many fibrils and presented results on how a fibrillar structure enhances adhesion.

In real life, most surfaces are actually rough, and surface roughness has been considered in many contact literatures. Meine *et al.* (2004) did experiments to find a correlation between the contact area and adhesion of rough surfaces, and the roughness was simulated by a structure. A simple method of using the JKR model to determine the interfacial adhesion between two ideal rough surfaces was demonstrated by Hodges *et al.* (2004) for individual asperity–asperity and asperity–flat contacts. ELS was also adopted by Hodges *et al.* (2004). An analytical model based on the JKR theory of adhesion was used by Hui *et al.* (2001) to study the contact mechanics and adhesion of a periodically rough surface.

In order to be convenient in studying the adhesion force or adhesion strength, most people have assumed that the fibrillar structure or the asperity of a rough surface is periodical and the force is homogeneously distributed on all the asperities or fibres, i.e.(1.1)where *P* is the pull-off force; *N* is the number of the asperities or fibres; and *F* is the pull-off force for single asperity or fibril contact.

Whether the pull-off force on each asperity or fibre attains the same value homogeneously and simultaneously (flaw insensitivity) was not investigated in the mentioned literatures. Are there any restrictions on the geometrical parameters to attain flaw insensitivity and satisfy ELS in fibrillar structures? The concept of flaw insensitivity has been adopted to understand the mechanics of biological systems (Gao *et al.* 2003; Persson 2003; Gao & Yao 2004; Glassmaker *et al.* 2004; Hui *et al.* 2004; Gao & Chen 2005; Gao 2006). In the state of flaw insensitivity, pre-existing crack-like flaw does not propagate or participate in the failure process so that the biological material failure occurs by uniform rupture at the limiting strength of the material. Examples are as follows.

The spatula in a gecko's attachment system was modelled by Gao *et al.* (2005) as an elastic cylinder with a flat tip in adhesive contact with a rigid substrate. The radius of the cylinder is *R* and the actual contact radius is , where . The rim of represents flaws or regions of poor adhesion. Gao *et al.* (2005) used the Griffith criterion and obtained the critical size of the radius of the cylinder as(1.2)at which the traction within the contact area uniformly reaches the theoretical strength *σ*_{th} and the pull-off force becomes . In the above equation, , where is a function varying in a narrow range between 0.4 and 0.5 for ; , where *E* and *ν* are the Young modulus and the Poisson ratio of the cylinder, respectively; and Δ*γ* is the van der Waals interaction energy.

If we let *α*=1, the critical radius of the cylinder becomes(1.3)which is identical to the result obtained by Persson (2003), where a rigid circular disc with radius *R* on a flat substrate with Young's modulus *E* and Poisson's ratio *ν* has been studied. Persson (2003) assumed that complete contact occurs at the interface and found that if , then the bond breaking may occur uniformly over the contact area and the pull-off stress is then independent of the size *R* and reaches the theoretical interface strength *σ*_{th}.

In this paper, the concept of ELS for a perfect interface containing many fibrils will be studied. All fibrils have radius not larger than the critical size of flaw insensitivity of a single fibre. The plan of this paper is as follows. We first study the problem of two-dimensional fibrillar structure, and then three-dimensional one. In the two-dimensional case, we first discuss the critical size of the fibril, the pull-off force and the pull-off stress required to pull off a single fibril under the condition of flaw insensitivity, and then use a plane-strain fibrillar interface model to study the restrictions on the number of fibrils and how to enhance the pull-off force. In the three-dimensional models, the corresponding aspects considered in the two-dimensional models will be investigated for different fibrillar distributing patterns, such as line array, square and hexagonal lattices.

## 2. Plane-strain models

Adhesive contact between elastic objects usually fails by the propagation of crack-like flaws initiated at poor contact regions around surface asperities, impurities, trapped contaminants, etc. Under this circumstance, the adhesion strength is not optimal because it is only a small fraction of material theoretical strength. From the robustness point of view, it would be best to design a material that allows the contact to fail not by crack propagation but by uniform detachment at the theoretical strength of adhesion, which is called flaw tolerance (Gao *et al.* 2003, 2005; Gao & Chen 2005) or flaw insensitivity (Glassmaker *et al.* 2004; Hui *et al.* 2004).

Inspired by the concept of flaw insensitivity, in this paper, we will assume that the fibrils with size not larger than the critical one are under ELS and the interface strength between each fibril and the substrate attains the theoretical strength. Thus, using the boundary condition that the maximum separation at the interface should not be larger than the effective interaction distance of the interface, we can obtain the relations between the various geometrical parameters, the pull-off force and the pull-off stress for different fibrillar configurations.

### 2.1 Critical size of a single fibril

Figure 1 shows a plane-strain model, in which a rigid plate with width 2*h* contacts an elastic substrate with Young's modulus *E* and Poisson's ratio *ν*. Perfect adhesion is assumed as that in Persson (2003) and the contact half-width is *h*. The Dugdale model will be used to obtain the critical half-width *h*_{cr} required by the condition of flaw insensitivity. When *h*≤*h*_{cr}, the tractions on the interface will uniformly attain the interface theoretical strength *σ*_{th}. The pulling stress is denoted by *σ* on the upper boundary, as shown in figure 1.

From the solution of a two-dimensional elastic half-space pulled by a homogeneous traction *σ* within the length region 2*h*, the normal displacement can be expressed as (Johnson 1985)(2.1)where *C* is an arbitrary constant.

The cohesive law in the Dugdale–Barenblatt model (Dugdale 1960; Barenblatt 1985) can be expressed as(2.2)where is the normal traction on the adhesion interface; *δ*_{0} is the effective range of cohesive interaction; and *δ*(*x*) is the separation between the two surfaces of the contact interface. The interfacial energy is(2.3)Consider the flaw insensitive concept. At the moment of pull-off, the maximum opening displacement at the contact edge should not be larger than the effective interaction range *δ*_{0}, i.e. . Thus, the normal traction on the adhesion interface uniformly attains the interfacial theoretical strength *σ*_{th}. The critical size *h*_{cr} of the above rigid plate can be obtained from(2.4)where(2.5)Substituting equation (2.1) into the critical criterion (2.4) yields(2.6)When *h*≤*h*_{cr}, the pull-off stress is the theoretical strength *σ*_{th} and has no relation to the width of the rigid plate.

The corresponding pull-off force *F*_{p} is(2.7)

### 2.2 Fibrillar structure

Hui *et al.* (2004) and Glassmaker *et al.* (2004) have studied the concept of ELS for a perfect interface containing many fibrils, in which they used the area fraction that fibrils cover and found enhanced adhesion compared with their non-fibrillar counterparts. Are there any other limits on the number of fibres to satisfy the condition of ELS in the regime of flaw insensitivity except for the characteristic fibril spacing that prevents fibrils' self-matting? How can we enhance adhesion by tuning the geometrical parameters?

A simple plane-strain fibrillar structure in adhesive contact with a substrate is shown in figure 2. The fibrils in figure 2 are assumed to be rigid and identical in width 2*h*≤2*h*_{cr} and ELS in this section, which would be valid, for example, if the structure in the figure is pulled uniformly upwards. The distance between neighbouring fibres is *L*. There are *N*=2*n*+1 fibrils consisting of the fibrillar interface with the central one numbered ‘0’ in figure 2.

Inspired by Hui *et al.* (2004), if the fibrils are elastic with Young's modulus *E*_{f} and length *l*, the energy per unit length in the direction of thickness needed to pull off a single fibril is(2.8)The work required to detach a unit area of the interface is(2.9)where is the area fraction that the fibres cover on the interface.

ELS and theoretical strength conditions have been satisfied above. One should note that another condition of flaw insensitivity is that the maximum separation of the interface should not be larger than the effective interaction range *δ*_{0}, which will lead to restrictions on the number of fibrils *N*.

Consider the model in figure 2, in which the fibrils are rigid. The normal displacement *u*_{z}(*x*) at an arbitrary point (*x*, 0) consists of the contributions of all the distributing tractions on the interface, and each part can be obtained from equation (2.1) as follows. For example,(2.10)denotes the displacement at the point (*x*, 0) contributed by the zeroth distributing traction, where *C*_{0} is a constant.(2.11)denotes the displacement at the point (*x*, 0) contributed by the first distributing traction.(2.12)denotes the displacement at the point (*x*, 0) contributed by the *i*th () distributing traction.

The total displacement at the point (*x*, 0) can be written as(2.13)where .

Note the model in figure 3. The maximum separation *δ* can be written as(2.14)Substituting equation (2.13) into (2.14) yields(2.15)Flaw insensitivity requires(2.16)The critical case yields the relation between the non-dimensional parameter *L*/*h* and the critical number *N*=2*n*+1, which is dependent only on the non-dimensional parameter .

The pull-off stress *σ*_{p} and the pull-off force *F*_{p} per unit length in the direction of thickness are(2.17)From the above, one can see that the normalized pull-off stress and pull-off force are identical to the area fraction and the number of fibrils, respectively.

## 3. Three-dimensional models

Three-dimensional models of fibrillar adhesive interface will be studied in this section. The anti-bunching condition for fibres of square cross-section in fibrillar structures has been investigated by Hui *et al.* (2002) and Gao *et al.* (2005). Glassmaker *et al.* (2004) have derived the critical length for bunching of cylindrical fibres. Owing to rigid fibrils considered in this paper, anti-bunching condition was not investigated in our model.

### 3.1 Critical size of a single fibril

A rigid cylindrical fibre with radius *R* in contact with an elastic substrate is shown in figure 4, which was analysed by Persson (2003) with energy method and Hui *et al.* (2004) with Maugis' (1992) solution. The contact zone is assumed to be perfect adhesion with contact radius *R*. If the bond break is flaw insensitive, then the pull-off stress should be the theoretical strength *σ*_{th}; thus the tractions on the interface are also *σ*_{th}.

The normal displacement at any point of a three-dimensional elastic half-space subjected to homogeneous pulling stress *σ*_{th} in an area with radius *R* can be obtained from Johnson (1985) as(3.1)and(3.2)where the functions are(3.3)and(3.4)The maximum separation *δ* can be obtained,(3.5)where(3.6)Then we have(3.7)Flaw-insensitivity condition requires(3.8)The critical contact radius can be derived from as(3.9)which means that if *R*≤*R*_{cr}, then the bond breaking may occur uniformly over the contact area and the pull-off stress is then independent of the size *R* of the cylinder.

The corresponding pull-off force is given by(3.10)The critical size in equation (3.9) is identical to that in Hui *et al.* (2004). Comparing equations (3.9) and (1.3), one can find that the critical size predicted by the Dugdale criterion is more conservative than that predicted by the Griffith criterion, which is similar to the flaw-insensitivity conditions for a thin strip with a central crack or double-edge cracks in Gao & Chen (2005).

### 3.2 Fibres distributed in a line array

We consider a fibrillar structure consisting of a line array of fibres as shown in figure 5, where a large number of identical fibrils are in perfect contact with an elastic substrate. We assume that the fibrils are rigid with radius *R*≤*R*_{cr} and the distance between neighbouring fibrils is *L*. The contact region is assumed to be a rectangular area with width 2*R* and length .

When a uniformly distributing stress acts to pull the fibrillar structure as shown in figure 5, we know that under the ELS and flaw-insensitivity assumptions the normal traction on the interface is the theoretical strength *σ*_{th}. The normal displacement at the point *x*=0 can be obtained from Johnson (1985) as(3.11)The normal displacement at the point is(3.12)Thus, the maximum separation of the fibrillar interface at the contact edge,(3.13)should not be larger than *δ*_{0}, i.e. *δ*≤*δ*_{0}. *δ*=*δ*_{0} denotes the critical case. For a given *h* and *L*, the constraint relation between the number of fibres and the non-dimensional parameter can be described.

The pull-off force and pull-off stress can then be written as(3.14)where *ρ* is the area fraction of the fibrillar interface in a line array pattern.

### 3.3 Fibres distributed in a hexagonal pattern

Figure 6 shows a hexagonal distributing pattern for a fibrillar structure, in which fibres with radius are assumed to be rigid and the substrate is an elastic half-space. ELS and flaw insensitivity are assumed in this model and the contact area is a circular region with radius .

The normal displacement at the centre point *r*=0 can be written as(3.15)where(3.16)

The corresponding displacement at is approximately written as(3.17)where(3.18)The maximum separation at the contact edge can be obtained as(3.19)The critical condition of flaw insensitivity for the fibrillar interface is(3.20)The above equation shows that the relation between the normalized parameter *L*/*R* and the number of fibrils *N* depends only on the non-dimensional parameter . The number of fibres is(3.21)The pull-off force *F*_{p} and pull-off stress *σ*_{p} in this case can be expressed as(3.22)

### 3.4 Fibres distributed in square pattern

Figure 7 shows the square distribution pattern of a fibrillar interface, in which rigid fibrils are assumed with distance *L* between neighbouring fibres. The contact region between the fibrillar structure and the substrate is a circular one with radius *nL*+*R*.

The normal displacement at *x*=0 can be expressed as (Johnson 1985)(3.23)

The corresponding displacement at is(3.24)where(3.25)(3.26)(3.27)and(3.28)

The maximum normal separation can be obtained as(3.29)which satisfies the flaw-insensitivity condition(3.30) denotes the critical case from which one can find the relation between the normalized parameter *L*/*R* and the number of all fibres *N* depending only on the non-dimensional parameter . The critical number of fibrils *N* that satisfies ELS and flaw-insensitivity conditions is(3.31)

The pull-off force *F*_{p} and pull-off stress *σ*_{p} can be expressed by *N* and *ρ*, respectively, as(3.32)where *ρ* is the area fraction of fibres in a square lattice pattern.

## 4. Numerical analysis

### 4.1 Two-dimensional fibrillar interface contact

In the plane-strain model of a rigid plate in contact with an elastic half-plane, the normalized pull-off force is equal to the critical number of fibres, which can be found in equation (2.17). Figure 8*a* shows the normalized pull-off force and the critical number of plates *N* as a function of the normalized space *L*/*h* to satisfy ELS and flaw sensitivity conditions for different values of non-dimensional parameter , from which one can see that the pull-off force remains almost a constant for a determined parameter when the space *L*/*h* increases. It means that the size of the plane-strain fibrillar structure can be very large and does not influence the pull-off force when the ELS and flaw sensitivity are satisfied and the half-width of each plate *h* is not larger than the critical size *h*_{cr}. On the other hand, for a determined space *L*/*h* of the fibrillar structure, increase in the net pull-off force can only be realized through decreasing , i.e. by increasing the effective Young's modulus of the substrate *E*^{*} or the effective interaction distance *δ*_{0}. In addition, for a fixed space *L*/*h*, a larger critical number *N* can be expected by means of increasing the value of *δ*_{0} and *E*^{*} or decreasing *σ*_{th}. According to equation (2.17), one can see that the normalized effective strength (pull-off stress) *σ*_{p}/*σ*_{th} and the area fraction *ρ* will decrease along with an increasing space *L*/*h* as shown in figure 8*b*. It should be of interest to design plane-strain fibrillar structures as adhesion-controlled devices, whose size or pull-off force can be regulated according to the requirements.

### 4.2 Three-dimensional fibrillar interface contact

#### 4.2.1 Line array pattern

In the case of adhesive contact between an elastic substrate and a rigid fibrillar structure with fibrils distributing in a line array, one can see that the normalized pull-off force is equal to the critical number *N* quantitatively in equation (3.14). Figure 9 shows the normalized pull-off force and the critical number of fibres *N* as a function of the normalized space of neighbouring fibres *L*/*R* for several different values of to satisfy the conditions of ELS and flaw insensitivity. From figure 9, one can see that the pull-off force and the critical number of fibrils are influenced significantly by the non-dimensional parameters and *L*/*R*. For a determined *L*/*R*, the pull-off force and the critical number increase when the parameter decreases. For a smaller value of , the pull-off force and the critical number *N* will decrease smoothly with an increasing value of *L*/*R*. For a larger value of , the pull-off force and the critical number will increase first along with *L*/*R* and then decrease, which means that the pull-off force can obtain a maximum value at an optimal *L*/*R* in the case with a larger . The pull-off stress will decrease along with increasing space *L*/*R* and the curve looks like that in figure 8*b*.

#### 4.2.2 Hexagonal lattice pattern

According to equations (3.20)–(3.22), the normalized pull-off force and the critical number *N* as a function of the normalized space *L*/*R* to satisfy ELS and flaw-insensitivity conditions are shown in figure 10 with fibrils distributing in a hexagonal lattice pattern. From figure 10, one can see that for a determined parameter , the normalized pull-off force increases first with the increasing normalized space *L*/*R* and then decreases for larger *L*/*R*. There exists a certain value of *L*/*R* in each case, at which the pull-off force and the critical number *N* attain maximum. For a fixed space *L*/*R*, the pull-off force and *N* are always increasing along with the decreasing parameter . As for the pull-off stress, it will decrease along with the increasing space *L*/*R*, which looks like the curve in figure 8*b*.

#### 4.2.3 Square lattice pattern

Figure 11*a*,*b* plots the curves of the normalized pull-off force (the critical number *N*) via the normalized space of neighbouring fibrils *L*/*R* to satisfy the ELS and flaw-insensitivity conditions in adhesive contact of fibrillar interface with fibrils distributing in a square lattice pattern. Owing to large difference between the quantities of the normalized pull-off force for different values of , we plot the relations in two figures (figure 11*a*,*b*) to make the curves more clear. From the two figures, one can see that the relation between the normalized pull-off force (the critical number *N*) and the ratio *L*/*R* is much more complex than that in the former cases for a given . For smaller non-dimensional value , the normalized pull-off force will increase first with the increasing ratio *L*/*R*, then decrease quickly after achieving a maximum value, from which one can see that the pull-off force is influenced significantly by tuning the geometrical parameter of fibrillar structures so that it is convenient to choose the space between neighbouring fibrils to obtain the required adhesion properties. When the value of increases, the relation between the normalized pull-off force and *L*/*R* consists of four regimes along with an increasing *L*/*R*: (i) the pull-off force (or the critical number *N*) increases at the initial stage, (ii) decreases at certain value of *L*/*R*, (iii) then increases again to a second top value, and (iv) after that, decreases once again. For a fixed ratio *L*/*R*, the pull-off force is always larger for a smaller . From the above, one can see that the pull-off force can be regulated by tuning the geometrical parameter of fibrillar structures. Furthermore, one can find easily that the effective adhesion strength (pull-off stress) decreases and asymptotically approaches zero with the increasing ratio *L*/*R*, and this variation looks very similar to that in figure 8*b*.

## 5. Discussion

In this paper, we study only the cases of rigid fibrillar structures in contact with elastic substrates to satisfy the ELS and flaw insensitivity, from which the pull-off force, the effective adhesion strength and the critical number of fibrils are influenced significantly by tuning the geometrical parameters of fibrillar structures. We can find an optimal space of neighbouring fibrils to achieve maximum pull-off force. The critical number of fibres is a criterion to justify whether the equation can be used correctly in studying periodical structures. For a determined space of neighbouring fibrils and the size of single fibril not larger than the critical one, if the number of fibrils *N*^{*} is larger than the critical number *N*, ELS assumption cannot be used any more and the idea of ‘contact splitting’ to increase the adhesion force becomes unsuitable. On the contrary, if , it is reasonable and very convenient to use ELS to analyse the adhesion characters of fibrillar structures.

From the analysis in this paper, one can see that if the number of fibrils satisfies ELS and flaw-insensitivity conditions, it needs only to tune the geometrical parameters to enhance adhesion.

As for the case of elastic fibrillar structures in contact with elastic or rigid substrates, Hui *et al.* (2004) has investigated this problem to some extent, in which they suppose that the pull-off of a fibril can be viewed as the growth of a pre-existing crack for the case of rigid fibrils in contact with substrates; however, this does not apply to an elastic fibril since the exterior of an elastic fibril is not an external crack. Details can be found in Hui *et al.* (2004). Yao & Gao (2006) have investigated the hierarchical fibrillar structures and made a comparison between their results and the observed hierarchical structure of the gecko. For cases of elastic fibrils, the elastic strain energy stored in any fibril should be considered in addition to the energy creating new surfaces during pull-off. Are there any constraints on the number of elastic fibrils and what effect will be produced on the pull-off force and adhesion strength? This more realistic mechanical mechanism adopted by bioadhesion systems will be left to future work.

## Acknowledgments

The work reported here was supported by NSFC (10672165) and KJCX2-YW-M04.

## Footnotes

- Received May 25, 2007.
- Accepted June 24, 2007.

- © 2007 The Royal Society

## References

## Notice of correction

The name of the second author is now correct. 9 August 2007