A double-Hertz model for adhesive contact between cylinders under inclined forces

Abstract

A generalized double-Hertz (D-H) model has been proposed to consider the adhesive contact between an elastic cylinder and an elastic half space under inclined forces. The normal traction is exactly the same as that in the conventional D-H model. The shear traction of finite value is distributed into a slipping zone and a non-slipping zone. In the slipping zone, the shear traction is proportional to the compressive pressure. With the model, adhesive contact behaviour between cylinders has been numerically illustrated. The shear-induced peeling has been demonstrated. The value of the ratio for shear traction to normal traction larger than friction coefficient has been found in part of the non-slipping zone. Those altogether are consistent with experiments.

1. Introduction

van der Waals force or alternatively adhesive interaction between two contacting solids plays a key role in determining the mechanical behaviour of small-scale systems. For example, adhesive force induces significant local stress in atomic force microscopy which may result in substantial wear and tip degradation [1–3], provides a significant contribution to the resistance of friction during the process of tangential separation and is closely related to the reliability of microelectromechanical systems and the attaching abilities of gecko and insects [4–6]. It is therefore imperative to capture the adhesive interaction, and in these contexts the effect of tangential stress cannot be ignored. Indeed, since Hertz's seminal work [7] on the contact of elastic spheres, several methods have been developed in the past decades to account for adhesive interactions in elastic contact problems. Adhesive contact models between elastic spheres proposed by Johnson et al. [8] and Derjaguin et al. [9], which are currently known as the JKR model and the DMT model respectively, have been widely used in bioadhesion and many other fields. However, their ranges of validity and the predicted magnitudes of the pull-off force are quite different. As shown by Tabor [10], the JKR and DMT models represent two limiting cases of adhesive contact with the ranges of validity assessed by a dimensionless parameter, i.e. Tabor parameter. The JKR model works well for soft solids with relatively high surface energy while the DMT model is more appropriate for hard materials with low surface energy. The transition range may be well characterized by the cohesive zone model established by Maugis [11], where the adhesive stress in the model is assumed to be a constant over the cohesive zone. It is usually referred to as the M–D model due to its analogy with the crack model by Dugdale [12] and appropriate to arbitrary values of the Tabor parameter. Another cohesive zone model was put forward by Greenwood & Johnson [13], known as the double-Hertz (D-H) model. This model is also applicable to arbitrary values of the Tabor parameter. In this model, the adhesive force within the cohesive zone is made up of two Hertzian pressure distributions of different contact radii. It has been found that results obtained from the D-H model can be very close to those of the M–D model. But the D-H model is analytically more tractable than the M–D model because the analysis relies on the classical Hertzian solutions. It has been well illustrated by Jin et al. [14], who have considered the adhesive contact between the cylindrical systems by the D-H model.

It is noted that investigations on adhesive contact with tangential forces are rather limited although in the absence of adhesion contact a problem with tangential forces can be tackled by introducing the Coulomb–Amonton law locally [15,16]. Based on the JKR model, Savkoor & Briggs [17] have taken into account the contribution of the tangential forces to the energy release rate and found the effect of shear-induced peeling, i.e. reduction of contact size with increase in tangential forces. It was verified by the experiment where rubber hemispheres were pressed against a flat glass counterface. Chen and his co-workers [18–20] have studied the effect under the assumption of non-slipping adhesive contact under inclined forces. However, the shear-induced peeling continues until the tangential force reaches a critical value. What will happen next is not very clear. In the work of Waters & Guduru [21], a phenomenological model for the dependence of the work of adhesion on mode-mixity is applied to describe the tangential loading experiments on PDMS. But the model can only be used for the initial stages of the loading process. To consider the interparticle sliding in the presence of adhesion, Thornton [22] proposed that slip may take place when the shear-induced peeling reduces the contact size to the radius within which compressive normal tractions of the Hertzian type were distributed. Papangelo & Ciavarella [23] established a model for partial slip contact where the classical Cattaneo–Mindlin problem for elastic half planes has been extended for a Griffith condition for the inception of slip, and otherwise following the standard Coulomb's law in the sliding zone. Singularities are involved in the normal and shear traction because they are obtained on the basis of the JKR solution for plane problems, which makes them different from the experimental results. Johnson [24] has established an adhesive contact model for smooth elastic spherical and a plane surface. Partial slip was introduced to this model to describe the inception of slip by assuming the shear traction to be a constant value in the sliding zone. Recently, Popov & Dimaki [25] have proposed a model for friction in an adhesive tangential contact where the adhesive force is described by the Dugdale model and the tangential force by Coulomb's law for friction.

Since the adhesive contact with tangential forces has been analysed mostly on the basis of JKR model and occasionally on by Dugdale model, one may wonder whether the shear-induced peeling is intrinsic to adhesive tangential contact. Indeed, both the JKR and the DMT models as mentioned above, however, do not have the adequate applicability for materials of arbitrary adhesive properties. How the adhesive interaction impacts the contact behaviour under inclined forces is not fully understood. Moreover, in many small-scaled experiments, inconsistent friction behaviours have been discovered [26–28], which may be better explained by considering adhesive contact under forces in any directions. Because of the singular stress concentration at the edge of the contact interface in JKR model, adhesive contact models involving no singularities may be more reasonably used instead to simulate the initiation process of friction. Given the general applicability of the D-H model, the present study aims to extend it to the situation of adhesive contact of cylinders under inclined forces. We expect the efforts may help to understand the contact behaviour and friction at small scales.

2. Double-Hertz model for adhesive contact between cylinders under inclined forces

Let us consider two cylinders in adhesive contact. They may be subjected to both normal and tangential forces as shown in figure 1. The loading direction is denoted by the angle θ between the normal of the contact interface and loading vector. Owing to the geometry and for infinitesimal deformation, the problem is equivalent to the plane strain contact problem between a rigid cylinder of radius R and an elastic half-plane with an equivalent Young's modulus E* [14], where

$$\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}$$ 2.1

and

$$\frac{1}{E^{\ast }}=\frac{1-{{\nu }_1}^2}{E_1}+\frac{1-{{\nu }_2}^2}{E_2}.$$ 2.2

Figure 1.

The D-H model for the adhesive contact between elastic cylinders under an inclined force F with a normal component P and a tangential one T: (a) schematic illustration of the contacting system, (b) illustration of the contact due to normal loading and (c) illustration of the slipping zone and non-slipping zone due to the tangential force. (Online version in colour.)

According to Greenwood & Johnson [13], who have considered adhesive contact under normal forces, the essential idea of double-Hertz model is to represent the adhesive traction by the combination of two Hertzian solutions. It results in the decomposition of the distribution of the normal surface traction into two parts with one Hertzian pressure acting on the contact zone and the other Hertz-like adhesive traction on a non-contacting zone and the contact zone, i.e. the adhesion interaction appears throughout the contact zone and extends into a non-contacting zone. The non-contacting zone is bounded by half-widths a and c as shown in figure 1b. For adhesive contact between a rigid cylinder and an elastic half-plane, Jin et al. [14] have given the distributions of the normal traction in the contact zone and the adhesive zone for D-H model that respectively may read

$$\begin{array}{cc}p(x)=\frac{(1+\mathrm{\lambda })E^{\ast }}{2R}{(a^2-x^2)}^{1/2}-\frac{\mathrm{\lambda }{E}^{\ast }}{2R}{(c^2-x^2)}^{1/2}\text{, }& 0\le |x|\le a\\ p(x)=-\frac{\mathrm{\lambda }{E}^{\ast }}{2R}{(c^2-x^2)}^{1/2}\text{, }& a\le |x|\le c\end{array},$$ 2.3

from which the adhesive traction can be separated as

$$\begin{array}{rlrl}{p}_a(x)& =\frac{\mathrm{\lambda }E\text{* }}{2R}{(a^2-x^2)}^{1/2}-\frac{\mathrm{\lambda }E\text{* }}{2R}{(c^2-x^2)}^{1/2}\text{,}& & 0\le |x|\le a\\ p_a(x)& =-\frac{\mathrm{\lambda }E\text{* }}{2R}{(c^2-x^2)}^{1/2}\text{,}& & a\le |x|\le c\end{array}$$ 2.4

In equations (2.3) and (2.4), λ is a parameter that can be related to the Tabor parameter μ by

$$\mu =2\mathrm{\lambda }{(c^2-a^2)}^{1/2}{\left(\frac{E^{\ast }}{{\pi }^2{R}^2\mathrm{\Delta }\gamma }\right)}^{1/3}.$$ 2.5

From equation (2.3), the total normal force can be obtained as

$$P=\frac{\pi E^{\ast }}{4R}[a^2-\mathrm{\lambda }(c^2-a^2)].$$ 2.6

It is easy to verify that the derivative of the surface normal displacement with respect to x has the following form

$$\begin{array}{rlrl}\frac{\text{d}{u}_z}{\text{d}x}& =-\frac{x}{R},& & \text{0}\le |x|\le a\\ \frac{\text{d}{u}_z}{\text{d}x}& =\frac{(1+\mathrm{\lambda })\sqrt{x^2-a^2}}{R}-\frac{x}{R},& & a\le |x|\le c.\end{array}$$ 2.7

So the geometric separation distance and its derivative of the rigid cylinder and the deformed half-plane surface in the adhesive zone are obtained as

$$h=-\delta +\frac{x^2}{2R}+u_z$$ 2.8

and

$$\frac{\text{d}h}{\text{d}x}=\frac{(1+\mathrm{\lambda })}{R}\sqrt{x^2-a^2}\text{, }a\le |x|\le c.$$ 2.9

In the model for normal loading, the surface energy is defined as the work of the normal load required to separate the two surfaces to infinity. So the surface energy of the contact can be defined by combining equations (2.3) and (2.9) as follows:

$$\mathrm{\Delta }\gamma =-{\int }_0^{\mathrm{\infty }}p(h)\hspace{0.17em}\text{d}h=-{\int }_a^{c}p(x)\frac{\text{d}h}{\text{d}x}\hspace{0.17em}\text{d}x.$$ 2.10

Nevertheless, because herein we are to consider the adhesive contact subject to an inclined force with a non-vanishing tangential component, the idea of the D-H model will be generalized such that the distribution of tangential traction can be decomposed into two parts, as schematically shown in figure 1c. Given that sliding may be initiated from the edge of the contacting zone, we denote the non-slipping region as [ − b, b], beyond which is the slipping zone with the shear traction assumed to be proportional to the normal traction minus the normal adhesive traction. Indeed, it is similar to the idea adopted by Papangelo & Ciavarella [23] except that in the present work, the D-H model has been resorted to account for the adhesion and the direct effect of normal adhesive traction on the shear traction has been excluded. The assumption is so made because in the classical Coulomb's law the friction is proportional to compressive normal force which is expected to be applicable locally herein. Thus, the shear traction in the slipping zone can be expressed as

$$q(x)=\frac{f{E}^{\ast }}{2R}{(a^2-x^2)}^{1/2}\text{, }b\le |x|\le a,$$ 2.11

with f being a constant identical to the friction coefficient. In the non-slipping zone, to make the problem tractable, the shear traction can be represented as the combination of the traction similar to that in the slipping zone and an extra shear traction q_e

$$q(x)=\frac{f{E}^{\ast }}{2R}{(a^2-x^2)}^{1/2}+q_e(x)\text{, }0\le |x|\le b.$$ 2.12

Obviously once q_e is solved, the shear traction is determined. So to obtain q_e, one may employ the non-slipping condition from which, according to Johnson [29], the following integral equation can be yielded:

$$\frac{\text{d}{u}_x}{\text{d}x}=\frac{-2}{\pi E^{\ast }}{\int }_{-a}^a\frac{q(s)}{x-s}\text{d}s=0\text{, }0\le |x|\le b.$$ 2.13

In deriving equation (2.13), the coupling effect between the normal and tangential tractions has been neglected. It strictly holds true only for the contacting objects of elastically similar materials [15,25], otherwise it may be a good approximation because according to Chen & Wang [18], the effect of coupling is not very significant. On substituting equation (2.11) and equation (2,12) into equation (2.13), one may find that the extra shear traction in the non-slipping zone has the following form

$$q_e(x)=-\frac{f{E}^{\ast }}{2R}{(b^2-x^2)}^{1/2}\text{, }0\le |x|\le b.$$ 2.14

And so altogether, the distribution of shear traction along the interface can be established as

$$\begin{array}{cc}q(x)=\frac{f{E}^{\ast }}{2R}{(a^2-x^2)}^{1/2}-\frac{f{E}^{\ast }}{2R}{(b^2-x^2)}^{1/2}\text{, }& 0\le |x|\le b\\ q(x)=\frac{f{E}^{\ast }}{2R}{(a^2-x^2)}^{1/2}\text{, }& b\le |x|\le a\end{array}.$$ 2.15

By integrating the shear traction along the interface, the tangential force can be obtained that is related to a and b as

$$\frac{\pi f{E}^{\ast }}{4R}(a^2-b^2)-T=0.$$ 2.16

It is worth mentioning that when under a force with a tangential component, the tangential force will do work due to slipping. In other words, the slipping zone in the current context should be understood as an adhesive zone for the tangential loading and the present model is, indeed, a special cohesive zone model similar to that of the mixed mode in fracture mechanics. So at equilibrium, the work of adhesion should be equal to the work due to both the normal force and the tangential force. Consequently, one may derive the following expression,

$$\mathrm{\Delta }\gamma =-{\int }_a^{c}p(x)\frac{\text{d}h}{\text{d}x}\hspace{0.17em}\text{d}x+{\int }_b^{a}q(x)\frac{\text{d}{u}_x}{\text{d}x}\hspace{0.17em}\text{d}x.$$ 2.17

To complete the calculation in equation (2.17), the derivative of the surface tangential displacement with respect to x can be obtained as

$$\frac{\text{d}{u}_x}{\text{d}x}=\frac{f\sqrt{x^2-b^2}}{R}.$$ 2.18

Upon substituting equations (2.3), (2.9), (2.16) and equation (2.18) into equation (2.17), one may finally get

$$\begin{array}{rl}\mathrm{\Delta }\gamma & =\frac{\mathrm{\lambda }(1+\mathrm{\lambda })E^{\ast }}{6R^2}\left[c(c^2+a^2)E\left(\frac{\sqrt{c^2-a^2}}{c}\right)-2c{a}^{2}K\left(\frac{\sqrt{c^2-a^2}}{c}\right)\right]\\ & +\frac{f^2{E}^{\ast }}{6R^2}\left[a(a^2+b^2)E\left(\frac{\sqrt{a^2-b^2}}{a}\right)-2a{b}^{2}K\left(\frac{\sqrt{a^2-b^2}}{a}\right)\right]\end{array}$$ 2.19

with E() and K() being the complete elliptical integrals of the first and second kinds.

Up to this point, the main equilibrium equations for the two-dimensional double-Hertz model under the combined action of normal load and tangential load have been established. It is worthwhile to point out that when the first term on right side in equation (2.17) or equation (2.19) vanishes, i.e. the work due to normal traction is equal to zero, the contact problem can be completely solved by using equations (2.6) and (2.16), which is exactly the same as the classical Cattaneo–Mindlin solutions. And in such a context, equation (2.19) or (2.17) seems redundant but may be understood as the work due to sliding friction.

3. Non-dimensional results

In this section, the above results are summarized in a dimensionless form by introducing the following non-dimensional parameters:

$$\begin{array}{rl}a\text{* }& =\frac{a}{\eta }\text{, }b\text{* }=\frac{b}{\eta },c\text{* }=\frac{c}{\eta },x\text{* }=\frac{x}{\eta },p^{\ast }=\frac{2pR}{\eta E^{\ast }},\\ q^{\ast }& =\frac{2pR}{\eta E^{\ast }},P^{\ast }=\frac{P}{P_c}\text{, }{T}^{\ast }=\frac{T}{P_c}\text{, }{F}^{\ast }=\frac{F}{P_c},\end{array}\}$$ 3.1

with

$$\eta =2{\left(\frac{R^2\mathrm{\Delta }\gamma }{\pi E^{\ast }}\right)}^{1/3},P_c={(\pi R{E}^{\ast }\mathrm{\Delta }{\gamma }^2)}^{1/3}.$$ 3.2

At equilibrium, the contact behaviour is governed by the following dimensionless expressions

$$\begin{array}{rl}& a^{\ast 2}(1+\mathrm{\lambda })-c^{\ast 2}\mathrm{\lambda }-P^{\ast }=0,\end{array}$$ 3.3

$$\begin{array}{rl}& f(a^{\ast 2}-b^{\ast 2})-T^{\ast }=0,\end{array}$$ 3.4

$$\begin{array}{rl}& 1=\frac{4\mathrm{\lambda }(1+\mathrm{\lambda })}{3\pi }\left[c^{\ast }(c^{\ast 2}+a^{\ast 2})E\left(\frac{\sqrt{c^{\ast 2}-a^{\ast 2}}}{c^{\ast }}\right)-2c^{\ast }{a}^{\ast 2}K\left(\frac{\sqrt{c^{\ast 2}-a^{\ast 2}}}{c^{\ast }}\right)\right]\\ & +\frac{4f^2}{3\pi }\left[a^{\ast }(a^{\ast 2}+b^{\ast 2})E\left(\frac{\sqrt{a^{\ast 2}-b^{\ast 2}}}{a^{\ast }}\right)-2a^{\ast }{b}^{\ast 2}K\left(\frac{\sqrt{a^{\ast 2}-b^{\ast 2}}}{a^{\ast }}\right)\right]\end{array}$$ 3.5

$$\begin{array}{rl}\text{and}& \mathrm{\lambda }=\frac{\mu \pi }{4{(c^{\ast 2}-a^{\ast 2})}^{1/2}}.\end{array}$$ 3.6

4. Numerical results and discussion

In this section, we will present numerical results to illustrate the adhesive contact behaviour given by the present model. For brevity, the friction coefficient f will be taken to be 0.6 unless otherwise stated explicitly.

(a). Contact behaviour predicted by the generalized D-H model

In many circumstances, adhesive contact may involve inclined forces, e.g. the pulling of cells by an oblique micro-pipette aspiration, climbing of insects along vertical surfaces and spraying nanoparticles onto substrates during depositing coatings. Although oblique contact in the absence of adhesion has long been considered, an account of adhesion under those conditions is rather limited. With the tangential force taken into account, the present model may be applicable to those situations under which we would like to illustrate the behaviours in this sub-section. By taking μ = 1, and the force direction θ = 0^°, 5^°, 25^°, 30^°, 60^°, 90^°, we have calculated the half contact width as a function of the applied force and the results are plotted in figure 2a,b where a negative value for the force represent pulling and a positive value for pushing. It can be seen that when θ is zero, i.e. adhesive contact under normal force, the behaviour is similar to the results by Jin et al. [14]. When the loading is slightly different from the normal loading, say for the value of θ ≤ 30^°, the half contact width increases with the applied force although the slope decreases with the increase in θ. Nevertheless, with further increase in the proportion of the tangential force, i.e. the increase in the value of θ, the half contact width may firstly increase and then decrease with the magnitude of the applied force. Such behaviour is due to the competitive effect of the compressive normal force and the tangential force because larger compressive normal force enhances contact while larger tangential force may undermine contact by inducing a larger slipping zone. Moreover, with the presence of the tangential component, failure in the adhesive contact due to pulling may be different from that in the conventional D-H model for which pull-off may be responsible. As is clearly demonstrated in figure 3, where the effect of a tangential force on the contact size under a given normal force P* = 1 has been presented, one can find that both half contact width a* and the half non-slipping zone b* decrease with the tangential force, although the size of the non-slipping zone decreases much faster. Thus, the shear-induced peeling effect still exists for the present model. Such an effect is actually attributed to the contribution of the work of tangential traction to the work of adhesion. Provided such a feature is taken into account, shear-induced peeling will arise in adhesive contact and irrespective of the models adopted.

Figure 2.

Variation of half contact width with the applied force in different directions for μ = 1. (Online version in colour.)

Figure 3.

Variation of a*, b* and c* with the tangential force for P* = 1, μ = 1. (Online version in colour.)

In figure 4a,b, distributions of the shear traction for the contact system in the regime of pulling and pushing have been plotted, respectively, with θ = 10^°and different values of F*. The normal traction is not of concern herein because its distribution is exactly the same as that in the conventional D-H model. At the boundary between the slipping zone and non-slipping zone, the shear traction achieves a peak value and varies drastically in the region adjacent to the boundary. The primary difference between figure 4a,b is due to the fact that pulling always decreases the contact size whereas pushing force may enhance the contact size so long as the tangential component is not very large enough. When θ becomes larger, as shown in figure 5a,b where distributions of the shear traction for θ = 60° and different values of F* have been presented, the shear traction overall is more evenly distributed in the slipping zone compared with that for smaller θ. Such behaviour is attributed to the larger slipping zone induced. All those distributions for the shear traction are qualitatively similar to those in Prevost et al.'s experiments [30].

Figure 4.

Distributions of the shear traction for the contact system in the regime of pulling (a) and pushing (b) with θ = 10^°. (Online version in colour.)

Figure 5.

Distributions of the shear traction for the contact system in the regime of pulling (a) and pushing (b) with θ = 60^°. (Online version in colour.)

In practice or experiments, as done by Fineberg's group [31–33], the shear traction and the normal traction may be determined although measurement of the respective adhesive stress and Hertzian pressure has seldom been reported. So the ratio of the shear traction to the normal traction has been calculated and demonstrated in figure 6 for μ = 1, θ = 60^°, F* = 0.4. To distinguish the non-slipping zone and the slipping zone, the shear traction and normal traction have also been given in the figure. It is interesting to observe that the ratio is not constant even in the slipping zone where it may possess singularity because the normal traction may be vanishing at some location. And in the non-slipping zone, the value may be much larger than the friction coefficient f in the region close to the slipping zone although mostly it is smaller than the friction coefficient in the rest of the zone. The results are not surprising because in the non-slipping zone the magnitude of the normal traction consisting of the compressive Hertzian pressure and the tensile adhesive stress is smaller than that of the Hertzian pressure. Such behaviour is consistent with the findings by Fineberg's group [31–33], who have found that the friction coefficient at a local point is not constant and even when the ratio of the shear traction to normal traction is larger than the friction coefficient, it may still be in the non-slipping state, which is in contrast to the conventional Coulomb's law. According to the present results, the behaviour can be attributed to the effect of adhesion.

Figure 6.

Distribution of the ratio of the shear traction to the normal traction. (Online version in colour.)

(b). The effect of Tabor parameter μ

As mentioned in the Introduction and many works, different values of the Tabor parameter μ may determine the applicability of the JKR model and the DMT model. Herein, we would like to consider its effect on the contact behaviours by the present model.

Figure 7a–c shows the variation of the dimensionless contact half width a* with the applied load F* for different values of the transition parameter μ and θ. Since the loading is merely slightly different from the normal loading, one can find the results are similar to those by Jin et al. [14]. Namely, for a larger value of μ, the relationship between a* and F* is similar to that by JKR model, whereas smaller value yields the relationship similar to that by DMT model. But with an increase in the value of θ, the proportion of the tangential force increases which makes the critical failure force at pulling different from that by the JKR model and the DMT model. In general, a larger value of μ results in a larger critical pulling force (negative in sign). In addition, in the regime of the pushing force, the contact may fail through overall sliding provided the tangential force is large enough. That is why only a much smaller portion is plotted for μ = 0.1 in figure 7c when compared with other curves.

Figure 7.

Effect of μ on the relationship between half contact width and the applied force for different loading directions (a) θ = 1^° (b) θ = 30^°and (c) θ = 60^°. (Online version in colour.)

Under a given normal force P* = 1, the effect of μ on the relationship between half contact width and the tangential force has been demonstrated in figure 8 from which one may find that the shear-induced peeling is more prominent for a larger value of μ.

Figure 8.

The effect of μ on the shear-induced peeling for P* = 1. (Online version in colour.)

Figure 9 gives the ratio of the shear traction to the normal traction for different values of μ for θ = 60^°, F* = 0.4. It shows that a larger value of μ gives rise to smaller regions with the ratio bigger than the friction coefficient. It can be easily understood because larger μ means higher adhesion.

Figure 9.

Effect of μ on the ratio of the shear traction to the normal traction. (Online version in colour.)

(c). Discussion

The present model, as shown above, can well capture the adhesive contact between cylinders under an inclined force. It has been demonstrated that the tangential component of the force usually decreases the contact size, which is exactly the same as predicted by JKR based models, i.e. shear-induced peeling arises in adhesive contact. However, in the JKR-based models, shear-induced peeling continues only until a critical shear force that is determined by the mathematics rather than physics as shown in [13]. Indeed, based on the JKR model, we have recently proposed that local sliding may occur well before the tangential force achieves that critical value [34]. By contrast, in the current context, the shear-induced peeling continues until the overall sliding occurs, which is physically reasonable. Since in the present model a slipping zone is introduced directly where the shear traction is proportional to the normal compressive traction, i.e. Amonton's law has been adopted, the topic of when the local slipping initiates has not been touched, which remains for further work. But in spite of this, the results given by the present model such as shear-induced peeling, distribution of the shear traction and the behaviour for the ratio of the shear traction to the normal traction are consistent with some experiments. It is worth mentioning that the recent model by Papangelo & Ciavarella [23] may also give the result that a local point with the ratio of the shear traction to the normal traction larger than the friction coefficient may be in a non-slipping state. But the present model is different from Papangelo & Ciavarella's model [23] mainly in two aspects. The traction here is regular while therein it is singular with the conventional square-root singularity. In the slipping zone, the shear traction in the present model is assumed to be proportional to the compressive part of the normal traction instead of the total normal traction. That is, the direct effect of the adhesive normal traction has been excluded in the shear traction, which is also different from that in Papangelo and Ciavarella's model. Since in experiments the friction coefficient is usually determined by the ratio of the shear traction to the total normal traction, its non-constant feature found by Fineberg's group [31–33] may be attributed partially to the adhesion.

5. Conclusion

In summary, a new adhesive contact model has been established in this paper by extending the normal D-H model to the situations where the external load may be applied in inclined directions but without moments at the contacting interface. In the model, both the shear traction and the normal traction are regular, i.e. no singularity is involved, although they are distributed in different zones. It is in contrast to those in JKR-based models and is applicable to systems with arbitrary values of the Tabor parameter. Numerical results from the present model have illustrated that the tangential component of the applied force may decrease the contact size which leads to the shear-induced peeling. Owing to the appearance of the tangential force, partial sliding may occur at the contact zone and it in turn gives rise to the peak value of the shear traction at the boundary between the slipping zone and the non-slipping zone. Moreover, the ratio of the shear traction to the normal traction may be larger than the friction coefficient even in the non-slipping zone and it is not constant in the slipping zone. Those results altogether are qualitatively consistent with some available experiments.

Data accessibility

This paper does not have any experimental data.

Authors' contribution

G.Y. conceived the mathematical models, interpreted the computational results and wrote the paper. J.F. implemented and performed most of the simulations and calculations in consultation with G.Y. and wrote the paper. All authors gave final approval for publication.

Competing interests

We declare we have no competing interests.

Funding

The present work is partially supported by the Natural Science Foundation of China (no. 11572216)