Adhesive contact on power-law graded elastic solids: The JKR–DMT transition using a double-Hertz model

doi:10.1016/j.jmps.2013.07.015

Journal of the Mechanics and Physics of Solids

Volume 61, Issue 12, December 2013, Pages 2473-2492

https://doi.org/10.1016/j.jmps.2013.07.015 Get rights and content

Abstract

A cohesive zone model of axisymmetric adhesive contact between a rigid sphere and a power-law graded elastic half-space is established by extending the double-Hertz model of Greenwood and Johnson (1998). Closed-form solutions are obtained analytically for the surface stress, deformation fields and equilibrium relations among applied load, indentation depth, inner and outer radii of the cohesive zone, which include the corresponding solutions for homogeneous isotropic materials and the Gibson solid as special cases. These solutions provide a continuous transition between JKR and DMT type contact models through a generalized Tabor parameter $μ$ . Our analysis reveals that the magnitude of the pull-off force ranges from $(3 + k) π R Δ γ / 2$ to $2 π R Δ γ$ , where $k$ , $R$ and $Δ γ$ denote the gradient exponent of the elastic modulus for the half-space, the radius of the sphere and the work of adhesion, respectively. Interestingly, the pull-off force for the Gibson solid is found to be identically equal to $2 π R Δ γ,$ independent of the corresponding Tabor parameter. The obtained analytical solutions are validated with finite element simulations.

Introduction

It is well known that surface adhesion can play a central role in microscale contact problems. Modeling adhesive contact between elastic solids can be, however, a challenging task. Although the classical non-adhesive contact theory for elastic bodies was proposed by Hertz (1882) as early as the 19th century, little effort has been made to address the adhesive contact problems until 1930s. Bradley (1932) firstly examined the effect of adhesion between two rigid spheres and found that the critical force for pull-off is $- 2 π R Δ γ$ , where $R$ is the equivalent radius of the two spheres and $Δ γ$ is the work of adhesion. Four decades later, two famous models for adhesive contact between elastic spheres were proposed by Johnson et al. (1971) (JKR model) and Derjaguin et al. (1975) (DMT model). In the JKR model, the adhesion forces outside the contact area are neglected, and the contact forces are infinite in the neighborhood of the contact edge and the resulting pull-off force is $- 3 π R Δ γ / 2$ . On the other hand, in the DMT model, by considering adhesion forces over an annulus surrounding the contact region, the pull-off force is predicted to be the same as that of Bradley's model. Tabor (1977) compared the two models and showed that their ranges of validity can be assessed by a dimensionless parameter (Tabor parameter) that measures the ratio between elastic displacements and the characteristic length for the influence range of surface forces. Furthermore, Tabor pointed out that the JKR model works well for soft materials with relatively high surface energy while the DMT model is more appropriate for hard solids with low surface energy (Muller et al., 1980, Greenwood, 1997, Johnson and Greenwood, 1997, Barthel, 2008).

The paradox between the JKR and DMT models was finally resolved by Maugis (1992), who developed a cohesive solution of adhesive contact based on the Dugdale (1960) model of surface interactions. In the Maugis–Dugdale model, a critical simplification that allows the problem to be treated analytically is that the adhesive traction within the cohesive zone is assumed to be constant. Soon afterwards, this model was extended to a two-dimensional contact (Baney and Hui, 1997), more general surface interactions (Barthel, 1998) and noncontact (Kim et al., 1998) cases.

Parallel to the Maugis–Dugdale theory, Greenwood and Johnson (1998) proposed an alternative simple and useful cohesive zone model known as the double-Hertz model. They observed that the difference between two Hertzian pressure distributions of different contact radii can be used to describe the adhesive tensile traction between two contacting surfaces. Although an ellipsoidal adhesive traction distribution is adopted inside the cohesive zone, the results are very close to those from the Maugis model. In fact, an obvious advantage of the double-Hertz model is that the analysis relies solely on Hertzian solutions, which makes it analytically more tractable than Maugis's approach. For this reason, the double-Hertz model is often favored in the studies of adhesion in more complex systems, such as contact on rough surfaces (Persson, 2002) and viscoelastic materials (Haiat et al., 2003). Recently, the double-Hertz model was reconsidered in a slightly different context using an auxiliary function method (Barthel, 2012).

In contrast to the above advances with respect to homogenous materials, adhesive contact involving functionally graded materials (FGMs) remains an open research topic in spite of the potential applications of these materials in many engineering and biological systems (Suresh, 2001, Sherge and Gorb, 2001, Yao and Gao, 2007, Yao and Gao, 2010). The enamel and dentin layers in teeth are joined over a region of graded material properties which is thought to be very important for reducing stress concentrations (Huang et al., 2007, Niu et al., 2009). Suresh and Needleman (1996) showed that the variation of Young's modulus with depth for some composites made of graded metal–ceramic layers can be well approximated as a power-law function with respect to the distance from the surface. The power-law graded material model has also been adopted in a series of indentation studies showing that elastic gradients could offer unprecedented opportunities for materials to achieve improved wear and failure resistance (Giannakopoulos and Suresh, 1997a, Giannakopoulos and Suresh, 1997b, Jitcharoen et al., 1998, Giannakopoulos and Pallot, 2000).

Recent studies have laid a solid foundation for more systematic studies in the adhesion of power-law graded materials. Specifically, the JKR-type models for power-law graded elastic materials have been developed by Chen et al., 2009a, Chen et al., 2009b for both axisymmetric and plane deformation cases, and further extended to non-slipping contact on pre-stretched substrates (Jin and Guo, 2010, Jin and Guo, 2012, Guo et al., 2011). More recently, a general analytical scheme for treating axisymmetric adhesive contact between a power-law graded elastic half-space and a rigid punch of any profile via superposition principle and Betti's reciprocity theorem has been proposed (Jin and Guo, 2013, Jin et al.,) and used to study the effects of surface roughness and punch profiles. Notably, existing studies on power-law graded materials are all based on the JKR-type adhesion model, where it is assumed that there are no adhesive interactions outside the contact area, and is therefore only applicable for soft bodies with relatively large Tabor parameters (Chen et al., 2009b). The corresponding DMT and Maugis type adhesive models, which can account for adhesive forces outside the contact region, are still unavailable for power-law graded materials. Up to now, it remains a challenge to develop a unified adhesive contact model that can include both JKR- and DMT-type models for FGMs.

The present study is aimed to establish a set of analytical solutions, based on the double-Hertz model of Greenwood and Johnson (1998), that are capable of capturing the JKR–DMT transition in power-law graded elastic solids. The paper will be organized as follows. We first extend the double-Hertz model to power-law graded elastic solids with a generalized Tabor parameter in Section 2. The main results of the paper are presented in dimensionless form in Section 3. Two limiting cases of small and noncontact cohesive zones are examined in Section 4. The noncontact extension of the double-Hertz model is also included. Section 5 shows that the JKR and DMT limits can be recovered for two extreme values of Tabor parameter. As a special case of particular interest, the solutions for the Gibson solid are obtained in Section 6. Finite element analyses are carried out to validate the obtained analytical results in Section 7. Some concluding remarks are provided in Section 8.

Section snippets

Double-Hertz model

Fig. 1 shows a rigid sphere of radius $R$ which is in adhesive, frictionless contact with a power-law graded elastic half-space under a normal force $P$ (negative when it is tensile). A cylindrical coordinate system ( $r, z$ ) is set up with origin at the center of the contact zone and $z$ -direction pointing into the half-space. The distribution of surface traction consists of two terms: the Hertz pressure $p_{H}$ acting on a circular contact region of radius $a$ and the adhesive tension $p_{A}$ acting on an

Non-dimensional results

In this section, the above results are summarized in a dimensionless form. By introducing the following non-dimensional parameters: $r^{⁎} = \frac{r}{η}, a^{⁎} = \frac{a}{η}, c^{⁎} = \frac{c}{η}, P^{⁎} = \frac{P}{2 π R Δ γ}, δ^{⁎} = \frac{δ R}{η^{2}},$ where $η = {(\frac{c_{0}^{k} R^{2} Δ γ}{E^{⁎}})}^{1 / (3 + k)},$ the adhesive traction can be rewritten as $\frac{p_{A}}{σ_{0}} = - {(\frac{c^{⁎ 2} - r^{⁎ 2}}{c^{⁎ 2} - a^{⁎ 2}})}^{(1 + k) / 2} + {(\frac{a^{⁎ 2} - r^{⁎ 2}}{c^{⁎ 2} - a^{⁎ 2}})}^{(1 + k) / 2}, 0 < r^{⁎} < a^{⁎},$ $\frac{p_{A}}{σ_{0}} = - {(\frac{c^{⁎ 2} - r^{⁎ 2}}{c^{⁎ 2} - a^{⁎ 2}})}^{(1 + k) / 2}, a^{⁎} < r^{⁎} < c^{⁎} .$

In addition, the dimensionless normal load and indentation depth can be obtained from Eqs. (2.12), (2.13) as $P^{⁎} = \frac{α_{1}}{3 + k} [a^{⁎ 3 + k} - λ (c^{⁎ 3 + k} - a^{⁎ 3 + k})]$ and $δ^{⁎} = \frac{1}{1 + k} [a^{⁎ 2} - λ (c^{⁎ 2} - a^{⁎ 2})],$

Two limiting cases

For various ratios of the total interaction zone size $c$ to contact zone size $a$ , the dependence of the adhesive traction on the surface separation within the cohesive zone is implicitly determined by combining Eqs. (2.11b), (2.15). In this section, we shall examine the traction–separation relation in two limiting cases: small cohesive zone and noncontact cohesive zone.

JKR limit

When the Tabor parameter $μ$ is large and the cohesive zone $(c - a)$ is small, it can be concluded from Eq. (3.7) that the factor $λ$ becomes large. By denoting $c = a + Δ l$ , the following Taylor expansions can be derived: $c^{3 + k} = a^{3 + k} + (3 + k) a^{2 + k} Δ l + \frac{(3 + k) (2 + k)}{2} a^{1 + k} Δ l^{2} + O (Δ l^{3}),$ $c^{1 + k} = a^{1 + k} + (1 + k) a^{k} Δ l + \frac{k (1 + k)}{2} a^{k - 1} Δ l^{2} + O (Δ l^{3}) .$ Substituting Eqs. (5.1a), (5.1b) into Eqs. (3.4), (3.5), (3.6) leads to $P^{⁎} = \frac{α_{1}}{3 + k} a^{⁎ 3 + k} - α_{1} a^{⁎ 2 + k} λ (c^{⁎} - a^{⁎}),$ $δ^{⁎} = \frac{1}{1 + k} [a^{⁎ 2} - 2 a^{⁎} λ (c^{⁎} - a^{⁎})],$ $λ (c^{⁎} - a^{⁎}) = α_{1}^{- 1 / 2} a^{⁎ - (1 + k) / 2} .$ Inserting Eq. (5.4) into Eqs. (5.2), (5.3) then

The Gibson solid

Taking the limit of $k = 1$ and $ν = 0.5$ in the general solution for a power-law graded materials, we can obtain the double-Hertz solution of the so-called Gibson solid (Gibson, 1967).

In the Gibson limit, the corresponding adhesive traction can be expressed as $p_{A} (r) = - σ_{0}, 0 < r < a,$ $p_{A} (r) = - σ_{0} (\frac{c^{2} - r^{2}}{c^{2} - a^{2}}), a < r < c .$ Note that the constant traction within the contact area is also the maximum traction: $σ_{0} = - p_{A} (a) = \frac{λ E_{0}}{3 c_{0} R} (c^{2} - a^{2}) .$

The separation between the contacting sphere and the deformed substrate surface is $h (r) = h_{c} r^{2} - a$

Finite element analysis

To validate the obtained analytical solutions, finite element analysis (FEA) was employed to simulate frictionless adhesive contact between a rigid sphere and a power-law graded elastic half-space. All simulations were carried out with the multi-purpose finite element program ABAQUS/Standard (version 6.10) using an implicit solver. A USDFLD subroutine was adopted to define the power-law graded materials.

Fig. 10 shows the finite element mesh of 53,586 four-node bilinear axisymmetric

Conclusions

The present paper represents a first attempt (to the best of our knowledge) to establish a cohesive zone model of adhesive contact in nonhomogeneous materials, which is often considered to be very difficult or even impossible for an analytical treatment. By employing the double-Hertz model of Greenwood and Johnson, a series of analytical solutions for power-law graded elastic materials such as the interfacial stress, surface displacement and equilibrium relations among applied load, indentation

Acknowledgments

The financial supports from the National Natural Science Foundation (10925209, 91216201 and 10932003), 973 Project of China (2010CB832703), Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT) are gratefully acknowledged. H.G. gratefully acknowledges support from the Center for Mechanics and Materials at Tsinghua University.

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In addition to the adhesive contact between an indenter and a semiinfinite domain, the graded coating-substrate structure has also been paid extensive attentions. Jin et al. (2013, 2021) investigated the adhesive contact between a rigid punch and a power-law graded elastic half-space by using the double-Hertz and a generalized MD model. Chidlow et al. (2013) solved the adhesive contact of an exponentially graded coating-substrate system.
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