## Abstract

A generalized thermomechanical model for adhesion was developed to elucidate the mechanisms of dissipation within the viscoelastic bulk of a hyperelastic hydrogel. Results show that in addition to the expected energy release rate of interface formation, as well as the viscous flow dissipation, the bulk composition exhibits dissipation due to phase inhomogeneity morphological changes. The mixing thermodynamics of the matrix and solvent determines the dynamics of the phase inhomogeneities, which can enhance or disrupt adhesion. The model also accounts for the time-dependent behaviour. A parameter is proposed to discern the dominant dissipation mechanism in hydrogel contact detachment.

## 1. Introduction

The contact between soft surfaces of a viscoelastic nature has been central in the recent study of biological systems in the areas of protective coatings for mechanical components under extreme environmental conditions, such as lubrication and biofouling. Soft materials exhibit adhesion-dominated contact and frictional characteristics at multiple scales. Continuum models have sought to connect the molecular effects described by the cohesive zone laws and the measured work of separation of contacting surfaces [1]. In particular, hydrogels are in use today due to their suitability in applications where biological tissues must be replaced with artificially produced materials. Improvements in the interrogation methods, as well as fabrication techniques, are allowing for the study of the properties of potential biomimetic materials suitable for tissue growth and replacement [2]. Applications that may benefit from a better understanding of the adhesion mechanisms in soft materials include the protection of surfaces against the corrosive maritime environment, self-cleaning and anti-biofouling, energy efficiency by the use of low frictional coatings, as well as medical applications, which seek to avoid pathogen surface colonization and tribological failure of diarthrodial joints [3–6]. The herein reported work seeks to construct a generalized model for the dissipation mechanisms involved in the adhesion observed in viscoelastic inhomogeneous hydrogel contacts.

## 2. Theoretical framework for generalized contacts

The modern treatment of contact mechanics is said to have started when Hertz formulated the first solution for the stress field within a half-space that was being impinged upon by a spherical body [7]. His solution was only applicable to the frictionless contact which exhibited no adhesion. At the time, efforts had been mostly concentrated in finding solutions to specific boundary-value problems using mathematical approximations [8]. Before this time, only rough estimates for the frictional behaviour of interfacing surfaces existed and it was not until works by Bowden & Tabor [9] that the effects of asperities and other molecular-scale variables of the shear strength of a contact started to be considered [9]. Independently, Archard studied the effects of loads on the real area of contacts produced by plastically deforming asperities, which developed wear upon experiencing shear stresses [10].

It was not until Johnson, Kendal and Roberts' model, known today as the JKR model, that a solution of the adhesive forces and contact area exhibited between contacting surfaces was produced [11]. In their work, a variational approach was used to calculate the area between the solids, which achieved a minimum state of energy with contributions from the surface energy. The JKR model remains the prevalent model used for soft elastic contacts. Derjaguin *et al.* [12] developed a similar model by a different approach, namely by emphasizing the difference between the Hertzian model and the area ring-bounding contact, deemed the effective area of action for the adhesive forces. Work in the area of viscoelastic contacts started earlier. Andrews & Kinloch [13,14] investigated the effects of viscous dissipation on the interfacial crack front as a result from studies by Gent & Petrich [15], and later Gent & Schultz [16]. In their work, Gent *et al.* documented the changes in peel force as a function of rate of extension and later energy release rate as a function of crack front speed along with a mixed adhesive and cohesive failure mode. The importance of the work by Gent & Petrich [15] correlating the energy release rate and the viscoelastic behaviour of the polymers cannot be exaggerated. An outstanding summary of this work, along with updated advances, was published by Maugis & Barquins [17]. Greenwood & Johnson [18] published a constitutive treatment of viscoelastic contacts, which also included the effects of the surface interactions. Today, the maximum adhesion energy predicted by such models remains of interest for most engineering applications. In the current work, the adhesive contact between swollen viscoelastic hydrogels was investigated. A model within thermodynamic and mechanical constraints was developed, which remains consistent with the previous models discussed while accounting for the bulk contributions to the adhesive energy absent from the said models.

## 3. Kinematics

The bodies in contact are mathematically defined as a configuration of points in a region (*R*) space bounded by a piecewise smooth boundary. For mathematical simplicity, the deformation will be confined to one of the bodies and will be described in terms of undeformed (*B*) and deformed body (*B*_{t}) configurations mapped by a motion (*χ*(**X**,*t*):*B*→*B*_{t}) described in the material frame. The motion (**x**=*χ*(**X**,*t*)) relating to the new position (**x**) of the material points (**X**) at time (*t*) is assumed as a regular (*C*^{2}) one-to-one mapping with an inverse (∃:*χ*^{−1}), where damage producing spatial discontinuities does not occur. The relationship between deformed and undeformed bodies will be given by the three-dimensional mapping defined by the convective derivative of the motion, which is known as the deformation gradient, denoted by: *B*_{t}=*χ*(*B*)) [19]. In order to produce the motion in the elastic body (*B*), a traction (**t**) is applied to the half-space, which produces a strain energy (*ψ*) stored within the elastic solid. The material's tendency to resist deformation may be quantified by the generalized first Piola–Kirchoff stress tensor as
^{ij} is the stretch factor of the deformation gradient tensor (equation (3.1)), λ^{α} is the first invariant; the compressibility term is *P*=*p*+2*J*∂*ψ*_{H}/∂*J* with *p* being an arbitrary function due to hydrostatic pressure or constraint boundary conditions during finite deformation [20,21]. In this work, the spatial transformations given by the metric tensors in undeformed (*G*^{IK},*G*_{IK}) and deformed (*g*^{ik},*g*_{ik}) states remain in the rectangular coordinate system, thereby, yielding a regular one-to-one transformation of the: *G*^{IK}=*g*^{ik}=*δ*^{ik}=*δ*^{IK}=*G*_{IK}=*g*_{ik}=*δ*_{ik}=*δ*_{IK} type, where *δ*^{ik} and *δ*_{ik} are the Kronecker delta. *ψ*_{H} is the stored mechanical energy function given by the working potential known as the Helmholtz free energy: *ψ*_{H}=*e*−*θη*, where the first term on the right represents the internal energy and the second term on the right is the change of total free energy for the polymer chain conformation in the solvent. The constitutive form of each term will be discussed in the following subsection. The counterpart of the first Piola–Kirchoff stress tensor in the deformed configuration can be derived by the relationship between the area element in the undeformed and deformed configurations, which is acted upon by the same force through Nanson's equation; *J* is the Jacobian of the deformation matrix, a measure of the volume strain and given the regular motion.

## 4. Balance laws for multiphase hyperelastic material

The relationship between the elastic strain energy and the rate at which work is done by the standard tractions (figure 1) is given by
*D*) [22]; *ρ*_{D} and *ρ* are the mass densities for *D* and the body (*B*), respectively; as described above, **n** and **N** are the vectors normal to ∂*B* and ∂*D*, respectively, and **v*** (**v**) for *B* and the material flow rate (

Eshelby's energy momentum tensor and the domain configurational body force may be written as functions of the free energy
**g** is the isotropic internal body force. In terms of the inhomogeneity domain surface stress, Eshelby's energy momentum tensor may be expressed as

## 5. Energy criteria for viscous–elastic contact detachment

The detachment of elastic contacts is an inherently dissipative process, where the Clausius–Duhem postulate for viscous–elastic contacts (equations (9.16) and (9.17)) provides the lower bound of entropy generation
*ρs*_{c}]∇*θ*=ℑ^{th} is a material thermal force per unit volume, *A*_{c} is the total area of contact, *G* is the energy release rate. The rate of dissipation is given by *G*_{Ic} is the mode I fracture energy release rate upon interfacial cracking during detachment given in terms of the interfacial energy between the two mating surfaces as *G*_{Ic}=2*γ*^{(s)} [24], and *G*_{eff} is the effective energy release.

## 6. Experimental design

### (a) Dimensional analysis for contact of viscous–elastic solids

As shown in §5, equation (5.2), the dissipation for a viscous–elastic contact during detachment has functional relationships [25], which for this set of variables are the following:
*E* and *υ* are the frequency-dependent elastic modulus and the viscosity of the fluid, respectively, and *γ*^{(inh)},*ξ*^{(inh)}) between the inhomogeneous phase and the bulk material. The material thermal forces are assumed to be small since the excess solvent in a hydrogel is free to flow through the matrix. The energy release rate in the normal mode is a function of the surface energy per unit area parameter (*γ*^{(s)}) as well as the micrometre- and nanoscale roughness (*H*(*x*_{1})).

### (b) Sample preparation

#### (i) Glycoprotein coating preparation

The lubricin (LUB) (PRG4) glycoprotein was purified from human synovial fluid as described by Jay [4]. Purified human LUB was diluted to concentrations of 10, 100, 200 and 250 *μ*g ml^{−1} in 0.9% NaCl (physiologic saline). Solutions were deposited on highly ordered pyrolytic graphite (HOPG) substrates to allow for thermodynamic equilibrium to be reached as the LUB molecules settled on the hydrophobic surfaces. After 10 min, the excess solution was removed with a pipette and the remaining fluid was extracted through capillary action, without disturbing the surfaces.

#### (ii) Direct grafting of polymer chains

Poly(acrylic acid) (PAA, greater than or equal to 99.5%), poly(styrene) (PS) and poly(styrene)-block-poly(acrylic acid) (PS-*b*-PAA), 3-(amino propyl) triethoxysilane (3-APTS) and *N*,*N*-dimethylformamide (DMF), polymers were grafted onto ultraflat (2 nm r.m.s. surface roughness) glass substrates initially cleaned with piranha solution to remove organic residues. Prior to coating with silane, the glass substrates were hydroxylated to enhance the density of available sites for silane reaction and improved the quality of surface modification. The slides were washed with DMF and methanol, and then dried under vacuum to remove the excess solvent.

### (c) Contact measurements

Adhesion measurements were made using the atomic force microscope (AFM) Dimension 3100 with Nanoscope IIIa controller (Veeco Metrology, Inc., Santa Barbara, CA, USA) in the pulse force mode (PFM) [2]. The PFM allows for the study of soft matter without disrupting its structural integrity by accessing feedback signals with digital data acquisition (WITec GmbH, Ulm Germany). A high-resolution PFM scan of the cross section was conducted prior to micrometre-scale stiffness measurements in order to assess the cross-sectional compositional morphology of the substrate [26].

Two types of AFM cantilevers were used; silicon nitride sharp cantilevers were used to obtain high-resolution images of the coatings, and fabricated colloidal probe cantilevers were used in order to apply a contact mechanics theoretical framework to the analysis. A MATLAB-based code was written in order to analyse the PFM curves. The main features gathered from the PFM curves include long-range forces during approach, van der Waals attractive forces during first contact, coating stiffness, maximum force exerted on the cantilever, maximum adhesive force and the work of adhesion during de-bonding.

### (d) Calibration method

The normal force calibration method used for the AFM measurements consists of a diamagnetic normal force calibration system or D-NFC system. A silicon cantilever bar is supported on a diamagnetic, highly oriented, pyrolitic graphite sheet levitated over four rare-earth magnets due to an induced opposing field. The cantilever used in the following measurements was calibrated with a 0.1% error [27] to have a stiffness of 7.1394 N m^{−1}.

## 7. Results and discussion

### (a) Phase inhomogeneity driven adhesion

In order to assess the adhesive properties of the hydrogel systems, the work of adhesion must be quantified. In a hydrogel mixture, the phase inhomogeneities may coalesce at the interface, in some instances, to a critical flaw size where the interface will be weakened (figure 2). This clustering effect may be modelled through image forces inducing an accumulation of defects at the interface where the local interfacial strength is reduced, as characterized by the sudden drop in adhesive force during retraction (figure 4). Due to the phase domain multiregional coalescence at the contact interface in a multiphase hydrogel, the contact between the AFM probe and the substrate surface is heterogenous in nature. Such a heterogeneous area of contact may be decomposed into the fractional terms comprising the contact between concentrated polymer and glass AFM probe (pg), and the area fraction consisting of the substrate and glass AFM probe (sg). Therefore, the effective work of adhesion is given by, *γ**=*γ*^{(e)}_{pg}(*A*_{pg}/*A*_{c})+*γ*_{sg}(*A*_{sg}/*A*_{c}), where *A*_{c} is the area of contact, and *γ* the energy of interaction or interfacial energy between the polymer layer or substrate and glass AFM probe. The asterisk (*) signifies that this is the measured surface energy from the energy release rate (*G*_{eff}≈2*γ**) for an elastic contact. The hydrogel polymer glass contact characterized by interfacial defects [30] may be further modelled as a periodic perturbation of the hydrogel concentration at the interface giving an effective surface energy
*ζ* is the defect distribution wavelength approximated by *ζ*=4*l*_{cr}, and *γ*_{pg} is the surface energy between the glass probe and the polymer solution. In spherical contacts, the contact line at the edge of the contact experiences tensile stresses which confine the interfacial crack propagation to the boundary length or circumference. In this case, the fractional boundary length may be related to the fractional area of coverage for each phase by employing a purely geometric argument, yielding the effective contact boundary length model (ECBL) that relates to the effective work of adhesion as a function of area covered by the inhomogeneities as
*β* is an analytical factor that depends on the periodicity of the surface roughness. It must be noted that the AFM adhesion measurements were done in liquid. Figure 3 shows the normalized work of adhesion for the hydrogel polymer brushes as a function of the inhomogeneity coverage function (1−*ϕ***A*_{pg}/*A*_{c}), where *ϕ** is the stable volume fraction estimated from the mixing chemical potential (*μ*^{(β)}) given by the Flory–Huggins mixing theory (∂*Ψ*_{mix}/∂*ϕ*=0) [31]. The lower and upper limits of the model signify a surface, where coverage is homogeneous (*β*=1) and characterized by a perturbed polymer–solvent concentration (*β*=*π*) pattern, respectively. In the low density coverage regime, there is not enough material to attain an inhomogeneous coverage unlike in the spinodal region where enough material is present. Note, from figure 3, the concentrated viscoelastic solution (triangle), as well as the elastic glass contacts (diamonds), exhibits behaviour consistent with ECBL parameter, *β*=1. The significance of the morphology at the contact may be explained as follows: as the flaw density at the interface increases, the normalized adhesive contact scaling parameter (*A*_{pg}/*A*_{c})*Φ**<0.4), defect clustering does not affect the interfacial energy cost of forming the new surface area and the effective adhesive contact scaling parameter approaches the behaviour of a surface wetting adhesive coating.

The typical retraction data of the AFM cantilever for the hydrogels (figure 4) exhibit instability points where the adhesive force drops drastically in a stepwise fashion [28,29,32]. This phenomenon within the contact modifies the effective work of adhesion by producing low adhesion areas where these instabilities allow for the interfacial crack to propagate with lower energy dissipation.

## 8. Viscoelastic adhesion

As shown by equation (5.2), viscoelastic dissipation is a function of the history-dependent plastic stress, *τ*_{p} [33], at the temperature of interest (*θ*) and some reference temperature if the relaxation time is the variable used to characterize the time-dependent behaviour of viscoelastic materials, and if the structural behaviour of the polymer solution is assumed to have the same dependence over a wide range of thermodynamic variables (temperature, pressures, concentrations, etc.). The ratio of viscous dissipation and the phase inhomogeneity dissipation is a function of the forcing frequency (*ω*), the phase inhomogeneity surface tension (*γ*^{(inh)}), the elastic modulus (*E*(*ω*)) and the reduced crack speed (*a*_{T}) [33] given by
*θ* is the temperature of interest, *θ*_{g} is the glass transition temperature (subscript g) and *ζ* is the frictional coefficient per unit monomer. The viscoelastic shift factor is given in terms of experimental factors as
*B* is a polymer-specific empirically approximated numerical factor, *ϕ*_{g} is the frictional coefficient per unit molecule and *α*_{f} is the thermal expansion of volume fraction. Equation (8.2) is derived from the Williams–Landel–Ferry (WLF) theory for dispersive response of a viscoelastic material [33,34].

A scaling parameter derived by applying dimensional similarity, and restricted by the dissipation given by the Clausius–Duhem postulate (equation (5.1)), and the constitutive equation of the stress for a material of fading memory given by equation (6.1) is given by
^{−3} has been added to convert the non-dimensional group to the order of unity. This document hereby refers to this parameter (equation (8.3)) as the Gent–Eshelby (Ge) number, where *R* is the contact radius and *a*_{c} is the radius of the contact area calculated from the JKR model [11] and is the ratio of viscoelastic dissipation and dissipation due to phase inhomogeneity motion.

Figure 5 shows that most of the coatings tested were of the hydrogel type (Ge≪1), pointing to the phase inhomogeneity as the dominating mechanism of dissipation. However, the polystyrene coatings in water (triangles), a combination of low miscibility, exhibit a Ge>1, which points to the viscous dissipation as the dominant mechanism. For Ge≫1, elastic brittle failure of the interface is the dominant mode of detachment. Note also, that the elastic contact for the unetched glass (diamond with Ge≫1) also falls under this category, while the hydroxylated (etched) glass (diamond with Ge∼1) exhibits some viscous drag due to the water bound to the surface hydroxide groups. This phenomenon is consistent with the usual practice of subjecting glass to a variety of chemical processes that produce the surface hydroxylation in order to improve adhesion, e.g. sulfuric acid/hydrogen-peroxide (piranha) cleaning solution, ultraviolet/ozone irradiation, plasma etching, etc. Although the layering of water at the surface produces structural (non-DLVO) repulsive forces measured on approach [1], this does not necessarily signify lower adhesion, as the oscillatory behaviour also implies an irreversible process with the energetic valleys that must be overcome during retraction in order to produce movement on the interfacial crack front. Nevertheless, the water layer increases adhesion by dissipating elastic energy, not only by increasing the intrinsic surface energy, but also through the exertion of a ‘drag’ on the crack front, represented by the first and second terms on the right-hand side of equation (5.2), respectively.

## 9. Conclusion

The effective energy release rate for a hydrogel hyperelastic contact has the functional form
*α*≅1/3 (figure 6) for the energy dissipated in the formation of an interface.

The rate dependence of adhesion strength was first reported by Gent & Petrich in 1969 as the viscoelastic transitions between a viscous interface failing at lower rates of peel, into a viscoelastic and subsequently purely elastic interface as the rates of peeling increase. All of these effects were also shown to be failure mode dependent. In their work, the WLF theory was used to reconcile the differences in crack front speed at a wide range of peel rates, thereby, elucidating the viscoelastic dissipation on the crack front. Previous work by Gent *et al*. [15] and Kaelble [35] had shown the congruence between this temperature and rate response for the bond strength, with the dispersive viscoelastic response of the polymeric adhesive.

Generally, an increase in adhesive dissipation was observed with peel rate or temperature increase to a specific maximum value followed by a decrease in adhesion due to a transition into the purely elastic material behaviour while maintaining the same failure mode throughout the temperature and peel rate range [15]. These transitions in viscoelastic behaviour were not thought then to be due to transitions from viscous to viscoelastic to purely elastic bulk behaviour, but transitions in viscolastic behaviour arising from the polymer chain dynamics. The first transition is associated with the frictional slip among polymeric chains and solvent flow, while the transition at high rates is due to individual polymer chains uncoiling. In the viscous regime, an increase in adhesion is commensurate with the increase in viscous dissipation as the viscous drag effects dominate the complex modulus. Cohesive failure is more likely to occur due to the cavitating effects of the triaxial stresses on the polymeric bulk [36]. The abrupt decay in contact stiffness within a narrow range of rates at the onset of elastic behaviour originates from energy dissipated exclusively in the reconstitution of the interface with little changes in the viscous modulus. Nevertheless, the increase in adhesive dissipation with increasing rates resumes as the viscous modulus increases and the cohesion returns to dominate the failure mode. Another less abrupt decay in contact stiffness within a range of rates occurs at the onset of elastic chain coiling induced elasticity and rotational drag, along with viscous flow dissipation.

For hydrogel contacts, the phase inhomogeneity-caused instabilities postulated in this work account for a mixed cohesive-interfacial failure with a commensurate increase in adhesion and deformation rates. As the deformation rate increases to produce a Gent–Eshelby number greater than unity (Ge>1), the crack speed through the interface (*ωγ*^{(inh)}) to accumulate at the crack front and increase the instability driven rupture. Although this process of transport increases the total dissipation by adding a new mechanism, namely, the dissipation due to phase inhomogeneity motion (second term on the right of equation (5.1)), that increase is countered by the reduction of the viscous dissipation at the crack front due to defect-enhanced instability, reducing the overall energy needed to form a new interface through crack opening.

Of all of the coatings herein studied, the glycoprotein, LUB coating, also known as PRG4, shows the lowest adhesion due to its complex failure mechanism, which includes cohesive as well as interfacial inhomogeneity weakening (figure 6) [4]. Therefore, coatings with the lowest possible adhesion must attain a multimodal failure mechanism. Ideally, in order to reduce adhesion, it is necessary to produce a coating which exhibits a hydrogel polymer surface structure, while allowing for low intrinsic surface energy and cohesive failure. As the coating fails cohesively, a brush structure must remain on the new surface.

The hydrogel adhesion model developed in this work has important implications for the micromechanics of boundary lubrication in the thin film rheological LUB –hyaluronate complex [4,37]. In such hydrogels, the glycoprotein LUB forms an inhomogeneous moiety, which provides the reduced adhesion in diarthrodial mammalian joints at high loads. In the absence of LUB, viscosity-supplementing hydrogels of hyaluronate will be less effective in improving boundary lubrication and only when grafted into the surface has hyaluronate shown a reduction in the coefficient of friction [38]. In the absence of LUB, ungrafted hyaluronate lacks the instability mechanism necessary for cohesive failure, which would yield a lower adhesion.

## Data accessibility

The experimental data used in this work has been made available to the publisher for public release.

## Authors' contributions

J.R.T. and K.S.K. conceived the mathematical models, and interpreted the experimental results. G.D.B. and G.D.J. fabricated and characterized the hydrogel coatings. J.R.T., G.D.B. and G.D.J. wrote the manuscript in consultation with K.S.K. J.R.T. performed most of the experiments and analysis calculations in consultation with K.S.K., G.D.B. and G.D.J.

## Competing interests

J.R.T., K.S.K., G.D.J. and G.D.B. have no competing interests. G.D.J. has authored and was granted a US patent ‘Tribonectin polypeptides and uses thereof’, no. 6743774 for the therapeutic use of LUB in joints. G.D.J. receives support from a Phase II STTR R42AR057276 for clinical translation efforts related to his patent. The present paper neither materially nor financially affects G.D.J.'s patent relating to rhPRG4.

## Funding

This work was supported by the National Institutes of Health (NIAMS RO1 AR050180, NCRR P20RR024484, NIAMS R42AR057276), the US Department of Defense (CDMRP PR110746), the Office of Naval Research ILIR, Maritime Sensing Program (ONR321 N0001411WX20311) and Coatings Biofouling Program (ONR332 N0001411WX20995); and the Chief Technology Office's Naval Undersea Warfare Center Fellowship.

## Acknowledgements

The authors would like to express their gratitude to Professors D. M. L. Meyer and Allan Bower for insightful suggestions and corrections, as well as to Ms Jillian Kiser and Mr Colin Murphy for assistance in laboratory experiments.

## Appendix A. Energy momentum tensor

The working balance after substituting the referential velocity and regrouping the terms for the body and the domain yields
**q**). Satisfying the *Principle of Virtual Work* and the energy conservation axiom (first Law) of thermodynamics yields the internal workings of the system
**h** is the heat flux vector through the control volume surface of the body (*B*). From the Helmholtz free energy (*η* is the entropy and *θ* is the absolute temperature [39], it may be seen that the configurational forces are a means of transforming the elastic energy provided by the external traction (**t**) into the movement of the domains within the body. In many systems, this is a dissipative process, which must satisfy the Clausius–Duhem postulate [40] which states
*B*), which is that dissipated on expanding the domain interface by the capillary force on the interface (

## Appendix B. Energy momentum tensor for multiphase solid

The energy momentum tensor given by equation (A 6) may be expressed in terms of the internal energy at the interface ([*e*]) substituting the Helmholtz free energy, where the brackets signify the interfacial jump conditions, then
*ρe*] and [*ρs*_{c}*θ*] are interfacial quantities, defined by jump conditions [*ϕ*]. The interfacial laws define the relationship of the energy balance for the bulk with respect to the interface as
**U** is a steady velocity field of the bulk, and *Laplace's constitutive equation* for a liquid droplet in equilibrium, *γ* is the surface tension and [*p*] is the scalar pressure difference between inside and outside of the phase [39,42]. Then the thermostatic energy balance at the interface may be written in terms of the interfacial stress (**S**), entropy and heat flux jump conditions as *μ*^{(β)} is the scalar chemical potential and *n*^{(β)} is the number of species beta. Further expressing the entropy in terms of its structural and configurational components
*ρηθ*]=−*μ*^{(β)}[Δ*n*^{(β)}]+[*ρs*_{mix}*θ*], then equation (9.11) substituted into equation (9.8) yields the Eshelby's energy momentum tensor in terms of the inhomogeneity domain surface stress as
*D*) [41] has a configurational surface tensile (*γ*) component, which originates when material is added to the interface as *D* increases in volume and the surface shearing (*ξ*) component as this same volume increases and the interface moves in the normal direction;

## Appendix C. Field relationships for dissipation in a hydrogel contact

The entropy generation is given by the Clausius–Duhem postulate as
**H**) and the mass flux due to chemical potential (*θ* is the absolute temperature and *ρs*_{c}]∇*θ*=ℑ^{th} is a material thermal force per unit volume (*F*/*L*^{3}), which acts similar to the inhomogeneous force (ℑ^{inh}) on the manifold (*T*_{p}*M*), [19,47] and drives a thermal flux [48]. Equation (9.17) shows that making the contact and applying the load produces dissipation, therefore, detachment will require an additional amount of energy which will be dissipated. The term *G* is the energy release rate. Therefore, the energy flux into the crack tip is the energy dissipated by crack opening and is given by

- Received December 31, 2015.
- Accepted April 19, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.