A chemo-mechano-thermodynamical contact theory for adhesion, friction, lubrication and (de)bonding reactions
AA chemo-mechano-thermodynamical contact theoryfor adhesion, friction, lubrication and (de)bonding reactions
Roger A. Sauer a , b , c , , Thang X. Duong a and Kranthi K. Mandadapu d , e a Aachen Institute for Advanced Study in Computational Engineering Science (AICES),RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany b Department of Mechanical Engineering, Indian Institute of Technology Kanpur, UP 208016, India c Faculty of Civil and Environmental Engineering, Gda´nsk University of Technology, ul. Narutowicza11/12, 80-233 Gda´nsk, Poland d Department of Chemical and Biomolecular Engineering, University of California at Berkeley,110 Gilman Hall, Berkeley, CA 94720-1460, USA e Chemical Sciences Division, Lawrence Berkeley National Laboratory, CA 94720, USA
Abstract
This work presents a self-contained continuum formulation for coupled chemical, mechanicaland thermal contact interactions. The formulation is very general and hence admits arbitrarygeometry, deformation and material behavior. All model equations are derived rigorously fromthe balance laws of mass, momentum, energy and entropy in the framework of irreversiblethermodynamics, thus exposing all the coupling present in the field equations and constitutiverelations. In the process, the conjugated kinematic and kinetic variables for mechanical, thermaland chemical contact are identified, and the analogies between mechanical, thermal and chemicalcontact are highlighted. Particular focus is placed on the thermodynamics of chemical bondingdistinguishing between exothermic and endothermic contact reactions. Distinction is also madebetween long-range, non-touching surface interactions and short-range, touching contact. Forall constitutive relations examples are proposed and discussed comprehensively with particularfocus on their coupling. Finally, three analytical test cases are presented that illustrate thethermo-chemo-mechanical contact coupling and are useful for verifying computational models.While the main novelty is the extension of existing contact formulations to chemical contact, thepresented formulation also sheds new light on thermo-mechanical contact, since it is consistentlyderived from basic principles using only few assumptions.
Keywords: chemical reactions, continuum contact mechanics, constitutive modeling, coupledproblems, irreversible thermodynamics, nonlinear field theories
List of main field variables and their sets identity tensor in R a α tangent vectors on contact surface S c ; α = 1 , b k prescribed, mass-specific body force at x k ∈ B k ; unit [N/kg] B k set of points contained in body k∂ B k set of points on the surface of body k∂ q B k ⊂ ∂ B k ; boundary of body k where heat flux ¯ q k is prescribed ∂ t B k ⊂ ∂ B k ; boundary of body k where traction ¯ t k is prescribed D k rate of deformation tensor at x k ∈ B k ; unit [1/s] D i k inelastic part of D k corresponding author, email: [email protected] a r X i v : . [ phy s i c s . c l a ss - ph ] D ec a k area element on the current surface S k ; unit [m ]d v k volume element in the current body B k ; unit [m ] η e k mass-specific external bulk entropy production rate at x k ∈ B k ; unit [J/(kg K s)] η i k mass-specific internal bulk entropy production rate at x k ∈ B k ; unit [J/(kg K s)] η ic internal contact entropy production rate density at x c ∈ S c ; unit [J/(K m s)] E k Green-Lagrange strain tensor at x k ∈ B k ; unit-free E e k elastic part of E k E i k inelastic part of E k E total energy of the two-body system; unit [J] φ non-dimensional bond state at x c ∈ S c ψ k mass-specific Helmholtz free bulk energy at x k ∈ B k ; unit [J/kg] Ψ k = ρ k ψ k ; Helmholtz free bulk energy density per undeformed volume of B k ; unit [J/m ] ψ Helmholtz free interaction energy between x ∈ S and x ∈ S ; unit [J] ψ c bond-specific Helmholtz free contact energy on surface S c ; unit [J] Ψ c = N ψ c ; Helmholtz free contact energy density per undeformed area; unit [J/m ] F k deformation gradient at x k ∈ B k ; unit-free g := x − x ; gap vector between surface points x ∈ S and x ∈ S ; unit [m] g e reversible (elastic) part of g associated with sticking contact g s irreversible (inelastic) part of g associated with sliding contact h heat transfer coefficient between body B and body B ; unit [J/(K m s)] h k heat transfer coefficient between body B k and the interfacial medium; unit [J/(K m s)] J k change of volume at x k ∈ B k ; unit-free J s k change of area at x ∈ S k ; unit-free k = 1 ,
2; body index K kinetic energy of the two-body system; unit [J] µ c chemical contact potential per bond at x c ∈ S c ; unit [J] M c chemical contact potential per current area at x c ∈ S c ; unit [J/m ] M c chemical contact potential per reference area at x c ∈ S c ; unit [J/m ] n k current density of potential bond sites at x k ∈ S k and time t >
0; unit [1/m ] N k = J s k n k ; initial density of potential bond sites at x k ∈ S k ; unit [1/m ] n b k current density of bonded bond sites at x k ∈ S k and time t >
0; unit [1/m ] n ub k current density of unbonded bond sites at x k ∈ S k and time t >
0; unit [1/m ] n b := n b1 in case n b2 = n b1 n k outward unit normal vector at x k ∈ S k P k subset of B k or S k p c contact pressure, i.e. normal part of t c ; unit [N/m ] q k heat influx per current area; unit [J/(m s)] q k heat flux vector per current area; unit [J/(m s)]¯ q k prescribed heat influx per current area at x k ∈ S k ; unit [J/(m s)] q c k contact heat influx per current area at x k ∈ S c ; units [J/(m s)] q cm := ( q c1 + q c2 ) /
2; mean contact heat influx into B and B q ct := ( q c1 − q c2 ) /
2; transfer heat flux from body B to body B ˜ q k entropy influx per current area; unit [J/(K m s)]˜ q k entropy flux vector per current area; unit [J/(K m s)]¯˜ q k prescribed entropy influx per current area at x k ∈ S k ; unit [J/(K m s)]˜ q c k contact entropy influx per current area at x c ∈ S c ; unit [J/(K m s)] ρ k current mass density at x k ∈ B k and time t >
0; unit [kg/m ] ρ k = J k ρ k ; initial mass density at x k ∈ B k ; unit [kg/m ]¯ r k prescribed, mass-specific heat source at x k ∈ B k ; unit [J/(kg s)] R c bonding reaction rate per current area; unit [1/(m s)]2 k Cauchy stress tensor at x k ∈ B k ; unit [N/m ] S k second Piola-Kirchhoff stress tensor at x k ∈ S k ; unit [N/m ] s k mass-specific bulk entropy at x k ∈ B k ; unit [J/(kg K)] S k = ρ k s k ; bulk entropy density per undeformed volume of B k ; unit [J/(K m )] s c bond-specific internal contact entropy on surface S c ; unit [J/K] S c = N s c ; internal contact entropy density per undeformed area; unit [J/(K m )] S total entropy of the two-body system; unit [J/K] S k ⊂ ∂ B k ; set of points defining the interacting surface of body B k S c set of points defining the shared contact surface; in case S = S =: S c t time; unit [s] t k surface traction per current area at x k ∈ S k ; unit [N/m ]¯ t k prescribed surface traction per current area at x k ∈ S k ; unit [N/m ]¯ t k prescribed surface traction per reference area at x k ∈ S k ; unit [N/m ] t c k contact traction per current area at x c ∈ S c ; unit [N/m ] t c k contact traction per reference area at x c ∈ S c ; unit [N/m ] t c := t c1 (and t c = t c ) in case t c2 = − t c1 t t tangential contact traction, i.e. tangential part of t c T k temperature at x k ∈ B k ; unit [K] T c temperature of the interfacial contact medium at x ∈ S c ; unit [K][[ T ]] := T − T ; temperature jump (or temperature gap) across the contact interface u k mass-specific internal bulk energy at x k ∈ B k ; unit [J/kg] U k = ρ k u k ; internal bulk energy density per undeformed volume of B k ; unit [J/m ] u internal interaction energy between x ∈ S and x ∈ S ; unit [J] u c bond-specific internal energy of the contact interface S c ; unit [J] U c = N u c ; internal contact energy density per undeformed area; unit [J/m ] U total internal energy of the two-body system; unit [J] U c internal energy of the contact interface S c ; unit [J] v k material velocity at x k ∈ B k ; unit [m/s][[ v ]] := v − v ; velocity jump across the contact interface; equal to the relative gap velocity v := v in case v = v at the common contact point x = x ∈ S c x k current position of a material point at time t > B k ; unit [m] X k initial position of a material point in body B k ; unit [m] x c current position of a material point at time t > S c ; unit [m] Many applications in science and technology involve coupled interactions at interfaces. An ex-ample is the mechanically sensitive chemical bonding appearing in adhesive joining, implantosseointegration and cell adhesion. Another example is the thermal heating arising in fric-tional contact, adhesive joining and electrical contacts. Further examples are the temperature-dependent mechanical contact conditions in melt-based production technologies, such as weld-ing, soldering, casting and additive manufacturing. A general understanding and description ofthese examples requires a general theory that couples chemical, mechanical, thermal and elec-trical contact. Such a theory is developed here for the first three fields in the general frameworkof nonlinear continuum mechanics and irreversible thermodynamics. Two cases are consideredin the theory: touching contact and non-touching interactions. While the former is dominatingat large length scales, the latter provides a link to atomistic contact models.There is a large literature body on coupled contact models. General approaches deal, however,with two-field and not with three-field contact coupling, such as is considered here.3eneral thermo-mechanical contact models have appeared in the early nineties, starting withthe work of Zavarise et al. (1992) that combined a frictionless contact model with heat con-duction. Following that, Johansson and Klarbring (1993) presented the full coupling for linearthermo-elasticity. This was extended by Wriggers and Miehe (1994) to large deformations usingan operator split technique for the coupling - a staggering scheme based on successively solvingmechanical and thermal subproblems. Oancea and Laursen (1997) extended this to a mono-lithical coupling formulation and proposed general constitutive models for thermo-mechanicalcontact. Following these initial works, many thermo-mechanical contact studies have appeared,studying the extension to wear (Str¨omberg et al., 1996; Stupkiewicz and Mr´oz, 1999; Molinariet al., 2001), rough-surface contact (Willner, 1999) multiscale contact (Temizer and Wriggers,2010; Temizer, 2014, 2016) and adhesion (Dittmann et al., 2019). Beyond that, there havealso been many advancements in the computational description of thermo-mechancial contact(Agelet de Saracibar, 1998; Laursen, 1999; Str¨omberg, 1999; Pantuso et al., 2000; Adam andPonthot, 2002; Xing and Makinouchi, 2002; Bergman and Oldenburg, 2004; Rieger and Wrig-gers, 2004; H¨ueber and Wohlmuth, 2009; Dittmann et al., 2014; Khoei et al., 2018; Seitz et al.,2019).General chemo-mechanical contact models go back even further – to the work of Derjaguin et al.(1975) that computed molecular, e.g. van der Waals, adhesion between an elastic sphere and ahalf-space. Argento et al. (1997) then extended this approach to a general surface formulationfor interacting continua. The work was then generalized to a nonlinear continuum mechanicalcontact formulation by Sauer (2006) and Sauer and Li (2007a,b). Subsequently, the frameworkwas applied to the study of cell adhesion (Zeng and Li, 2011), generalized to various surfaceinteraction models (Sauer and De Lorenzis, 2013), and combined with sliding friction models(Mergel et al., 2019). In a strict sense, the chemo-mechanical contact models mentioned sofar are not coupled models. Rather, the chemical surface interaction is described by distance-dependent potentials. Hence, the resulting problem is a single-field problem that only dependson deformation. A second field variable representing the chemical contact state is not used.This is different to the debonding model of Fremond (1988). There a state variable is introducedin order to describe irreversible damage during debonding. It is essentially a phenomenologicaldebonding model, where the bond degrades over time following a first order ordinary differentialequation (ODE). Raous et al. (1999) extended the model to sliding contact and implemented itwithin a finite element formulation. Its finite element implementation is also discussed in Wrig-gers (2006) in the framework of large deformations. Subsequently the model has been extendedto thermal effects (Bonetti et al., 2009), applied to multiscale contact (Wriggers and Reinelt,2009) and generalized to various constitutive models (Del Piero and Raous, 2010), among oth-ers. Even though the approach is a coupled two-field model, it only describes debonding andnot bonding. It therefore does not provide a general link to chemical contact reactions.The adhesion and debonding models mentioned above are similar to cohesive zone models (CZMs). Those propose phenomenological traction-separation laws for debonding, that areoften derived from a potential, like the seminal model of Xu and Needleman (1993), thus ensur-ing thermodynamic consistency. CZMs have been extended to thermo-mechanical debondingthrough the works of Hattiangadi and Siegmund (2004); Willam et al. (2004); Fagerstr¨om andLarsson (2008); ¨Ozdemir et al. (2010); Fleischhauer et al. (2013) and Esmaeili et al. (2016),among others. Recently, CZMs have also been coupled to a hydrogen diffusion model in orderto study fatigue (Busto et al., 2017). CZMs usually have a damage/degradation part that some-times follows from an evolution law, e.g. see Willam et al. (2004). This makes them very similarto the Fremond (1988) model. Like the Fremond model, CZMs have not yet been combinedwith chemical contact reactions, and so this aspect is still absent in general contact models.Adhesion models based on chemical bonding and debonding reactions have been developed byBell and coworkers in the late seventies and early eighties in the context of cell adhesion (Bell,4978; Bell et al., 1984). Many subsequent works have appeared based on these models, forexample to study substrate adhesion (Hammer and Lauffenburger, 1987), strip peeling (Demboet al., 1988), nanoparticle endocytosis (Decuzzi and Ferrari, 2007), sliding contact (Deshpandeet al., 2008), cell nanoindentation (Zhang and Zhang, 2008), cell migration (Sarvestani andJabbari, 2009), cell spreading (Sun et al., 2009), focal adhesion dynamics (Olberding et al.,2010) and substrate compliance (Huang et al., 2011), among others. Similar chemical bondingmodels have also been used to describe sticking and sliding friction, see Srinivasan and Walcott(2009). The Bell model is an ODE for the chemical reaction. Assuming separation of chemicaland mechanical time scales, this can be simplified into an algebraic equation that describeschemical equilibrium (Evans, 1985). Even though Bell-like models have been combined withcontact-induced deformations in several of the works mentioned above, the contact formula-tions that have been considered in those works are not general continuum mechanical contactformulations as will be considered here.Another related topic is the field of tribochemistry that is concerned with the growth of so-called tribofilms during sliding contact. Chemical evolution models are used to describe thetribofilms in the framework of elementary contact models, e.g. see Andersson et al. (2012) andGhanbarzadeh et al. (2016). There is also a review article on how mechanical stresses can affectchemical reactions at the molecular scale (Kochhar et al., 2015). But none of these works usegeneral continuum mechanical contact formulations.Chemical contact reaction can be described by a state φ ( x , t ) that follows from an evolutionlaw of the kind ˙ φ = f ( φ ). Mathematically, they are thus similar to the description of contactageing (Dieterich, 1978; Rice and Ruina, 1983; Ruina, 1983), contact wear (Str¨omberg et al.,1996; Str¨omberg, 1999; Stupkiewicz and Mr´oz, 1999) and contact debonding (Fremond, 1988;Raous et al., 1999). The thermodynamics is, however, very different.None of the present chemo-mechanical models is a general contact model accounting for the gen-eral contact kinematics, balance laws and constitutive relations. This motivates the developmentof such a formulation here. To the best of our knowledge, it is the first general thermo-chemo-mechanical contact model that accounts for large-deformation contact and sliding, chemicalbonding and debonding reactions, thermal contact and the full coupling of these fields. It isderived consistently from the general contact kinematics, balance laws and thermodynamics,introducing constitutive examples only at the end. It follows the framework of general irre-versible thermodynamics established in chemistry (Onsager, 1931a,b; Prigogine, 1961; de Grootand Mazur, 1984), thermo-mechanics (Coleman and Noll, 1964), and chemo-mechanics (Sahuet al., 2017). Its generality serves as a basis for computational formulations and later extensionssuch as membrane contact or thermo-electro-chemo-mechanical contact.The novelties of the proposed formulation can be summarized as follows. The formulation • derives a self-contained, fully coupled chemical, mechanical and thermal contact model, • accounts for general non-linear deformations and material behavior, • captures consistently all the coupling present in the interfacial balance laws, • obtains the general material-independent constitutive contact relations, • provides several interface material models derived from the 2. law of thermodynamics, • is illustrated by three elementary contact solutions.The remainder of this paper is organized as follows: Sec. 2 presents the generalized continuumkinematics characterizing the mechanical, chemical and thermal contact behavior. The kine-matics are required in order to formulate the general balance laws in Sec. 3 that govern thecoupled contact system. Based on these laws the general constitutive relations are derived inSec. 4. Sec. 5 then gives several coupled constitutive examples satisfying these relations. In5rder to illustrate these, Sec. 6 provides three analytical test cases for coupled contact. Thepaper concludes with Sec. 7. This section introduces the kinematic variables characterizing the mechanical, thermal andchemical interaction between two deforming bodies B k , k = 1 ,
2. The primary field variablesare the current densities ρ k = ρ k ( X k , t ), current positions x k = x k ( X k , t ), current velocities˙ x k =: v k = v k ( X k , t ) and current temperatures T k = T k ( X k , t ) that are all functions of spaceand time. Here, the spatial dependency is expressed through the initial position X k = x k (cid:12)(cid:12) t = 0 ,and the dot denotes the material time derivative˙ ... = ∂...∂t (cid:12)(cid:12)(cid:12) X k = fixed . (1)The two bodies can come into contact and interact mechanically, thermally and chemically ontheir common contact surface S c := ∂ B ∩ ∂ B , see Fig. 1. In case of long-range interaction,like van der Waals adhesion, they may also interact when not in direct contact. In this case,interaction is considered to take place on the (non-touching) surfaces regions S k ⊂ ∂ B k . a. b. c. Figure 1:
Continuum contact description: a. bodies before contact; b. bodies in contact;c. free-body diagram for contact. Here, n k , v k and T k denote the bond density, velocity andtemperature, respectively, on the contact surface of body B k ( k = 1 ,
2) at time t . Duringcontact, φ denotes the degree of bonding, [[ v ]] the velocity jump, [[ T ]] the temperature jump, T c the contact temperature, t c k the contact tractions, q c k the heat influx, µ c the chemical contactpotential and s c the contact entropy. T c and s c are associated with an interfacial medium.The deformation within each body is characterized by the deformation gradient F k := ∂ x k ∂ X k , (2)from which various strain measures can be derived, such as the Green-Lagrange strain tensor E k := (cid:0) F T k F k − (cid:1) / . (3) In principle, long-range interaction can also take place in the volume. The present theory can be extendedstraightforwardly to account for such interactions. Alternatively, one may also project them onto the surfacefollowing Sauer and Wriggers (2009). E k = E e k + E i k . (4)An example for an inelastic strain is thermal expansion. From F k also follows the quantity J k := det F k , (5)which governs the local volume change d v k = J k d V k (6)at x k ∈ B k between undeformed and deformed configuration. Similarly, the local surface areachange d a k = J s k d A k (7)at x k ∈ S k is governed by the quantity J s k := det s F k , (8)where det s ( ... ) is the surface determinant on S k , e.g. see Sauer et al. (2019).Time dependent deformation is characterized by the symmetric velocity gradient D k := (cid:0) ∇ v k + ∇ v T k (cid:1) / , (9)also known as the rate of deformation tensor. The velocity also gives rise to the identities˙ J k J k = div v k (10)and ˙ J s k J s k = div s v k , (11)where div s ( ... ) is the surface divergence on S k .Mechanical interaction between the two bodies is characterized by the gap vector g := x − x , (12)that is defined for all pairs x ∈ S and x ∈ S . Its material time derivative is the velocityjump ˙ g = v − v =: [[ v ]] . (13)Two cases have to be distinguished: If the bodies are in contact, [[ v ]] is tangential to the commoncontact surface S c at momentarily touching points x = x . Otherwise (for bodies not yet incontact, or bodies moving out of contact), [[ v ]] is a general vector. In the former case, thevelocity can be written as v k = v αk a α (with summation over α ∈ { , } ), where a and a area set of tangent vectors on S c , and v k and v k are the tangential components of v k . This leadsto ˙ g = (cid:0) v α − v α (cid:1) a α =: L g , (14) Strictly, the additive decomposition of the Green-Lagrange strain tensor in the form of (4) is only valid forsmall deformations. However, for large deformations other strain tensors can be identified that admit an additivedecomposition in the form of (4). Typically the tangent vectors of one of the surfaces, say S , are chosen for this. See also footnote 6. Lie derivative of the gap vector. Analogous to (4), the gap can bedecomposed into elastic and inelastic parts, g = g e + g s , (15)where the later is associated with dissipative (sliding) contact and hence denoted g s . It isnoted that both g e and g s have normal and tangential components as in the unified contactformulations of Wriggers and Haraldsson (2003) and Duong and Sauer (2019).Thermal contact is characterized by the temperature jump[[ T ]] := T − T , (16)also denoted as the thermal gap (Agelet de Saracibar, 1998) and the contact temperature T c that is associated with an interfacial medium, e.g. a lubricant or wear particles, see Fig. 2. Itis assumed that this medium is very thin, such that T c is constant through the thickness of themedium and can only vary along the surface. Figure 2:
Thermal contact: Assumed temperature profile T ( g n ) across the contact interface(along coordinate g n ). Thermal contact between two bodies is characterized by the temperaturejump [[ T ]] := T − T and the contact temperature T c associated with an interfacial medium.[[ T ]] leads to the transfer heat flux q ct = − h [[ T ]] identified in Sec. 4.3.In order to characterize chemical contact, the bonding state variable φ = φ ( x , t ) is introducedon the contact surface, as discussed in the following section. It varies between φ = 0 for nobonding and φ = 1 for full bonding, and can be associated with a chemical gap (e.g. defined as1 − φ ).Tab. 1 summarizes the kinematic contact variables and their corresponding kinetic counterpartsthat will be introduced in later sections.field kinematic variable kinetic variablechemical bonding state φ chemical contact potential µ c mechanical velocity jump [[ v ]] contact traction t c thermal (gap) temperature jump [[ T ]] contact heat influx q c thermal (medium) contact temperature T c contact entropy s c Table 1:
Energy-conjugated contact pairs.8
Balance laws
This section derives the chemical, mechanical and thermal balance laws for the two-body system.The derivation is based on the following three mathematical ingredients. The first is Reynold’stransport theorem for volume integrals,dd t (cid:90) B k ... d v k = (cid:90) B k (cid:16) ˙( ... ) + div v k ( ... ) (cid:17) d v k . (17)It follows from substituting (6) on the left, using the product rule, and then applying (10).Applied to surfaces, Reynold’s transport theorem simply adapts todd t (cid:90) S k ... d a k = (cid:90) S k (cid:16) ˙( ... ) + div s v k ( ... ) (cid:17) d a k . (18)It follows from substituting (7) on the left, using the product rule, and then applying (11).The second ingredient is the divergence theorem, (cid:90) ∂ B k ... n k d a k = (cid:90) B k div ( ... ) d v k , (19)where n k is the outward normal vector of boundary ∂ B k . The third ingredient is the localizationtheorem, (cid:90) P ... d v = 0 ∀ P ⊂ B ⇔ ... = 0 ∀ x ∈ B , (20)that can be equally applied to surface integrals.Some of the field quantities appearing in the following balance laws can be expressed per mass,bond, current volume, current area, reference volume or reference area. The notation distin-guishing these options is summarized in Tab. 2.quantity per mass/bond per curr. vol./area per ref. vol./area in totalinternal energy u k , u c U k , U c U k , U c U Helmholtz free energy ψ k , ψ c Ψ k , Ψ c Ψ k , Ψ c –entropy s k , s c S k , S c S k , S c S reaction rate r c R c R c –chemical potential µ c M c M c –body force / traction ¯ b k ¯ t k , t c ¯ t k , t c –heat source / flux ¯ r k ¯ q k , q c ¯q k , q c – Table 2:
Notation for various quantities: • k are bulk quantities expressed per mass or volume ofbody k , while • c are contact surface quantities expressed per bond or area. All a k ( a = u, ψ, s )satisfy A k = ρ k a k and A k = J k A k = ρ k a k , while all a c ( a = u, ψ, s, r, µ ) satisfy A c = n a c and A c = J s1 A c = N a c . Not all combinations are required here: the reaction rate, chemicalpotential, contact tractions and contact heat flux are only defined on the contact surface, whilethe body force and heat source is only defined in the bulk. Assuming no mass sources, the mass balance of each body is given by the statementdd t (cid:90) P k ρ k d v k = 0 ∀ P k ⊂ B k . (21)9pplying (17) and (20), this leads to the local balance law˙ ρ k + ρ k div v k = 0 ∀ x k ∈ B k . (22)Due to (10), this ODE is solved by ρ k = ρ k /J k , where ρ k := ρ k (cid:12)(cid:12) t =0 is the initial mass density.In order to model chemical bonding, the interacting surfaces are considered to have a certainbinder density n k (the number of potential bond sites per current area) composed of bondedsites and unbonded sites, i.e. n k = n b k + n ub k . (23)The number of bonding sites is considered to be conserved, i.e.dd t (cid:90) P k n k d a k = 0 ∀ P k ⊂ S k , (24)implying ˙ n k + n k div s v k = 0 ∀ x k ∈ S k , (25)due to (18) and (20). Due to (11), this ODE is solved by n k = N k /J s k , where N k := n k (cid:12)(cid:12) t =0 isthe initial bond density. Two cases will be considered in the following, see Fig. 3:(a) Long-range interaction (between non-touching surfaces), where each bonding site on onesurface can interact with all other bonding sites on the other surface.(b) Short-range interactions (between very close or even touching surfaces), where each bondingsite on one surface can only bond to a single bonding site on the other surface.a. b. Figure 3:
Chemical contact: a. Long-range interactions between all bonding sites of twonon-touching surfaces. Here n d a = 3 and n d a = 4. b. Short-range interactions betweenneighboring bonding sites of two close surfaces. Here n b1 d a = n b2 d a = 4 bonds have formed,while n ub1 d a = 3 and n ub2 d a = 2 sites are unbonded.In the first case we assume n ub k = 0 such that n k = n b k . Then no further equation is needed.In the second case a further equations is needed for n b k . Since in this case the total number ofbonded sites is the same on both surfaces, we must have (cid:90) S n b1 d a = (cid:90) S n b2 d a . (26)If the two surfaces truly touch, i.e. S = S =: S c , this implies that n b1 = n b2 =: n b due tolocalization (since (26) is still true for any subregion P c ⊂ S c ). This case is considered in the Long-range interactions are understood to be interactions between non-touching surfaces S (cid:54) = S here. Theseinclude electrostatic and van-der-Waals interactions, even if the latter are classified as short-range in other works. R c such that the mass balance for thebonded species (considering no surface diffusion) is given bydd t (cid:90) P c n b d a = (cid:90) P c R c d a ∀ P c ⊂ S c , (27)which gives ˙ n b + n b div s v = R c ∀ x ∈ S c , (28)where v := v = v is the common velocity required to enable chemical bonding. For theunbonded species on the two surfaces, we have the two balance lawsdd t (cid:90) P k n ub k d a k = − (cid:90) P k R c d a k ∀ P k ⊂ S k , k = 1 , n ub k + n ub k div s v k = − R c ∀ x k ∈ S k , (30)where still v = v (as long as n b > k ) is redundant.For convenience, the non dimensional phase field φ := n b n (31)is introduced such that (28) can be rewritten into n ˙ φ = R c ∀ x ∈ S c , (32)using Eq. (25). Eq. (32) is the evolution law for the chemical contact state. Remark 3.1:
As we are focusing on solids, the derivation of (32) assumes no surface mobility(or diffusion) of the bonds. In the context of fluidic membranes, the surface mobility of bondsis usually accounted for, e.g. see Brochard-Wyart and de Gennes (2002) and Freund and Lin(2004).
The linear momentum balance for the entire two-body system is given by (cid:88) k =1 dd t (cid:90) B k ρ k v k d v k = (cid:88) k =1 (cid:20) (cid:90) B k ρ k ¯ b k d v k + (cid:90) ∂ t B k ¯ t k d a k (cid:21) , (33)where ¯ b k and ¯ t k are prescribed body forces and surface tractions. The latter are prescribedon the Neumann boundary ∂ t B k ⊂ ∂ B k that is disjoint from contact surface S k . The bondingevents, described by Eq. (32), are not considered to affect the momentum of the system.If the two bodies are cut apart, the additional contact interaction traction t c k needs to be takeninto account on the interface (see Fig. 1). The individual momentum balance for every part P k ⊂ B k ( k = 1 ,
2) then readsdd t (cid:90) P k ρ k v k d v k = (cid:90) P k ρ k ¯ b k d v k + (cid:90) ∂ P k t k d a k ∀ P k ⊂ B k , (34) Since the designation of body 1 and 2 is otherwise arbitrary, Eq. (31) thus provides meaning to body 1 asthe master body – the body with respect to which contact is formulated (Hallquist et al., 1985). t k is the traction on surface ∂ P k that contains the cases t k = ¯ t k on ∂ t B k , t k = t c k on S k . (35)Applying (17), (19) and (20) leads to the corresponding local form ρ k ˙ v k = div σ k + ρ k ¯ b k ∀ x k ∈ B k , (36)where σ k is the Cauchy stress at x k that is defined by the formula σ k n k = t k . (37)Using (34), Eq. (33) simplifies to (cid:90) S t c1 d a + (cid:90) S t c2 d a = . (38)This simply states that the net contact forces are in equilibrium. It is the correspondingstatement to Eq. (26) for mechanical contact. If the two surfaces touch, i.e. S = S =: S c andd a = d a , (38) implies that t c1 = − t c2 =: t c ∀ x ∈ S c . (39)Hence there is no jump in the contact traction. Such a jump only arises if an interface stress(e.g. surface tension) is considered.The angular momentum balances for the individual bodies and the entire two-body system canbe written down analogously to (34) and (33), respectively. As long as no distributed bodyand surface moments are considered, the angular momentum balance of each body has thewell-known consequence σ T k = σ k (Chadwick, 1976), while global angular momentum balanceimplies (cid:90) S x × t c1 d a + (cid:90) S x × t c2 d a = , (40)analogously to (38). This statement is relevant for separated surfaces ( S (cid:54) = S ), but in thecase of touching surfaces ( S = S ), it leads to the already known traction equivalence (39) atcommon contact points x = x . The energy balance for the entire two-body system isd E d t = (cid:88) k =1 (cid:20) (cid:90) B k ρ k ¯ r k d v k + (cid:90) ∂ q B k ¯ q k d a k + (cid:90) B k ρ k v k · ¯ b k d v k + (cid:90) ∂ t B k v k · ¯ t k d a k (cid:21) , (41)where the total energy in the system, E = K + U , (42)is given by the kinetic energy K = 12 (cid:90) B ρ v · v d v + 12 (cid:90) B ρ v · v d v (43)12nd the internal energy U = (cid:90) B ρ u d v + (cid:90) B ρ u d v + U c , (44)which accounts for the individual energies u k = u k ( x k ) in B k and the contact energy U c . U c can describe long range surface interactions, as discussed in Remark 3.4, or it can correspondto the energy of a third medium residing in the contact interface, e.g. a thin film of lubricantsor wear particles. In the latter case, assuming a sufficiently thin film, U c can be expressed asthe surface integral U c := (cid:90) S c n u c d a , (45)where u c is the contact energy per bond on master surface S . Further, ¯ r k and ¯ q k in (41)denote external heat sources in B k and external heat influxes on ∂ q B k ⊂ ∂ B k , respectively.The Neumann boundary ∂ q B k is considered disjoint from contact surface S k . An external heatsource on the interface is not required here. As will be seen later, the present setup alreadyaccounts for the heat from interfacial friction and interfacial reactions.If the two bodies are cut apart, the additional contact heat influx q c k needs to be taken intoaccount on the interface (see Fig. 1), leading to the individual energy balances for each bodydd t (cid:90) P k ρ k e k d v k = (cid:90) P k ρ k ¯ r k d v k + (cid:90) ∂ P k q k d a k + (cid:90) P k v k · ρ k ¯ b k d v k + (cid:90) ∂ P k v k · t k d a k ∀ P k ⊂ B k , (46)where e k := u k + v k · v k / t k satisfies (35). Further, q k is the heat influx on surface ∂ P k that contains the cases q k = ¯ q k on ∂ q B k ,q k = q c k on S k . (47)Introducing the heat flux vector q k at x k ∈ B k that is defined by Stokes formula − q k · n k = q k , (48)and using (37), the divergence theorem (19) can be applied to obtain (cid:90) ∂ P k q k d a k = − (cid:90) P k div q k d v k (49)and (cid:90) ∂ P k v k · t k d a k = (cid:90) P k (cid:0) v k · div σ k + σ k : D k (cid:1) d v k , (50)where D k is the symmetric velocity gradient from (9). Using (17), (20) and (36), the local formof (46) thus becomes ρ k ˙ u k = ρ k ¯ r k − div q k + σ k : D k ∀ x k ∈ B k . (51)With this and Eq. (36), the combined balance statement (41) simplifies to˙ U c + (cid:90) S (cid:0) q c1 + t c1 · v (cid:1) d a + (cid:90) S (cid:0) q c2 + t c2 · v (cid:1) d a = 0 , (52)which is the corresponding statement to Eqs. (26) and (38) for thermal contact. Here˙ U c = (cid:90) S c n ˙ u c d a , (53)13ollows from (45) for bond conservation. If the two surfaces are very close or even touch, i.e. S = S =: S c and d a = d a (i.e. for very thin interfacial media), this implies that (since S c can be considered an arbitrary subregion of the contact surface) n ˙ u c + q c1 + q c2 + t c · ( v − v ) = 0 ∀ x ∈ S c , (54)due to (39). Eq. (54) states that the energy rate ˙ u c and the mechanical dissipation t c · ( v − v )cause the heat influxes q c k on S k . Eq. (54) is equivalent to Eq. (6.63) of Laursen (1999). Remark 3.2:
For sticking contact (e.g. during chemical bonding), v = v . Hence, the me-chanical dissipation t c · ( v − v ) disappears. Remark 3.3:
Note, that contrary to the contact tractions governed by (39), the contact heatflux according to (54) can have a jump across the interface. This has also been recently exploredby Javili et al. (2014).
Remark 3.4:
For long-range interactions between two surfaces, U c can be written as U c := (cid:90) S (cid:90) S n n u d a d a , (55)where u = u ( x , x ) is an interaction energy defined between points on the two surfaces.For bond conservation this leads to˙ U c = (cid:90) S (cid:90) S n n ˙ u d a d a , (56)in (52). Localization in the form of (54) is not possible for long-range interaction so that Eq. (52)remains the only governing equation now. The entropy balance for the entire two-body system can be written as d S d t = (cid:88) k =1 (cid:20) (cid:90) B k ρ k (cid:0) η e k + η i k (cid:1) d v k + (cid:90) ∂ q B k ¯˜ q k d a k (cid:21) + (cid:90) S c n η ic d a , (57)where the total entropy in the system, S := (cid:90) B ρ s d v + (cid:90) B ρ s d v + (cid:90) S c n s c d a , (58)accounts for the individual entropies s k in B k and the contact entropy s c that is associated withan interfacial medium. Further, η e k and η i k are the external and internal entropy productionrates in B k , ¯˜ q k is the entropy influx on the heat flux boundary ∂ q B k , and η ic is the internalentropy production rate of the interface. An external entropy production rate is not needed forthe interface, as long as no heat source is considered in the interface. Like u c in (45), s c and η ic are bond-specific. In Laursen (1999) the contact traction t c and heat fluxes q c k are defined with opposite sign, and (54) is writtenper unit reference area. e.g. by coarse-graining the molecular interactions across the two surfaces (Sauer and Li, 2007b) where – as in Eq. (31) – body B is the chosen master body, see footnote 6
14f the two bodies are cut apart, the additional contact entropy influx ˜ q c k needs to be taken intoaccount on the interface, leading to the individual entropy balances ( k = 1 , t (cid:90) P k ρ k s k d a = (cid:90) P k ρ k (cid:0) η e k + η i k (cid:1) d v + (cid:90) ∂ P k ˜ q k d a ∀ P k ⊂ B k , (59)where ˜ q k is the entropy influx on surface ∂ P k that contains the cases˜ q k = ¯˜ q k on ∂ q B k , ˜ q k = q c k on S k . (60)Introducing the entropy flux ˜ q k in body B k defined by − ˜ q k · n k = ˜ q k , (61)theorems (17), (19) and (20) can be applied to (59) to give the local form ρ k ˙ s k = ρ k η e k + ρ k η i k − div ˜ q k ∀ x k ∈ B k . (62)Plugging this equation into the total entropy balance (41), lets us simplify this balance into (cid:90) S c n (cid:0) ˙ s c − η ic (cid:1) d a + (cid:90) S ˜ q c1 d a + (cid:90) S ˜ q c2 d a = 0 . (63)This is the corresponding entropy statement to Eqs. (26), (38) and (52). If the two surfacestouch, i.e. S = S =: S c , it implies that n ˙ s c = n η ic − ˜ q c1 − ˜ q c2 ∀ x ∈ S c . (64)The second law of thermodynamics states that η i k ≥ , η ic ≥ . (65)Eq. (64) then becomes equivalent to Eq. (6.66) of Laursen (1999). This section derives the general constitutive equations for the two bodies and their contactinterface – as they follow from the internal energy and the second law of thermodynamics. Ingeneral, the internal energy is a function of the mechanical, chemical and thermal state of thesystem that is characterized by the deformation, the bond state and the entropy. Introducingthe Helmholtz free energy, the thermal state can be characterized by the temperature insteadof the entropy.
For each B k , the Helmholtz free energy (per mass), ψ k = u k − T k s k (66)is introduced as the chosen thermodynamic potential. Thus, T k ˙ s k = ˙ u k − ˙ ψ k − ˙ T k s k . (67)15nserting (51), then leads to T k ρ k ˙ s k = σ k : D k + ρ k ¯ r k − div q k − ρ k (cid:0) ˙ ψ k + ˙ T k s k (cid:1) . (68)For the interface, the Helmholtz free energy (per bond) is introduced as ψ c = u c − T c s c , (69)where T c is the interface temperature introduced in Sec. 2 (see Fig. 1). Thus, T c ˙ s c = ˙ u c − ˙ ψ c − ˙ T c s c . (70)Inserting (54), then leads to n T c ˙ s c = − t c · ( v − v ) − q c1 − q c2 − n (cid:0) ˙ ψ c + ˙ T c s c (cid:1) . (71)In this study, the Helmholtz free energy within B k is considered to take the functional form ψ k = ψ k (cid:0) E e k , T k (cid:1) , (72)where E e k is the elastic Green-Lagrange strain tensor introduced by Eq. (4). For the interface,the Helmholtz free energy is considered to take the form ψ c = ψ c (cid:0) g e , T c , φ (cid:1) , (73)where g e is the reversible (i.e. elastic) part of the gap as introduced by Eq. (15). From Eqs. (72)and (73) follow ˙ ψ k = ∂ψ k ∂ E e k : ˙ E e k + ∂ψ k ∂T k ˙ T k (74)and ˙ ψ c = ∂ψ c ∂ g e · ˙ g e + ∂ψ c ∂T c ˙ T c + ∂ψ c ∂φ ˙ φ . (75)The following subsections proceed to derive the general constitutive equations for the two bodiesand their interaction based on this. Inserting (62) and (68) into (65.1), gives ρ k η i k = ρ k (cid:18) ¯ r k T k − η e k (cid:19) − div (cid:18) q k T k − ˜ q k (cid:19) − q k · grad T k T k + σ k : D k T k − ρ k T k (cid:0) ˙ ψ k + ˙ T k s k (cid:1) ≥ . (76)Since this is true for any ¯ r k , q k , T k , v k and ˙ ψ k , we can identify the relations η ek = ¯ r k T k and ˜ q k = q k T k (77)for the external entropy source and the entropy flux. We are then left with the well knowdissipation inequality ρ k η i k = σ k : D k T k − q k · grad T k T k − ρ k T k (cid:0) ˙ ψ k + ˙ T k s k (cid:1) ≥ , (78)which, in view of (4), (74) and σ k : D k = S k : ˙ E k /J k , (79)16here S k := J k F − k σ k F − T k is the second Piola-Kirchhoff stress, becomes T k ρ k η i k = 1 J k (cid:18) S k − ∂ Ψ k ∂ E e k (cid:19) : ˙ E e k − ρ k (cid:18) s k + ∂ψ k ∂T k (cid:19) ˙ T k + σ k : D i k − q k · grad T k T k ≥ . (80)Here Ψ k := J k ρ k ψ k = ρ k ψ k is the Helmholtz free energy per reference volume, and D i k is theinelastic part of D k that is related to E i k via (79). Since (80) is true for any E k and T k , we findthe constitutive relations S k = ∂ Ψ k ∂ E e k , s k = − ∂ψ k ∂T k , σ k · D i k ≥ , q k · grad T k ≤ . (81)Inserting (74) and (81) back into (68) then leads to the entropy evolution equation T k ρ k ˙ s k = σ k : D i k + ρ k ¯ r k − div q k . (82)Here, the term σ k : D i k + ρ k ¯ r k can be understood as an effective heat source.The equations in (81) are the classical constitutive equations for thermo-mechanical bodies.They happen to be very similar to the contact equations obtained in the following section. Remark 4.1:
The above derivation considers elastic and inelastic strains according to (4),which are analogous counterparts to the elastic and inelastic gap, see (15), required for a generalcontact description. On the other hand, a stress σ k decomposition is not considered, whichcorresponds to a Maxwell-like model for the elastic and inelastic behavior. For other constitutivemodels, that are not considered here, elastic and inelastic stress contributions can appear andneed to be separated. Next we examine the case that there is a common contact interface S = S = S c . Inserting(64) and (71) into (65.2) and using (13) and (77.2), we find n η ic = (cid:18) T − T c (cid:19) q c1 + (cid:18) T − T c (cid:19) q c2 − t c · ˙ g T c − n T c (cid:0) ˙ ψ c + ˙ T c s c (cid:1) ≥ . (83)Further inserting (15) and (75), and introducing the nominal contact traction t c := J s1 t c , wefind n η ic = (cid:18) T − T c (cid:19) q c1 + (cid:18) T − T c (cid:19) q c2 − T c J s1 (cid:18) t c + ∂ Ψ c ∂ g e (cid:19) · ˙ g e − t c · ˙ g s T c − n T c (cid:18) s c + ∂ψ c ∂T c (cid:19) ˙ T c − n T c µ c ˙ φ ≥ , (84)where Ψ c := J s1 n ψ c = N ψ c and where µ c := ∂ψ c ∂φ (85)denotes the chemical potential associated with the interface reactions. Since inequality (84) istrue for any T k , g e , g s and φ , we find the constitutive relations t c = − ∂ Ψ c ∂ g e , t c · ˙ g s ≤ s c = − ∂ψ c ∂T c (87)for the interfacial entropy, µ c R c ≤ R c , and (cid:18) T − T c (cid:19) q c1 + (cid:18) T − T c (cid:19) q c2 ≥ , (89)for the contact heat fluxes. Multiplying by T , T and T c (that are all positive), the laststatement can be rewritten into T (cid:0) T c − T (cid:1) q c1 + T (cid:0) T c − T (cid:1) q c2 ≥ . (90)Since this has to be satisfied for any q c k , setting either q c2 = 0 or q c1 = 0 yields the two separateconditions ( k = 1 , (cid:0) T c − T k (cid:1) q c k ≥ , (91)for q c k . They are, for example, satisfied for the simple and well-known linear heat transfer law q c k = h k (cid:0) T c − T k (cid:1) , (92)where the constant h k > B k and the interfacialmedium. Introducing the mean influx into B k , q cm := q c1 + q c2 , (93)and the transfer flux from B to B (see Fig. 2), q ct := q c1 − q c2 , (94)such that q c1 = q cm + q ct and q c2 = q cm − q ct , one can also rewrite inequality (90) into (cid:0) ( T + T ) T c − T T (cid:1) q cm + (cid:0) T − T (cid:1) T c q ct ≥ . (95)Since this also has to be true for q cm = 0, we find the further condition (cid:0) T − T (cid:1) q ct ≥ , (96)which is, for example, satisfied by q ct = h (cid:0) T − T (cid:1) , (97)where the constant h > B and B .From (54) and (13) we can further find that the mean influx is given by q cm = − (cid:0) t c · ˙ g + n ˙ u c (cid:1) , (98)which in view of (15), (70), (75), (86.1) and (87) becomes q cm = − (cid:0) t c · ˙ g s + µ c R c + n ˙ s c T c (cid:1) , (99)18.e. the mean heat influx is caused by mechanical dissipation (from friction), chemical dissipation(from reactions), and entropy changes at the interface. The three terms are composed of theconjugated pairs identified in table 1. While the first two terms are always positive (due to(86.2) and (88)) and thus lead to an influx of heat into the bodies, the third term can bepositive or negative. It thus allows for a heat flux from the bodies into the interface, wherethe heat is stored as internal energy, see the example in Sec. 6.2.2. If there is no mechanicaldissipation we can classify q cm > < Remark 4.2:
A special case is s c = 0 ∀ t , see the example in Sec. 6.2.1. In this case ˙ s c = 0,and 2 q cm = − t c · ˙ g s − n µ c ˙ φ . Remark 4.3:
Interface equation (99) is analogous to the bulk equation (82). However, (82)contains an explicit heat source, while (99) contains a chemical dissipation term. In principle,an explicit heat source can also be considered on the interface, while a chemical reaction canalso be considered in the bulk. This would lead to a complete analogy between (99) and (82).
Remark 4.4:
In the works of Agelet de Saracibar (1998), Pantuso et al. (2000) and Xing andMakinouchi (2002) the heat flux is not split into mean and transfer fluxes. Instead, laws areproposed for the relative contributions going into the two bodies based on the separate transfercoeficients h and h for temperature jumps ( T − T c ) and ( T c − T ). Pantuso et al. (2000) alsoaccount for a loss of the interface heat to other bodies, like an interfacial gas. Remark 4.5:
The nominal traction t c := J s1 t c is not a physically attained traction. Onlythe traction t c = − n ∂ψ c /∂ g e is. Since n (cid:54) = n ( g e ), one can also write t c = − ∂ Ψ c /∂ g e , forΨ c := n ψ c . Remark 4.6:
Multiplying (99) by the area change J s1 yields q cm := J s1 q cm = − (cid:0) t c · ˙ g s + M c ˙ φ + ˙ S c T c (cid:1) , (101)where S c := N s c and M c := N µ c are the contact entropy and chemical potential per referencearea. Due to Ψ c := N ψ c , (85) and (87), they follow directly from S c = − ∂ Ψ c ∂T c , M c = ∂ Ψ c ∂φ . (102) In case of non-touching (long-range) interactions, the two surfaces remain distinct, i.e. S (cid:54) = S .For simplification we consider that the surfaces have the uniform temperatures T and T , andthat the space between between S and S has no mass, temperature or entropy, i.e. s c ≡ φ ≡
0. The Helmholtzfree energy of the interface then is ψ = u , (103)so that ˙ u = ˙ ψ . The Helmholtz free energy is now considered to take the form ψ = ψ (cid:0) x , x ) , (104)19uch that ˙ ψ = ∂ψ ∂ x · v + ∂ψ ∂ x · v . (105)Inserting this into (52) yields (cid:90) S (cid:18) t c1 + n ∂ψ c2 ∂ x (cid:19) · v d a + (cid:90) S (cid:18) t c2 + n ∂ψ c1 ∂ x (cid:19) · v d a + (cid:90) S q c1 d a + (cid:90) S q c2 d a = 0 (106)where ψ c1 := (cid:90) S n ψ d a , ψ c2 := (cid:90) S n ψ d a . (107)Noting that v k and q c k are arbitrary and can be independently taken as zero, we find t c1 = − n ∂ψ c2 ∂ x , t c2 = − n ∂ψ c1 ∂ x , (108)for the interaction tractions, and (cid:90) S q c1 d a + (cid:90) S q c2 d a = 0 , (109)for the interaction heat fluxes. At the same time (63) and (65.2) yield (cid:90) S q c1 T d a + (cid:90) S q c2 T d a ≥ . (110)If the temperature is constant, one can multiply this by T T and insert (109) to get( T − T ) (cid:90) S q c1 d a ≥ , (111)which is the corresponding statement to (96) for long-range interactions. The traction laws in(108) are identical to those obtained from a variational principle (Sauer and De Lorenzis, 2013). This section lists examples for the preceding constitutive equations for contact and long-rangeinteraction accounting for the coupling between mechanical, thermal and chemical fields. Theexamples are based on the material parameters defined in Tab. 3.
A simple example for the contact potential (per unit reference area) is the quadratic function Ψ c ( g e , T c , φ ) = 12 g e · E c g e − C c T ( T c − T ) + K c φ − , (112)with E c = E n n ⊗ n + E t i , (113) where n = n is the surface normal of the master body E n , E t normal & tangential contact stiffness N/m C c interfacial heat capacity J/(K m ) K c bond energy density J/m T reference temperature K t maxn , t maxt normal & tangential bond strength N/m g , ψ reference distance & reference energy m, J µ , µ static and dynamic friction coefficient 1 η dynamic interface viscosity N s/m h heat transfer coefficient J/(K s m ) C r reaction rate constant 1/(J s m ) Table 3:
Material parameters for the contact interface.where i := − n ⊗ n (114)is the surface identity on S c . Here, E n and E t denote the normal and tangential contact stiffness(e.g. according to a penalty regularization), C c denotes the contact heat capacity and K c denotesthe bond energy. (112) corresponds to an extension of the models of Johansson and Klarbring(1993); Str¨omberg et al. (1996) and Oancea and Laursen (1997) to chemical bonding. Theparameters E • , C c and K c are defined here per undeformed surface area. They can be constantor depend on the contact state variables, i.e. E • = E • ( T c , φ ), C c = C c ( g e , φ ) and K c = K c ( g e , T c ).If they are constant, (86.1) and (102) yield the contact traction, entropy and chemical potential(per reference area) t c = E c g e , S c = C c (cid:0) T c /T − (cid:1) , M c = K c ( φ − . (115)If E • , C c and K c are not constant, further terms are generated from (86.1) and (102). Anexample is given in (145). Remark 5.1:
The first term in (112) is zero in an ideal setting with locally perfect contact. Inthat case, the elastic gap g e is zero, while E c approaches infinity. If there is no mass associatedwith the contact interface, its heat capacity C c , and hence the second term in (112), is also zero.On the other hand, non-zero g e can be used to capture surface roughness during contact, whilenon-zero C c can be used to capture the heat capacity of trapped wear debris (Johansson andKlarbring, 1993). Remark 5.2:
The last part in (112) corresponds to a classical surface energy. In the unbondedstate, ( φ = 0) the free surface energy is K c / Remark 5.3:
Choice (112) has minimum energy at full bonding ( φ = 1). Since 0 ≤ φ ≤ µ c ≤ K c >
0. Thus R c ≥ φ ≥ φ is monotonically increasing over time. Onlydebonding, addressed in Sec. 5.2, leads to a decrease in φ . See examples in Sec. 6. Remark 5.4:
One can modify the last term in (112), such that the minimum energy state isat φ = 0 (full debonding) or at 0 < φ <
1. The latter case implies that chemical equilibrium isa balance of binding and unbinding reactions, like in the model of Bell (1978).21 emark 5.5:
The contact traction t c = t c /J s1 can be decomposed into the contact pressure p c = − n · t c and tangential contact traction t t = i t c (such that t c = − p c n + t t ). From (113)-(115.1) and (cid:15) • := E • /J s1 thus follow p c = − (cid:15) n g n and t t = (cid:15) t g t , where g n := n · g e is the (elastic)normal gap and g t := i g e the (elastic) tangential gap. Remark 5.6:
The normal contact contribution in (112) and (115.1) is only active up to adebonding limit. Beyond that it becomes inactive, i.e. by setting E n = 0. This is discussed inthe following section. An adhesion or debonding limit implies a lower bound on the contact pressure p c , i.e. − p c ≤ t maxn , (116)where the tensile limit (or bond strength) t maxn can depend on the contact state, i.e. t maxn = t maxn ( T c , φ ). An example is t maxn = φ T T c t max0 . (117)According to this, the bond strength is increasing with φ and decreasing with T c .Once the limit is reached, sudden debonding occurs, resulting in φ = 0. This makes debonding(mechanically) irreversible (unless t maxn = 0). Fig. 4a shows a graphical representation of this. a. b. Figure 4:
Normal contact behavior: a. Irreversible debonding according to (115.1) and (116);b. Reversible adhesion according to (118); Figure b is adopted from Sauer (2016b).Debonding can also be described in the context of cohesive zone models. An example is theexponential cohesive zone modelΨ c = − t max (cid:0) g e + g (cid:1) exp (cid:18) − g e g (cid:19) , g e := (cid:107) g e (cid:107) , (118)where t max and g are material parameters that can depend on T c and φ . Eq. (118) is anadaption of the model of Xu and Needleman (1993). It leads to the contact traction t c = − t max exp (cid:18) − g e g (cid:19) g e g , (119) Debonding can also be modeled as compliant, such that the irreversible path C → D has a finite negativeslope. t max is a function of φ thatdrops to zero beyond some g e .Model (118) is similar to adhesion models for non-touching contact discussed next. An example for non-touching contact interactions according to Sec. 4.4 is the interaction po-tential ψ (¯ g k ) = ψ (cid:20) (cid:18) g ¯ g k (cid:19) − (cid:18) g ¯ g k (cid:19) (cid:21) , ¯ g k := (cid:107) x k − x (cid:96) (cid:107) > , (120)where ψ and g are constants and where either k = 1 and (cid:96) = 2 or k = 2 and (cid:96) = 1. From(107) and (108) then follows t c k ( x k ) = − n k (cid:90) S (cid:96) n (cid:96) ∂ψ ∂ x k d a (cid:96) , (121)with − ∂ψ ∂ x k = ψ g (cid:20) (cid:18) g ¯ g k (cid:19) − (cid:18) g ¯ g k (cid:19) (cid:21) ¯ g k , ¯ g k := x k − x (cid:96) ¯ g k . (122)These expressions are valid for general surface geometries and arbitrarily long-range interactions.For short-range interactions between locally flat surfaces, these expressions can be integratedanalytically to give (Sauer and De Lorenzis, 2013) t c k ( x k ) = 2 π n n g n ψ ( g n ) n p , (123)where g n is the normal gap between point x k and surface S (cid:96) and n p is the surface normal of S (cid:96) at x p . Remark 5.7:
The surface interaction potential (120) can be derived from the classical Lennard-Jones potential for volume interactions (Sauer and Li, 2007b; Sauer and Wriggers, 2009).
Remark 5.8:
Example (120) is only valid for separated bodies (¯ g k > ψ that admit penetrating bodies (with neg-ative distances). Examples, such as a penalty-type contact formulation, are given in Sauerand De Lorenzis (2013). Computationally, (121) leads to the so-called two-half-pass contactalgorithm , which is thus a thermodynamically consistent algorithm. Remark 5.9:
Eq. (121) also applies to the Coulomb potential for electrostatic interactions(Sauer and De Lorenzis, 2013). However, in order to account for the full electro-mechanicalcoupling, the present theory needs to be extended.
Remark 5.10:
We note again that for non-touching contact, the tractions t c1 and t c2 generallyonly satisfy global contact equilibrium (38), but not local contact equilibrium (39). Remark 5.11:
Eq. (123) is a pure normal contact model that does not produce local tangentialcontact forces. Tangential contact forces only arise globally when t c k acts on rough surfaces. Similar to the debonding limit (116), a sticking limit implies a bound on the tangential traction t t , i.e. (cid:107) t t (cid:107) ≤ t maxt , (124)23here the limit value t maxt is generally a function of contact pressure, temperature and bondingstate, i.e. t maxt = t maxt ( p c , T c , φ ). It is reasonable to assume that it is monotonically increasingwith p c and φ and decreasing with T c , e.g. t maxt = µ p c , (125)where µ = (cid:0) φ µ b0 + (1 − φ ) µ ub0 (cid:1) T T c (126)is a temperature- and bonding state-dependent coefficient of sticking friction based on theconstants µ b0 and µ ub0 describing the limits for full bonding and full unbonding, respectively.Once the limit is reached, the bonds break ( φ = 0) and tangential sliding occurs, as is discussedin the following section. Fig. 5 shows a graphical representation of this. Figure 5:
Tangential contact behavior: Sticking, debonding and sliding according to (124)-(126) and (128). The latter two are irreversible.
The simplest viscous (hydrodynamic) friction model satisfying (86.2) is t t = − η ˙ g s , (127)where η is the positive definite dynamic viscosity tensor. In the isotropic case, η = η .One of the simplest dry friction models satisfying (86.2) for p c ≥ t t = − µ p c ˙ g s (cid:107) ˙ g s (cid:107) , (128)where µ is the positive definite coefficient tensor for sliding friction. In the isotropic case, µ = µ . For adhesion, where p c ≥ − t maxn , this extends to t t = − µ (cid:0) p c + t maxn (cid:1) ˙ g s (cid:107) ˙ g s (cid:107) . (129)In the adhesion literature this extension is often attributed to Derjaguin (1934), whereas in soilmechanics it is usually referred to as Mohr-Coulomb’s law. Note that µ t maxn can be taken as Often just referred to as Amontons’ law or Coulomb’s law.
24 new constant. The application of (129) to coupled adhesion and friction in the context ofnonlinear 3D elasticity has been recently considered by Mergel et al. (2019, 2021).
Remark 5.12:
A transition model between dry (rate-independent) and viscous (rate-dependent)sliding friction is Stribeck’s curve, e.g. see Gelinck and Schipper (2000).
Remark 5.13:
The above sliding models can be temperature dependent. An example for atemperature dependent friction coefficient µ = µ ( T c ) is given in Laursen (2002). Temperaturedependent viscosity models are e.g. discussed in Roelands (1966) and Seeton (2006). Remark 5.14:
The present formulation assumes fixed bonding sites that break during slidingfriction. Sliding friction is thus independent from φ . Bonding sites can, however, also be mobileand hence stay intact during sliding. Sliding friction may thus depend on φ . This requires amobility model for the bonds, e.g. following the model of Freund and Lin (2004). Remark 5.15:
The friction coefficient can be considered dependent on the surface deformation,as has been done by Stupkiewicz (2001).
Remark 5.16:
Friction coefficients that are dependent on the sliding velocity and (wear) statehave been considered in the framework of rate and state friction models (Dieterich, 1978; Ruina,1983). Those models are usually based on an additive decomposition of the shear traction t t instead of an additive slip decomposition as is used here. Remark 5.17:
Note that t c causes the mechanical dissipation D mech = − t c · ˙ g s that leads toheating of the contact bodies due to Eq. (99). The mechanical dissipation is D mech = ˙ g s · η ˙ g s for model (127) and D mech = p c ˙ g s · µ ˙ g s / (cid:107) ˙ g s (cid:107) for model (128); see Sec. 6.1 for an example. The simplest heat transfer model satisfying condition (96) is the linear relationship alreadygiven in (97), where h > h → ∞ implies T − T →
0. In general, h can be a function of the contactpressure, gap or bonding state, i.e. h = h ( p c , g n , φ ). Various models have been considered inthe past. Those usually consider h to be additively split into a contribution coming from actualcontact and contributions coming from radiation and convection across a small contact gap.During actual contact ( g n = 0, p c >
0) a simple model is the power law dependency on thecontact pressure, h = h + h c (cid:18) p c H (cid:19) q , (130)where h , h c , H and q are positive constants (Wriggers and Miehe, 1994; Laschet et al., 2004).This model is a simplification of the more detailed model of Song and Yovanovich (1988) thataccounts for microscopic contact roughness. Another model for h is the model by Miki´c (1974).A new model has also been proposed recently by Martins et al. (2016). The dependency of(130) on the bonding state can be, for example, taken as h = φ h b0 + (1 − φ ) h ub0 ,h c = φ h bc + (1 − φ ) h ubc , (131)i.e. assuming that h and h c increase monotonically with φ . Here h b0 , h ub0 , h bc and h ubc are modelconstants. 25ut of contact ( g n > p c = 0, φ = 0), the heat tranfer depends on the contact gap. Consideringsmall g n , Laschet et al. (2004) propose an exponential decay of h with g n according to h = h rad + h gas + ( h − h rad − h gas ) exp( − C trans g n ) , (132)where h rad = c rad ( T + T ) ( T + T ) ε − + ε − − h gas = k gas ( T c ) g n + g ( T c ) (134)correspond to the heat transfer across the gap due to radiation and gas convection, respectively.Here, C trans , c rad , ε and ε are constants, while k gas and g depend on the contact temperature T c .We note that all the above models are consistent with the 2. law as long as h > The simplest reaction rate model satisfying condition (88) is the linear relationship, R c = − C r µ c , (135)where the constant C r ≥ C r = C r ( p c , T c ). Writing C r = n c r , the reaction equation in (32) thus becomes˙ φ = − c r µ c . Alternatively (and in consistency with (88)), c r ≥ R c = C r R T c (cid:18) − exp µ c R T c (cid:19) , (136)where R is the universal gas constant. Remark 5.18:
Plugging example (115.3) into (135) yields R c = c r k c ( n − n b ) with k c = K c /N ,which is a pure forward reaction (see also Remark 5.3). k f := c r k c is then the forward reactionrate coefficient. An example how this could depend on the contact gap is given in Sun et al.(2009). This section presents the analytical solution of three elementary contact test cases: thermo-mechanical sliding, thermo-chemical bonding and thermo-chemo-mechanical debonding. Thosetest cases are for example useful for the verification of a computational implementation. Alltest cases consider two blocks with initial height H and H brought into contact. The energyand temperature change in the contacting bodies is then computed assuming instantaneous heattransfer through the bodies ( q k = ), no heat source in the bulk (¯ r k = 0), no boundary heat fluxapart from q c k (¯ q k = 0), equal heating of the bodies ( T = T = T c ), quasi-static deformation( D k = ), homogeneous deformation ( E k (cid:54) = E k ( x k )), uniform contact conditions ( g (cid:54) = g ( x c )),and fixed contact area ( J sk = 1), such that one can work with the Helmholtz free contact energy26 c = n ψ c . Under these conditions the temperature change within each body is governed bythe energy balance ρ k H k ˙ u k = q cm , (137)that follows directly from integrating (51) over the reference volume, writing d V k = H k d A k on the left hand side and applying the divergence theorem on the right. Considering the bulkenergy ψ k = ψ mech − c k T (cid:0) T k − T (cid:1) , (138)where c k is the heat capacity per unit mass, leads to u k = u mech + c k T (cid:0) T k − T (cid:1) , (139)due to (66) and (81). For quasi-static deformations then follows ˙ u k = c k T k ˙ T k /T . Insertingthis into (137) then gives C k T T c ˙ T c = q cm , (140)where C k := H k ρ k c k is the heat capacity per unit contact area satisfying C = C . The first test case illustrates thermo-mechanical coupling by calculating the temperature risedue to mechanical sliding. The test case is illustrated in Fig. 6a: Two blocks initially atmechanical rest and temperature T are subjected to steady sliding with the relative slidingvelocity magnitude v := (cid:107) ˙ g s (cid:107) . Following Str¨omberg et al. (1996) and Oancea and Laursen a. b. Figure 6:
Sliding thermodynamics: a. model problem; b. temperature rise during sliding(1997), the contact energy Ψ c = 12 g e · (cid:15) g e − C c T (cid:0) T c − T (cid:1) (141)is used. Eq. (141) leads to the entropy given in (115.2). The mean influx, given by (99) and(101), then becomes 2 q mc = D mech − C c T T c ˙ T c , (142) J s = 1 leads to U c = U c , S c = S c , Ψ c = Ψ c , etc. D mech = ηv , according to model (127) and D mech = µpv according to model (128).During stationary sliding ( v = const.) the mechanical deformation is time-independent. From(140) thus follows T c ˙ T c = T τ , with τ := C + C + C c D mech T , (143)where τ is the time scale of the temperature rise. Integrating this from the initial condition T c (0) = T leads to the temperature rise T c ( t ) = T (cid:112) t/τ , (144)which is shown in Fig. 6b. The second test case illustrates thermo-chemical coupling by calculating the temperature changedue to chemical bonding. The test case is illustrated in Fig. 7a: Two blocks initially unbondedand at temperature T are bonding and changing temperature. Now, the contact energy a. b. Figure 7:
Bonding thermodynamics: a. model problem; b. bonding function φ ( t ).Ψ c = K c ( T c )2 ( φ − , with K c ( T c ) = K + K T T , (145)is used, which, due to (102), leads to the chemical potential and entropy M c = K c ( T c ) ( φ − ,S c = − K T c T ( φ − . (146)This in turn leads to the internal energy U c = 12 ( φ − (cid:18) K − K T T (cid:19) , (147)and the bonding ODE ˙ φ = 1 τ (1 − φ ) , with τ = n c r ( T c ) K c ( T c ) , (148)according to (32), (69) and (135). ODE (148) can only be solved with knowledge about T c = T c ( t ). 28 .2.1 Exothermic bonding Exothermic bonding occurs for K = 0 (which implies U c = Ψ c ). Considering c r = const.,the bonding ODE becomes independent of T c and can be integrated from the initial condition φ (0) = 0 to give φ ( t ) = 1 − e − t/τ , (149)where τ = n / ( c r K ) is the time scale of the exothermic bonding reaction. Solution (149) isshown in Fig. 7b. The mean influx, given in (99) and (101), then becomes q cm = K τ e − t/τ . (150)It satisfies the exothermic condition q cm > C k T c ˙ T c = K T τ e − t/τ , (151)which can be integrated from the initial condition T c (0) = T to give the temperature rise T c ( t ) = T (cid:114) κ (cid:16) − e − t/τ (cid:17) , with κ := K T C k , (152)shown in Fig. 8b for various values of κ . a. b. Figure 8:
Bonding thermodynamics: a. heat influx q cm ( t ) and b. temperature rise T c ( t ) duringexothermic bonding Endothermic bonding occurs for K = 0 in the above model (which implies U c = − Ψ c accordingto (145) and (147)). The mean influx, given in (99) and (101), now becomes q cm = K T (1 − φ ) (cid:16) T c ˙ T c − τ T (cid:17) . (153)In order to solve (148), a temperature dependent reaction rate is considered in the form c r = c r0 T /T with c r0 = const. Thus (149) is still the solution of bonding ODE (148). The timescale now becomes τ = n / ( c r0 K ). From (140) now follows˙ T c T c = 1 τ κ e − t/τ κ e − t/τ − , with κ := K T C k . (154)29ntegrating this from the initial condition T c (0) = T gives the temperature drop T c ( t ) = T (cid:114) − κ − κ e − t/τ , (155)where the parameter κ must be smaller than unity for the temperature to remain physical.According to (153), this now leads to q cm ( t ) = − K τ − κ (1 − κ e − t/τ ) e − t/τ , (156)and satisfies q cm <
0. Fig. 9 shows the energy outflux and temperature drop of the contactinterface for the endothermic case. a. b.
Figure 9:
Bonding thermodynamics: a. heat influx q cm ( t ) and b. temperature drop T c ( t ) duringendothermic bonding The last test case illustrates thermo-chemo-mechanical coupling by calculating the temperaturechange due to mechanical debonding. The test case is illustrated in Fig. 10a: Two blocks initiallybonded and at temperature T are pulled apart leading to debonding and rising temperature.Now the contact energy densityΨ c = − C c T ( T c − T ) + K φ − (157)is used. When the bond breaks, elastic strain energy is converted into surface energy and kineticenergy. The former corresponds to the stored bond energy (and is given by the second termin (157)). If viscosity is present, the latter then transforms into thermal energy. Waiting longenough, the kinetic energy K completely dissipates into heat. The temperature rise can thenbe calculated from the energy balance. Setting the energy (per contact area) before and afterdebonding equal, gives U mech + C tot T T = C tot T T + K , (158)where U mech is the mechanical energy at debonding and C tot := C + C + C c is the totalheat capacity of the system. Considering linear elasticity gives U mech = ( t maxn ) /k eff , where30 eff = ( H /E + H /E ) − is the effective stiffness of the two-body system based on height H k and Young’s modulus E k . This then leads to the temperature change T c = T (cid:112) − κ + ( t maxn /t ) , (159)with the positive constants κ := K / ( T C tot ) and t := T C tot k eff . It is shown in Fig. 10b forvarious values of κ and t maxn /t . Increasing the bond energy (represented by κ ) leads to lower a. b. Figure 10:
Debonding thermodynamics: a. model problem; b. temperature change duringdebonding.final temperatures, while higher bond strengths t maxn lead to higher final temperatures. As aconsequence, the final temperature can either be higher or lower than the initial temperature.But note that t maxn /t > κ − T c >
0. This implies that for large κ >
This work has presented a new continuum theory for coupled nonlinear thermo-chemo-mechanicalcontact as it follows from the fundamental balance laws and the principles of irreversible ther-modynamics. It highlights the analogies between the different physical field equations, andit discusses the coupling present in the balance laws and constitutive relations. Of particularimportance is (99), the equation for the mean contact heat influx. It identifies how mechanicaldissipation, chemical dissipation and interfacial entropy changes lead to interfacial heat gener-ation. This is illustrated by analytically solved contact test cases for steady sliding, exothermicbonding, endothermic bonding and forced bond-breaking.The proposed theory applies to all contact problems characterized by single-field or coupledthermal, chemical and mechanical contact. There are several applications of particular interestto the present authors that are planned to be studied in future work. One is the pressure-dependent curing thermodynamics of adhesives (Sain et al., 2018). A second is the study ofthe bonding thermodynamics of insect and lizard adhesion based on the viscolelastic multiscaleadhesion model of Sauer (2010). A third is the local modeling and study of bond strength andfailure of osseointegrated implants (Immel et al., 2020). There are also interesting applicationsthat require an extension of the present theory. An example is electro-chemo-mechanical contactinteractions in batteries. Therefore the extension to electrical contact is required, e.g. followingthe framework of Weißenfels and Wriggers (2010). Another example is contact and adhesion ofdroplets and cells. Therefore the extension to surface stresses, surface mobility of bond sites,31ontact angles and appropriate bond reactions is needed, e.g. following the framework of Sauer(2016a) and Sahu et al. (2017).
Acknowledgements
RAS and TXD acknowledge the support of the German Research Foundation through projectsSA1822/8-1 and GSC 111. The authors are also grateful to the ACalNet for supporting RAS fora visit to Berkeley in 2018, and they thank Katharina Immel and Nele Lendt for their commentson the manuscript.
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