Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 3, 39, 2001 VARIATIONAL DESCRIPTIONS OF WEARING OUT SOLIDS AND WEAR PARTICLES IN CONTACT MECHANICS Alfred Zmitrowicz Institute of Fluid-Flow Machinery, Polish Academy of Sciences, Gdańsk e-mail: azmit@imp.gda.pl Experimental studies demonstrated the central role playedbywearparticles in sliding contacts. In this contribution, the classical variational formula- tion of contact problems is extended to wearing out solids by including the interfacial layer of wear debris between contacting bodies. Variational for- mulations are developed starting from strong forms of governing relations for the wearing solids and the layer. An implementation of a finite element representation of the contact problem is presented. The Lagrange multi- pliermethod is applied to determine contact forces and to satisfy kinematic contact constraints. Discretized forms of the variational functionals lead to equations which can be useful in numerical analysis. Key words: contact mechanics, friction, wear, variational methods 1. Introduction Analysis of the interaction problemof solid bodies generally requires deter- mination of stresses and strains within the individual contacting bodies, toge- ther with the information about distribution of displacements and stresses in the contact region, for the given boundary and initial conditions. An effective way to solve the contact problems is given by variational formulations and approximations with the aid of the finite element method, see Telega (1988), Szefer et al. (1994), Wriggers (1995), Stupkiewicz andMróz (1999). Easily observable results of a wear process can be identified as an increase of the gap between contacting bodies (worn contours) and as formation and circulation of the wear debris in contact interfaces. Wear particles trapped between two surfaces can be a source of significant mechanical phenomena. Development of a thin, uniform and almost continuous transfer film (interme- diate layer) of the wear debris between contacting bodies belongs to themost 792 A.Zmitrowicz important phenomena resulting from thewear process.The thin layer between the bodies (called sometimes the third body) is treated as a continuum con- sisting of solid particles withmechanical properties different from themating bodies. In the published literature, various continuummechanics-basedmodels for the third body have been proposed. Godet (1988) suggested modelling the third body (i.e. wear debris or a solid powder lubricant film) as a viscous fluid film. InGodet’s opinion the load carryingmechanismcanexistwithin the third body, as it occurs in the hydrodynamic fluid lubrication. Elrod (1992), Berker andVanArsdale (1992) proposedphenomenologicalmodels for the third body materials taking into account granularmedia. In order to specialize themecha- nics of granularmedia to the third body, it can be assumed that the interfacial layer consists of isolated discrete particles (spherical in shape, similar in size, non-cohesive and incompressible). Hou et al. (1997) proposed an elasto-plastic rheologicalmodel for the thirdbodyconsisting of fine oxidizedwear debris and contaminants. It was assumed that the wear and other debris, under traction and extreme pressure, form a uniform solid incompressible layer. The shear stresses in the layer are elastic below a critical shear stress and plastic above it. In the studies by Zmitrowicz (1987), the effect of wear particles was consi- dered in the form of a two-dimensional interfacial layer. Various constitutive models have been analysed for the interface layer: micropolar thermoelastic, micropolar fluid and thermoviscous fluid. Stupkiewicz and Mróz (1999) mo- dified the classical wear law taking into account the third body (oxides, wear particles) between tool and workpiece surfaces in metal forming processes. In the paper by Szefer et al. (1994), the interface layer was assumed to be bonded to contacting bodies. Problems of bonding of two solids by a thin ad- hesive layer have been extensively studied in the literature, see e.g.Ganghoffer et al. (1997). A typical model contains two bodies joined by a thin layer, i.e. the thin layer adheres to the bodies. The aim of this paper is to present a complete set of governing relations describing solids subject to wear process.We start with a local strong formu- lation of the contact problem being based on differential equations, then we pass to variational forms of the problemand to finite element approximations. 2. Experimental observations of wear particles Friction process of elements operating in contact conditions always involves frictional heat and wear of their surfaces. Wear is often defined as gradual Variational descriptions of wearing out solids... 793 removal of the material from contacting and rubbing surfaces of solids during their relative sliding. Taking into account the amount of the removedmaterial from the body, the measures of wear have been formulated with respect to changes of the following quantities: mass, volume and dimensions of the body (e.g. depth of the removed material). Wear of machine component parts and tools is often identified with irreversible changes of the shapes and sizes of rubbing elements (so called wear profiles). A significant increase of stresses have been observed in worn profiles of the elements. Many mechanical, physical and chemical phenomena are responsible for wear of materials. Several types of wear have been recognized: adhesive, abra- sive, contact fatigue, fretting, oxidation, corrosion, erosion. Inmachinery,wear occursmost frequently as the abrasive or contact fatigue process.Most of the wear processes are the consequences of the interfacial rubbingprocess.Wear in sliding systems is usually a very slow process, but it is steady and continuous. The removal of the material takes the form of small particles which may come off in loose form. The wear particles are detached from solids during sliding by the following microscopic mechanisms: microcutting of adhesive junctions between surfaces,mechanical failure of contacting asperities, surface spalling, plastic deformations of the surfaces in a formofgrooves and scratches, nucleation andpropagation of the surface and subsurface cracks andvoids, etc. There are numerous studies in the field of wear revealing details about the size and nature of wear particles, and a large body of empirical data collected, see Dowson et al. (1992), Zanoria et al. (1995). The shape, size and number are the main parameters characterizing morphological properties of the wear particles. For example, when twometal surfaces slide against each other under a load, thewear debris is produced in the form of bothmacroscopic (size from a few to several micrometers) and microscopic (size from submicrometer to a few micrometers) particles. It was found that the number of macroscopic wear particles was small and the number ofmicroscopic debris was great. The generatedwear debris can assumedifferent shapes (flakes, chips, thin platelets, filings, powder-like particles). The entrapped wear particles remain within the wear track, and they be- come a part of the interfacialmedium. Some of themare processed further e.g. crushed into finer particles, compacted and agglomerated into large debris. It is often observed that the wear debris in the form of flakes and chips etc. can be ”rolled over” into balls, cylinders or needles. It takes place especially, when the contacting bodies are in oscillating sliding andwhen the number of sliding cycles increases. For instance, large cylindrical rolls are formed when using a rubber eraser. 794 A.Zmitrowicz Roll-likeweardebris are created by rubbing forces.Each roll is subjected to opposing tangential forces at its top andbottom surfaces.The torque resulting fromthese forcesmakes thedebris roll.Theaxes of the roll particles are aligned perpendicularly to the direction of sliding, indicating that they are actually rolling on thewear track (or that they are generated by rolling in the contact), and suggesting that theparticle circulation is common in the contact. The rolls can grow as snow balls by collecting more andmore finer wear particles. Aparticularly intriguing typeof awearparticle, often observedand subject tomuch interest, is the spherically shapedwear particle (Dowson et al., 1992). The spherical particles ranging in the size from 1 up to 80µm have been observed. Small (less then 1µm) spherical particles have also been seen by some investigators. Rolls and spheres were obtained in a very wide range of runningconditions, and for verydifferentmaterials (steels, ceramics, polymers, etc.). Thematerial whichwas removed from surfaces during the course of awear process is not in fact the same either structurally or chemically, as the parent material. Instead, it is very fine-grained material which may be derived from both parts of the contacting system and may include contaminants from the environment as well. Physical properties of the wear particles include compo- sition, microstructure, density, thermal expansion, thermal conductivity, mel- ting temperature, etc. Depending on the type of the bulk material different kinds of wear particles are created: metals, metal oxides, alloys, plastics, cera- mic materials, etc. Formation of loose particles is the key of a wear process. Wear debris generally accompanies wear of any sliding system. Practical importance of wear particles in mechanical devices depends on the sliding system – it is small for an automobile tire rolling on a road; it is important for bearings and for knee and hip joints prostheses. Numerous experimental investigations and everyday practice indicate the formation of a wear debris layer on the sliding interface (i.e. an intermediate layer between the sliding surfaces) practically immediately after the rubbing process starts, transforming the contact of two bodies into a contact of two bodies with the interfacial layer. The surfaces slide on each other being separated by particles of the wear debris. The particles can slide and roll in contacting surfaces and in relation to each other. In the case of the wear debris layer, the friction occurs partially between the surfaces of the contacting bodies, partially between the surfaces of each body and the wear particles, and inside the layer between the wear debris. The resistance to rolling (i.e. rolling friction) follows from rotations of single particles with respect to the neighboring particles. Variational descriptions of wearing out solids... 795 Friction phenomenon is very sensitive to changes of sliding conditions. The wear particles entrapped between the sliding surfaces can significantly affect the frictional and wear behavior. The presence of debris implies modification of the friction coefficient andwear rate. The friction coefficient increases when the particles are accumulated and decreases when the particles are removed from the sliding interface. The steady-state wear rate may be larger in some sliding conditionswhen thewear particles cannot be removed fromthe contact area and act as abrasive particles. It has also been reported that as a result of formation of spherical and cylindrical particles, the coefficient of friction undergoes transition to lower values up to a factor of three and the wear decreases by several orders of magnitude (Zanoria et al., 1995). It has been suggested that the cylindrical debris can act as miniature roller bearings so that the sliding friction can be reduced when this debris forms. Wear particles will be active in a rubbing system until they are removed from the contact. The ejection of particles fromboth the contact and thewear track (sliding path) is predominant in most situations. It can be helpful to eliminate thewear debris froma sliding system. Sometimes, wear particles are removed from the sliding interface by brushing off (the case of lubricated or openmechanical devices). Sometimes this canbedonebyfiltering a circulating lubricant. Complete separation with third bodies is not a rare occurrence. An inter- facial layer can also be formed by hard particles and contaminants entrapped into a sliding system from the operating environment (e.g. airborne debris of dust and sand, combustion products such as fly ash, corrosion scale) or solid body particles specially introduced into the sliding interface with respect to their beneficial role during the friction process (e.g. typical lamellar solid lu- bricants such as carbon graphite powder, molybdenum disulphide, PTFE) or polishing slurry composed of hard abrasive particles (diamond, silicone carbi- de, alumina) in material-finishing operations. A considerable effort has beenmade to characterize themechanical proper- ties of the third bodies andhow theybecomedeformed in the rubbing contact. The transfer material and debris trapped in the interface deform when sub- jected to high compressive and shear stresses. The stiffness of the third body can influence the deformation of the rubbing system.Thewear debris give rise to a unique stress pattern and deformation behavior in the regions near the contact surface. The layer has certain characteristic propertieswhich dependupon the phy- sical andmechanical properties of the wear particles and the layer as a whole. Identification and understanding of the third body mechanical behaviour is 796 A.Zmitrowicz difficult since it requires direct observations of the interface during the wear process, i.e. in real working conditions. It might be some promise in treating the third bodyas a single continuum.For this purpose, one needs some equiva- lent but inscrutable properties to characterize the solid third body particles. Then, typical strength tests should be done, i.e. investigating the strength with respect to: tension, compression, shear, bending, torsion, creep, fatigue, hardness. Such fundamental tests have not been undertaken yet for the wear debris layer. 3. Governing equations 3.1. Wearing out of solids Thepresence of a thin layer of thewear debris separating contacting solids is an extremely important topic of the present analysis. Therefore, the model of rubbing and wearing out solids consists of two contacting solids A,B and the interfacial layer S of wear debris. Elasticmaterials and linearized theories are taken into account. Motion of the wearing body A is governed by the following equilibrium equation divσA − f̂A =0 in VA (3.1) where σA – Cauchy’s stress tensor f̂A – vector of body forces. The constitutive relations of the elastic body are as follows σA =EAεA εA(uA)= 1 2 (graduA + grad ⊤uA) (3.2) where EA – tensor of elasticity εA – (right) Cauchy-Green’s strain tensor uA(xA) – displacement function, xA ∈VA. The boundary conditions for the body A are given by uA(xA)= ûA(xA) on ΩuA σAnA = q̂A on ΩqA σAn + =pA on ΩcA (3.3) Variational descriptions of wearing out solids... 797 where nA – unitvectornormal to thebodyboundaryoutof the contact region n+ – unit vector normal to the boundary of the body A in the region of contact ûA, q̂A – displacements and loads pA – contact stresses ΩcA – contact region of the body A. The contact conditions with respect to the contact stresses (contact trac- tions) are following pAn ¬ 0 on ΩcA |pAt | ¬µA|p A n | on ΩcA (stick) pAt =−µA|p A n | V AS |V AS| on ΩcA (slip) (3.4) where µA – friction coefficient pAt ,p A n – tangential andnormalcomponents of thecontact stresses, i.e. pAt =(1−n +⊗n+)(σAn +) (stick) (3.5) pAn =(n +⊗n+)(σAn +)≡ pAnn + V AS is the relative sliding velocity between the body A and the layer S V AS =(1−n +⊗n+)(u̇A − u̇S)≡ u̇ A t − u̇ S t (3.6) The power of the friction force is non-positive in any sliding direction, i.e. pAt V AS ¬ 0, ∀V AS. Various friction laws can be included in this formulation (Zmitrowicz, 1998). The contact conditionswith respect todisplacements in the contact regions are as follows dn = gn −u A n +u + n +u B n −u − n ­ 0 on ΩcA ∪ΩcB (3.7) The total clearance gap (dn) between the two contacting bodies A and B consists of: initial gap (gn), increase of the gap as a result of the wear process of the bodies A and B (u+n ,u − n) and elastic deformations of the bodies in the normal direction to the contact (uAn ,u B n ). 798 A.Zmitrowicz The depth worn out – wear profiles (u+n ,u − n) – are evaluated by the inte- grals of wear velocities v+ and v− for the interval of time < t0, t1 > taken as the range of the integration, i.e. u+n(xA, t1)= t1∫ t0 v+ dt u−n(xB, t1)= t1∫ t0 v− dt (3.8) The velocities normal to solid boundaries at the contact region (v+,v−) are dependent variables of the constitutive relations of wear, and they are given by the following equations v+ =−iA|p A n ||V AS| v − =−iB|p B n ||V BS| (3.9) where iA, iB – wear intensities V BS – relative velocity between B and S. Relations (3.9) define the velocities at which the surfaces of the sliding bodies ”travel” into other surfaces because of their wearing out. Wear equ- ations (3.9) differ from the classical Archard law ofwear in the omission of the term directly representing hardness of materials. Nevertheless, the effect can be incorporated in the termdefining thewear intensity coefficients iA, iB. Fur- thermore,we donot establish the principalwear relationship for the volume of the removedmaterial but for thewear velocity.With the aid of equations (3.8), the volume and the mass of the removed material can be easily calculated. 3.2. The wear debris layer Perhaps the most difficult step in the analysis of a wear process is the development of an acceptable model of thewear debris between a pair of solid bodies as the wear debris exhibits properties quite distinct from those asso- ciated with interiors of the bodies. Let us consider the following assumptions: a) It is natural to assign to wear debris the continuum models with micro- structure which capture fundamental kinematical features of the wear particles. In particular, microrotational effects can be depicted in such models in a natural straightforward way. Themicropolar layer can sup- port stress and body force couples. The separate structure of the wear debris is characterized by constitutive equations independent of those characterizing the parent materials. b) The evolution of displacements and stresses transmitted by wear debris is a complicated phenomenon in general case, since it implies that thewear Variational descriptions of wearing out solids... 799 debris can undergo large displacements or/and large strains. Therefore, a simplified model of the wear debris is needed, by specifying the kine- matical andmechanical features of thewear debris, in order tomake the problem tractable from both analytical and numerical points of view. We assume that the layer particles undergo only small displacements andmicrorotations and the deformations of the layer are elastic. c) The fact that thewear debris layer can be regarded as thinmakes it possi- ble to introduce simplifying assumptions in the continuum description. The governing equations of the wear layer are simplified in order to eli- minate the dependence on the through-the-thickness coordinate, thus considering the layer as a two-dimensional material continuum. The mass continuity equation and equilibrium equations with respect to translational and rotational degrees of freedomof amicropolar elastic layer S (Zmitrowicz, 1987) are given by ∂ρS ∂t + divs ( ρS ∂uS ∂t ) −mA −mB =0 divsσS − f̂S −mAV AS −mBV BS =p A t +p B t (3.10) divsµ+(mA +mB)jS ∂ψ ∂t = c where ρS – mass density uS,ψ – vectors of the displacements and independent microrota- tions of the wear debris σS – Cauchy’s stress tensor f̂S – body force pAt ,p B t – friction forces between the bodies A,B and the layer S, µ – stress couple tensor in the layer jS – inertia tensor of the wear debris c – couple of friction forces. The terms mA and mB determine the mass fluxes of the wear debris supplied to the layer, mAV AS and mBV BS define the momentum supplied to the layer by the wear debris, and (mA +mB)jS∂ψ/∂t is the moment of momentum of the supplied wear debris. 800 A.Zmitrowicz The boundary conditions for equilibrium equations (3.10)2,3 are as follows uS(l)= ûS(l) on Lu σSν = q̂S on Lq ψ(l)= ψ̂(l) on Lψ µν = m̂ on Lm (3.11) and the boundary and initial conditions for mass continuity equation (3.10)1 are ρS(l, t)= ρ̂(t) on Lr ρS(l, t0)= ρ0(l) in S (3.12) where ûS, q̂S,ψ̂,m̂ – boundary displacements, forces, microrotations and couples ν – unit vector normal to the layer boundary ρ̂ – boundarymass efflux of the debris removed from the contact region ρ0 – initial mass intensity. The elastic micropolar layer S is characterized by the constitutive equ- ations independent of those characterizing the bulk response properties of the parent bodies A and B σS =ESεS εS(uS)= 1 2 (gradsuS + grad ⊤ s uS) µ=DSκ κ(ψ)= gradsψ (3.13) where εS – infinitesimal strain tensor κ – couple strain measure in S ES,DS – tensors of elasticitywith respect to thedisplacements and microrotations, respectively grads(·) – surface (two-dimensional) gradient operator. Here, ρS(l, t), uS(l), ψ(l), l ∈ S, t ∈ I, I ⊂ R are functions of mass intensity, displacement andmicrorotation. The wear debris mass fluxes (from the bodies A and B to the layer S) are mA = ρAv + mB = ρBv − (3.14) Variational descriptions of wearing out solids... 801 where ρB is themass density. Thewear velocities of the bodies A and B are defined by (3.9). The couple of friction forces acting on a single particle in the layer is given by c= ζA|p A n | ( n +× V AS |V AS| ) +ζB|p B n | ( n −× V BS |V BS| ) (3.15) where ζA,ζB – constants n− =−n+ – unit vectors in the contact region of the body B. The constraints of the energy dissipated in the contact region are as follows pAt V AS +p B t V BS = cψ̇+q+βA +βB +β ∗ (3.16) Equation (3.16) shows that the friction force power goes into the power of the frictional couple cψ̇, the frictional heat fluxes entering into the solids and the layer q and energies spent on wear process of the solids and the layer βA, βB, β ∗. The energies spent in the wear process are defined by: βA =−mAǫA, βB = −mBǫB, β ∗ = −mcrǫS where ǫA, ǫB, ǫS are the internal energies consumed by formation of a unit mass of wear debris, mcr is the mass of the wear debris crushed into finer particles. 4. Variational formulations 4.1. Displacements in the contact system Anexact solution determining all unknownsof the contact problemcannot be found since the complex system of coupled differential equations and the boundary conditions must be taken into account. Modern methods of analy- sis are based on a variational approach and numerical techniques. We apply the principle of stationary total potential energy to state the equilibrium con- ditions of the contact system. The total potential energy Π defined in a deformed configuration of the contact system consisting of the two contacting solids A, B and the interfacial layer S of wear particles with translational and rotational degrees of freedom is given with the stored energy Πint and the energy of external loads Πext as Π =Π(uA,uB,uS,ψ)=Π int +Πext → stationary point Π(uA,uB,uS,ψ)= 1 2 ∫ VA σA : εA dV + 1 2 ∫ VB σB : εB dV+ 802 A.Zmitrowicz + 1 2 ∫ S (σS : εS +µ :κ) dS− ∫ S (mAV AS +mBV BS)uS dS− (4.1) − ∫ S [(mA +mB)jSψ̇]ψ dS− ∫ VA f̂AuA dV − ∫ VB f̂BuB dV − − ∫ S (f̂SuS +cψ) dS− ∫ ΩqA q̂AuA dA− ∫ ΩqB q̂BuB dA− ∫ Lq q̂SuS dL− − ∫ Lm m̂ψ dL− ∫ ΩcA pAuA dA− ∫ ΩcB pBuB dA− ∫ S (pAt +p B t )uS dS The principal difficulty in contact problems is characterized by unknown potential contact regions ΩcA,ΩcB and unknown contact forces pA,pB – the quantities which depend on the solution to the problem.To solve the problem, step-by-step solution procedures must be applied (incremental formulation, iterative procedures).We search for such uA,uB,uS and ψ which guarantee stationarity of Π at the given step of an incremental approach and iteration process, then the current contact regions and current contact forces are as- sumed to be given. This is a simplified local weak formulation of the contact problem. We search for such uA, uB, uS and ψ which guarantee stationarity of Π, i.e. δΠ(uA,uB,uS,ψ)= 0 δΠ = [ ∂Π ∂uA ,δuA ] A + [ ∂Π ∂uB ,δuB ] B + [ ∂Π ∂uS ,δuS ] S + [∂Π ∂ψ ,δψ ] S (4.2) ∂Π ∂uA =0 ∂Π ∂uB =0 ∂Π ∂uS =0 ∂Π ∂ψ =0 where [·, ·]A, etc. mean scalar products (represented by integrals over the corresponding volumes and surface parts, respectively) and ∂Π/∂uA, etc. are gradients of the functional Π, see Bufler (1979). We derive equations which are a basis of approximated solutions to the contact problem.Let us consider thediscretization bymeans of finite elements. Finite element approximations of displacements, strain and stress fields over Variational descriptions of wearing out solids... 803 an element e in A are as follows uA(xA)=NA(xA)u e A εA(xA)=BA(xA)u e A (4.3) σA(xA)=EABA(xA)u e A where NA – shape functions BA – derivatives of the shape functions ueA – displacements at the nodes of the finite element. Furthermore, we approximate the local fields uB, εB, σB in the body B and uS, ψ, εS, κ, σS, µ in the layer S. The total potential energy Πe(uA,uB,uS,ψ)defined for thefinite elementshas the similar formas (4.1)2. The stationarity condition of the total potential energy for any finite element leads to equations of the displacements in the body A ∂Πe ∂ueA =KeAu e A + q̃ e A (4.4) where K e A = ∫ V e A B ⊤ AEABA dV q̃ e A =− ∫ V e A N ⊤ Af̂A dA− ∫ Ωe qA N ⊤ Aq̂A dA− ∫ Ωe cA N ⊤ ApA dA the displacements in the layer S ∂Πe ∂ueS =KeS1u e S + q̃ e S +m̃ e 1 (4.5) where K e S1 = ∫ Se B ⊤ SESBS dS m̃ e 1 =− ∫ Se N ⊤ S(mAV AS +mBV BS) dS q̃ e S =− ∫ Se N ⊤ S f̂S dS− ∫ Se N ⊤ S(p A t +p B t ) dS− ∫ Leq N ⊤ S q̂S dL and themicrorotations in the layer S ∂Πe ∂ψe =KeS2ψ e + c̃e +m̃ e 2 (4.6) 804 A.Zmitrowicz where K e S2 = ∫ Se B ⊤ SDSBS dS m̃ e 2 = ∫ Se N ⊤ S(mA +mB)(jSψ̇) dS c̃ e =− ∫ Se N ⊤ Sc dS− ∫ Lem N ⊤ Sm̂ dL and m̃ e 1, m̃ e 2 define the momentum and moment of momentum of the wear debris supplied to the layer S; c̃e is the frictional moment and the boundary moment in the layer. Notice, thatdiscretizing thedeformationandthe loadhistory, thevelocities are replaced by increments and the subsequent integrals by sums. 4.2. Mass continuity in the wear debris layer Amass continuity functional of the layer S can be written as Π̂(ρS)= ∫ S [1 2 divs(ρSu̇S)−mA −mB + ∂ρS ∂t ] ρS dS (4.7) Themass intensity function ρS(l, t) satisfies the following relation δΠ̂ = [∂Π̂ ∂ρS ,δρS ] S =0 (4.8) The finite element approximation of the mass intensity field in S is given by the function ρS(l, t)=NS(l)ρ e S(t) (4.9) Let the functional Π̂e be defined for any finite element, then the stationarity condition is as follows ∂Π̂e ∂ρe S =Ceρρ̇ e S +K e ρρ e S +m̃ e 3 (4.10) where C e ρ = ∫ Se N ⊤ SNS dS K e ρ = ∫ Se N ⊤ S u̇ e SBS dS m̃ e 3 =− ∫ Se (mA +mB)NS dS and ρeS is themass density at the nodes of the finite element. The vector m̃ e 3 defines mass of the wear debris supplied to the layer. Variational descriptions of wearing out solids... 805 4.3. Active solution strategies The variational formulation may be restricted not so strongly as we have assumed in (4.1)2.Apowerful formulation of contact problems canbeobtained by making use of the principle of stationary potential energy or the principle of virtual work to derive variational formulations and so the called constraint methods, e.g. Lagrangemultipliermethods and their generalizations. They are called the active strategies. The principle of virtual work in the case of a discretized contacting single body A (neglecting friction) has the following form δL(uA)= δu ⊤ A(KAuA +qA)=0 (4.11) where qA is the global external load vector for simplicity the body and bo- undary loads are denoted by this letter. The functional (Lagrangian) is given by L(uA)= 1 2 u ⊤ AKAuA +u ⊤ AqA (4.12) Kinematic contact condition (3.7) is a constraint on the displacements of the body in the contact region. Let us assume that the contact takes place in m nodes of the considered discretization. Then, the discretized kinematic contact condition, Eq (3.7), in the contact region (reduced to the single body) can be written as AuA −g ¬0 (4.13) where A – compliance matrix g – initial gap vector in the nodes being in contact. Notice, that any component of the constraint vector (4.13) is nonpositive. Theequalitymeans that condition (4.13) is related to thenodesbeingcurrently in the state of sticking. Then, the contact problem reduces to theminimization of functional (4.12)with thekinematic contact constraint on thedisplacements of the bodies, see (4.13). This method is designed to fulfill the constraint equation in the direction normal to the contact area. An approach to friction problems can be introduced using the tangential and normal Lagrange multipliers, see Wriggers (1995). We distinguish the tangential gap (get) ⊤ = [gt1,gt2] and the normal gap g e n = gn+u + n , where gt is the relative sliding displacement. Then, the discretized kinematic contact conditions (in the equality version) can be written as A1uA −gn =0 A2uA −gt =0 (4.14) 806 A.Zmitrowicz where A1,A2 are the compliance matrices. Due to satisfying of contact conditions (4.14), we extend the definition of the Lagrangian (4.12). Then, the Lagrange multiplier method in the equality version is to search for a saddle point of the Lagrangian (4.12) changed as follows L(uA,λn,λt)= 1 2 u ⊤ AKAuA +u ⊤ AqA +(A1uA −gn) ⊤ λn +(A2uA −gt) ⊤ λt (4.15) The Lagrange multipliers contain the following additional unknowns associa- ted with sticking nodes: (λet) ⊤ = [λt1,λt2] and λ e n = λn. λn denotes the Lagrangian multiplier which can be identified as the contact pressure pn. λt is associated with the tangential forces in stick or slip motion pt. In the stick the relative tangential slip gt is equal to zero and λt is the reaction. In the case of sliding λt is determined by the friction constitutive law. We calculate the minimum of the Lagrangian (4.15) with respect to three independent variables, i.e. we search for such uA, λn, λt which guarantee stationarity of L(uA,λn,λt) δL(uA,λn,λt)= 0 (4.16) δL= ∂L ∂uA δuA + ∂L ∂λn δλn + ∂L ∂λt δλt where ∂L ∂uA =KAuA +qA +A ⊤ 1 λn +A ⊤ 2 λt =0 (4.17) ∂L ∂λn =A1uA −gn =0 ∂L ∂λt =A2uA −gt =0 Then(4.17) yield the following systemof linear equations for thedisplacements and the Lagrange multipliers   KA A ⊤ 1 A ⊤ 2 A1 0 0 A2 0 0     uA λn λt  =   −qA gn gt   (4.18) In the active solution strategies, the contact stresses are determined toge- ther with the displacements of the contact system during solving the problem, see (4.18). The contact region can be calculated in an iterative process. Variational descriptions of wearing out solids... 807 5. Conclusions • Variational formulations of displacements in rubbing andwearing solids are derived with the aid of the principle of stationary potential energy in a deformed configuration of a contact system. The definition of the gap between two contacting bodies is modified taking into account wear profiles. • Classical variational formulations of contact problems are extended by including an interfacial layer of wear debris between wearing bodies. Fi- nal equations are presented in a discretized formusing the finite element method. • Friction and wear phenomena are very sensitive to changes of sliding conditions. Wear particles between sliding surfaces can affect frictional and wear behaviour very significantly. In everyday life, one can easily observewearprofiles andweardebrisduringabrasion, e.g. pencil drawing marks on a paper, a piece of chalk writing on a blackboard, a rubber eraser rubbing out pencil marks on a paper, etc. Acknowledgements Thefinancial supportwasprovidedby theStateCommittee forScientificResearch (KBN) under grant No. 8T07A03420. References 1. Berker A., Van Arsdale W.E., 1992, Phenomenological Models of Third Body Rheology, In: Wear Particles: From the Cradle to the Grave, edit. D.Dowson et al., Elsevier, Amsterdam, 337-346 2. Bufler H., 1979, Zur Struktur der Gleichungen der Elastokinetik und Elasto- stabilität, Z. Angew. Math. Mech., 59, 2, 73-78 3. DowsonD.,TaylorC.M.,ChildsT.H.C.,GodetM.,DalmazG. (edit.), 1992, Wear Particles: From the Cradle to the Grave, Proceedings of the 18th Leeds-Lyon Symposium on Tribology, Elsevier, Amsterdam 4. Elrod H.G., 1992, Granular Flow as a Tribological Mechanism - a First Lo- ok, In: Wear Particles: From the Cradle to the Grave, edit. D.Dowson et al., Elsevier, Amsterdam, 75-88 5. Ganghoffer J.F., Brillard A., Schultz J., 1997, Modelling of the Me- chanical Behaviour of Joints Bonded by a Nonlinear Incompressible Elastic Adhesive,Eur. J. Mech, A/Solids, 16, 2, 255-276 808 A.Zmitrowicz 6. Godet M., 1988,Modeling of Friction andWear Phenomena, In: Approaches to Modeling Wear, edit. F.F.Ling, C.H.T.Pian, Springer, Berlin, 12-36 7. HouK., Kalousek J., Magel E., 1997,RheologicalModel of Solid Layer in Rolling Contact,Wear, 211, 1, 134-140 8. Stupkiewicz S., Mróz Z., 1999, A Model of Third Body Abrasive Friction andWear in HotMetal Forming,Wear, 231, 1, 124-138 9. SzeferG., JasińskaD., SalamonJ.W., 1994,Conceptof aSingularSurface in ContactMechanics,Arch. Mech., 46, 4, 581-603 10. Telega J.J., 1988,Variational Inequalities inContactProblemsofMechanics, In:Contact Mechanics of Surfaces, edit. Z.Mróz,Ossolineum, 51-165, in Polish 11. Wriggers P., 1995, Finite Element Algorithms for Contact Problems,Archi- ves of Computational Methods in Engineering, 2, 4, 1-49 12. Zanoria E.S., Danyluk S., Mc Nallan M.J., 1995, Formation of Cylin- drical Sliding-Wear Debris on Silicon in Humid Conditions and Elevated Tem- peratures,Tribology Transactions, 38, 3, 721-727 13. Zmitrowicz A., 1987, A Thermodynamical Model of Contact, Friction and Wear: I Governing Equations; II Constitutive Equations for Materials and Li- nearizedTheories; III Constitutive Equations for Friction,Wear and Frictional Heat,Wear, 114, 2, 135-168, 169-197, 199-221 14. Zmitrowicz A., 1998, ConstitutiveModelling of Non-Homogeneous andAni- sotropic Friction of Materials – Sliding Path Curvature Effects, In: Computa- tional Mechanics. New Trends and Applications, CD-ROM, edit. S.Idelsohn, E.Oñate, E.Dvorkin, CIMNE, Barcelona, Spain, 1-21 Wariacyjne opisy w mechanice kontaktu zużywających się ciał stałych i cząstek zużycia Streszczenie Badania doświadczalne wskazały na istotną rolę, którą odgrywają cząstki zuży- cia znajdujące się w obszarach styku. W pracy uogólniono klasyczne sformułowa- nia wariacyjne zagadnień kontaktowych na przypadek zużywających się ciał stałych uwzględniając warstwę cząstek zużycia między stykającymi się ciałami. Opisy waria- cyjnewyprowadzonowychodząc z różniczkowychpostaci równań stanu dla zużywają- cych się ciał i warstwy cząstek zużycia. Przedstawiono implementację dometody ele- mentów skończonych.WykorzystanometodęmnożnikówLagrange’aw celu określenia sił kontaktowych oraz spełnienia kinematycznych więzów styku. Dyskretne postacie funkcjonałówwariacyjnychprowadządo równań, któremogąbyćużytecznewanalizie numerycznej. Manuscript received November 10, 2000; accepted for print January 24, 2001