FEM Method for Contact Problems

8/3/2019 FEM Method for Contact Problems

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Tribo logy International Vol. 29, No. 8, pp. 651-658, 1996Copyr Ight 0 1996 Elsevier Science Ltd

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ELSEVIER SO301-679X(96)00011-4SCIENCE:

ite element methods forcontact problems with frictionP. WriggersThe numerical treatment of contact problems involves theformulation of the geometry, the statement of interface laws, thevariational formulation and the development of algorithms. In thispaper an overview with regard to the numerical simulation offrictional problems is presented when general constitutiveequations are formulated in the contact interface. To be mostgeneral we will apply a geometrical model and its discretizationfor contact which is valid for large deformations. Furthermore thealgorithms to integrate the interface laws will be discussed for thetangential stress components. Special emphasis is laid on thedeveiopment of algorithms which allow an efficient treatment offrictional contact problems. Copyright 0 1996 Elsevier Science LtdKeywords: dry friction, large deformations, finite element method

IntroductionBoundary value problems involving contact are of greatimportance in industrial applications in mechanical andcivil engineering. The range of application includesmetal forming processes, drilling problems, bearingsand crash analysis of cars or car tires. Other applicationsare related to biomechanics where human joints,impla-Its or teeth are of consideration. Due to thisvariety contact problems are today often combinedeither with large elastic or inelastic deformationsincluding time dependent responses.In this article we will restrict ourselves mainly to finiteelement techniques for the treatment of contactproblems despite many other numerical schemes andanalytical approaches that could be discussed as well.Furthermore we like to note that the description ofthe mechanical behaviour of the bodies coming intocontact will not be investigated in detail, although thisis of great importance. This article thus concentrateson the behaviour in the contact interface for which acontir uum formulation, discretization within the finiteelement method and the associated algorithms arediscussed.

lnstitut fiir Mechanik, TH Darmstadt, Hochschu lstr. 1, D-64289Darmsladt, Germany

Due to the precision which is needed to resolve themechanical behaviour in the contact interface, differentapproaches have been used in the literature to modelthe mechanical behaviour in contact area. Two mainlines can be followed to impose contact conditionsin normal direction: these are the non-penetrationcondition as geometrical constraints and constitutivelaws for the micromechanical approach within thecontact area.Here we restrict our discussions o the micromechanicalview. Constitutive equations for the normal contactcan be developed by investigating the micromechanicalbehaviour within the contact surface, seee.g. Kragelskyet al..The interfacial behaviour in the tangential direction(frictional response) is even more complicated. Themost frequently used constitutive equation is theclassical law of Coulomb. However, other frictionallaws are available which take into account local,micromechanical phenomena within the contact inter-face, see e.g. Woo and Thomas2. An extensiveoverview may be found in Oden and Martins3; for thephysical background see e.g. Tabor4. During thefew last years frictional phenomena have also beenconsidered within the framework of the theory ofplasticity. This leads to non-associative slip rules.Different relations have been proposed for frictionalproblems by e.g. Michalowski and Mroz or Curnier6.

Tribology International Volume 29 Number 8 1996 651


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Fini te element methods for contact problems: P. Wriggers

The application of constitutive equations for frictionwithin finite element calculation can be found in e.g.Fredriksson and Wriggers et al.*. The major advantageof this formulation is related to the so called returnmapping algorithms, known from plasticity9 and itsapplications to frictionlO.The weak formulation of contact problems leads tovariational inequalities. Different possibilities exist forthe numerical solution of these problems. The mostfrequently used algorithms are based on active setstrategies which are applied in combination withLagrangian multip lier or penalty techniques. Each ofthe methods has its own advantages and disadvantages.Due to its simplicity the penalty method is favouredby most finite element software developers.A combination of the penalty and the Lagrangianmultip lier techniques leads to the so called augmentedLagrangian methods which try to combine the meritsof both approaches. These techniques are neededwhen constitutive interface laws are employed, sinceoften ill-conditioning of the problem occurs, seeWriggers and Zavarise.

Fig 1 Contact geometry and geometrical approach

For the general case of contact includ ing large defor-mations, arbitrary sliding of a node over the entirecontact area has to be allowed. This behaviour iscaptured by a discretization often called the node-to-segment approach. The associated matrix formulationfor large deformations can be found in Wriggers andSimoi for two-dimensional problems. The formulationfor frictional contact is discussed in Wriggers et aZ.s.A continuum based approach to contact problems hasbeen derived in Laursen and Sirno.

tangent vectors of the contact surface area, : = a:,([, t2) and Ai : = XTa( 0c2) gN + enters for gN+ > 0 as a local kinematicalvariable the constitutive function for the contactpressure.

We assume that two bodies which undergo largedeformations can come into contact. Let 987, y = 1,2, denote the two bodies on interest and & mapspoints Xy E 37 of the reference configuration ontopoints xy = cp:(X) of the current configuration.Motivated by micromechanical investigations of contactproblems we view the mechanica l approach of the twocontact surfaces as a microscopical penetration of thecurrent mathematical boundaries @(ICY). In thisformulation IY,YC cBJ~ are possible contact surfacesof the bodies By; see Fig 1 for an illustration of thisconcept. In what follows we denote VP: Ii) as thecurrent slave surface which penetrates in the case ofcontact into the current master surface &(I:) (movingreference surface). We parametrize the master surfaceIf in its reference and current configuration by thenatural parameters p, p; i.e. we consider materialcurves X2 = X2 (


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-a and 5 are the first and second fundamentalf::rn of tl$ deformed surface, well known fromdifferential geometry. With this we can define thetangential relative velocity function on the currentslave surface pi (ry ):.Yv g, := ta2, (5)

Equation (5) determines per definition the evolutionof the tangential slip g, which enters as a localkinematical variable the constitutive function for thecontact tangential stress, see next section. By exchang-ing th- velocities with the variations of the tangentialslip it can be stated as:6gT = &$a &gv) so (11)with the special case of classical Coulombs modelf ;( tT ,pN) = I l tT l i - PpN 5 O.

The constitutive evolution equation for the plastic orfrictional slip can be stated in a general form of a sliprule for large deformations in the contact zone asfollows:Zv g+ = A$ = An, with nT = ~

T l;t:II& = A (12)where Equation (12) describes the evolution of theeffective slip which is defined as:

A is a parameter which describes the magnitude of theplastic slip. Equations (lo), (11) and (12), along withthe loading-unloading conditions in Kuhn-Tucker form:AZO, fssOo, hfs=O (13)

establish the constitutive framework for the tangentialslip-stick behaviour. The algorithmic treatment will bediscussed ater.Boundary value problem, global solutionstrategiesFor a numerical solution of the nonlinear boundaryvalue problem we will use the finite element method.Thus we need the weak form of the local balanceequations. Due to the fact that the constraint condition(7) is represented by an inequality we obtain in

Finite element methods for contact problems: P. Wriggers



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general a variational inequality. Algorithms for solvingvariational inequalities are: mathematical program-ming, active set strategies or sequential quadraticprogramming methods. Here we will apply the activeset strategy which is implemented in many existingfinite element codes.Within an active set strategy we can write the weakform as an equality since we know the active set ofthe contact constraints within an incremental solutionstep:

t @N&N +fT . %TjdA =O (14)Here Py is the first Piola-Kirchhoff stress tensor andFY denotes the deformation gradient of the body City.VY is the virtual displacement which is zero at theboundary I; where the deformations are prescribed.In Equation (14) the constitutive equation (8) and(10) have to applied for the determination of pN andtT . The variation Of the normal gap function gN+ andthe tangential slip gT is given by (3) and (6).A major problem associated with the numericaltreatment of the penalty method and the contactinterface laws is the ill-conditioning which arises whenthe stiffnesses due to law (8) are combined withstiffnesses of the bodies within the fini te elementformulation. A method to overcome this problem isthe augmented Lagrangian technique, well known inoptimization theory. This leads to the following formfor the contact contribution in Equation ( 14)r1:

subject toc+(~y,PN) = gN+ - [i- @N)] = o

The fulfillment of the nonlinear interface law (8) willbe practically accounted for by an update formula forthe Lagrangian multip lier pN

with the known quantities { ...}o,d from the previousstate.Augmented Lagrangian techniques have also beenapplied to frictional problemsr6~r7. The latter paperalso investigates the possibility of an algorithmicsymmetrization of the frictional part of the tangentmatrix.Discretization techniques within the contactareaThe discretization of the domain contributions of thebodies being in contact in (14) is not objective of this654 Tribology International Volume 29 Number 8 1996

Fig 2 Node-to-segment element


work. We just state the matrix formulation of theweak form of (14):G(v) = i BT7dV- NT fy dVy=l

- jr:,TiydA} (17)where the matrix N contains the shape functions andthe so-called B-matrix contains the derivatives of theshape functions. Any standard finite element bookwill provide the necessary details, see e.g. Zienkiewiczand Tay1ors.A genera1 discretization of the contact interface whichallows for large tangentia l sliding is given by the setupdepicted in Fig 2. This discretization is called node-to-segment contact element and is widely used innonlinear finite element simulations of contact prob-lems.Due to its importance we will consider this contactelement in more detail. Assume that the discrete slavepoint (s) comes into contact with the master segment(l)-(2), then the kinematical relations can be directlycomputed using the equations given earlier. With theinterpolation for the mater segment, a(t) =x: + (x s - xj)[, the tangent vector of the segmentfollows as: a: = a([),, = (x2 - x:). It is connectedto an orthonormal base vector af = if/ l withI = IIxf - x:/l being th e current length of the mastersegment. Now the unit normal to the segment (l)-(2)can be defined as n7 = e3 x a:.2 and gN are given by the solution of the min imaldistance problem (2), i.e. by the projection of theslave node x,~ in (s) onto the master segment (l)-(2)

From these equations and the local formulation (3)we compute directly the variation of the gap function8gSN+ on the straight master segment (l)-(2):


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Equation (4) yields the expression for 62. With theinterpolation for the variation, +j2(Q = v: + e(q$- $), on the strakht master segment (l)-(2) wespecialize H,, and RI in (4) which leads to:Sg, = IS?= [7ji - (1-j)gf - f+] . a:

+ yyq; - q:] . n2 (20)Equations (18) to (20) characterize the main kinemat-ical relations of the contact element.In what follows, we compute the contribution of thenode-to-segment element to the weak form (14). Weassume that we know the normal forceP N 5 = pN .A b and the tangential force TT 5 = fT s Asat the discrete contact point (s) of the contact elementunder consideration, where A, denotes the area of thecontact element. Both forces, PN s and TT d, can beobtairted from the constitutive relations discussedearlie!:. This leads to:

i (PN h + fT %r)dr =iI.,. c (PNJ &NT + T T ~ %T s)

s-1(21)

PNs follows directly from (8) multiplied by the areaA, of the contact element. For the tangential forceT.r s we have to perform an algorithmic update whichis described in the next section.The contributions of one contact element in (21) cannow Ike cast into a matrix formulation. For the normalpart (19) and the tangential part (20) we have:

Q:,.s = v?N.s 6gTs = qT t T, +g*N,,,I 1 (22)with the definitions:

3-, = (n 2, -(l-j)n, -i$n2)T,3 os = (0, -n2, n)P (23)

andTH, = (af, -(l- j)af, -&I:)~T,P os = (0, -al, aT)T (24)

Thus the virtual mechanical work (21) of the contacteleme:lt can be written as qTGS with the contactelement residual vector:GF=P,.N, + TTs t

gN sT, +pNos1 1 (25)Often a Newton-Raphson iteration is used to solvethe global set of equations. Then the linearization of(25) is needed to achieve quadratic convergence nearthe solution point. The associated derivation is a littlecumbersome and thus only the final results will besummarized. Details of the frictionless case can befound in Wriggers and Sirno and for contact includingfriction in Wriggers et a1.8.

The tangent matrix for the normal contact is derivedfrom the t_erm 6gN 5 P, ~ in (21). Observe that thechange in 5 has been considered as well as the changeof the normal n2. We obtain with (8) the tangentmatrix:

(26)Note that in a geometrically linear case all termsvanish which are multiplied by g,,. This results in theIllatriX Kkc, = (df,.J/d&,Tr)N, NT.For the tangential contributions in the contact areawe have to linearize the term 6gT s TT s in (21) whichleads for the elastic response (stick condition) to:TK+.=cT gN FT, +--Nos gN I1 T, +--No,I

+gy N,,N: +N,N;, -T,.T,T -T,T;f,- Zgy(N,.T6., + T,,,N;,) Ii (27)

Also in this case all terms containing gN .s disappearin a geometrically linear situation which yieldsK+ : = c,T,~TT. The case of frictional slip leads to anadditional contribution in (27) which will be discussedin the next section.

Algorithms for contact problemsThe algorithm which is applied in many standard finiteelement programs is related to the active set strategyin which the problem is solved for a chosen active setof constraints. Before we state this algorithm wecombine (17) and (25) to the global set of equations:G?(v) = G(v) + i G,(v) = 0 (28)s=l

where G(v) denotes the contributions of the bodiesdue to the weak form (17). Now the algorithm issummarized in Box 1.

Initialize algorithmset: v, = 0LOOP over iterations: i = 1, . . . convergenceCheck for contact: g, Si 5 0 7 active nodeSolve: GJVi) = G(v;) + Uzs, Gz(vi) = 0Check for convergence: /GJvi)jl 5 TOL +END LOOPEND LOOPBox I Contact algorithm using the penaltymethod

When ill-conditioning occurs the augmented Lagrang-ian technique has to be applied see e.g. Wriggers andZavarisell.




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Within the global algorithm the algorithmic update ofthe tangential stress tT n+l due to friction is performedby a return mapping algorithm based on an objective(backward Euler) integration of the evolution equat ion(12) for the plastic slip, see e.g. Wriggers orGiannokopoulos19. The results can be summarized asfollows: integration of (5) gives the increment of thetotal slip within the time step AtntlAg T,z+l = (?:+I - khz+l (29)

The total slip has to be decomposed into an elasticand a plastic part, see Equation (10). Then we cancompute the elastic tria l state from (10) and evaluatethe slip criterion (11) at time trZ+,:ttr t n+l := CT(gTn-1.1 - gk,,) = h-n + CT A&n+,Ef,l,z-f1 := Il%-l~~ -FPN ntl (30)

If this state is elastic Cftsrll+i 5 0) then no frictiontakes place and we have to use the elastic relation(10). In the case that firn+, > 0 we have to performthe return mapping. Using the impl icit Euler scheme,(12) yields:

g+n+1 = !&-II + An Tnilg, n il = g II +A (31)

With the standard arguments regarding the projectionschemes, see e.g. Simo and Taylor9, we obtain:t - ttrTn+l - t w- i - ACT n,- ,+,

nTFz+l =n% T-l+1 (32)The multip lication of (32) by nT n+l yields the conditionfrom which h can be computed:

where & can be a nonl inear function of h. Thus ingeneral we need an iterative scheme to solve K(A). Incase of Coulombs model we can solve directly for h:A = $ Wn+l II - PPN n+l)

Knowing A the stress update follows from (32) andthe frictional slip from (31). Again we state the explici tresults for Coulombs model:

Fig 3 Purzch problem, jkite element mesh Fig 5 Deformed meshes for cases (I), (2) and (3)

Punch ProblemPlane StrainLoad Indentation Curve

,,I,

4

0.4 0.8 1.2 1.6Maximal IndentationFig 4 Total punch force versus maximal displacement

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whick completes the algorithm for the frictionalinterface law.The tangent matrix which is needed within a Newtoniteration can be derived by lineariz ing the termwhich- appears in the weak form with respect to thedispkcement field. The explicit matrix form resultsfrom the term 6gr .s,l+ 1 T, arz-ml nd can be stated forCoulombs model as follows:

K;.c, = K dfN-r >p-- %N s T,++$, NTi (36)with K$- J from (27). This matrix is unsymmetric whichcorresponds to the non-associativity of Coulombsfrictional law. Note that the non-symmetry of thetangent matrix has to be considered to obtain a robustalgorithm.Numerical exampleTo demonstrate the applicability of the derived numeri-cal method we consider a punch problem in which a

STRESS 5Min = O.OOE+OOMax = 1.90E+OO

9.90E-01l.O9E+OOl.l9E+OO1.3OE+OO.._ . 1.40E+OO150E+OO

Current ViewMin = O.OOE+OOX = 4,99E+OOY = 3.00E+OOMax = 1.90E+OOx = 6.86E+OOY =-9.81 E-01STRESS 5Min = O.OOE+OOMax = 1.86E+OO

I 9.90E-01

Current ViewMin = O.OOE+OOX = 5.00E+OOY = 3.00E+OOMax = 1.86E+OOx = 6.62E+OOY =-7.36E-01

Fig 6 Plastic zoue for cases (I) and (2)Tribology International Volume 29 Number 8 1996 657


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rigid punch, as depicted in Fig 3, is pressed against abody made of steel. Large deformations and elasto-plastic behaviour are considered. Three different casesare examined: (1) frictionless contact, (2) contact withCoulomb friction (p = 0.2), and (3) frictionless contactwhen the surface of the body is coated with a materialof much higher yield strength.The finite element mesh and the boundary conditionsare shown in Fig 3. The material data for the bodyare: bulk modulus K = 165, shear modulus G = 80,yield strength Y0 = 0.45 and linear isotropic hardeningmodulus H = 0.15. Within the calculations the dis-placement of the punch is prescribed. The maximaldisplacement of 2 is reached within 40 load steps.Within each load step the above-described Newtonalgorithm needs between six and 11 iteration steps forconvergence. This includes the nonlinearities dueto elasto-plasticity, large deformations and frictioncontact. The difference of the total number of iterationsper step stems from the fact that Newtons methodconverges only quadratically near the solution point.Thus, if the number of nodes being in contact changesduring a load step then the additional iterations areneeded to establish the new contact area. Figure 4shows the total force versus the maximal punchdisplacement. We observe that cases 1) and (2) depictalmost the same curve. Only in the range of largeindentations do the curves deviate and case 2) presentsa more stiff behaviour. This can be explained by thefact that the frictional forces only contribute to thepunch force in case of a large indentation, as depictedin Fig 5 (2). For case (3) we obtain immediately adifferent solution since the coating distributes theloading due to the punch over the body which resultsin a considerable stiffening. The deformed meshes fora punch displacement of 2 show clearly in Fig 5 thedifferent behaviour of the three cases n the contactarea. Due to the plastic incompressibility the materialis squeezed out from under the punch in the frictionlesscase (1). This is prevented by the frictional forces incase (2). In Fig 6 the plastic zone is depicted for cases(1) and (2) when the same punch displacement isapplied. Here we observe a different elasto-plasticbehaviour of the system under the punch.ConclusionThis overview summarized some of the current researchwork in computational contact mechanics for frictionalproblems. Due to the broadness of contact formulationsand algorithms and the limitation of space not allpromising new approaches have been discussed indetail. One numerical example, which included largedeformation, has been presented. However this couldnot stand for all mentioned topics. More exampleshave been omitted, since this overview was aimed at

providing the reader with the underlying theoreticalderivations.References

I.

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FEM Method for Contact Problems

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