Submitted to: International Journal of Mechanical Sciences Revised paper: SR/2000/6329

An enriched finite element algorithm for numerical

computation of contact friction problems

A.R. Khoei * and M. Nikbakht

Center of Excellence in Structural and Earthquake Engineering, Department of Civil Engineering,

Sharif University of Technology, P.O. Box. 11365-9313, Tehran, Iran

Abstract. In this paper, the extended finite element method is employed to model the presence of discontinuities caused by frictional contact. The method is used in modeling strong discontinuity within a standard finite element framework. In XFEM technique, the special functions are included in standard FE method to simulate discontinuity without considering the boundary conditions in meshing the domain. In this study, the classical finite element approximation is enriched by applying additional terms to simulate the frictional behavior of contact between two bodies. These terms, which are included for enrichment of nodal displacements, depend on the contact condition between two surfaces. The partition of unity method is applied to discretize the contact area with triangular sub-elements whose Gauss points are used for integration of the domain of elements. Finally, numerical examples are presented to demonstrate the applicability of the XFEM in modeling of frictional contact behavior. Keywords: Contact friction, Extended FEM, Partition of unity, Theory of friction.

1. Introduction

The numerical modeling of engineering contact problems is one of the most difficult and demanding tasks in computational mechanics. Frictional contact can be observed in many problems; such as: crack propagation, metal forming operation, drilling pile etc. In metal forming operations the required shape changes are obtained by either of forming process, such as pressing, hammering, rolling or extruding the material between the tools which are much stiffer than shaped material. Because of large difference between deformability of the tool and material, relative movements occur in contact area. These relative movements produce the normal and tangential stresses, which have important role on metal flow and may cause serious inhomogeneities in works products. * Corresponding author. Tel. +98-21-66005818; Fax: +98-21-66014828. Email address: arkhoei@sharif.edu (A.R. Khoei)

Therefore, it is not surprising that much attention has been given to both the experimental and numerical research aspects of this complex problem. From the finite element point of view the modeling of interface friction has been categorized into three routes; the traction boundary condition, contact node algorithm, and interface element technique. In the method of 'traction boundary condition', the frictional forces are appended to the external force vector as a traction boundary force. In this technique, the contact requirements appear in the variational formulation as constraints. The ‘contact node algorithm’ is an iterative implementation of the boundary conditions on velocity or displacement depending on the nodal forces at the interface friction boundary. This method is a pointwise algorithm, which uses the contact nodes as indications of the contact conditions. The ‘interface element technique’ is an alternative to contact node algorithm, which can be applied by implementation of thin elements having very high aspect ratio. Early studies on contact problems were largely related to a linear geometry and often involved node-to-node contacts when two boundaries come into contact (Fracavilla and Zienkiewicz [1], Hughes et al. [2] and Beer [3]). Such a node-to-node model can only be applied to problems in which relative sliding displacements of the two contact boundaries are sufficiently small. Once significant non-linear contact deformations were introduced, methods tended to switch to node-on-segment contact (Wriggers and Simo [4], Parisch [5], Papadopoulos and Taylor [6] and Goncalves et al. [7]). There are important links between the finite element contact problem and mathematical programming techniques. Indeed, because the contact problems usually involve inequality constraints (varying contact areas), the mathematics can be related to the method of variational inequalities. Basically, two main constraint methods of solution have been employed in the finite element solution of contact problems; the method of Lagrangian multipliers and the penalty approach. In Lagrangian multipliers approach, the contact forces are taken as primary unknowns and the non-penetration condition is enforced (Simo et al. [8], Chaudaray and Bathe [9] and Gallego and Anza [10]). In penalty method, the penetration between two contacting boundaries is introduced and the normal contact force is related to the penetration by a penalty parameter (Curnier and Alart [11] and Peric and Owen [12]). The aim of this study is to present a model for simulation of frictional contact problem using the extended finite element method based on the penalty approach.

The eXtended Finite Element Method (XFEM) is a numerical approach used to simulate the discontinuity within the standard finite element framework. In this approach, the standard displacement based approximation is enriched by incorporating discontinuous fields through a partition of unity method (Melenk and Babuska [13]). The method was developed by Dolbow et al. [14, 15] to model cracks, voids and inhomogeneities. This allows for the entire crack geometry to be modeled independently of the mesh, and completely avoids the need to remesh as the crack grows. A methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh was developed by Daux et al. [16] in modeling crack discontinuities. A technique for modeling arbitrary discontinuities in finite elements was presented by Belytschko et al. [17]. In this method, the discontinuous approximation was constructed in terms of a signed distance functions and the level sets were used to update the

position of the discontinuities. An algorithm which couples the level set method with the extended finite element method was proposed by Stolarska et al. [18]. They applied a discontinuous function based on the Heaviside step function in modeling two-dimensional linear elastic crack-tip displacement fields. Recently, a numerical technique was developed by Sukumar et al. [19] for three-dimensional fatigue crack growth simulations. This technique couples the extended finite element method to the fast marching method using the partition of unity method.

In the present paper, an extended finite element method (XFEM) is developed to simulate the

frictional behavior of contact problems. The classical finite element approximation is enriched by employing additional terms based on the Heaviside step function. These terms depend on the contact conditions between two surfaces and model the stress-strain relationship in contact area. As the contact region has discontinuities in domain, special techniques are implemented in the XFEM. The partition of unity method (PUM) is applied to discretize the contact area with triangular sub-elements whose Gauss points are used for integration of the domain of elements. The conditions that describe frictional contact are formulated as a non-smooth constitutive law on the interface of contact and the iterative scheme is implemented to solve the nonlinear boundary value problem. Finally, numerical examples are presented to demonstrate the applicability of the extended finite element method in modeling of frictional contact behavior.

2. Continuum model of friction The objective of the mathematical theory of friction is to provide a theoretical description of motion at the interface of bodies in contact. The plasticity theory of friction can be achieved by an analogy between plastic and frictional phenomena. In order to formulate such a theory of friction several requirements have to be considered (Curnier [20], Rodic and Owen [21] and Peric and Owen [12]). These requirements, which are similar to the requirements considered in the theory of elasto-plasticity, are as follows;

i) Stick (or adhesion) law; a mathematical description of the stress state under sticking (elastic) conditions,

ii) Stick-slip law; a theoretical description of the relationship between stress and stick-slip (elasto-plastic) conditions,

iii) Wear and tear rule; a hardening and softening rule during sliding,

iv) Slip criterion; a yield criterion indicating the stress level at which relative slip motion occurs,

v) Slip rule; a flow rule indicating the relationship between stress and slip motion. Consider two bodies, a master (target) and a slave body, as shown in Figure 1, with initial configurations denoted by SΩ and TΩ . The relative displacement from the point x on the contact surface of the slave body to the point y on the contact surface of the target may be defined as

)()()( on )()()( TSC N ,g Ω∂Ω∂=Ω∂⋅−= χχχχχ INyxyx (1)

where Ng is the gap between the two bodies, )(xχ and )(yχ are the configuration mapping of the slave and target bodies, )( SΩ∂χ and )( TΩ∂χ are the slave and target body surfaces respectively,

)( CΩ∂χ denotes a surface where contact between the two bodies occurs and N is the unit outward normal vector on the target surface. During the contact and sliding of the bodies, we define Np and

Tp as the normal and tangential load acting on the point x , respectively. The contact conditions may be expressed in the standard Kuhn-Tucker form as 00 0 =≤≥ ⋅ NNNN gppg (2)

which is best suited for a variational formulation. Consider that there is no gap between the two bodies in the sliding contact problem (Figure 1), the normal displacement is assumed to be zero and a tangential displacement is only considered, which consists of stick and slip decompositions and is in principle the same as the decomposition of elastic and plastic behavior. Thus, the decomposition of the tangential displacement at the contact surface can be given as

pT

eTT uuu += (3)

where Tu is the tangential part of the displacement described by uu N)NI ( ⊗−=T and, eTu and p

Tu are the elastic and plastic components of tangential displacement.

In relation (2), the kinetic constraint of impenetrability of two bodies can be satisfied as well as the static condition of compressive normal load. To resolve the resulting unilateral contact problem the Lagrange multiplier method, or the penalty method, are typically used. In the case of the Lagrange multiplier method, however, a large number of additional unknown variables need to be included owing to incorporation of Np as new variables. On the contrary, the penalty method needs no additional variable, because the impenetrability condition is approximately satisfied (by constraint via embedding very stiff springs on the contact surface). Consequently, the normal load

Np can be obtained from multiplication of the penalty factor Nk and the displacement in the normal direction Nu . Similarly, the stick (or elastic) component of the tangential load may be obtained by multiplying the penalty factor Tk and elastic part of the displacement in the tangential direction e

Tu . The penalty factors Nk and Tk can be considered as being the normal stiffness constant and shear stiffness constant, respectively. Constitutive laws for the contact loads can now be summarized as

( ) eN

efN uDp = (4)

( ) eT

efT up D= (5)

where ( )N

efD and ( )

TefD are the normal and tangential parts of the elastic modulus tensor for friction

defined as

( ) ( )NND ⊗−= NNef k (6)

( ) ( )NNID ⊗−−= TTef k (7)

In order to perform the additive decomposition of displacement into adherence and slip, a slip criterion must be introduced. To this end the slip surface fF is postulated in the contact stress space on which slip will occur. The slip criterion is expressed based on the Coulomb law as

( ) ( ) ( )⎩⎨⎧<=

−−= adherence0

graporslip0,, fNNfTf cwpwF ppp µ (8)

where fc denotes the cohesion between two bodies and the Coulomb friction coefficient is defined as ff ϕµ tan= , with fϕ denoting the tool friction angle. The direction of slip is governed by an appropriate slip rule, which can be derived from the gradient of a convex potential Z . If the potential Z is replaced by the slip criterion fF , the slip rule becomes associated. Although in plasticity the flow rules associated with the standard criteria prove realistic for relatively large classes of materials, the slip rule associated to the usual friction criterion (8) is not acceptable. Indeed, the relative movement at the interface derived from the associated potential fFZ = results in the creation of gaps (separation). Thus, in order to avoid the slave body separation from the contacting surface, a non-associated flow rule is typically adopted (Curnier [20]). Hence, the slip potential Z is introduced as a cylinder with radius TP for isotropic frictional contact and the slip direction is defined as the outward normal to the slip potential Z . Consequently, the plastic part of the tangential displacement p

Tu , in equation (3), can be defined by the definition of the slip rule as

Tp

u λλ d d d =∂∂

=T

pT

Z (9) where λd is a constant expressing the collinearity of the slip increment with the outward normal to the potential Z , and

T

Tp

pT

= is the unit tangential vector.

The constitutive model for the contact problem is conveniently described by equations (2−9).

Following the standard arguments of elasto-plasticity, the continuum tangent tensor for the contact problem with non-linear frictional evolution can be derived using the consistency condition as (Khoei [22])

( ) ( ) ( )

0d,

d,

d,

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂w

wwFwFwF f

NN

fT

T

f pp

pp

ppp

(10)

substituting λd into the constitutive law, a linearized equation is obtained in the incremental form as

uDp d d epf= (11)

where the continuum tangent tensor epfD is

( ) ( )

( ) ( )

( ) ( )NTp

p

TTp

TTNNINND

⊗⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+−

⊗⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

⊗−⊗−−⊗−=

N

fNNfN

fN

T

NfT

TNepf

wpk

wkwp

k

kk

µµ

β

µµβ

,1

,1 2 (12)

where ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+=

wkwp f

NT

Nf µµβ 2,

1 p .

In equation (12), the first term of the continuum tangent tensor ep

fD indicates the stiffness in the normal direction to the contact surface. The second term denotes the adhesion stiffness perpendicular to the sliding direction on the contact surface. The third and forth terms indicate the adhesion and slip stiffness with hardening, or softening, in the sliding direction, respectively. In the case of frictional slip without hardening, or softening, equation (12) can be simplified as

( ) ( ) ( )NTTTNNINND ⊗−⊗−⊗−−⊗−= NfTNepf kk k µ (13)

Evidently, the non-associative slip rule (9) results in non-symmetry of the slip modulus tensor, which is defined by (13) under the conditions of frictional slip. 3. The extended finite element method The enriched finite element methods are powerful and accurate approaches to model discontinuities without considering their geometries. In these methods, the discontinuities are not considered in mesh generation operation and special functions which depend on the nature of discontinuity are included into finite element approximation. One of these approaches is extended finite element method, which has been extensively employed in numerical modeling of crack. The aim of this method is to simulate the discontinuity with minimum enrichment. In XFEM, the external boundaries are only consideration in mesh generation operation and internal boundaries, such as cracks, voids or contact surfaces, have no effect on mesh configurations. This method has proper applications in problems with moving discontinuities, such as consolidation, phase changing, crack propagation, and shear banding. In order to introduce the concept of discontinuous enrichment, consider that cΓ be a contact surface between two bodies in domain Ω , as shown in Figure 2(a). We are interested in the construction of a finite element approximation to the field Ω∈u which can be discontinuous along contact surface cΓ . The traditional approach is to generate the mesh to conform to the line of contact surface as shown in Figure 2(b), in which the element edges align with cΓ , and implement the contact elements in the line of contact surface. While this strategy certainly creates a

discontinuity in the approximation, it is cumbersome if the line cΓ evolves in time, or if several different configurations for cΓ are to be considered. In this study, we intend to model the discontinuity along contact surface cΓ with extrinsic enrichment, in which the uniform mesh of Figure 2(c) is capable of modeling the contact surface in u when the circled nodes are enriched with functions which are discontinuous across cΓ . The standard FE approximation can be enriched with additional functions by using the notion of partition of unity (Melenk and Babuska [13]). The enriched approximation in modeling of contact surface cΓ can be expressed in following form gji

jjj

iii

h nnfNN n nauu ∈∈+= ∑∑ andfor )()( )()( xxxx (14)

The first term of above equation denotes the classical finite element approximation and the second term indicates the enrichment function considered in XFEM. In this equation, iu is the classical nodal displacement, ja the nodal degrees of freedom corresponding to the enrichment functions,

)(xf the enrichment function, and )(xN the standard shape function. In equation (14), n is the set of all nodal points of domain, and gn the set of nodes of elements located on discontinuity, i.e. 0 ,: ≠Γ∈= cjjjg nn Iω nn (15)

In the above equation, )(supp jj n=ω is the support of the nodal shape function )(xjN , which consists of the union of all elements with jn as one of its vertices, or in other words the union of elements in which )(xjN is non-zero. It must be noted that the enrichment varies from node to node and many nodes require no enrichment, which is an application of the partition of unity concept. Different techniques may be used for the enrichment function; these functions are related to the type of discontinuity and its influences on the form of solution. These techniques are based on the signed distance function, branch function, Heaviside jump function, level set function, and etc. The signed distance function is applicable to crack problem, which is discontinuous across the crack line (Belytschko et al. [17]). The function can be viewed as an enrichment with a windowed step function, where )(xN is the window function. The window function localizes the enrichment so that the discrete equations will be sparse. For cracks which are not straight, a mapping is required to align the near-tip discontinuities with the crack edges. In this case, a near-tip function, or branch function, can be constructed in terms of the distance function, which enables the discontinuity to be curved or piecewise linear (Dolbow et al. [15]). This function spans the near-tip asymptotic solution for a crack, and gives very good accuracy for these problems. The level set method is a numerical scheme developed by Sethian [23] for tracking the motion of interfaces. In this technique, the interface is represented as the zero level set of a function of one higher dimension. Recently, the technique of fast marching method, which was first introduced by Sethian [24], was coupled with XFEM to model crack growth (Sukumar et al. [19] and Chopp and Sukumar [25]). The method

computes the crossing time map for a monotonically advancing front in an arbitrary number of spatial dimensions.

4. Modeling frictional contact with the X-FEM

Numerical simulation of frictional contact in FEM can be achieved by employing contact elements. Although these elements have wide application in simulation of contact problems, the modeling of evolving contact surfaces with the finite element method is cumbersome due to the need to update the mesh topology to match the geometry of the contact surface, and implement those elements between two different bodies (Figure 2-b). The extended finite element method alleviates much of the burden associated with mesh generation by not requiring the finite element mesh to conform to contact surfaces, and in addition, provides a seamless means to use higher-order elements or special finite elements without significant changes in the formulation. The essence of X-FEM lies in sub-dividing a model problem into two distinct parts; mesh generation for the geometric domain in which the contact surface is not included, and enriching the finite element approximation by additional functions that model the geometric of contact surface (Figure 2-c). Consider that cΓ be a contact surface between two bodies in domain Ω with n denoting the normal vector to cΓ . The displacement and traction are then introduced on the contact surface by u and t . In order to obtain an appropriate form that is suitable for numerical treatment of contact behavior, the weak form of equilibrium equation of elasto-plasticity can be written as

dΓdΓddcΓtΓ

)( ututubu ∫∫∫∫ ++Ω=Ω:ΩΩ

εσ (16)

in which the last term represents the concept of energy, dissipated in the relative motion of the contact surface. It is important that the displacement field of domain and the displacement on contact surface be kinematically admissible. The interfacial constitutive law on the contact surface is expressed by equation (8) that indicates the stress level at which relative slip motion occurs. The goal is to obtain the stress and displacement fields on the contact surface which satisfy the equilibrium and consistency conditions.

For an arbitrary contact displacement field, equation (14) can be rewritten as

gjij

jji

iih nnHNN n nauu ∈∈+= ∑∑ andfor )()( )()( xxxx (17)

where )(xH is the Heaviside jump function. In above relation, the contact surface is considered to be a curve parameterized by the curvilinear coordinate s , as shown in Figure 3. Considering a point x in the domain, we denote *x the closest point to x on the contact surface. At point *x , we construct the tangential and normal vectors to the curve, se and ne , with the orientation of ne taken such that zns eee =∧ . The Heaviside jump function )(xH is then given by the sign of the scalar

product nexx )( *− , in which the function )(xH takes the value of +1 ‘above’ the contact surface, and −1 ‘below’ the contact surface, i.e.

⎩⎨⎧

−≥−+

= otherwise

f

10 )( i 1)( nH exxx

*

(18)

On substituting the trial function of equation (17) into the weak form of equilibrium equation of elasto-plasticity (16), and using the arbitrariness of nodal variations, the discrete system of equations can be obtained as fdK = , where d is the vector of unknowns of iu and ja at the nodal points, and K and f are the global stiffness matrix and external force vector, defined as

⎥⎥⎦

⎤

⎢⎢⎣

⎡= aa

ijauij

uaij

uuij

ij KKKK

K , Tai

uii fff = (19)

where

) , ,()( )(

aude j

epTiij =Ω= ∫Ω

βαβααβ BDBK

) ,(

audNdNee iii =Ω+Γ= ∫∫ ΩΓ

αααα btf (20)

where epD is the elasto-plastic constitutive matrix. In equation (20), ii NN ≡α is for a finite element displacement degree of freedom, and HNN ii ≡α for an enriched degree of freedom. The matrices

αiB and β

jB include the shape function derivatives defined as

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

xiyi

yi

xiui

NNN

N

,,

,

,

00

B

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

xiyi

yi

xiai

HNHNHN

HN

,,

,

,

)()()(0

0)(B

(21)

For the elements cut by the contact surface, the standard Gauss quadrature points are insufficient for numerical integration, and may not adequately integrate the discontinuous field. If the integration of the discontinuous enrichment is indistinguishable from that of a constant function, the system of equations may be rank deficient. Thus, it is necessary to modify the element quadrature points to accurately evaluate the contribution to the weak form for both sides of the contact surface. In what follows, we present the modifications made to the numerical integration scheme for elements cut by a contact surface, which was applied by Daux et al. [16] in modeling of cracks with multiple branches.

The discrete weak form is normally constructed with a loop over all elements, as the domain is

approximated by

e

m

eΩ=Ω

=1U (22)

where m is the number of elements and eΩ is the element sub-domain. For elements located on contact surface, an appropriate procedure is performed. For those elements, we divide the element into triangular sub-domains sΩ with boundaries aligned with the contact surface geometry, i.e.,

s

m

se

s

Ω=Ω=1U (23)

where sm denotes the number of sub-polygons of the element. The Gauss points of sub-triangles are used for numerical integration across the contact surface, as shown in Figure 4. Different algorithms may be applied to generate these sub-polygons, based on sub-triangles and sub-quadratics. In this study, sub-triangles are implemented for numerical integration. It is essential to mention that these sub-polygons only generated for numerical integration and no new degrees of freedom are added to system. In the construction of the matrix equations, the element loop is replaced by a loop over the sub-triangles for those elements cut by the contact surface. 4.1. Friction tangent matrix Evaluation of the stiffness matrices for the elements located on contact surface requires linearization of the governing equations for the frictional contact problem. Strict mathematical linearization results in non-symmetric constitutive matrices ep

fD , defined in equation (13), due to the non-associated slip rules employed, i.e.

⎥⎦

⎤⎢⎣

⎡=

2221

1211

DDDDep

fD (24)

In order to preserve the symmetry of the numerical formulation the off-diagonal term in epfD , which

represents the coupling between the normal and tangential stresses at the interface, is neglected, so that its effect is brought into the formulation via residual 'pseudo loads'. In this way, the problem is artificially decomposed into a pure contact in the normal direction and frictional resistance in the tangential direction, which are linearized separately as

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

f

fepf E

G

0

0D (25)

In an incremental manner, the stress components are related to the strains through the material property matrix ep

fD by εσ ∆=∆ epfD . The stress vector at each Gauss point located in specified

distance from both sides of contact surface, i.e. the contact bond, is assumed to have only two components; the normal stress nσ and shear stress τ , where T

nστ ∆∆=∆ , σ and Tnγ ε∆∆=∆ , ε .

The material property matrix epfD needs to capture the details of the physical processes taking place

such as asperity contact, adhesion and the consequent 'stick-slip' behavior. Since there is no volume

change due to shearing strains, the shear and normal components of deformation are therefore uncoupled. The normal strains are measured only to monitor the normal stress by means of the linear equation, i.e. nfn E εσ = , where fE is chosen as an arbitrary large number for numerical convenience. Note that only compressive normal stresses are allowed, i.e. 0≤nσ . Since the tangential micro-shifts due to adherence are negligible in comparison to the micro-slips due to sliding, the adherence does not require a very precise numerical treatment. The incremental form of the shear stress-strain relationship under adherence is defined as γτ ∆=∆ AG , where AG is proportional to the stick shear modulus of contact surface. The frictional shear force, however, is limited by the slip criterion. The shear stress-strain relationship is presented in Figure 5, where

nf σµ=T is considered to be a known state variable [22]. The frictional non-linearity is modeled by an appropriate variation of fG which can be obtained from the shear stress-strain )( γτ − relationship for the stick and slip region (Figure 5). It must be noted that the shear modulus fG is divided into two components, the first where the body has not moved but there is a rapid buildup in load. Under this condition, a stick shear modulus Af GG = is derived directly from the slope of the

γτ − curve. On commencement of movement, the slip shear modulus condition Bf GG = is now appropriate and this is derived similarly. Finally, the stiffness matrix of equation (20) can be evaluated at each Gauss point located on contact bond by

) , ,()( )()( aude j

epf

Tiijf =Ω= ∫Ω

βαβααβ BDBK (26)

4.2. Computational algorithm In order to construct the integrals on the contact surface, it is necessary to implement the material property matrix ep

fD at the Gauss points located on contact bond. In traditional finite element discretization, the integrals are evaluated using the Gauss integration points of contact elements. Here, since the contact surface and mesh geometry are independent, we replace equation (17) by (23) employing the material property matrix defined in (22) at the Gauss integration points located on contact bond. According to numerical procedures described in preceding sections, a computational algorithm is performed based on the Newton-Raphson method. For iteration i within the time step t∆ ( nn ttt −=∆ +1 ) and for each active Gauss point located on contact bond, the following algorithm is set up;

a) Evaluate the stiffness matrix of element ifK cut by the contact surface using the appropriate

shear modulus fG of the Gauss point (In the first iteration of first time step, set Af GG = . In subsequent iterations, fG is calculated from stage (f)),

b) Solve the global system of equations iii dd fuK = ,

c) Compute the increment of nodal displacement as iii d uuu 1 +∆=∆ − , where 00

=ud ,

d) Evaluate the increment of strain in 1 +∆ε and total strain i

nnin 11 ++ ∆+= εεε ,

e) Calculate the stress at the normal direction according to nfn E εσ = and evaluate nf σµ=T at 1+nt (Note that only non-positive values of normal stress are allowed),

f) Evaluate the frictional shear stress τ and fG for the next iteration, i.e.

inAn

in γG 11 ++ ∆+=ττ and Af GG = (27)

g) If T>τ at 1+nt then correct the current values of τ and fG according to

γγ

∆∆

= Tτ , γ

G f ∆−

=τT (28)

h) Evaluate the out of balance, or residual forces. Computational steps (a) to (h) are repeated until the norm of residual forces and maximum

residual are both less than prescribed tolerances.

5. Numerical simulation results In order to illustrate the accuracy and versatility of the extended finite element method in frictional contact problem, several numerical examples, including: sliding of two bodies, pull-out of pile, torsion of steel rod in concrete, and compaction process are presented. The examples are solved using both FEM and XFEM techniques, and the results are compared.

5.1. Sliding of two bodies In the first example, the contact friction behavior between two elastic bodies which are sliding relative to each other is investigated, as shown in Figure 6. When the top block is sliding over the surface of bottom block, the stick-slip motion can be observed due to friction. This motion is characterized by a periodic switching between sticking and slipping. In this study, an extended finite element approach is employed to simulate the contact friction behavior between two sliding surfaces. The problem statement for this example is shown in Figure 6. The block is constrained at the bottom while the uniform horizontal and vertical loadings of mKg105.2 3×=xw and

mKg101 4×=yw are imposed at the top. The blocks are assumed to be elastic with the Young’s modulus of 210 mKg102× and Poisson’s coefficient of 0.3. The contact behavior between two blocks is modeled by the Coulomb friction law with 0=fc and 3.0=fµ . Two meshes corresponding to the FEM and XFEM techniques are considered, as shown in Figure 7. In FEM mesh, the finite elements are combined with the interface elements along the contact surface. While in XFEM mesh, the contact surface passes through the center of elements in

which the nodal points of relevant elements are enriched with the Heaviside jump function. For those elements which are intersected by the contact surface, the concept of the partition of unity is used to generate the sub-triangles. The Gauss points of sub-triangles are then employed to evaluate the tangent stiffness matrix of friction. The numerical simulation is performed using 200 six-noded triangular elements in FEM mesh and 361 four-noded rectangular elements in XFEM mesh. The convergence tolerance is set to 410− and the analysis was performed using 20 increments. In Figures 8(a)-(d), the distribution of shear stress contours at two different values of shear modulus, i.e. 8101×=fG and 210 mKg101× are presented for both XFEM and FEM techniques. Remarkable agreements can be observed between two different methods. Figure 9 presents the variation of horizontal displacement of top edge with the shear modulus of contact surface using two different approaches. A convergence study is conducted in Figure 10 using various values of contact bond width, i.e. the region whose Gauss points have similar property as contact surface. In this figure, the variation of uniform shear loading with horizontal displacement of top edge is plotted at three values of contact bond width, i.e. 0.5, 0.7, 1.0. It can be observed from the figure that the contact bond width of 1.0 gives a remarkable improvement in accuracy of horizontal sliding.

5.2. Pull-out of pile The second example demonstrates the accuracy of XFEM technique in numerical simulation of pile foundation. Pile foundations are widely used in highway construction, buildings and other structures. Accurate and reliable determination of pile capacity is very important for proper design, construction and estimation of the cost of these foundations. The capacity of pile is strongly dependent on frictional interaction between soil and pile. When the pile is subjected to gradually increasing load, slip is induced at the interface and propagated from top to bottom of the pile. In this example, the pull-out of pile is modeled by XFEM technique, as shown in Figure 11. The simulation was performed by Lei [26] using the FE method and employing interface element to present the performance of integration schemes in evaluation of the interpolation function matrix.

The material properties chosen for the concrete pile are; 29 mKg102×=cE , 3.0=υ and 33 mKg105.2 ×=ρ . The pile is placed in clay soil with material properties of 28 mKg102×=sE ,

25.0=υ and 33 mKg102×=ρ . The contact friction behavior between pile and soil is modeled by the Mohr-Coulomb law with 2cmKg5.0=fc , 58.0=fµ and the maximum tensile stress of

2max cmKg30=tσ . The soil is restrained at the bottom and right hand edges, and a tension force of mKg105 4×=q is imposed on the upper nodes of the pile. On the virtue of symmetry, the pile is

analyzed for half of space, as shown in Figure 12. A uniform FE mesh of four-noded rectangular elements of 20 x 20 is considered in order to model the contact surface with the enriched nodal points shown in this figure. The convergence tolerance is set to 410− and the computation was carried out using 100 increments. For elements located on contact surface, the sub-triangles are generated based on the concept of the partition of unity to construct the tangent stiffness matrix of friction at the integration points located on contact bond. The shear stress distribution of pile along

the contact surface due to pull-out force is presented in Figure 13(a). This result can be compared with those obtained by Lei [26] using the interface element technique in Figure 13(b).

5.3. Concrete reinforced with steel under torsion The next example is chosen to present the capabilities of XFEM technique in modeling of contact behavior between steel rod and concrete. The concrete reinforced with steel is an alternative reinforcement for concrete structures, particularly in environments capable of chemical and electromagnetic corrosion. Simulation of contact phenomenon results in better application of the steel rods in construction of modern concrete structures. The bond behavior is important in perception of the nature of local failures and the amount of energy dissipation in the structural elements. In this example, a numerical simulation of contact friction of steel rod with concrete is presented using FEM and XFEM techniques. A steel rod with a radius of m 4.0 is subjected to a tortional moment of ton.m96.0 at the center of a square concrete of 2 x 2 m, as shown in Figure 14. The contact behavior between rod and concrete is modeled by the Coulomb friction law with

0=fc and 10=fµ . The material properties chosen are; 29 mKg102×=cE and 210 mKg102×=sE . The numerical simulation is carried out using 192 quadrilateral bilinear elements in FEM mesh and 400 four-noded rectangular elements in XFEM mesh. The convergence tolerance is set to 410− . In order to assess the accuracy of the use of XFEM technique for modeling steel rod with concrete, we compare the finite element solution employing contact element formulation to that obtained by the new technique. Figure 15 presents the corresponding meshes to the FEM and XFEM analyses. In XFEM, the weak form is integrated appropriately by partitioning the elements that are intersected by the contact surface. For the Gauss points located on contact bond, the integrals are evaluated using the friction material property matrix. In Figures 16(a)-(f), the stress distribution contours of xσ , yσ and xyt on a cross section subjected to torsion are presented for both XFEM and FEM techniques. Remarkable agreements can be observed between two different methods. Also plotted in Figures 17(a)-(d) are the horizontal and vertical displacements contours for both XFEM and FEM techniques. These results demonstrate that how the XFEM technique can be efficiently used to model the contact friction behavior between steel rod and concrete.

5.4. Compaction process Compaction process considers the methods of producing commercial products from metal powders by pressure. The influence of powder-tool friction on the mechanical properties of the final product is significant in pressing metal powders. Friction between the powder and tool affects the density distribution in the compact. A non-homogeneous density distribution induces cracks and residual stresses during compaction and sintering which is detrimental to the strength of the component. Friction also affects the final density, pressing and ejection forces and die wear. Thus, it is important to predict the behavior of the powder-tool friction. In traditional approach, the finite element formulation is characterized by the use of interface elements in which the plasticity theory of friction is incorporated to simulate sliding resistance at the powder-tool interface (Khoei and Lewis [27]). In the present study, we illustrate the performance of XFEM technique in modeling

powder-die friction. Due to large movements of punches, the updated Lagrangian description is incorporated into the XFEM formulation. The initial geometry of uncompacted powder in its position before compaction is presented in Figure 18, where mm 1001 =h , mm 502 =h , mm 3001 =w and mm 302 =w . The material properties chosen for powder are 25 mKg102×=E and 3.0=υ . The rigid die-wall is assumed to be elastic with the Young’s modulus of 210 mKg102× and Poisson’s coefficient of 0.3. The powder-tool friction is modeled by the Coulomb law with 0=fc and 3.0=fµ . On the virtue of symmetry, the process is modeled for half of specimen, as shown in Figure 19. In order to present the accuracy of the XFEM technique, a comparison is performed with the finite element solution for a mesh with approximately the same number of unknowns. The numerical simulation is performed using 240 six-noded triangular elements in FEM mesh and 400 four-noded rectangular elements in XFEM mesh. The convergence tolerance is set to 410− and the analysis was carried out using 20 increments. The final deformed mesh for both techniques is presented in Figure 20 at the top punch movement of mm 43 . In Figures 21(a)-(d), the horizontal and vertical displacements contours are presented for both XFEM and FEM approaches. Also plotted in Figures 22(a)-(f) are the stress distribution contours of xσ , yσ and xyt for two different techniques. Remarkable agreements can be observed between two different methods. The performance of XFEM in the case of large sliding is presented in Figure 23 by a detailed plot of traction on the contact surface at the volume reduction of 34 percents. This figure clearly presents the variation of the shear and normal stresses on the surface of rigid die-wall with the distance from bottom on the contact surface between the die and powder. As can be observed, the values of shear and normal stresses are approximately zero on the portion of the boundary that is free at the top punch movement of mm 43 , in which the two bodies have already slide past each other. The evolution of top punch vertical reaction force with its vertical displacement is depicted in Figure 24(a). Complete agreements can be observed between two methods. It must be noted that on using the finite element simulation, no further advance of the punch movement is obtained, however – the volume reduction of 90 percents can be observed with the same computational algorithm in XFEM technique, as plotted in Figure 24(b). This example adequately presents the applicability of XFEM technique in modeling of powder-tool friction in compaction pressing of powder.

6. Conclusion In the present paper, a method was developed based on the extended finite element method in modeling the discontinuity caused by frictional contact. The classical finite element approximation was enriched by employing additional terms based on the Heaviside step function. The partition of unity method was applied to discretize the contact area with triangular sub-elements whose Gauss points were used for integration of the domain of elements. For the elements cut by contact surface, the integration of stiffness matrix was performed by employing the material property matrix of contact surface at the integration points located on contact bond. The frictional contact behavior was

formulated by a non-smooth constitutive law on the interface of contact and the iterative scheme was implemented to resolve the nonlinear boundary value problem. Finally, numerical examples were presented to demonstrate the accuracy and capability of the XFEM in modeling the frictional contact behavior. It is shown how the XFEM technique can be effectively used to model 2D contact problems. In a later work, the implementation of XFEM technique will be presented in modeling 3D contact problems. Acknowledgement Parts of this research were conducted by the first author on his sabbatical leave at the Laboratorie de Mecanique des Solides, Ecole Polytechnique, France (2002). The author would like to appreciate the collaboration and guidance of Professor Claude Stolz.

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Figure Legend: Figure 1. Definition of sliding contact between two bodies

Figure 2. Modeling of contact surface between two bodies a) Problem definition, b) The FE mesh which conforms to the geometry of contact together with employed contact elements, c) A uniform mesh in which the circled nodes have additional degrees of freedom and enrichment functions

Figure 3. Illustration of normal and tangential coordinates for a contact surface in the case of jump function 1)( −=xH ; *x is the closest point to x on the contact surface

Figure 4. The sub-triangles associated with elements cut by contact surface in XFEM

Figure 5. Frictional shear stress-strain relationship

Figure 6. Sliding of two bodies; Problem statement

Figure 7. Sliding of two bodies, a) The FEM mesh, b) The XFEM mesh

Figure 8. Sliding of two bodies; The shear stress contours corresponding to; a) The FEM mesh with 10101×=fG , b) The XFEM mesh with 10101×=fG , c) The FEM mesh with

8101×=fG , d) The XFEM mesh with 8101×=fG

Figure 9. Sliding of two bodies; The variation of horizontal displacement of top edge with the shear modulus of contact surface using the FEM and XFEM techniques

Figure 10. Sliding of two bodies; The variation of uniform shear loading with horizontal displacement of top edge at three values of contact bond width

Figure 11. The pull-out of pile; Problem description

Figure 12. The pull-out of pile; The XFEM mesh

Figure 13. The pull-out of pile; The shear stress distribution of pile along the contact surface; a) The XFEM model, b) Lei [24]

Figure 14. Torsion of FRP rod with concrete; Problem statement

Figure 15. Torsion of FRP rod with concrete; a) The FEM mesh, b) The XFEM mesh

Figure 16. Torsion of FRP rod with concrete; a) The stress xσ contour using FEM analysis, b) The stress xσ contour using XFEM analysis, c) The stress yσ contour using FEM analysis, d) The stress yσ contour using XFEM analysis, e) The shear stress xyτ contour using FEM analysis, f) The shear stress xyτ contour using XFEM analysis

Figure 17. Torsion of FRP rod with concrete; a) The horizontal displacement contour using FEM analysis, b) The horizontal displacement contour using XFEM analysis, c) The vertical displacement contour using FEM analysis, d) The vertical displacement contour using XFEM analysis

Figure 18. The compaction process; Problem description

Figure 19. The compaction process; a) Initial FEM mesh, b) Initial XFEM mesh

Figure 20. The compaction process; a) Deformed FEM mesh, b) Deformed XFEM mesh

Figure 21. The compaction process; a) The horizontal displacement contour using FEM analysis, b) The horizontal displacement contour using XFEM analysis, c) The vertical displacement contour using FEM analysis, d) The vertical displacement contour using XFEM analysis

Figure 22. The compaction process; a) The stress xσ contour using FEM analysis, b) The stress xσ contour using XFEM analysis, c) The stress yσ contour using FEM analysis, d)

The stress yσ contour using XFEM analysis, e) The shear stress xyτ contour using FEM analysis, f) The shear stress xyτ contour using XFEM analysis

Figure 23. The compaction process; The variation of the shear and normal stresses on the surface of rigid die-wall with the distance from bottom of the contact surface at the top punch movement of mm 43 using the XFEM technique

Figure 24. The compaction process; The evolution of top punch vertical reaction force with its vertical displacement; a) A comparison between two different approaches at the top punch movement of mm 43 , b) The XFEM technique at the top punch movement of

mm 09

Figure 1. Definition of contact between two bodies

Figure 2. Modeling of contact surface between two bodies a) Problem definition, b) The FE mesh which conforms to the geometry of contact together with employed contact elements, c) A uniform mesh in which the circled nodes have additional degrees of freedom and enrichment functions

(a)

(b)

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Figure 3. Illustration of normal and tangential coordinates for a contact surface in the case of jump function 1)( −=xH ; *x is the closest point to x on the contact surface

Figure 4. The sub-triangles associated with elements cut by contact surface in XFEM

Figure 5. Frictional shear stress-strain relationship

Figure 6. Sliding of two bodies; Problem statement

(a) (b)

Figure 7. Sliding of two bodies, a) The FEM mesh, b) The XFEM mesh

(a) (b)

(c) (d)

Figure 8. Sliding of two bodies; The shear stress contours corresponding to; a) The FEM mesh with 10101×=fG , b) The XFEM mesh with 10101×=fG , c) The FEM mesh with 8101×=fG , d) The

XFEM mesh with 8101×=fG

Figure 9. Sliding of two bodies; The variation of horizontal displacement of top edge with the shear modulus of contact surface using the FEM and XFEM techniques

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Figure 10. Sliding of two bodies; The variation of uniform shear loading with horizontal displacement of top edge at three values of contact bond width

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Figure 11. The pull-out of pile; Problem description

Figure 12. The pull-out of pile; The XFEM mesh

(a) (b)

Figure 13. The pull-out of pile; The shear stress distribution of pile along the contact surface;

a) The XFEM model, b) Lei [24]

Figure 14. Torsion of FRP rod with concrete; Problem statement

(a) (b)

Figure 15. Torsion of FRP rod with concrete; a) The FEM mesh, b) The XFEM mesh

(a) (b)

(c) (d)

(e) (f)

Figure 16. Torsion of FRP rod with concrete; a) The stress xσ contour using FEM analysis, b) The stress xσ contour using XFEM analysis, c) The stress yσ contour using FEM analysis, d) The stress yσ contour using XFEM analysis, e) The shear stress xyτ contour using FEM analysis, f) The shear stress xyτ contour using XFEM analysis

(a) (b)

(c) (d)

Figure 17. Torsion of FRP rod with concrete; a) The horizontal displacement contour using FEM analysis, b) The horizontal displacement contour using XFEM analysis, c) The vertical displacement contour using FEM analysis, d) The vertical displacement contour using XFEM analysis

Figure 18. The compaction process; Problem description

(a) (b)

Figure 19. The compaction process; a) Initial FEM mesh, b) Initial XFEM mesh

(a) (b)

Figure 20. The compaction process; a) Deformed FEM mesh, b) Deformed XFEM mesh

(a)

(b)

(c)

(d) Figure 21. The compaction process; a) The horizontal displacement contour using FEM analysis, b) The horizontal displacement contour using XFEM analysis, c) The vertical displacement contour using FEM analysis, d) The vertical displacement contour using XFEM analysis

(a) (b)

(c) (d)

(e) (f)

Figure 22. The compaction process; a) The stress xσ contour using FEM analysis, b) The stress xσ contour using XFEM analysis, c) The stress yσ contour using FEM analysis, d) The stress yσ contour using XFEM analysis, e) The shear stress xyτ contour using FEM analysis, f) The shear stress xyτ contour using XFEM analysis

Figure 23. The compaction process; The variation of the shear and normal stresses on the surface of rigid die-wall with the distance from bottom of the contact surface at the top punch movement of

mm 43 using the XFEM technique

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Figure 24. The compaction process; The evolution of top punch vertical reaction force with its vertical displacement; a) A comparison between two different approaches at the top punch movement of mm 43 , b) The XFEM technique at the top punch movement of mm 09

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