Finite element analysis of three-dimensional contact and impact in large deformation problems

Finite element analysis of three-dimensional contact andimpact in large deformation problems

Eduardo Bittencourt *, Guillermo J. Creus

Centro de MecaÃnica Aplicada e Computacional ± CEMACOM, Curso de PoÂs Graduac° aÄo em Engenharia Civil, Universidade Federal

do Rio Grande do Sul, Av. Osvaldo Aranha, 99, 3o andar, 90035-190, Porto Alegre, RS, Brazil

Received 15 January 1997; accepted 8 October 1997

Abstract

An algorithm that models three-dimensional contact with friction, including cases where large deformations occuris proposed. The algorithm is an extension to three-dimensions of an algorithm for contact previously proposed by

Ponthot and Hogge [1]. The present algorithm can be used in implicit or explicit codes, when contact between arigid and a deformable body or among deformable bodies (multibody contact) is to be determined. The penaltymethod is used as the contact±friction formulation. Constitutive relations for contact±friction and integration ofthese relations in a corotational reference system are detailed. In order to provide a good convergence rate, the

contact±friction sti�ness matrices used are consistent with the time integration process. A simple and fully explicitform was obtained for these matrices. For contact problems involving rigid and deformable bodies,interpenetrations are detected using an iterative procedure. For frictionless multibody contact, a new element that

simpli®es the gap calculation, eliminating iterative procedures, is proposed. Comparison with benchmark results andthe solution of practical examples demonstrate the e�ectiveness of the algorithm. # 1998 Elsevier Science Ltd. Allrights reserved.

Keywords: Contact; Friction; Large deformation; Finite element; Impact; Elasto±plasticity; Metal-forming; Three-dimensional

1. Introduction

In the last years, ®nite element algorithms for the

treatment of two-dimensional (2D) contact with fric-

tion in explicit and implicit codes have been

developed [1, 2] and a better understanding of contact±

friction physics has been achieved [3]. Furthermore,

the use of the analogy between friction and plasticity

helped to develop better algorithms [4], as, for

example, in the application of the return mapping

method for time integration of the frictional

tractions [5].

In this paper we are concerned with the extension of

contact and friction analysis to three-dimensional (3D)

large deformation problems that occur, for instance, in

metal-forming processes. This extension is not

straightforward [6±10] and some new di�culties have

to be faced. One of these di�culties appears when the

penalty method is applied for contact±friction pro-

blem, using an implicit solution method. In this case

an iterative searching algorithm and an appropriate

tangent sti�ness matrix are necessary. The di�culty is

that the classical continuous tangent matrix for con-

tact±friction is not consistent with the process of con-

tact traction integration [6, 11], and the use of this

matrix may lead to convergence problems, specially

when high penalty values are employed. This problem

happens solely in 3D cases.

Moreover, the contact tangent matrix must contain

correction terms due to the rigid rotation of the con-

tact surface and these terms are much more di�cult to

handle in 3D cases, as we can see in Refs. [10, 12].

Computers and Structures 69 (1998) 219±234

0045-7949/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved.

PII: S0045-7949(98 )00008-X

PERGAMON

* Corresponding author.

Little information is available in the literature on

methods to precisely de®ne a reference system where

the contact tractions are to be calculated. The de®-

nition of this system is a key point for integration of

time derivatives of contact tractions. For instance,

some researchers [13] use a ®xed reference system. In

this case, to be objective, the derivative of the contact

traction must contain correction terms due to rigid ro-

tation. In most recent papers, researchers determine

the frictional components directly on the contact sur-

face avoiding the calculation of these correction terms.

In this case, the reference system is attached to the sur-

face and acts as a corotational system. Some authors

use the local coordinates of the contact

elements [7, 10, 15, 16] while other de®ne axes tangent

to the contact surface [14, 17, 18] as the reference sys-

tem. Unfortunately, details on these algorithms are not

usually given and the procedure to update the refer-

ence system when the contact point moves on the sur-

face is not clear. The use of the local coordinates of

the contact element may cause numerical di�culties

when the contactor node slide from one contact el-

ement to another.

As in 2D, integration of time derivatives of contact

tractions must be objective. The same techniques used

for integration of the time derivative of Cauchy stres-

ses can be used. A complete treatment of the problem

can be found in Ref. [19].

Finally, much more complex calculations are necess-

ary in 3D to determine the gap among bodies. In

many metal-forming situations, the tools (die, punch,

blanckholder, etc.) can be considered as rigid. Only the

workpiece must be modeled with ®nite elements. The

rigid tool shape can be described by complex

functions [20] or by a set of plane (generally triangu-

lar) elements [15, 17]. The gap is calculated projecting

orthogonally the ®nite element nodes onto the tool sur-

face. When the surface is represented by complex func-

tions, the projection requires an iterative procedure. A

di�erent approach is used in multibody contact. In this

case, the external faces of the ®nite elements de®ne the

contact surface. Using linear isoparametric hexahedral

elements (the most common case), the faces are usually

non-planar after contact whether or not they had been

planar during mesh generation. So, again, an iterative

method is necessary to determine the gap [7, 8, 12]. For

very large contact problems, as in crashworthiness ana-

lyses, these iterations can make the contact algorithm

too time-consuming. An alternative is the use of con-

tact elements that are representative of, but separate

from, the external faces. A very interesting idea is used

in Refs. [9, 21], where the external faces are replaced

by a set of spheres.

Some new methods to deal with these di�culties are

developed in the next sections.

1.1. Notation

In this paper the deformable body, modeled by ®niteelements, will be called slave and the rigid body will becalled target (some authors use also the expression

master). The corresponding nodes in the ®nite elementmodel are referred to as slave nodes and target nodes.Bold face letters will identify matrix and vectors.

When they are referred to global system of axes, theywill be identi®ed by uppercase and when referred torotated local system they will be referred by lowercase

variables. A dot upon a variable indicates derivationwith respect to time. Superscripts in parenthesis indi-cate the time or iteration where the variable is calcu-lated. Superscripts without parenthesis identify the

node or body related to the variable (exception is theuppercase T that indicates transpose of vector ormatrix). Subscripts indicate the components of the vec-

tor or matrix. Single and double dot indicates scalarproduct of vectors and matrices, respectively.

2. Basic formulation

We employ here the penalty method to deal with thecontact±friction problem. In this case, the weak form

of the governing equations can be written as Eq. (1).The contact constraints are taken in account, in thiscase, simply adding the virtual work done by contact

tractions.�Osss:�@dU@X

�dV�

�Or �U � dU dV �

�OB � dU dV

��GF

F � dU dS��GC

T � dG dS �1�

A rigorous proof of the validity of Eq. (1) can be

found in Ref. [10]. It is expressed over the currentdomain O that follows the body; the boundary of thedomain is G, which consists of the prescribed displace-

ment boundary GU, the prescribed traction boundaryGF and the contact boundary GC. We have also inEq. (1):

U, G= Displacement ®eld in the body (O) and onthe contact boundary (GC) respectively; both requiredto be C0.F, B = Prescribed traction vector and body forces,

respectively.ss = Cauchy stress tensor ®eld, required to be C1.T = Contact tractions on the contact boundary

(GC).r = Material density.Pre®x d designates an arbitrary, virtual and compati-

ble variation.The contact traction T is related to a penalty par-

ameter and the results will be dependent on its value.

E. Bittencourt, G.J. Creus / Computers and Structures 69 (1998) 219±234220

This is a well-known drawback of the penalty method.Besides, as GC is unknown a priori, an incremental/

iterative method of solution is necessary to solveEq. (1), as the Newton±Raphson method. Despitethese di�culties, the bene®ts of establishing the virtual

work as above is based on its general applicability andstraightforwardness. These are very welcome propertiesbecause we try to solve here complex and general situ-

ations were large deformations, non-linear materialbehavior, contact among deformable bodies and fric-tion can occur. Moreover, there is no increase in the

number of degrees of freedom as the Lagrangian mul-tiplier technique.The formulation proposed in Eq. (1) is extensively

used in the literature (see e.g. Refs. [2, 8, 16, 10]). We

can ®nd also interesting alternative formulations as inRefs. [22±24].After de®ning a ®nite element model for the body

and eliminating virtual nodal displacement, Eq. (1) canbe rewritten as the set of nonlinear ordinary di�eren-tial equations below:

MU� Fint ÿ Fext � 0 �2�where Fint and Fext represent internal and externalforces acting on the ®nite element nodes, respectively.

U are the nodal accelerations and M, as obtained fromthe second term of Eq. (1), is the consistent massmatrix. Implicit (e.g. Newmark) or explicit (e.g.

Central Di�erence) methods can solve this system (seethe Refs. [25, 26] for detailed discussion). If inertiale�ects are not important, the ®rst term in Eq. (2) dis-appears and we have

Fint � Fext �3�The set of nonlinear equations above can be solved bythe Newton±Raphson method. In this case (and for

implicit solution methods in general) a tangent matrixhas to be obtained. The convergence velocity of thesolution depends on the adequateness of this matrix.

Although we will not consider in this discussion im-plicit solutions for dynamic cases, the theory developedfor contact±friction can be applied also to this case.

In the next Sections we focus on the determinationof the contact traction T and discuss the de®nition oftangent sti�ness matrix for contact±friction.

3. De®nition of a contact reference system

As a ®rst step, we will de®ne a contact reference sys-tem (CRS) where all contact variables are de®ned.This system is based on a plane tangent to the target

surface at the contact point. If n, the outward normalvector to the surface at the contact point is known, itis possible to de®ne the tangent plane and the CRS.

We will de®ne the CRS as three mutually orthogonal

axes represented by the normal vector n and the tan-

gent vectors q and r. When a node contacts the surface

for the ®rst time, q and r are de®ned arbitrarily. For

the next iterations of the solution, a relation is estab-

lished with the preceding values as follows:

r�i � � n�i � � q�iÿ1�

kn�i � � q�iÿ1�k �4�

q�i � � n�i � � r�i �

kn�i � � r�i �k �5�

where (i) is the present iteration, (iÿ 1) the previous

one and � represents the vector product. So, we have

in Eqs. (4) and (5) a procedure to update the CRS

when the node moves onto the contact surface.

The relation between the CRS and the global system

(x, y, z), can be established once the global com-

ponents of q, r and n are known, by the rotation

matrix below

Z�t� �qx qy qzrx ry rznx ny nz

24 35 �6�

This relation is obviously dependent on the position of

the node on the contact surface. Then, each slave node

in contact will have its own CRS, which changes with

time.

The CRS can be considered as a corotational system.

So, an observer linked to the CRS will be able to cal-

culate the frictional tractions, i.e. the tangent com-

ponents of contact traction, directly from the

constitutive relation without concern with rotation.

Moreover, the above de®nition is general. It can be

applied to both cases studied here: contact between

rigid and deformable bodies and multibody contact.

Although we will not apply the CRS for multibody

contact in the present discussion, this case could be

treated without any additional di�culty because CRS

is not de®ned in terms of the local coordinates of the

target. Therefore, no special procedures would be

necessary when the slave node slides from one element

to another.

In the CRS the vector t, that represents the contact

traction, has components (tq, tr, tn) in the above men-

tioned directions. For convenience, we also de®ne, on

the tangent plane, a vector tt, which is related solely to

frictional e�ects and has components (tq, tr) along q

and r respectively.

The vector n is de®ned projecting orthogonally the

slave node onto the target surface. This projection is

studied in detail in Section 4.

E. Bittencourt, G.J. Creus / Computers and Structures 69 (1998) 219±234 221

4. Calculation of gap between bodies

The gap (gn) between target and slave surfaces isde®ned as

gn � �ws ÿ wt� � n �7�where wt is the deformation of the target surface and ws

is the deformation of the slave surface. The non-pene-trating condition is

gnr0: �8�When the penalty method is used to calculate traction,some interpenetration is allowed, relaxing the con-dition in Eq. (8). In this case, we determine the normal

traction (tn) using the expression

tn � kngn �9�where kn is the normal penalty parameter. This value

can be seen as the sti�ness of the surface topenetration [3].The determination of gn for a slave node, with re-

lation to a target surface, will be seen in the next two

sub-sections.

4.1. Determination of the gap in the contact between arigid and a deformable body

Rigid bodies are here de®ned by means of a set of

target surfaces. The ®rst step of gap calculation con-sists in identifying which surface the slave node willcontact. To perform this identi®cation, boxes can be

de®ned on the target surfaces using its extreme coordi-nates. A search for each surface is done to de®newhether a given slave node is located inside the box.

Then, the surface is a candidate for contact with thenode. The gap will be the minimum distance betweenthe slave node and the candidate surface. This distance

can be determined projecting orthogonally the node

onto the surface. This projection has two basic steps:

(1) Search of an initial guess of the solution (point

P* on the surface, as we can see in the Fig. 1). It is

necessary to create a cloud of points on the surface.

The distance from each of these points to the slave

node (S) is calculated, and the closest point P* is cho-

sen.

(2) Minimization process: As in general the vector

joining P* and S is not orthogonal to the target sur-

face, an iterative procedure is necessary. If a formula

relating the coordinates of a generic point on the sur-

face to the local coordinates of the surface (x,Z) is

known, we can write the equations:

�XP* ÿ XS� � @XP*

@x� Rx �10�

�XP* ÿ XS� � @XP*

@Z� RZ �11�

where (@XP*/@x) and (@XP*/@x) are vectors tangent to

the surface at XP* and Rx and RZ are residual values

that must be minimized using an iterative method, as,

for example, the Newton±Raphson method. The iter-

ations will end when the residual values are less than a

prescribed tolerance. The converged value of P* is

called P.

The gap is then

gn � �XS ÿ XP� � n: �12�Note that if, during deformation process, the slave

node remains on the same surface, then P can be used

as the initial guess P* in the next iteration, avoiding

step 1.

The procedure above can be applied to any surface

representation. In this paper we use Coons surface [27]

to map complex shapes. For simpler shapes, iterative

Fig. 1. Projection of a slave node S onto the target surface.


gap calculation is not necessary. For instance, if the

target surface is planar, the gap is simply

gn � �XS ÿ XI� � n �13�where XI is the position vector of a target corner. If

the target surface is a sphere the gap is

gn � �XS ÿ Rn� � n �14�where R is the sphere radius and n is calculated as:

n � XS ÿ X0

kXS ÿ X0k �15�

X0 is the vector position of the center of the sphere.

4.2. Determination of the gap for multibody contact

In the case of contact among deformable bodies, the

contact surface is de®ned using the contact elements ofthe ®nite element mesh. These contact elements arede®ned as the external faces of the ®nite elements onthe contact surface (GC). In this work the ®nite el-

ements used are linear hexahedrons (8-node brick)resulting contact elements with four nodes.As in the case discussed before, boxes enclose the

target elements and if the slave node is inside one par-ticular box, the corresponding element is candidate tocontact the node.

If the contact element is planar, the gap can be cal-culated by Eq. (13). However, as the target is now de-formable, after application of the contact forces theface will probably no longer be a plane, invalidating

this equation. An iterative method, similar to thatdescribed for rigid/deformable bodies, could be used todetermine the gap, as usually proposed in the literature

(see e.g. Refs. [7, 8, 12]).In this paper, we propose an approximation that

avoids the iterations and still gives good precision. We

create a contact element that keeps the planar form,even when the original external face of the elementdoes not. This new contact element is located on a

plane at the mean position among the target nodes as

can be seen in Fig. 2. The new contact element will be

here called planed contact element (PCE).

Eq. (13) is then modi®ed to

��gn �1

4

X4I�1

gI � ��n �16�

where

gI � �XS ÿ XI� �17�and

��n � 1

4

X4I�1

mI � pI �18�

XI is the position vector of the target node I and mI

and pI are vectors joining this node with its neighbors

(see Fig. 3). So ��n can be seen as the mean outward

normal of the target element, and ��gn is the mean gap

resulting.

The PCE is similar to the pinball algorithm proposed

in Ref. [9] because, in both algorithms, the actual com-

Fig. 2. Actual contact surface (from the ®nite element mesh) and the planed contact element (PCE).

Fig. 3. De®nition of a normal vector at the target node 1.


plex contact surface is replaced by a set of simple geo-

metrical forms: planes in our case and spheres in

Ref. [9]. In the called splitting pinball algorithm [21],

when the gap is detected inside a sphere, it is replaced

by a set of smaller spheres that are closer and more

like the actual surface. This step is not done in the pre-

sent algorithm and we still have a good precision, as

will be seen in Section 9.

As seen in Eq. (8), when interpenetration occurs, the

following condition must be true:

��gn<0 �19�Finally, it is necessary to con®rm if the slave node

really lies on the target element studied. To do so, we

de®ne a vector linking the projection of the slave node

on the PCE surface and a target node (vector v in

Fig. 3). Using the vectors m and p previously de®ned,

the following conditions must be true for each target

node (the same criterion was proposed in Ref. [7])

�mI � vI� � �mI � pI�r0 �mI � vI� � �vI � pI�r0 �20�If the conditions above are true, the searching process

ends. Otherwise, a new target element has to be tested.

In practice the conditions given in Eq. (20) are relaxed

and accepted for small negative values. This allows us

to handle situations where the slave node is in a con-

vex region, as shown in Fig. 4 below.

Afterwards, we begin the search with the target el-

ement that contacted the slave node in the previous

step. If contact is not detected in this element, a new

search is begun on the elements around it. If contact is

still not detected, the slave node is considered to be no

longer in contact. This procedure is obviously more

e�cient than a search on all target elements (known as

the brute force algorithm).

Finally, to avoid asymmetries and unexpected inter-

penetrations, at each iteration we swap the de®nition

of slave and master (known as the two-pass algorithm).

It means that all contact nodes, of the target and slave

body, are tested with its candidate contact elements.

5. Constitutive relations for friction

In this Section, constitutive relations for friction will

be de®ned in reference to the CRS. Therefore, the re-

lations bellow are totally frame indi�erent and similar

to that in small deformations. The sliding is modeled

with the Coulomb surface but the procedure below can

be applied to more complex surfaces.

The Coulomb relation for friction is written

f � kttk ÿ mjtnj � 0 �21�where m is the friction coe�cient and f represents the

Coulomb surface. Following [4], we use the theoretical

framework of perfect elasto±plasticity for the contact±

friction problem. Thus, Eq. (21) represents a yield (or

slide) condition. For f< 0 an elastic or reversible

(sticking) behavior is indicated. It is de®ned by

t � kg �22�where

k �kt 0 00 kt 00 0 kn

24 35: �23�

The components of the contact displacement vector g

in the CRS are gq, gr and gn. kt is the tangential pen-

alty parameter and physically represents the sti�ness of

the surface to tangential displacement [3].

For f= 0 an irreversible (sliding) behavior is indi-

cated. In this case, a rate formulation of the constitu-

tive law of friction is necessary. The deformation rate

is de®ned as

_g � _gr � _gir �24�the indexes r and ir indicate reversible and irreversible

parts respectively.

The irreversible deformation rate is determined using

the non-associated plasticity relation [2].

_gir � _l@ f*

@ t�25�

_l is a scalar to be determined and

f* � kttk: �26�For the irreversible case, the traction rate can be calcu-

lated as

_t � k�__gÿ _gir�: �27�To complete the basic relations, we consider the con-

sistency condition

_f � @ f@ t

_t � 0 �28�

From Eqs. (28), (27) and (25), we have

Fig. 4. Slave node (S) in a convex target region.


_l � �@ f=@ t�k_g

�@ f=@ t�k�@ f*=@ t� : �29�

Using Eqs. (27) and (25) we ®nally obtain

_t � y_g �30�where

y �ÿkt�t̂q�2 � kt ÿkt t̂q t̂r mkn t̂q

ÿkt t̂q t̂r ÿkt�t̂r�2 � kt mkn t̂r0 0 kn

264375 �31�

where t̂q � tq=kttk and t̂r � tr=kttk. A return mapping

algorithm can be used to integrate Eq. (30) in time and

to determine the frictional components. The algorithm

has two steps:

First, in the elastic predictor phase, we assume a lin-

ear path of the node on the contact surface and a re-

versible behavior. In this case, for time t+ Dt, we

have

te�t�Dt�t � t�t�t ÿ ktDgt �32�where the negative signal indicates that tangential trac-

tion components oppose the movement. The linear

path is an approximation and it can be a rough one

depending on the size of the time-step. Trial runs show

that, in practice, steps are small enough to make the

approximation valid.

Considering, on the target surface, a slave node that

has a position vector XS(t) in the reference con®gur-

ation and a position vector XS(t + Dt) in the present

con®guration, we may calculate the components of Dgtas

Dgq � �XS�t�Dt� ÿ XS�t�� q�t�Dt� �33a�

Dgr � �XS�t�Dt� ÿ XS�t�� r�t�Dt� �33b�As an application of these expressions, we consider in

Fig. 5 a case where the slave node moves elastically

(reversible friction) on the plane of the paper. The

node moves from A to B (time t1 = t + Dt) and after

that from B returning to A (time t2 = t1 + Dt), per-forming a closed cycle. (This occurs, for instance,

when we have a reversion of external loads.)

As expected, at the end of this cycle we have Dgt=0and Dtt=0, because q and r do not change in the tan-

gent plane (see Eq. (4) and Eq. (5)).If t, as found using the elastic predictor Eq. (32),

satis®es the condition f< 0, then we have reversible

displacement. The process has been terminated, andthe value of friction is de®ned as

t�t�Dt�t � t

e�t�Dt�t : �34�

In the second step, the plastic corrector phase, if t, asfound using the elastic predictor, leads to f>0, we

must return to the yield surface. We write

t�t�Dt�t � t

e�t�Dt�t ÿ tcorrt �35�

where the correction ttcorr is given by

tcorrt � kt

�t�Dtt

_girt dt �36�

and (see Eq. (25)),

_girt � _ltet �37�Assuming that the vector normal to the sliding surface

remains constant between t and t+ Dt, we have

tcorrt � ktDl̂te

t �38�where Dl is (see Eq. (29))

Dl � t̂e

qDgq � t̂e

rDgr ÿ mt�t�Dt�n =kt: �39�Using the expression above in Eq. (35), we obtain

t̂�t�Dt�q � mt�t�Dt�n t̂

e

q �40a�and

t̂�t�Dt�r � mt�t�Dt�n t̂

e

r �40b�that are the tangential components of the traction cal-culated in the corotational axes q and r, respectively.

6. Determination of contact forces

In the previous Sections, the integration of traction t

in the interval (t, t+ Dt) was performed in the CRS.

This system, however, as seen in the next section,changes in time with relation to the global system.In order to obtain the global components of traction

(T), we apply the rotation matrix given in Eq. (6)

Tx

Ty

Tz

8<:9=;�t�Dt�

�qx rx nxqy ry nyqz rz nz

24 35�t�Dt� tqtrtn

8<:9=;�t�Dt�

: �41�

So, after integration of tractions in the CRS, assuminga linear path of displacement during the time inte-gration, we apply an instantaneous rotation using the

Fig. 5. Elastic displacement of a slave node from point A to

B (time t1 = t + Dt) and after from B to A (time

t2 = t1 + Dt).


®nal position of the CRS (calculated in t+ Dt). Thisscheme is similar to the incrementally objective pro-cedure to integrate Cauchy stresses (seeRefs. [11, 19, 28]), known as the ®nal instantaneous ro-

tation method.The traction in Eq. (41) must be integrated over the

contact element to give nodal forces. Convergence pro-

blems have been reported in the literature [29] whenusing Gauss quadrature to perform this integration.To overcome this di�culty, the traction calculated for

a slave node is simply considered to be constant on thearea corresponding to one quarter of each contact el-ement around the contact node, as we can see in Fig. 6.Nodal forces can be calculated by

Fc � TAc �42�where Ac is the contact area.

7. The tangent matrix for contact

In the Newton±Raphson method, the contribution

of contact forces to the tangent matrix is given by

K TG,Cij � @F

Ci

@Gj�43�

where the vector Gj represents the displacement at thecontact zone. Rewriting Eq. (42):

F Ci � TiA

C �44�Ti can be obtained as the integration in time of the

equation bellow (see e.g. Refs. [13, 14, 30])

_Ti � Tri � yijTj �45�where yij is an anti-symmetric matrix that measures therotation rate of the CRS at the contact point. Tri is

called a corotational rate traction and, when we have

sliding, it is associated with the constitutive relation as:

Tri � Yij_Gj �46�

where

Yij � ZkiZljykl �47�Zki is the rotation matrix according to Eq. (6). The

traction rate can be written as

_Ti � ZkiZljykl _Gj � yijTj: �48�The last term above is the correction due to rigid ro-

tation of CRS. However, we noticed that this term

have little in¯uence on convergence rate of Newton±

Raphson method. As we will show later, this conver-

gence remain quadratic and practically unchanged,

with or without the correction, for values of the pen-

alty parameters and time-steps usually employed in

metal-forming simulations. So, if we neglect the last

term of Eq. (48) and using Eq. (44), the time derivative

of contact force can be calculated as:

_FC

i � ZkiZljykl _GjAC: �49�

If we consider the contact element area constant

during integration, the equation above enable us to

determine the variation of contact force with relation

the contact displacement, that corresponds to the con-

tact tangent matrix:

K TG,Cij � ZkiZljyklA

C: �50�

This matrix is similar to the matrix used in small de-

formation. We observe also that the constitutive re-

lation ykl used above is valid for in®nitesimally small

integration steps, resulting, in Eq. (50), a continuous

tangent operator. However, during the time inte-

Fig. 6. Area AC, where traction t is considered as constant.


gration, the step is discrete and the matrix above can

not guarantee a good convergence rate.

Below, we try to deduce a constitutive relation con-sistent with the integration process and adequate to

the Newton±Raphson method.

During application of the return mapping, tractionst are scaled as

tqtrtn

8<:9=;�t�Dt�

� bteqter0

8<:9=;�

(00tn

)�51�

where tqe and tr

e are the elastic predictor tractions and bis a scale factor. Di�erentiating Eq. (51)

_tq_tr_tn

8<:9=;�t�Dt�

� _bteqter0

8<:9=;� b

_teq

_ter

0

8><>:9>=>;�

(00tn

)� yij* _gi �52�

where yij* is the new relation searched. The scale factor

b can be determined substituting Eq. (51) into theyield (slide) condition (21)

b � mjtnjktetk

�53�

and

_b � bj_tnjjtnj ÿ

k_tetkk_tetk

!: �54�

Substituting the equations above into Eq. (52), wedetermine yij* (the expression for t and its time deriva-

tives can be obtained in Eq. (22))

yij* �ÿkt*�t̂q�2 � kt* ÿkt*t̂q t̂r mkn t̂q

ÿkt*t̂q t̂r ÿkt*�t̂r�2 � kt* mkn t̂r0 0 kn

264375�55�

and

kt* � ktb: �56�The matrix in Eq. (55) is consistent with the return

mapping method applied and must replace ykl in

Eq. (50) as the constitutive relation for irreversible fric-tion, in order to guarantee a good convergence rate in

the Newton±Raphson method. Then, the consistentcontact tangent matrix can be written as

K TG,Cij � ZkiZljykl*A

C: �57�We notice that this matrix is not symmetric. If the fric-tion coe�cient is not too high (e.g. less than 0.1), the

matrix in Eq. (57) can be symmetrized without largereduction of the convergence rate. This symmetrization

can be done in di�erent ways: using only the upper (or

the lower) part of the matrix or taking the mean valueof the two parts. In this case the standard solvers can

be used and we can save time and computer memory/disk. However, for higher values of the friction coe�-

cient, important asymmetries arise that can destroy thegood convergence of the method or even make theanalysis impracticable. In this case, the complete

equation system should be solved.If the friction is sticking, we have the symmetric tan-

gent matrix below:

K TG,Cij � ZkiZljkklA

C: �58�

8. Overview of contact force calculation

For the sake of clarity, the steps needed to calculatecontact forces at each iteration in the two cases con-sidered (contact involving rigid and deformable bodies

and multibody contact) will be brie¯y reviewed inTables 1 and 2 below.

Table 1

Contact involving rigid and deformable bodies

1. Loop over all contact nodes2. Determination of tool surfaces candidates to contact

with the slave node. (A surface will be a candidate ifthe node is inside a box surrounding the surface. Thisbox is de®ned using the extreme coordinates of the sur-

face).3. Calculation of orthogonal projection on all candi-date surfaces. The ®rst surface searched will be the sur-

face where contact occurred in the last step.3.1. Determination of a point (P*) on the tool surface,near the slave node. (If the slave node was in contact

with this surface in the previous iteration, P* will bethe orthogonal projection at that iteration.)3.2. Determination of the point (P) on the tool surfaceclosest to the slave node, calculating the orthogonal

projection with a Newton±Raphson procedure(Eqs. (10) and (11)).4. Calculation of the gap using Eq. (12).

5. Calculation of the normal contact traction usingEq. (9).6. If friction is present, tangential displacement is cal-

culated using Eqs (33a and b).7. For sticking friction, Eq. (32) is used to determinetangential contact traction. If sliding friction is present,Eqs (40a and b) are used.

8. Rotation of tractions is performed using Eq. (41).Tractions multiplied by the contact area will givenodal forces, according to Eq. (42).


9. Numerical examples

This Section shows trial runs of the algorithm

described using the code METAFOR [19] as a base. Inall examples studied, the penalty method was used.

Therefore, an arbitrary choice of the penalty par-ameters (kn and kt) was necessary. The practice showedthat values approximately equal to the Young modulus

are large enough to assure satisfactorily the applicationof contact constraints. In the examples below, the

value of the Young modulus was used as the lowerbound for the penalty parameters. In all cases, brick

elements of 8 nodes were used. (More examples anddetails can be found in Ref. [31].)

Two quasi-static cases of contact between a rigidand a deformable body were studied ®rstly. In these

cases we emphasized the correct application of fric-tional formulation and the good convergence achieved

by the Newton±Raphson method.The last two examples show the capabilities of the

new contact element created (PCE) for multibody con-tact. Both examples were solved by the explicit centraldi�erence method and were considered frictionless.

The stability of central di�erence method is dependenton the time increment used, which is a function of the

material constants and ®nite element dimensions [32].In the present cases, we apply a reduction factor of

0.75 on the calculated time increment, due to presenceof contact [7].

9.1. Upsetting of a billet

In this example a numerical 3D simulation of the

upsetting of a cylindrical billet is presented to compare

the behavior of the consistent contact tangent matrix

Eq. (57), exact in this case (no rotation in the contact

zone), with the non-consistent matrix Eq. (50). The

material parameters used are:

Young modulus E = 200 kN/mm2

Yielding stress sv=0.7 kN/mm2

Linear hardening modulus H = 0.3 kN/mm2

Poisson coe�cient u = 0.3

Due to symmetry, only 1/8th of the billet was mod-

eled. The initial mesh is shown in Fig. 7 and consists

of 60 brick elements. The dimensions of the billet are:

radius 10 mm; length 30 mm.

A rigid plane tool was used to reduce 64% of the

billet length. The deformed mesh is shown in Fig. 8.

Table 2

Multibody contact

1. Loop over all contact nodes2. Determination of elements candidate to contact the

slave node. (A element will be a candidate if the nodeis inside a box surrounding the element). All contactelements are searched unless the node was in contact

in the previous iteration; in which case candidate el-ement will be the previous element where the contactoccurred as well as the elements that surround it.3. Calculation of the orthogonal projection on all can-

didate elements, using Eq. (16). If Eq. (19) is not true,the element is not a candidate anymore (the slave nodeis outside the body on the element region).

4. If condition expressed by Eq. (20) is true, the projec-tion of the slave node lies on the target element andthe search is considered ®nished.

5, 8. Idem the previous case.(Steps 6±7 concerning friction, were not implementedfor multibody contact, but would be the same as those

for contact involving rigid and ¯exible bodies.)

Fig. 7. Initial mesh.

Fig. 8. Mesh after 64% height reduction.


A friction coe�cient (m) of 0.1 was adopted and slid-ing (irreversible friction) occurred. An automatic time

stepping scheme [19], that determines the time-step asa function of the rate of convergence achieved in thepreceding time-steps, was ®rst used. Table 3 shows a

comparison between the results obtained using the con-sistent and the non-consistent tangent matrix.Using a ®xed displacement step, the results obtained

with the non-consistent matrix deteriorates: when the

step reaches 2% of total height reduction, only the useof the consistent matrix leads to convergence.It is important to remark that di�erences observed

in convergence velocity are strongly dependent on thekt value. In the examples above, kt=kn=3.5 � 105.Using smaller values of kt, the di�erences in conver-

gence between the two cases are also smaller.

9.2. The Wagoner case

This is a well known benchmark [33] used for sheetmetal-forming simulations and was chosen here to testthe algorithm for friction. The material parameters

used are:

Young modulus E = 69004 N/mm2

Poisson coe�cient u = 0.3Yielding stress sv=589(10ÿ4+�ep)

0.216 N/mm2

where �ep is the equivalent plastic strain.

The dimensions of the specimen are: radius 66 mm;thickness 1 mm. Due to symmetry, only one-quarter ofthe specimen was modeled. The ®nite element model is

shown in Fig. 9.

All nodes on the external boundary are ®xed in thethree directions. The tool dimensions are furnished in

Fig. 10 and the set of tools is depicted in Fig. 11. The

®xed die is toroidal and was modeled using a Coons

surface; the punch was modeled using a spherical sur-face.

The loading was applied through a displacement of

40 mm of the punch in the z direction. The penalty

parameters used were kt=kn=1 � 105 and m = 0.3.

Mesh deformation and equivalent plastic strain are

shown in Figs. 12 and 13, respectively, for the ®nalpunch position. As expected, the maximum value of

plastic strain is not at the sheet center due to frictional

Table 3

Convergence characteristics of tangent matrices

Equation Iterations Time-steps

57 (consistent) 53 38

50 (non-consistent) 80 43

Fig. 9. Initial mesh.

Fig. 10. Dimensions of the tools [33].

Fig. 11. 3D scheme of the tools used: 1 ± punch; 2 ± speci-

men; 3 ± die.


constraints in this region. The maximum value occurs

on a circumference with a radius of approximately

20 mm, where tangential forces are greater and the

sliding can occur. This can be seen better in Fig. 14,

where logarithm meridional strains are shown. These

values are compared with mean values taken from

Ref. [33].

Again an automatic time-stepping was used. The

Table 4 shows a comparison among three di�erent

ways to calculate the tangent matrix for contact±fric-

tion: with the approximated expressions in Eq. (57)

and Eq. (50) and with the exact tangent matrix,obtained from Eq. (43), numerically di�erentiated.

We see that the rotation terms, neglected in Eq. (57),have little importance on the convergence velocity. Onthe other hand, a consistent de®nition of constitutive

relations seems to have again an important e�ect onconvergence.To complete the convergence data, the Table 5

shows the decrease of the residue using the matrix ofEq. (57). We see that this decrease is approximatelyquadratic up to very low residue values, where equili-

brium can be considered fully satis®ed. These data cor-respond to a step where the displacement of the punchis approximately 0.6 mm or 1.5% of the total process.For greater displacement values, the matrix in Eq. (57)

will probably fail as a tangent operator, due to the ap-proximations done. However, in metal-forming simu-lations this is not too serious a drawback because the

continuum is elasto±plastic and the time-step is, ine�ect, limited by the constitutive equations of the ma-terial.

9.3. Impact between two elasto±plastic tubes

Here an application of frictionless multibody contact

is presented using an explicit solution method. Thegeometry of the problem is depicted in Fig. 15. 1800brick elements were used and the PCE (Section 4.2)

was employed to apply the contact constraints.The material parameters are:

Fig. 12. Final mesh con®guration.

Fig. 13. Final equivalent plastic strain.


Young modulus E = 25 � 109 N/m2

Yielding stress sv=2 � 106 N/m2

Linear hardening modulus H= 630 � 106 N/m2

Poisson coe�cient u= 0.3Density r = 7840 kg/m3

The tube at the left side had an initial velocityvy=50 m/s and tube at the right had an initial velocity

vy=ÿ 50 m/s. Deformed con®gurations are shown in

the Figs. 16 and 17.

In this case, we were not especially concerned with

the behavior of the solid element, which is not the bestfor this case because of considerable bending. The goal

here was simply to show the performance of the con-

tact element. Absence of visible interpenetrations or

asymmetries seem to indicate that the PCE performed

as expected.

9.4. Application to an engineering case

To show the capability of the code to solve indus-

trial problems, the analysis of impact between a car-

tridge and the cylindrical chamber of a handgun,

during a shot, is presented. The problem involves

impact and contained plastic strain. An additional pro-

blem here is the fact that we have contact between

di�erent materials, which could be a troublesome task

for the penalty method (see commentaries in Ref. [8]).

Here, again, the PCE and an explicit solution method

were used.

The initial mesh is depicted in Fig. 18. Only 1/6th of

the actual structure was modeled. The rest of the struc-

Fig. 14. Logarithm meridional strains � radius (mm).

Table 4

Convergence characteristics of tangent matrices

Equation Iterations Steps

43 (exact/

numerical)

170 83

57 (consistent) 176 87

50 (non-consistent) 219 95

Table 5

Convergence for a typical time-step (Eq. (57))

Iteration Mean residue* Maximal residue

0 0.1694 0.2650 � 10+2

1 0.2336 � 10ÿ2 0.6084

2 0.1120 � 10ÿ4 0.2804 � 10ÿ2

3 0.2441 � 10ÿ9 0.8926 � 10ÿ7

* Euclidean norm of residue divided by Euclidean norm of

external forces.


ture was replaced by springs. The cylinder is built of a

SAE 4140 steel and cartridge is of a SAE 70 A brass.

A uniform pressure is applied inside the cartridgeand simulates the explosion during the shot. Its value

changes linearly from 0 (time 0 s) to 265 MPa at time

0.002 s.During the explosion, the cartridge su�ers a signi®-

cant deformation and impacts the chamber of the

handgun at di�erent zones. (The contact zone propa-gates from the free side of the cartridge up to its bot-

tom). In Fig. 19 undeformed mesh and con®gurationat time 0.002 s are compared and visible deformations

can be observed.

In this case, the Young modulus of the cylinder ma-terial (steel) is approximately twice the value of the

cartridge (brass). Even in this case we used the ``rule'',

established at the beginning of this Section, to set thepenalty parameter (its value can not be less than theYoung modulus). So, the penalty parameter adopted

here had the same value of the biggest Young modulus(steel). We did not observe interpenetrations in thecontact zone or instabilities, indicating the corrective-ness of the penalty method and the PCE implemented.

10. Final remarks

In this work we proposed procedures to work with

contact±friction in presence of large deformation andnon-linear materials. For contact between rigid and¯exible bodies, a new corotational system was de®ned

Fig. 15. Geometry of impact between two tubes (dimensions in meters).

Fig. 16. Deformed con®guration at 0.001 s. Fig. 17. Deformed con®guration at 0.002 s.


that permits easily the use of small deformation consti-tutive relations for contact±friction in large defor-mation. We showed also that a simpli®ed contact

tangent matrix can be as e�cient as the exact tangentmatrix, due to relatively small time-steps used in theanalysis of elasto±plastic materials. To objectively inte-

grate contact tractions, procedures similar to that usedto deal with Cauchy stresses in large deformation wereused.

In the presence of multibody contact, a simpli®edcontact element was proposed and it showed to be suf-®ciently precise to apply contact constraints in twocomplex examples. The implementation of friction in

this contact element can be done using the same coro-tational system de®ned for contact between rigid/de-formable body. This case will be treated in a

forthcoming publication.

Acknowledgements

The ®rst author (E. B.) worked at the AerospaceLaboratory of LieÁ ge University (Belgium) and wishes

to express his deep gratitude to Professor MichelHogge and Dr. Jean-Philippe Ponthot. CNPq andPROPESP/UFRGS gave ®nancial support. Theexamples were run in a Cray YMP/2 at CESUP

(National Center of Super Computing).

References

[1] Ponthot JP and Hogge M. The use of Eulerian±

Lagrangian ®nite element method in metal forming

including contact and adaptive mesh. Proc. ASME

Winter Annual Meeting, Atlanta 1±6 Dec., 1991:1±16.

[2] Wriggers P, Vu Van T, Stein E. Finite element formu-

lation of large deformation impact-contact problems with

friction. Comput. Struct. 1989;37:319±31.

[3] Oden JT, Martins JAC. Models and computational

method for dynamic friction phenomena. Comput. Meth.

Appl. Mech. Engng 1985;52:527±634.

[4] Fredriksson B. Non-linearities in structural mechanics

with special emphasis to contact and fracture mechanics

problems. Comput. Struct. 1976;6:281±90.

[5] Giannokopoulos AG. The return mapping method for

the integration of friction constitutive relations. Comput.

Struct. 1989;32:157±68.

Fig. 18. Initial mesh: 1 ± Cylinder; 2 ± Cartridge.

Fig. 19. (a) Initial mesh; (b) Con®guration at time 0.002 s.


[6] Peric D, Owen DRJ. Computational model for three-

dimensional contact problems with friction based on the

penalty method. Int. J. Num. Meth. Engng

1992;35:1289±310.

[7] Benson DJ, Hallquist JO. A single surface contact algor-

ithm for the post buckling analysis of shell structures.

Comput. Meth. Appl. Mech. Engng 1990;78:141±63.

[8] Zong ZH. Finite element procedures for contact-impact

problems. Oxford: Oxford University, 1993.

[9] Belytschko T, Neal MO. Contact-impact by the pinball

algorithm with penalty and Lagrangian methods. Int. J.

Num. Meth. Engng 1991;31:547±72.

[10] Laursen TA, Simo JC. A continuum-based ®nite element

formulation for the implicit solution of multibody, large

deformation frictional contact problems. Int. J. Num.

Meth. Engng 1993;36:3451±85.

[11] Nagtegaal JC. On the implementation of inelastic consti-

tutive equation with special reference to large defor-

mation problems. Comput. Meth. Appl. Mech. Engng

1982;33:469±84.

[12] Parisch H. A consistent tangent sti�ness matrix for

three-dimensional non-linear contact analysis. Int. J.

Num. Meth. Engng 1989;28:1803±12.

[13] Chen JH, Kikuchi N. An incremental constitutive re-

lation of unilateral contact friction for large deformation

analysis. J. Appl. Mech. 1985;52:639±48.

[14] Van Der Lugt J. A ®nite element method for the simu-

lation of thermo±mechanical contact problems in form-

ing processes. Ph.D. thesis, Twente University, 1988.

[15] Habraken A-M, Radu J-P, and Charlier R. Numerical

approach of contact with friction between two bodies in

large deformations. Proc. Contact Mech. Int. Symp.,

1992:391±408.

[16] Wriggers P, Miehe C. Contact constrains within coupled

thermomechanical analysis ± A ®nite element model.

Comput. Meth. Appl. Mech. Engng 1994;113:301±19.

[17] Marques MJMB, Martins PAF. Three-dimensional ®nite

element contact algorithm for metal forming. Int. J.

Num. Meth. Engng 1990;30:1341±54.

[18] Surdon G, Chenot JL. Finite element calculation of

three-dimensional hot forging. Int. J. Num. Meth. Engng

1987;24:2107±17.

[19] Ponthot JP. Traitement uni®eÂ de la meÂ canique des mili-

eux continus solides en grandes transformations par la

meÂ thode des eÂ leÂ ments ®nis. Ph.D. thesis, LieÁ ge

University, 1995.

[20] Shiau YC, Kobayashi S. Three-dimensional ®nite el-

ement analysis of open die forging. Int. J. Num. Meth.

Engng 1988;25:67±86.

[21] Belytschko T, Yeh LS. The splitting pinball method for

contact-impact problems. Comput. Meth. Appl. Mech.

Engng 1993;105:375±93.

[22] BjoÈ rkman G, Klarbring A, SjoÈ din B, Larsson T,

RoÈ nnqvist M. Sequential quadratic programming for

non-linear elastic contact problems. Int. J. Num. Meth.

Engng 1995;38:137±65.

[23] Eterovic AL, Bathe KJ. On the treatment of inequality

constraints arising from contact conditions in ®nite el-

ement analysis. Comput. Struct. 1991;40:203±10.

[24] Oden JT, Carey GF. Finite element ± Special problems

in solids mechanics, Vol. 5. Englewood Cli�s: Prentice

Hall, 1984.

[25] Bathe KJ. Finite element procedures. Englewood Cli�s:

Prentice-Hall, 1996.

[26] Hogge M, Ponthot JP. E�cient implicit schemes for

transient problems in metal forming situation. Proc.

NUPHYMAT'96, Paris, 5±7 June, 1996:1±11.

[27] Farin G. Curves and surfaces for computer aided geo-

metric design. Academic Press, 1988.

[28] Braudel HJ, Abouaf M, Chenot JL. An implicit incre-

mentally objective formulation for the solution of elasto-

plastic problems at ®nite strain by the FEM. Comput.

Struct. 1986;24:825±43.

[29] Kikuchi N. A smoothing technique for reduced inte-

gration penalty methods in contact problems. Int. J.

Num. Meth. Engng 1982;18:343±50.

[30] Baaijens FPT, Brekelmans WAM, Veldpaus FE,

Starmans FJM. A constitutive equation for frictional

phenomena including history dependency. Proc.

NUMIFORM'86, Gothenburg, 25±29 Aug, 1986:91±96.

[31] Bittencourt E. Treatment of contact impact problem in

large deformation by the ®nite element method. Doctoral

Thesis (in portugese), Universidade Federal do Rio

Grande do Sul, 1994.

[32] Flanagan DP, Belytschko T. Eigenvalues and stable time

steps for the uniform strain hexahedron and quadrilat-

eral. J. Appl. Mech. 1984;51:35±40.

[33] Lee JK, Wagoner RH, Nakamashi E. A benchmark test

for sheet forming analysis. Rep. ERC/NSM-S-90-22, The

Ohio State University, 1990.


Documents

Finite element analysis of three-dimensional contact and impact in large deformation problems