COMPUTATIONAL MECHANICSNew Trends and Applications

S. Idelsohn, E. Onate and E. Dvorkin (Eds.)c©CIMNE, Barcelona, Spain 1998

AN EXTENSION OF THE RADIAL RETURN ALGORITHMTO ACCOUNT FOR VISCO-PLASTICITY AND

RATE-DEPENDENT EFFECTS IN FRICTIONAL CONTACT

J.P. Ponthot

University of LigeLTAS - Milieux Continus et Thermomcanique

21 rue E. Solvay B4000 Lige, Belgium

Key Words: Large Deformation, Frictional Contact, Rate-dependent friction coefficient,Visco-Plasticity

Abstract. This paper deals with unified elastoplastic and elastic-viscoplastic constitutiveequations for metals submitted to large deformations. We present here a newly developedtime integration algorithm which is an extension to the viscoplastic range of the classicalradial return algorithm for plasticity. The resulting implicit algorithm is both efficient andvery inexpensive. This algorithm is also subsequently extended to frictional problems whereit allows a generalization of the classical Coulomb dry friction criterion to a criteriondepending on the relative slip velocity such as those encountered in hydrodynamic regime.Moreover, a consistent tangent modulus for frictional contact is also presented. Thus,the proposed law can be implemented very efficiently in both explicit and implicit finiteelement codes.

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J.P. Ponthot

1 INTRODUCTION

Radial return is now extensively used in finite element codes for large-scale computationsof elastoplastic behaviour. This integration scheme is both inexpensive and accurate. Inaddition, it allows to write down a closed-form expression for the so-called consistentelastoplastic tangent modulus. Use of this consistent modulus (and not the continuummodulus) for the establishment of the global tangent stiffness matrix is essential in pre-serving the quadratic rate of convergence in Newton’s procedure required by implicitalgorithms.

However, regarding elastic-viscoplastic modelling of material behaviour, the situationis completely different. At the present time, many different algorithms have been devel-oped in order to integrate elastic-viscoplastic equations, see e.g. Golinval1 for a valuablediscussion. However, none of them actually exhibits the same level of performance asthe radial return algorithm for plasticity. Moreover, none of the schemes described inGolinval1 is amenable to consistent linearization. This fact is highly penalizing and pre-cludes an efficient treatment of viscoplastic problems in large-scale finite element or finitedifference codes.

In the present paper, we describe a new algorithm which does not suffer from the previ-ously cited drawbacks. This algorithm treats the elastoplastic and the elasto-viscoplasticproblem in an unified way. It is a simple generalization to rate-dependent plasticity prob-lems of the radial return algorithm for rate-independent plasticity. Therefore, it exhibitsall its (good) properties.

On the other hand, contact interactions exist in almost all mechanical systems andthough many efforts have been devoted to contact modelling, attention is often restrictedto the classical Coulomb model where the coefficient of friction does not depend on relativeslip velocity. In practise, use of a constant friction coefficient is much too restrictive.In order to overcome that limitation, we propose, based on the analogy with the bulkconstitutive law, an extension of the Coulomb friction law which can take into account,both the adhesion and the relative slip velocity. The resulting time integration algorithmof the proposed frictional law can also be regarded as an extension to frictional contact ofthe radial return algorithm. The new proposed criterion is rather general and particularcases such as the classical Coulomb friction law, or a friction law where the friction forcesonly depend on slip velocity can be easily recovered.

2 CLASSICAL J2 RATE-INDEPENDENT PLASTICITY

2.1 Introduction

It is generally assumed that an hypoelastic stress-strain relation can be written as

5σij= Hijkl(Dij − Dp

ij) (1)

where

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J.P. Ponthot

5σ is an objective rate of Cauchy stress tensor,H is the Hooke stress-strain tensor,D is the deformation rate tensor,Dp is the plastic part of the def. rate tensor,De is the elastic part of the def. rate tensor,

Moreover, we assume the existence of a yield function f given, for a J2 von Misesmaterial with isotopic hardening, by

f(σ, σv) = σ − σv = 0 (2)

where

σ is the effective stress, i.e. σ =√

32s : s;

s is the deviator of the stress tensor;σv is the current yield stress.

The admissible stress states are constrained to remain on or within the elastic domain (f ≤0). The notion of irreversibility, linked to the plastic part, is built into the formulationby introducing nonsmooth equations of evolution for Dp (flow rule) and σv (hardeninglaws).

2.2 -Flow rule:

When plastic deformation occurs, f = 0 and one can write

Dp = ΛN where N ij = sij/√

sklskl (3)

is the unit outward normal (N : N = 1) to the yield surface and Λ is a positive parametercalled the consistency parameter. It is classically determined by expressing the so-calledconsistency condition, i.e.

•f (σ, σv) = 0 (4)

where the superposed dot denotes time differentiation.

2.3 -Hardening law:

They are given by:

•σv=

√2

3hΛ (5)

with h is called the hardening coefficient and corresponds to the slope of the effectivestress vs. effective plastic strain curve under uniaxial loading conditions. Equation (5)can also be rewritten as

•σv= h

•ε p where

•ε p =

√2/3Dp : Dp (6)

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J.P. Ponthot

is the rate of effective plastic strain.

3 EXTENSION TO VISCOPLASTICITY

Contrarily to the case of rate independent plasticity, the effective stress σ is no longerconstrained to remain less or equal to the yield stress but one can have σ ≥ σv . Thereforewe define the overstress as

d = 〈σ − σv〉 (7)

where 〈x〉 denotes the Mac Auley brackets defined by 〈x〉 = 1/2(x + |x|). Clearly, aninelastic process can only take place if, and only if, the overstress d is positive i.e., inthat case, f ≥ 0. For example, classical viscoplastic models of the Perzyna2 type may beconsidered as penalty regularization of rate-independent plasticity where the consistencyparameter has been replaced by an increasing function of the overstress e.g.

Λ =

√2

3

⟨σ − σv

η(εvp)n

⟩m

(8)

wheren is a hardening exponentm is a viscosity exponentη is a viscosity parameter.

In that case, the evolution equations are still of the form

Dvp = ΛN (9)

•εvp =

√2

3Λ or

•σv= h

•εvp (10)

which are quite similar to the rate-independent case. Combining equations (6), (8) and(9) gives

•εvp =

√2

3Dvp : Dvp =

⟨σ − σv

η(εvp)n

⟩m

(11)

so that, in the viscoplastic range, one can define a new constraint

f = σ − σv − η(εvp)1/n(•εvp)1/m = 0 (12)

This criterion is a generalization of the classical von-Mises criterion f = 0 for rate-dependent materials. The latter can simply be recovered by imposing η = 0 (no viscosity

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J.P. Ponthot

effect). In the elastic regime, both f and f are equivalent since, in that case•εvp = 0 and

σ ≤ σv so that one has f ≤ 0.Moreover, from relation (8), it can be noted, that as viscosity η goes to zero (rate-

independent case), the consistency parameter Λ remains finite and positive (though inde-terminate) since σ − σv also goes to zero. The extended criterion (12) will play a crucialrole in the integration algorithm described hereafter.

4 TIME INTEGRATION PROCEDURE

4.1 Elastic Predictor

To integrate these equations in time, we will rely on the general methodology of elastic -predictor /plastic - corrector (return mapping algorithm), see e.g. Hughes and Simo.3, 4

Following a time integration procedure as described in details in Ponthot 5, 6 an elasticpredictor can be obtained as

σtr = R [σ0 + H : C] RT (13)

where

σ0 is the stress tensor at t0C = 1

2ln(F T F ) is the natural strain tensor

R is the rotation matrix.C and R both result from the polar decomposition F = RU of the deformation gradientover the considered timestep.

4.2 Viscoplastic Corrector

If f(σtr, σ0v) ≤ 0, the process is clearly elastic and the trial state is in fact the final state.

If, on the other hand, f(σtr, σ0v) > 0 which, in turn, implies that f > 0 (since εvp = 0),

the Kuhn-Tucker loading/unloading conditions are violated by the trial state which nowlies outside the generalized criterion. Consistency will be restored by a generalizationof the radial return algorithm. The viscoplastic corrector problem may be rephrased as(the objective rates reduce to a ”simple” time derivative due to the fact that the globalconfiguration is hold fixed) :

•σcorr = −H : Dvp = −2GΛN (14)

•εvp =

√2

3Λ or

•σv=

√2

3hΛ (15)

These equations have to be integrated over the time interval [t0, t1], with initial condi-tions given by(σ0, ε

vp0 , σ0

v) . The basic idea of the integration procedure is very similar tothe radial return procedure of plasticity. The tensor N(t) is approximated by

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J.P. Ponthot

N =str√

str : str

(16)

so that the final values are given by

σ1 = σtr − 2G Γ N (17)

εvp1 = εvp

0 +

√2

3Γ (18)

•εvp =

εvp1 − εvp

0

∆t=

√2

3

Γ

∆t(19)

where the (unkonwn) scalar parameter Γ stands for

Γ =

∫ t1

t0

Λ dt (20)

This parameter is simply determined by the enforcement of the generalized consistencycondition, f = 0, at time t = t1 i.e.

f(Γ) =

√3

2[str − 2G Γ N ]2 − σ1

v − η(εvp0 +

√2

3Γ)

1n (

√2

3

Γ

∆t)

1m = 0

This scalar equation is a nonlinear expression where the only unknown parameter isΓ. It can be easily solved by a local Newton-Raphson iteration. In the particular casewhere n = ∞ (no multiplicative hardening), m = 1 (linear dependence between overstressand viscoplastic rate of deformation), and h = constant (linear hardening) a closed formsolution of this equation is given by

Γ =1

2G

√str : str −

√23σ0

v

1 + 13G

(h + η∆t

)(21)

so that it is now obvious that the present algorithm is a generalisation to the rate-dependent case of the classical radial return algorithm. This one is exactly recovered(with no numerical difficulty) by setting η = 0 (no viscosity effect). In the viscouscase, one can see that the rate-dependent solution (60) is equivalent to rate-independentsolution with a fictitious hardening given by h∗ = h + η∆t.

4.3 Consistent Elasto-Viscoplastic Tangent Moduli

The present algorithm is amenable to exact linearization leading to a closed-form expres-sion for the so-called consistent elastoviscoplastic tangent moduli. Use of the consistentmoduli - as opposed to the continuum moduli - is essential in preserving the asymptoticrate of quadratic convergence in Newton-Raphson method for the global finite elementproblem. The final results and details pertaining to this linearization procedure are givenis Ponthot.6

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J.P. Ponthot

5 A GENERALIZED LAW FOR CONTACT WITH FRICTION

Generally, when dealing with frictional contact, the Coulomb criterion for dry friction isused. This one can be written :

f = ‖tT‖ − A − µ|tN | ≤ 0 (22)

wheretT is the tangential force due to contact;tN is the normal force due to contact;µ is the Coulomb friction coefficient;A characterizes the adhesion.

Following standard formalism of the theory of elastoplasticity, additive decompositionof the tangential velocity at the contact interface is adopted, Curnier,7 Ponthot,6 i.e. :

•gT =

•g

el

T +•g

in

T (23)

where gT is the tangential gap (similarly gN is the normal gap or penetration - ”in”stands for inelastic). Further a non-associative slip rule is generally used to formulate thefrictional interface law such that

•g

in

T = 0 if f < 0 (24)•g

in

T = Λ tT

‖tT ‖ if f = 0 (25)

where Λ is determined from the so-called consistency condition•f= 0, see Ponthot6 for

details. By analogy with the elasto-viscoplastic case and relation (8), we propose for Λ aform given by

Λ =

⟨‖tT‖ − A − µ|tN |η(gin

T )1/n

⟩m

(26)

where•g

in

T =

åg

in

T · •g

in

T so that the following generalized slip criterion f can be deduced

f = ‖tT‖ − A − µ|tN | − η(•g

in

T )1n (

•g

in

T )1m = 0 (27)

By analogy with (12), η plays the role of a viscosity parameter and classical Coulombcriterion can be recovered by imposing η = 0. This criterion is obviously depending onthe relative inelastic slip velocity and therefore, it is able to take into account lubricated

7

J.P. Ponthot

friction where this slip velocity is an important parameter. For example, by choosingn = ∞ and µ = A = 0, one has

f = ‖tT‖ − η(•g

in

T )1/m = 0 ⇒ ‖tT‖ = η(•g

in

T )1/m (28)

This criterion can also be easily generalized in order to take into account a dependenceof A and µ on the pressure, the cumulated slip and/or its rate, see Ponthot6 for details.

6 A RADIAL RETURN ALGORITHM FOR LUBRICATED FRICTION

6.1 Elastic Predictor (Sticking Contact)

The trial normal force (which is also the final one) and trial tangential forces are computedfrom the increments of displacement. They are given by (contact is supposed to besticking) :

ttrN = CNg1

N ttrT = t0

T + CT ∆gT (29)

whereCN is the normal (negative) penalty coefficient;CT is the tangential (positive) penalty coefficient;g1

N is the (total) normal gap or penetration;∆gT is the increment of tangential gap;t0T is the tangential force vector in the reference configuration.

Moreover, in the predictor phase we suppose•g

in

T = 0. If f = ‖ttrT ‖−A−µ|tN | < 0, the

contact is sticking and the trial state is the final state. If not, the elastic predictor mustbe modified in order to restore compatibility with the slip criterion.

6.2 Visco-Plastic Corrector (Sliding Contact)

The final expression for t1T and g1

T can be written:6

t1T = ttr

T − CTΓT with T =ttrT

‖ttrT ‖

(30)

g1T = g0

T +•g

1

T ∆t with•g

1

T =Γ

∆t(31)

where Γ is identical to the expression in the visco-plastic case. Once again, this parameteris simply determined by enforcement of the generalized consistency condition, f = 0 attime t1 , i.e.

f = ‖ttrT − CT ΓT ‖ − A − µ|t1

N | − η(g0T + Γ)1/n(Γ/∆t)1/m = 0 (32)

In the particular case where n = ∞ and m = 1, a closed-form solution is given by

Γ =‖ttr

T ‖ − A + µt1N

CT + η∆t

(33)

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J.P. Ponthot

Moreover, in the general nonlinear case, the present algorithm for contact is amenable toexact linearization leading to a closed-form expression for the consistent tangent moduli.This expression can be found in Ponthot.6

6.3 Consistent Tangent Operator

6.3.1 Introduction

Formely the consistent tangent operator is the tensor C such that

dt = Cdg (34)

where

dt =⟨

dtT1 dtT2 dtN⟩T

is the differential of contact forces;

dg =⟨

dgT1 dgT2 dgN

⟩Tis the differental of gap (tangent & normal).

6.3.2 Sticking Contact

In case of sticking contact, C is trivially given by (for isotropic frictional behaviour):

Cstick =

CT 0 0

0 CT 00 0 CN

(35)

6.3.3 Sliding Contact

In case of sliding contact, the expression for C is no longer trivial, as far as the frictionalforces are concerned. Actually, the contact forces resulting from the integration proceduredescribed in section 6.2 are given by relation (30), so that one can write (we suppose herethat the penalty parameters are constant):

dt1T = dttr

T − CT dΓT − CT ΓdT (36)

which has to be computed as a function of the gap increments dgN and dgT . After somerather involved algebra, the final results are given by, see Ponthot6 for details:

Cslip =

CT [1 − (1 + α)T1T1] −CT (1 + α)T1T2 −CN µT1

−CT (1 + α)T2T1 CT [1 − (1 + α)T2T2] −CN µT2

0 0 CN

(37)

with

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J.P. Ponthot

CT = βCT (38)

β = ‖t1T‖/‖ttr

T ‖ = 1 − CT Γ/‖ttrT ‖ (39)

µ =µ − µt1

N − A

1 + A′CT

+ µ′|t1N |

CT+ V T

CT

A′ =∂ A

∂ Γµ′ =

∂ µ

∂ Γ(40)

α =1

1 + A′CT

+ µ′|t1N |

CT+ V T

CT

A =∂ A

∂ tNµ =

∂ µ

∂ tN(41)

α = (α − 1)/β (42)

VT = η(g1T )

1n (

•g

1

T )1m

1

n g1T

+1

m ∆t•g

1

T

(43)

where V T stands for “Viscous Terms”.From this general expression, one can retrieve some particular cases. First, let’a assume

that the adherence A and the friction coefficient µ are constants. As a result, one hasA′ = A = µ′ = µ = 0 so that

µ =µ

1 + V TCT

(44)

α =1

1 + V TCT

(45)

In this case, we can see that the Viscous terms modify the friction coefficient similarly asthey did for the hardening parameter in visco-plasticity.

The rate-independent case can be recovered by setting η = 0, which implies thatTV = 0, so that µ = µ, α = 1 and α = 0. In this particular case, (37) is exactly equalto the consistent tangent operator previously derived by Peric & Owen8 and Bittencourt9

for the classical Coulomb friction law.Let’s now start again from the general consistent case and let’s suppose that there is no

time dependency, i.e. η = 0 and that A and µ are independent from the sliding velocity.Under those hypothesis, the Continuum operator can be deduced from the consistent one.Therefore, one just has to set β = 1, so that CT = CT and α = α − 1. In this case, theContinuum Operator can be written:

Ccontinuumslip =

CT [1 − αT1T1] −CT αT1T2 −CN µT1

−CT αT2T1 CT [1 − αT2T2] −CN µT2

0 0 CN

(46)

When the friction coefficient µ and the adherence A are constant, µ = µ and α = 1. Inthis case, (46) is totally equivalent to the one given by Charlier.10

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J.P. Ponthot

Another interesting observation can be done in 2D cases (plane strain and axisymmetriccases). In such a case, the sliding direction is constrained in the considered plane. As aconsequence, one can take, without any loss of generality, T1 = 1 and T2 = 0. The 2DConsistent Tangent operator thus reduces to:

Cconsistentslip2D =

[CT [1 − α] −CN µ

0 CN

](47)

In this case, the Consistent tangent operator, if it exists, is identical to the Continuumone! In the rate-independent case, it is identical to the one determined by Wriggers etal.11 by the chain rule.

7 NUMERICAL ILLUSTRATION

7.1 Superplastic forming of an aeronautical component

This first example pertains to the industrial aerospace engineering. It was previouslyproposed in Argyris12 who modelled the material as a Stokesian fluid (rigid viscoplasticmaterial), whereas here, elasticity effects are taken into account. It consists of a com-plex forming process simulation of a titanium alloy component. The initial geometry isillustrated in figure 1. The material properties are

Figure 1: Initial geometry

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J.P. Ponthot

Young Modulus E = 12000N/m2

Poisson ratio ν = 0.25

Viscoplastic behaviour σ = 0.0296 + 5267.62(•εvp)0.85N/m2

Table 1: Material properties for superplastic forming

Figure 2: Comparison of shape evolution for µ = 0.3 andµ = 0 (frictionless)

Figure 3: Comparison of final shapes for µ =0.15 and µ = 0 (frictionless)

The loading consists of an applied pressure. This one is automatically updated soas to keep a maximum value for the equivalent viscoplastic strain rate equal to 1.5−4

(superplastic range). The problem is isothermal but the the forming temperature isequal to 925oC. The finite element mesh consists of 78 quadrilateral finite elements withconstant pressure in order to avoid locking, see Ponthot6 for details.

The die is supposed to be rigid and three different friction coefficients were used (µ =0.0; 0.15 and 0.3) . Figure 3 clearly illustrates the strong influence of this parameter onthe total forming time (which takes several hours).

The presented procedure is quite efficient as is shown in table 2. The results wereobtained on a Vaxstation Dec/3000 Model 500 (125 SPECmark) and the CPU time isquite reasonnable.

µ forming time steps iter. CPU (sec)0.0 4h1/4 140 344 910.15 5h 162 396 1750.30 7h 208 529 235

Table 2: Costs for obtaining the solution

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J.P. Ponthot

7.2 Conical Extrusion

The considered problem is the quasi-static conical extrusion of an aluminium billet. Ithas been previously studied by Laursen & Simo.13 It consists of an initially cylindricalbillet (length = 25.4 cm & diameter = 10.16 cm) that is pushed by a ram through aconical rigid die. The total displacement of the ram is 17.8 cm and a friction coefficientof 0.1 is assumed between the rigid die and the aluminium. Contact constraints areimposed through the penalty formulation described above. The penalty parameters usedare respectively CN = 108N/cm and CT = 107N/cm. The aluminium is supposed tobehave like a J2 elastic-plastic material with linear isotropic hardening whose materialproperties are given in table 3.

Young Modulus E = 68956MPaPoisson ratio ν = 0.32Yield stress σ0

v = 31MPaHardening h = 261.2MPa

Table 3: Material properties for conical extrusion

The initial mesh consits of 80 (4 x 20) axisymmetric bilinear elements with constantpressure. The initial configuration, as well as deformed configurations corresponding toa ram displacement of 4.45 cm; 8.9 cm; 13.5 cm and 17.8 cm are depicted in figure 4. Inthe final configuration, the cumulated slip of the first contact node is larger than 25 cmwhereas the maximum penetration due to the penalty treatment is smaller than 1/500of the initial radius. The force applied by the ram as a function of ram displacementis plotted in figure 5. The results obtained by Laursen & Simo13 are also plotted. Theagreement is excellent.

The final configuration was obtained in 115 timesteps and 264 iterations in the presentwork, whereas Laursen & Simo’s algorithm (which is generally considered as very effec-tive), required 140 timesteps and 980 iterations while using the penalty formulation.However, to obtain their results, Laursen & Simo used penalty coefficients equal toCN = 1013N/cm and CT = 1010N/cm. As it is well known that higher penalty coefficientslead to higher CPU times, we have also repeated the calculation with those coefficients.For this latter case, exactly the same results were obtained but in 193 timesteps and 408iterations, which is still a much better convergence rate than in Laursen & Simo.13

7.3 Elbow forming

This last example has been proposed by El Mouatassim14 as a benchmark problem forcontact algorithms. A plane billet (length = 12 mm, width = 4 mm) is guided through arigid tube (inner radius = 1.5 mm, outer radius = 5.5 mm) as described in figure 6. Thevertical top row of nodes has an imposed vertical displacement of 12 mm (El Mouatassimproposed 10 mm). Plane strain conditions are assumed and the material is supposed to

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J.P. Ponthot

Figure 4: Initial mesh and deformed configurations for a ram displacements of 4.45 cm; 8.9 cm; 13.5 cmand 17.8 cm

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J.P. Ponthot

02468

101214161820

0 2 4 6 8 10 12 14 16 18

Forc

e(M

N)

Ram Displacement (Cm)

Conical extrusion

Present workLaursen & Simo

Figure 5: Conical extrusion process: Force applied by the ram as a function of its displacement

behave like an elastic perfectly plastic material whose properties are given in table 4.

Young Modulus E = 200000MPaPoisson ratio ν = 0.3Yield stress σ0

v = 400MPa

Table 4: Material properties for elbow forming

The finite element mesh consists of 48 bilinear/constant pressure elements, initially squareand identical. Figure 7 displays the initial mesh as well as deformed configurations andeffective plastic strains after a top displacement of 4, 8 and 12 mm. The final configurationhas been obtained in 123 steps and 319 iterations (7.5 seconds in CPU) which, once again,shows the efficiency of the proposed formulation. For comparison purposes, El Mouatassimannounced 10 000 timesteps, i.e. a vertical displacement of 0.001 mm per step for a totaldisplacement of 10 mm. This is an average timestep 1000 times lower than the one wehave used.

8 CONCLUSIONS

We have briefly described a new algorithm which is an extension of the classical radialreturn scheme to the viscoplastic domain on the one hand, and to frictional contact onthe other hand. This scheme is very inexpensive. It is generalizable to more involved cri-terion than Von-Mises plasticity or Coulomb friction law. This can be easily implementedby considering the more general class of so-called closest-point projection algorithm asdescribed in Simo.4 Many other examples illustrating the effectiveness of the presentalgorithm can be found in Ponthot.6

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J.P. Ponthot

Figure 6: Initial mesh and deformed configurations for an imposed displacement of 4 mm, 8 mm and12 mm

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J.P. Ponthot

0

69.5

139

208.6

278.1

347.6

417.1

486.6

556.2

625.7

695.1VALEUR * 1.E -3

0

94.4

188.7

283.1

377.5

471.9

566.2

660.6

755

849.4

943.6VALEUR * 1.E -3

0.002

0.105

0.209

0.312

0.416

0.52

0.623

0.727

0.83

0.934

1.037

Figure 7: Initial mesh and map of effective plastic strains in deformed configurations for an imposeddisplacement of 4 mm, 8 mm and 12 mm

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J.P. Ponthot

REFERENCES

[1] J.C Golinval. Calculs par elements finis de structures elasto-viscoplastiques soumisesa des chargements cycliques a haute temperature. PhD thesis, University of Liege,Liege, Belgium, (1988).

[2] P. Perzyna. Fundamental problems in visco-plasticity. In Advances in Applied Me-chanics, pages 243–377. Academic Press, (1966).

[3] T.J.R. Hughes. Numerical implementation of constitutive models: Rate-independentdeviatoric plasticity. In S. Nemat-Nasser, R.J. Asaro, and G.A. Hegemier, editors,Theoretical Foundation for Large-Scale Computations of Nonlinear Material Behav-ior. Martinus Nijhoff, (1983).

[4] J.C. Simo and T.J.R Hughes. General return mapping algorithms for rate-independent plasticity. In Constitutive Laws for Engineering materials : Theoryand Applications. Elsevier Science Publishing Co, (1987).

[5] J.P. Ponthot. Radial return extensions for visco-plasticity and lubricated friction. InM. Rocha, editor, SMIRT-13 Internatinal Conference on Structural Mechanics andReactor Technology, pages 711–722, Porto Alegre, Brazil, (1995).

[6] J.P. Ponthot. Traitement unifie de la Mecanique des Milieux Continus solides engrandes transformations par la methode des elements finis. PhD thesis, University ofLiege, Liege, Belgium, (1995).

[7] A. Curnier. A theory of friction. International Journal of Solids and Structures,20(7), 637–643 (1984).

[8] D. Peric and D.R.J. Owen. Computational model for 3-d contact problems withfriction based on the penalty method. International Journal of Numerical Methodsin Engineering, 35, 1289–1310 (1992).

[9] E. Bittencourt. Tratamento do Problema de Contacto e Impacto em Grandes Defor-macoes pelo Metodo dos Elementos finitos. PhD thesis, Universidade federal do Riogrande do Sul, Porto Alegre, Brazil, (1994).

[10] R. Charlier. Approche Unifiee de quelques problemes de la Mecanique des MilieuxContinus par la methode des elements finis. PhD thesis, Universite de Liege, Belgium,(1987).

[11] P. Wriggers, T. Vu Van, and E. Stein. Finite element formulation of large deformationimpact contact problems with friction. Computers & Structures, 37, 319–331 (1990).

[12] J.H. Argyris. A primer on super plasticity in natural formulation. Computer Methodsin Applied Mechanics and Engineering, 46, 83–13 (1984).

[13] T.A. Laursen and J.C. Simo. Algorithmic symmetrisation of coulomb frictional prob-lems using augmented lagrangian. Computer Methods in Applied Mechanics andEngineering, 108, 133–146 (1993).

[14] M. El Mouatassim. Modelisation en grandes transformations des solides massifs parelements finis. PhD thesis, Universite Technologique de Compiegne, France, (1989).

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