Simulation Of Rate Dependent Plasticity Of Polymers

Ahmed Shaban and Rolf Mahnken∗

Chair of Engineering Mechanics (LTM), University of Paderborn, Warburger Street 100, D-33098 Paderborn, Germany

Polycarbonate is an amorphous polymer which exhibits nonlinear deformation before failure. It shows a pronounced strength-differential effect between compression and tension. Strain rate influences the mechanical response of the polycarbonate. Inparticular, the yield stress is increased with increasing strain rate. The concept of stress mode dependent weighting functionis used in the proposed model to simulate the asymmetric effects for different loading speeds. In this concept, an additivedecomposition of the flow rule is assumed into a sum of weighted stress mode related quantities. The characterization of thestress modes is obtained in the octahedral plane of the deviatoric stress space in terms of the mode angle, such that stress modedependent scalar weighting functions can be constructed. The resulting evolution equations are updated using a backwardEuler scheme and the algorithmic tangent operator is derived for the finite element equilibrium iteration. The numericalimplementation of the resulting set of constitutive equations is used in a finite element program for parameter identification.The proposed model is verified by showing a good agreement with the experimental data.

1 Introduction

Polycarbonate (Lexan 104R) specimens have been tested under tension and compression for different strain rates at 20◦C.Fig.1a shows that the yield stress is increased with increasing strain rate.

tension

compression

tension

compression

tension

compression

ð3

1ð

3

2ð ð

3

4ð

3

52ð0

1ó�

2ó�

3ó�

è

3

ð

è

a)

b)

c)

� �èwi

1.0

0.0

w1

w2

1.0

0.0

w1

w2

-100.00

-80.00

-60.00

-40.00

-20.00

0.00

20.00

40.00

60.00

80.00

100.00

-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08

Strain (-)

Str

es

s(M

Pa

)

Exp. 8.3e-02 (1/s)

Num. 8.3e-02 (1/s)

Exp. 8.3e-03 (1/s)

Num. 8.3e-03 (1/s)

Exp. 8.3e-04 (1/s)

Num. 8.3e-04 (1/s)

Fig. 1 a) Stress strain curve for polycarbonate at different strain rates and 20◦C. b) Octahedral plane in the deviatoric stressspace where σ1,σ2,σ3 denote the principal deviatoric stresses. c) Weighting functions for tension and compression modes.

It is also observed that the yield stress under compression is higher than that under tension where applying a pressure givesthe molecular chains less room in which to move which reduces the mobility of the chains. Therefore the material needs extraenergy under compression to yield.

2 Constitutive Modeling

The existance of a difference between the yield stress under tension and compression is called asymmetric effect. Thiseffect is simulated using the concept of stress mode dependent weighting functions [1]. In that concept, the second invariant

∗ Corresponding author: e-mail: rolf.mahnken@ltm.uni-paderborn.de, Phone: +49 5251 60 2283, Fax: +49 5251 60 3483

PAMM · Proc. Appl. Math. Mech. 6, 409–410 (2006) / DOI 10.1002/pamm.200610185

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

J2 = 121 : σ2

dev and the third invariant J3 = 131 : σ3

dev are defined, where σdev represents the deviatoric part of the stresstensor. The so-called stress mode angle θ is formulated as follows:

θ =13

arccos[ξ], where ξ = 272

J3J2

. (1)

A graphical illustration of θ is given in Fig.1b, which demonstrates that θ has different values for tension and compressionmodes. In the proposed constitutive framework, the inelastic strain rate tensor is decomposed additively into a sum of twostress mode related quantities:

εin =2∑

i=1

wiFidi, where Fi = Fi[σ, q, T, ...,κ], di = di[σ, q, ...,κ], q = q[σ, q, T, ...,κ]. (2)

Here, the flow factor Fi and the flow direction di are dependent on the stress tensor σ, a set of internal variables q and avector of the material parameters κ. Both Fi and q are dependent on the temperature T . Furthermore, the stress mode relatedweighting functions illustrated in Fig.1c have the following forms:

w1[ξ] =12(1 + ξ) for tension, and w2[ξ] = 1

2 (1 − ξ) for compression. (3)

For further details of this constitutive theory and its algorithmic implementation, refer to Shaban et al. [2].

3 Results and Discussion

After the development of the model, a parameter identification is done to determine the parameters of the model. The results ofthe simulation using the optimized parameters are given in Fig.1a, which shows a good agreement with the experimental data.After that, the proposed model has been numerically incorporated into the finite element code ABAQUS to simulate the laserthrough-transmission welding in which laser beams are used to join two parts, one is transparent and the other is absorbant(for more details, see [3]). The results of the simulation are shown Fig.2 where Fig.2a shows the temperature distribution afterabout 13 seconds at which the maximum equivalent inelastic strain occurs as shown in Fig.2b. The capability of the proposedapproach is assessed by specifying the position of the maximum temperature. Fig.2c shows the temperature distribution at thehorizontal center line for selected welding times. From the later figure, the required position is found to be shifted to the rightof the vertical center line which means that the maximum temperature is not located in the joining zone but in the absorbingmaterial (see [3]).

0

40

80

120

160

200

240

280

320

360

400

440

480

0 2 4 6 8 10 12

Distance (mm)

Tem

pera

ture

(°C

)

t = 00.000 sec

t = 01.337 sec

t = 05.053 sec

t = 10.190 sec

t = 13.140 sec

a)

b) c)0

40

80

120

160

200

240

280

320

360

400

440

480

0 2 4 6 8 10 12

Distance (mm)

Tem

pera

ture

(°C

)

t = 00.000 sec

t = 01.337 sec

t = 05.053 sec

t = 10.190 sec

t = 13.140 sec

a)

b) c)

Fig. 2 a) Temperature distribution after about 13 sec. b) Equivalent inelastic strain after about 13 sec. c) Temperaturedistribution in the horizontal center line at different times.

Acknowledgements All experiments on polycarbonate specimens have been performed at the Institute of Plastics (KTP), University ofPaderborn by Dr.-Ing. H. Ridder from 3 Pi Consulting & Management GmbH.

References

[1] R. Mahnken, Int. J. Solids Struct. 40, 6189–6209 (2003).[2] A. Shaban and R. Mahnken, in preparation.[3] H. Potente et al., in: Proceedings of Annual Technical Conference, Chicago, Illinois, 2004, p. 306.

Section 6 410

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim