PAMM · Proc. Appl. Math. Mech. 11, 411 – 412 (2011) / DOI 10.1002/pamm.201110197

A material model of nonlinear fractional viscoelasticity

Sebastian Müller1,∗, Markus Kästner1, Jörg Brummund1, and Volker Ulbricht1

1 Technische Universität Dresden, Institute for Solid Mechanics, D-01062 Dresden

Multiscale methods are frequently used in the design process of textile reinforced composites. In addition to the models for

the local material structure it is necessary to formulate appropriate material models for the constituents. While experiments

have shown that the reinforcing fibers can be assumed as linear elastic, the material behavior of the polymer matrix shows

certain nonlinearities.

These effects are mainly due to strain rate dependent material behavior. Fractional order models have been found to

be appropriate to model this behavior. Based on experimental observations of Polypropylene a one-dimensional nonlinear

fractional viscoelastic material model has been formulated. Its parameters can be determined from uniaxial, monotonic

tensile tests at different strain rates, relaxation experiments and deformation controlled processes with intermediate holding

times at different load levels. The presence of a process dependent function for the viscosity leads to constitutive equations

which form nonlinear fractional differential equations. Since no analytical solution can be derived for these equations, a

numerical handling has been developed. After all, the stress-strain curves obtained from a numerical analysis are compared

to experimental results.

c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The macroscopic properties of composite materials often show nonlinearities, which are, among others, driven by damage

effects and the inelastic material behavior of the polymeric matrices. The authors have applied multiscale modeling and

simulation techniques [1, 2] to predict the effective linear elastic stiffness properties as well as the macroscopically nonlinear

material behavior of textile reinforced polymers using only the properties of the individual constituents and their geometrical

arrangement in the composite. To this end, suitable constitutive relations are required to model the inelastic material behavior

of the polymer matrix. The present paper addresses the modeling of strain rate dependent effects through constitutive relations

of fractional order. The phenomena observed during the experimental characterization of the polymeric matrix material

Polypropylene thereby motivated the formulation of a process dependent viscosity function. While section 2 summarizes the

experimental observations for Polypropylene, section 3 presents a material model of nonlinear fractional viscoelasticity. After

all certain aspects of the parameter identification as well as selected simulation results are shown in section 4.

2 Nonlinear Material Behavior of Polypropylene

In order to characterize the material behavior of Polypropylene, displacement controlled experiments on dog bone shaped

specimens have been carried out. The stress-strain curves obtained in tensile tests at different strain rates indicate a nonlinear

material behavior with a clear strain rate dependence (cf. Fig. 1). Relaxation experiments allow for the separation of the strain

rate dependent and independent fraction of stress response. It can be seen in Fig. 2 that after a pronounced relaxation of the

stress at the beginning, the time rate of stress decreases until the relaxation seems to stop after approximately 48 hours. For

this reason the termination point is assumed to be a state of equilibrium. A multitude of these states, identified during loading

and unloading experiments with intermediate holding times (cf. Fig. 3), indicate a non vanishing equilibrium relation.

Fig. 1 Stress-strain curve for tensile tests

at three distinct velocities

Fig. 2 Stress-time curve of relaxation ex-

periment Fig. 3 Stress-strain curve for cyclic load-

ing and unloading experiment

∗ Corresponding author: Email Sebastian.Mueller@tu-dresden.de, phone +49 351 463 31929, fax +49 351 463 37061

c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

412 Section 6: Material modeling in solid mechanics

3 Nonlinear Fractional Viscoelastic Material Model

According to the experimental observations a viscoelastic material model has been formulated, which is based on an additive

decomposition of the overall stress σ = σeq + σov into a strain rate independent equilibrium stress σeq and the strain rate

dependent overstress σov. While the equilibrium relation is described by a linear elastic spring element σeq = E0ε, the

strain rate dependent overstress is modeled by the parallel connection of two fractional MAXWELL elements [4]. Each is

characterized by the fractional evolution equation

Dαk

kσov +

1

τ̃αk

k

kσov = EkD

αkε, k = 1...2, (1)

with the relaxation strength Ek, the fractional derivative order αk ∈ (0; 1) and a process dependent relaxation time τ̃k. The

strong nonlinear slope in the stress-strain curve during the loading process (cf. Fig. 1) required the extension of the model

by an appropriate viscosity function. In the present model the viscosity of the fractional MAXWELL element is controlled by

adapting the characteristic relaxation time

τ̃k = [τk(sk ǫ̂+ 1)ck ] exp (−s0|qs|) , k = 1...2, (2)

according to the average strain rate

ǫ̂ =1

t

t∫

0

|ε̇(ξ)|dξ, for t > 0, ǫ̂ = 0 for t = 0 (3)

and the structural variable qs, which accounts for changes within the molecular structure of the material [3]. It has an initial

value of qs(0) = 0 and the evolution is described by the differential equation

q̇s = cqε̇(1− |qs|)−1

τq

qs. (4)

4 Parameter Identification and Simulation

While the parameters of the equilibrium relation as well as of the fractional MAXWELL elements have been identified from the

stress-time curve of the relaxation experiment, the parameters of the viscosity function are obtained from the nonlinear stress-

strain curves of the tensile tests [5]. The simulation of the previously described experiments show, that the formulated model is

capable of representing the observed strain rate dependent material behavior (cf. Fig. 4, 5). However, the prediction partly fails

for loading and unloading experiments (cf. Fig. 6), which is due to the simple linear elastic approach for equilibrium relation.

Based on the uniaxial constitutive relations the model has been generalized for the multiaxial case [5] and implemented into a

finite element code.

Fig. 4 Approximation of tensile tests at

three distinct velocities

Fig. 5 Approximation of the relaxation

experiment Fig. 6 Approximation of the cyclic load-

ing and unloading experiment

Acknowledgements The present study is supported by the German Research Foundation (DFG) within the Collaborative Research Center

(SFB) 639, subproject C2. This support is gratefully acknowledged.

References

[1] P.P. Camanho, C.G. Dávila,S.T. Pinho, J.J.C. Remmers (eds.), Mechanical Response of Composites (Springer, 2008), chap. 13.[2] M. Kästner, G. Haasemann, and V. Ulbricht, Int. J. Numer. Meth. Engng. 86 (4-5), 477-498 (2011).[3] P. Haupt, and K. Sedlan, Archive of Applied Mechanics 71 (2), 89-109 (2001).[4] R.C. Koeller, Journal of Applied Mechanics 51 (2), 299-307 (1984).[5] S. Müller, M. Kästner, J. Brummund, and V. Ulbricht, Computational Materials Science, (2011).

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