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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 [email protected], 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).
c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.gamm-proceedings.com