The viscoelastic material behavior is modelled here by a spring-dashpot model. Such a phenomenological modeling approach does not take into consideration the real molecular structure of the material but it may be able to model accurately the macroscopic material re-sponse.
Considering the material behavior of rubber and rubber-like materials the generalized Maxwell-model (several Maxwell-models connected in parallel) connected parallel to a spring can be a reasonable choice as a material model because it is a widely accepted constitutive law and available in the most, commercial FE software packages. At the same time, it must be mentioned that its applicability depends strongly on the number of Maxwell elements. Even in case of 15-term Maxwell-model, there is a strong fluctuation in the loss factor-frequency curve that may cause inaccurate modeling results, especially when the excitation frequency varies in a broad frequency range. According to the experience of the author of this thesis a 40-term Maxwell model is able to describe with high accuracy the time dependent material behavior of viscoelastic materials in a wide frequency range.
Description of linear viscoelasticity can be found, among others, in the books of Ferry [99], Christensen [100], Findley et al. [101], and Aklonis and MacKnight [102]. Linear visco-elastic deformation is commonly described using the Boltzmann integral representation (con-volution integral) in its relaxation form (stress relaxation) as [99]:
time-dependent material behavior is involved in the tensile relaxation modulus (E(t)). In other words, E(t) is the relaxation modulus at time t. This equation can be rewritten as
t E
te
tt
ddtt dtwhere e(t) is the normalized relaxation modulus, defined as e
t E t /E0. Here
0
0 E t
E is the instantaneous (or glassy) modulus. The normalized relaxation modulus is defined (approximated with desirable accuracy) as a Prony series (sum of exponentials) and can be formulated as
Similarly, the relaxation modulus is defined as
ratio is assumed to remain unchanged over time. Consequently, it does not appear in the stress relaxation formulation. The physical interpretation of time independent Poisson’s ratio is that relaxation processes in mutually orthogonal directions proceed in the same manner. Although, in many cases, this assumption is a good approximation it must be mentioned that, in real pol-ymers, the Poisson’s ratio usually exhibits time- and temperature dependent behavior as re-ported by Tschoegl et al. [103] and Pandini and Pegoretti [104].This representation of the relaxation modulus emphasizes that the relaxation occurs not at a single time, but at many, different time instants (spectrum of relaxation times). The specification of the linear viscoelastic model by a Prony series is extremely useful for the FE prediction of hysteresis friction because, in most finite element software packages, this speci-fication is used. The above equation of stress relaxation (reduction of stress under constant strain) can be represented graphically by using a generalized Maxwell-model connected paral-lel to a spring. Such a model is depicted in Fig. A1.
Fig. A1. Schematic representation of a generalized Maxwell-model connected parallel to a spring [106]
It is worth noting that there is no consensus in the literature regarding the precise name of the viscoelastic model. In FE software packages, research papers dealing with FE model-ling and many papers investigating the viscoelastic response of polymers or elastomers (for example in the paper of Bardenhagen et al. [105]) the material model seen in Fig. A1 is termed generalized Maxwell-model. In this thesis, the term generalized Maxwell-model indi-cates several Maxwell-models (a linear spring and dashpot in series) connected parallel. One of the possibilities available in MSC.Marc to model linear viscoelasticity is to use the general-ized Maxwell-model connected parallel to a spring. As dictated by its graphical representation
the stress in such a viscoelastic model is the sum of stresses in the Maxwell-elements and in the spring E. The model consists of linear springs and dashpots. In case of linear spring, the modulus is constant while the stress-strain relationship is linear. Contrary to this, in case of linear dashpot, the stress-strain rate relationship is linear while their ratio, the viscosity, is constant (stress or strain rate independent viscosity).
Denoting the modulus of the separate spring by E (relaxed modulus) the interrelation among the normalized moduli can be written as
,
The i-th relaxation time of the generalized Maxwell model (i) is defined as
i i
i E
, where
i is the dynamic viscosity of the i-th dashpot.
As the viscoelastic model connected parallel to a spring is used for hysteresis friction prediction (prediction of friction force due to the hysteresis) in this thesis the dynamic modu-lus of rubber has primary importance. If a linear viscoelastic material is subjected to a sinus-oidally varying strain having an amplitude 0 and angular frequency ω then the steady-state response (stress) will also be sinusoidal with the same frequency (ω) but some time delay.
This time delay or phase shift is represented by the so-called phase angle δ. In case of elastic material 0, while in case of totally viscous material 90. For rubbers the phase angle is greater than 0° but smaller than 90° (0 90) as reported by Bodor and Vas [107]. The stress response can be written as
t
t
t
t 0sin 0cos sin 0sin cos , (A.7)
where 0 is the stress amplitude. According to the latter expression the stress response can be divided into two parts. One part of the stress (0cos
sint 0'sin
t ) is in phase with the strain (elastic stress) while its other part (0sin
cost 0"cos
t ) is out of phase with a phase angle of 90 (viscous stress). Considering the stress as a complex quantity (*) having a norm of 0it can be written that strain). The strain energy associated with the in-phase stress and strain is the energy stored in the material (reversible strain energy). Contrary the strain energy associated with the out of phase stress and strain is the energy converted into heat (irreversible strain energy) [107]. The complex dynamic modulus E* is defined as the stress amplitude divided by the strain ampli-tude i.e.
2 2where 'E is the storage (elastic) modulus (real part of the complex dynamic modulus) and "E is the loss (viscous) modulus (imaginary part of the complex dynamic modulus). As reported by Roylance [108] the storage modulus represents the energy stored (elastic portion) and the loss modulus represents the energy dissipated as heat (viscous portion).
As it was mentioned the N-term generalized Maxwell-model consists of N Maxwell models in parallel. With regard to the serial connection of the linear spring and the linear dashpot of a Maxwell-model the resultant deformation of the model is given as the sum of deformations of the spring and the dashpot. This can be written in mathematical form as
E
, (A.10)
where denotes the resultant strain rate, E denotes the modulus of the linear spring, and denotes the dynamic viscosity of the dashpot. Based on this equation both the frequency de-pendent storage and loss moduli of the Maxwell-model can be derived as presented by Pahl et al. [109].
The loss factor (tan(δ)) is defined as the ratio of the loss and storage modulus thus it can be formulated as Maxwell-models are connected in parallel as done by Wiechert [110]. In the case of generalized Max-well element, the relationship for storage and loss modulus can be written as the sum of sepa-rate Maxwell models (elements) i.e.
where N denotes the number of Maxwell elements and i denotes the relaxation time of the i-th Maxwell element.
Finally, consider the viscoelastic model where a finite number of Maxwell elements and a separate linear spring (E) are connected in parallel (see Fig. A1). The linear spring E is responsible for the time-independent elastic part of the deformation while the finite number of Maxwell elements represents the time-dependent elastic (viscoelastic) part of the deformation. Derivation of the frequency dependent complex dynamic modulus of the viscoe-lastic model can be found, among others, in studies of Kaliske and Rothert [111], Achenbach
and Frank [22], and Achenbach and Streit [98]. Using these results the storage and loss continuous relaxation spectrum can be approximated with any desirable accuracy as the above expressions show. Contribution of the different Maxwell elements to the cumulative (result-ant) frequency-dependent material behavior of the viscoelastic model is shown in Fig. A2. In many cases, components of the dynamic modulus are expressed in function of the cyclic fre-quency f. As 2 f the cyclic frequency dependence of the storage modulus can be writ-ten as
The above shows that the time dependence of relaxation modulus and frequency dependence of storage and loss modulus can be captured by a spring-dashpot model with a sufficient number of elastic and viscous elements [99].
Fig. A2. Contribution of Maxwell elements to the cumulative (resultant) normalized storage
(
) of a 14-term generalized Maxwell model connected
parallel to a separate spring [111]
Experimental data from either creep or relaxation tests can be used to identify the model parameters with the software’s built-in algorithm. In other words, more sophisticated
commercial FE software packages are able to compute the model parameters (ei, ) automat-i ically from experimental data. However, most of the software packages have limited capabil-ity for “calibrating” n-term Maxwell model. In most cases, the number of terms is limited anywhere between ten and fifteen. As an example MSC.Marc is able to identify automatically the model parameters from a fit to creep or stress relaxation tests but cannot to do this if DMTA data are available only. Additionally, the number of terms in the fitted viscoelastic model is limited to ten. Finally, it is worth to mention that in case when model parameters are specified using the graphical user interface of the software the number of Maxwell-terms is limited to fifteen. As it is pointed out in Appendix C the models calibrated by without over passing the capability of the applied, commercial FE software have strongly limited accuracy when they are used for hysteresis friction prediction.