Prestrain-dependent viscosity of a highly filled elastomer: experiments and modeling

Abstract

Highly filled elastomers exhibit a complex microstructure made up of rigid fillers bounded by a thin layer polymeric matrix. The interactions between the fillers and the binder amplify locally the applied strains and induce a nonlinear viscoelastic behavior. The aim here is to analyze the influence of prestrain on the viscoelastic behavior. This paper proposes a prestrain-dependent viscoelastic constitutive model. The model is a superposition of three relaxation spectra, each corresponding to a family of polymer chains, and can be regarded in either its continuous or discrete expression. More specifically, one of these relaxation spectra is modified to assure the prestrain sensitivity. The parameters of the discrete model are identified from relaxation and DMA experiments performed on a solid propellant, and the obtained predictions match closely the experiments. The novelty of the analysis proposed in this paper is threefold. On the one hand, we report a new series of experimental measures, performed for a large range of frequencies for the DMA experiment and relaxation times for the relaxation experiment, and, on the other hand, we propose a constitutive law compatible with the principles of thermodynamics, which predicts closely the measurements. Finally, the analysis is performed comparing both relaxation and DMA experiments using the spectrum of relaxation times. A peculiarity of the present discussion is the novel identification method used, which identifies directly the relaxation times. This technique leads to models with smaller and optimum numbers of parameters than classical methods based on a logarithmic distribution of relaxation times.

Appendix:  Discretization of the continuous spectrum of relaxation time

The generalized Maxwell model, represented as a Prony series is a discrete approximation of the continuous spectrum of relaxation H(τ). The standard discretization method starts with the discretization of the time axis into a finite set of instants (τ _i ), separated by equidistant intervals on the logarithmic scale. Then, the associated elastic modulus is computed using the formula $E_{i} = H(\tau_{i}) \ {\rm ln}(\sqrt{\frac{\tau_{i+1}}{\tau_{i-1}}})$ and leads to the complete Prony series (τ _i ,E _i ), i=1,…,n, with n the number of viscoelastic elements of the model. The associated viscosity is defined as the ratio of elastic moduli and characteristic time, that is, η _i =E _i /τ _i . A complete description of the method is given, for example, in Baumgaertel and Winter (1992).

The method discussed next is based on the general remark that a continuous function can be approximated by a staircase function of variables widths (see Kurtz et al. 2004). The method will compute the set relaxation times τ _i based on a nonlinear optimization method minimizing the integral distance between the continuous relaxation function and its staircase approximation. A detailed presentation of the method is given in Jalocha et al. (2015a).

The discretized spectrum, represented as a staircase function S(τ) with n elements (see Fig. 13), is expressed as

$$ S(\tau)=\sum_{i=1}^n H( \tau_i) U_{(t_{i-1},t_i)}(\tau) $$

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with $\tau_{i}=\frac{t_{i-1}+t_{i}}{2}$ and

$$U(t_{i-1},t_i) = \begin{cases} 1&\mbox{if}\ t_{i-1} \leq\tau\leq t_i,\\ 0&\mbox{otherwise,} \end{cases} $$

also denoted as the unit box function. The relative distance between the continuous kernel and the discretized models, H(τ) and R, respectively, is given by the formula

$$ R = \frac{\| \int_{\tau_{c1}}^{\tau_{c2}} H(\tau) - S(\tau) \, d\tau \| }{\| \int_{\tau_{c1}}^{\tau_{c2}} H(\tau) \, d\tau \|}. $$

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The optimal distribution of the relaxation times τ _i is defined in terms of the endpoints of each unit box function $\tau_{i}=\frac {t_{i-1}+t_{i}}{2}$, i=1,…,n, for given n. Moreover, it will minimize the residual function R. The end points of the time series t ₀ and t _n are supposed to be a priori chosen. See Fig. 13 for a schematic view.

Fig. 13

An example of continuous spectrum of relaxation time H(τ) and a staircase function S(τ). Computation principle of τ _i