Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers

Abstract

The nonlinear viscoelastic behavior of the composites of rubber filled with carbon black, silica, carbon nanotube (CNT), clay and surface-modified nanosilica were studied. The behavior of carbon black-filled rubber is thoroughly analyzed with the intention of developing a constitutive model able to reproduce both static and dynamic material responses. Several nonlinear viscoelastic models have been examined thoroughly and for each of them advantages and disadvantages are highlighted. A series of experiments concerning both static and dynamic tests were performed aimed at measuring all the relevant nonlinear effects. Temperature and strain rate dependencies were investigated and discussed. The standard methodology was applied to perform both tensile and compressive quasi-static tests. Some shortcomings of this procedure, resulting in a unreliable stress-strain constitutive curve around the undeformed configuration, were identified. This lead to the design a non-standard cylindrical specimen able to bear both tensile and compressive loading. Consequently, the influence of the shape factor was removed and the same boundary conditions, in tension and compression, were applied. This allowed the stiffness around the undeformed configuration to be evaluated in detail. The quasi-static experimental results also allowed the influence of the Mullins effect on the quasi-static response to be investigated: during the loading cycles, there is a significant reduction in the stress at a given level of strain, which is a consequence of the internal material rearrangement, i.e., the Mullins effect. This damage phenomenon is sometimes reported to induce transverse isotropy in the material, which is usually assumed to be isotropic. The Payne effect becomes more pronounced at higher silica loading. The filler characteristics such as particle size, specific surface area, and the surface structural features were found to be the key parameters influencing the Payne effect. A nonlinear decrease in storage modulus with increasing strain was observed for unfilled compounds also. The results reveal that the mechanism includes the breakdown of different networks namely the filler-filler network, the weak polymer-filler network, the chemical network, and the entanglement network. The model of variable network density proposed by Maier and Goritz has been applied to explain the nonlinear behavior. The model fits well with the experimental results. The interaction between epoxidized elastomeric matrix and silica as filler was extremely improved, even in the presence of very low content of epoxy groups into the polymer chain.

In memory to my parents Gordana Marković

Dynamic Moduli of Nonlinear Viscoelastic Models

In Eq. (161) f, l and h are nonlinear functions of the strain and the strain rate. Under weak

$$ f(t)=f\left[{E}_0+{E}_1 \sin \left(\omega t\right),{E}_1\omega \cos \left(\omega t\right)\right]=\frac{f_0}{2}+{\displaystyle \sum_{i=1}^{\infty}\left[{f}_i^S \sin \left(i\omega t\right)+{f}_i^C \cos \left(i\omega t\right)\right]} $$

(162)

Where

$$ {f}_0={f}_0\left({E}_0,{E}_1\right),\kern1em {f}_i^S={f}_i^S\left({E}_0,{E}_1,\omega \right),\kern1em {f}_i^C={f}_i^C\left({E}_0,{E}_1,\omega \right) $$

regularity assumption, their Fourier series are uniformly convergent, e.g.,

The Fourier coefficients of l and h will be denoted as l ^S _i , l ^C _i and h ^S _i , h ^C _i , respectively.If the series (162) are also absolutely convergent, by means of the Cauchy formula, the following expression of the stationary stress is recovered

$$ {\sigma}_s(t)=\frac{f_0}{2}+\frac{H_0{l}_0}{4}+{\displaystyle \sum_{i=1}^{\infty}\left[\left({f}_i^S+\frac{l_0}{2}{H}_i^S+\frac{l_i^S}{2}{H}_0\right) \sin \left(i\omega t\right)+\left({f}_i^C+\frac{l_0}{2}{H}_i^C+\frac{l_i^C}{2}{H}_0\right) \cos \left(i\omega t\right)\right]}+ $$

$$ \begin{array}{l}+{\displaystyle \sum_{i=1}^{+\infty }{\displaystyle \sum_{n=1}^i{l}_n^S\left[{H}_{i-n+1}^S \sin \left[\left(i-n+1\right)\omega t\right]+{H}_{i-n+1}^C \cos \left[\left(i-n+1\right)\omega t\right]\right]}} \sin \left(n\omega t\right)+\\ {}+{\displaystyle \sum_{i=1}^{+\infty }{\displaystyle \sum_{n=1}^i{l}_n^C\left[{H}_{i-n+1}^S \sin \left[\left(i-n+1\right)\omega t\right]+{H}_{i-j+1}^C \cos \left[\left(i-n+1\right)\omega t\right]\right]}} \cos \left(n\omega t\right).\end{array} $$

(163)

By projecting σ_s(t) over sin(ωt) and cos(ωt), Eqs. (3.94)-(3.95) of the storage and loss moduli are recovered.In order to evaluate the derivatives ∂S/∂ω and ∂L/∂ω, ∂H ^S _i /∂ω and ∂H ^C _i  /∂ω can be derived from Eqs. (3.91)-(3.93), i.e.,

$$ \frac{\partial {H}_i^S}{\partial \omega }=\frac{\partial {h}_i^S}{\partial \omega }{\displaystyle {\int}_0^{+\infty}\dot{k}(s)} \cos \left(i\omega s\right)ds-{h}_i^S{\displaystyle {\int}_0^{+\infty } is\dot{k}(s)} \sin \left(i\omega s\right)ds $$

$$ -\frac{\partial {h}_i^C}{\partial \omega }{\displaystyle {\int}_0^{+\infty}\dot{k}(s)} \sin \left(i\omega s\right)ds-{h}_i^C{\displaystyle {\int}_0^{+\infty } is\dot{k}(s)} \cos \left(i\omega s\right)ds $$

(164)

$$ \frac{\partial {H}_i^S}{\partial \omega }=\frac{\partial {h}_i^S}{\partial \omega }{\displaystyle {\int}_0^{+\infty}\dot{k}(s)} \sin \left(i\omega s\right)ds-{h}_i^S{\displaystyle {\int}_0^{+\infty } is\dot{k}(s)} \cos \left(i\omega s\right)ds $$

$$ -\frac{\partial {h}_i^C}{\partial \omega }{\displaystyle {\int}_0^{+\infty}\dot{k}(s)} \cos \left(i\omega s\right)ds-{h}_i^C{\displaystyle {\int}_0^{+\infty } is\dot{k}(s)} \sin \left(i\omega s\right)ds $$

(165)

provided that the integrability condition (3.96) holds. As a consequence, when assessed at low frequencies, the result is

$$ \frac{\partial {H}_i^S}{\partial \omega}\left|{}_{\omega =0}\right.=\left({k}_{\infty }-{k}_0\right)\frac{\partial {h}_i^S}{\partial \omega}\left|{}_{\omega =0}\right.-{h}_i^C\left|{}_{\omega =0}\right.i{\displaystyle {\int}_0^{+\infty }s\dot{k}(s)ds} $$

(166)

$$ \frac{\partial {H}_i^C}{\partial \omega}\left|{}_{\omega =0}\right.=\left({k}_{\infty }-{k}_0\right)\frac{\partial {h}_i^C}{\partial \omega}\left|{}_{\omega =0}\right.-{h}_i^S\left|{}_{\omega =0}\right.i{\displaystyle {\int}_0^{+\infty }s\dot{k}(s)ds} $$

(167)

and, therefore, these sensitivities depend upon the sensitivities of the constitutive functions h ^S _i and h ^C _i , respectively. To evaluate these quantities, let us assume that the functions f, l and h are analytic with respect to E₁. By expanding h in Taylor series, as in Eq. (156), and by projecting over sin(iωt) and cos(iωt), the Fourier coefficients h ^C _i and h ^S _i are obtained, e.g.,

$$ {h}_i^S={\displaystyle \sum_{p=0}^{\infty }{\displaystyle \sum_{q=0}^p\frac{\omega^{p-q}{E}_1^p}{q!\left(p-q\right)!}}}{h}^{\left(q,p-q\right)}\left({E}_0\right){{\displaystyle {\int}_{-1}^1 \sin \left(\pi s\right)}}^q \cos {\left(\pi s\right)}^{p-q} \sin \left( i\pi s\right) ds $$

(168)

$$ {h}_i^C={\displaystyle \sum_{p=0}^{\infty }{\displaystyle \sum_{q=0}^p\frac{\omega^{p-q}{E}_1^p}{q!\left(p-q\right)!}}}{h}^{\left(q,p-q\right)}\left({E}_0\right){{\displaystyle {\int}_{-1}^1 \sin \left(\pi s\right)}}^q \cos {\left(\pi s\right)}^{p-q} \cos \left( i\pi s\right) ds $$

(169)

When assessing h ^C _i and h ^S _i at ω → 0, the only term not vanishing in the infinite summations (168) and (169) are those corresponding to n = m, that is

$$h_i^S\left| {_{\omega = 0}} \right. = \mathop \sum \limits_{p = 0}^\infty \frac{{E_1^p}}{{p!}}{h^{\left( {p{\rm{,}}0} \right)}}\left( {{E_0}} \right)\mathop \smallint \limits_{ - 1}^1 \sin {\left( {\pi s} \right)^p}\sin \left( {i\pi s} \right)ds\left\{ {\begin{array}{*{20}{c}} { = 0,}&{iEven}\\ { \ne 0,}&{iOdd} \end{array}} \right.$$

(170)

$$h_i^C\left| {_{\omega = 0}} \right. = \mathop \sum \limits_{p = 0}^\infty \frac{{E_1^p}}{{p!}}{h^{\left( {p{\rm{,}}0} \right)}}\left( {{E_0}} \right)\mathop \smallint \limits_{ - 1}^1 \sin {\left( {\pi s} \right)^p}\cos \left( {i\pi s} \right)ds\left\{ {\begin{array}{*{20}{c}} { = 0,}&{iOdd}\\ { \ne 0,}&{iEven} \end{array}} \right.$$

(171)

with the corresponding derivatives given by

$$\frac{{\partial h_i^S}}{{\partial \omega }}\left| {_{\omega = 0}} \right. = \mathop \sum \limits_{p = 0}^\infty \frac{{E_1^p}}{{\left( {p - 1} \right)!}}{h^{^{\left( {p - 1,1} \right)}}}\left( {{E_0}} \right)\mathop \smallint \limits_{ - 1}^1 \sin {\left( {\pi s} \right)^{p - 1}}\cos \left( {\pi s} \right)\sin \left( {i\pi s} \right)ds\left\{ {\begin{array}{*{20}{c}} { = 0,}&{iOdd}\\ { \ne 0,}&{iEven} \end{array}} \right.$$

(172)

$$\frac{{\partial h_i^C}}{{\partial \omega }}\left| {_{\omega = 0}} \right. = \mathop \sum \limits_{p = 0}^\infty \frac{{E_1^p}}{{\left( {p - 1} \right)!}}{h^{^{\left( {p - 1,1} \right)}}}\left( {{E_0}} \right)\mathop \smallint \limits_{ - 1}^1 \sin {\left( {\pi s} \right)^{p - 1}}\cos \left( {\pi s} \right)\sin \left( {i\pi s} \right)ds\left\{ {\begin{array}{*{20}{c}} { = 0,}&{iEven}\\ { \ne 0,}&{iOdd} \end{array}} \right.$$

(173)

As a result, the sensitivities ∂H ^S _i /∂ω and ∂H ^S _i /∂ω can be assessed at ω = 0, i.e.

$$\frac{{\partial H_i^S}}{{\partial \omega }}\left| {_{\omega = 0}} \right. = \left\{ {\begin{array}{*{20}{c}} { = 0,}&{iOdd}\\ { \ne 0,}&{iEven} \end{array}} \right.$$

(174)

$$\frac{{\partial H_i^C}}{{\partial \omega }}\left| {_{\omega = 0}} \right. = \left\{ {\begin{array}{*{20}{c}} { = 0,}&{iEven}\\ { \ne 0,}&{iOdd} \end{array}} \right.$$

(175)

Equations (168)–(173) allow the sensitivities of f ^S _i , f ^C _i , l ^S _i and l ^C _i to be estimated and, consequently, Eq. (161) to be recovered.