Linear rheology of compressible soft nanocomposites

Abstract

The influences of interfacial tension and compressibility to the linear viscoelastic properties of nanocomposite and nanoporous materials are considered theoretically. The effective bulk and shear moduli of the systems are calculated within the generalized composite sphere model which takes into account the effect of interfacial tension. It is found that frequency dependence of the effective dynamic shear and bulk moduli of nanocomposites with the compressible elastic matrix and viscous inclusions may be represented in terms of the Zener model comprising of the viscoelastic Kelvin element in series with the elastic spring. The relations of the Zener model parameters with the material characteristics are revealed. The physical interpretation of the frequency behavior of the dynamic shear and bulk moduli against the interfacial tension, component compressibility, viscosity, and inclusion volume fraction is discussed.

Appendix

The components of the stress tensor can be obtained by substitution of Eqs. 1, 19, and 20 to the relationship $\sigma _{kl} = C_{klpq} \varepsilon _{pq}$. In the arbitrary point of the shell, they can be represented as follows:

$$\begin{array}{*{20}l} {\sigma _{rr}^{\left( m \right)} = 2G_m \left( {2A_1^{\left( m \right)} - A_2^{\left( m \right)} r^2 + 4\frac{{v_m - 5}}{{5 - 4v_m }}\frac{{A_3^{\left( m \right)} }}{{r^3 }} - 8\frac{{A_4^{\left( m \right)} }}{{r^5 }}} \right)\left( {\cos ^2 \theta - \frac{1}{2}\sin ^2 \theta } \right)} \hfill \\ {\sigma _{r\theta }^{\left( m \right)} = 2G_m \left( { - 3A_1^{\left( m \right)} - \frac{{2v_m + 7}}{{2v_m }}A_2^{\left( m \right)} r^2 - 6\frac{{1 + v_m }}{{5 - 4v_m }}\frac{{A_3^{\left( m \right)} }}{{r^3 }} - 8\frac{{A_4^{\left( m \right)} }}{{r^5 }}} \right)\sin \theta \cos \theta } \hfill \\ \end{array} .$$

(32)

In the inclusion, the stresses are:

$$\begin{array}{*{20}l} {\sigma _{rr}^{\left( i \right)} = 2\overline G _i \left( {2A_1^{\left( i \right)} - A_2^{\left( i \right)} r^2 } \right)\left( {\cos ^2 \theta - \frac{1}{2}\sin ^2 \theta } \right)} \hfill \\ {\sigma _{r\theta }^{\left( i \right)} = 2\overline G _i \left( { - 3A_1^{\left( i \right)} - \frac{{2\overline v _i + 7}}{{2\overline v _i }}A_2^{\left( i \right)} r^2 } \right)\sin \theta \cos \theta } \hfill \\ \end{array} .$$

(33)

Substitution of Eqs. 14, 32, and 33 to the boundary conditions 12 and 13 gives the following set of equations for the unknown coefficients $A_p^{\left( m \right)}$ and $A_p^{\left( i \right)}$:

$$\begin{array}{*{20}l} {A_1^{\left( m \right)} R + A_2^{\left( m \right)} R^3 + \frac{{A_3^{\left( m \right)} }}{{R^2 }} + \frac{{A_4^{\left( m \right)} }}{{R^4 }} = R} \hfill \\ { - 3A_1^{\left( m \right)} R + A_2^{\left( m \right)} \left( {2 - \frac{7}{{2v_m }}} \right)R^3 + 6\frac{{A_3^{\left( m \right)} }}{{R^2 }}\frac{{2v_m - 1}}{{5 - 4v_m }} + 2\frac{{A_4^{\left( m \right)} }}{{R^4 }} = - 3R} \hfill \\ {A_1^{\left( m \right)} a + A_2^{\left( m \right)} a^3 + \frac{{A_3^{\left( m \right)} }}{{a^2 }} + \frac{{A_4^{\left( m \right)} }}{{a^4 }} - A_1^{\left( i \right)} a - A_2^{\left( i \right)} a^3 = 0} \hfill \\ { - 3A_1^{\left( m \right)} a + A_2^{\left( m \right)} \left( {2 - \frac{7}{{2v_m }}} \right)a^3 + 6\frac{{A_3^{\left( m \right)} }}{{a^2 }}\frac{{2v_m - 1}}{{5 - 4v_m }} + 2\frac{{A_4^{\left( m \right)} }}{{a^4 }}} \hfill \\ { + 3A_1^{\left( i \right)} a - \left( {2 - \frac{7}{{2\bar v_i }}} \right)A_2^{\left( i \right)} a^3 = 0} \hfill \\ {G_m \left( {2A_1^{\left( m \right)} a - A_2^{\left( m \right)} a^3 + 4\frac{{A_3^{\left( m \right)} }}{{a^2 }}\frac{{v_m - 5}}{{5 - 4v_m }} - 8\frac{{A_4^{\left( m \right)} }}{{a^4 }}} \right) - \overline G _i \left( {2A_1^{\left( i \right)} a - A_2^{\left( i \right)} a^3 } \right) = 0} \hfill \\ {G_m \left[ { - 6A_1^{\left( m \right)} a - A_2^{\left( m \right)} \left( {2 + \frac{7}{{v_m }}} \right)a^3 - 12\frac{{A_3^{\left( m \right)} }}{{a^2 }}\frac{{1 + v_m }}{{5 - 4v_m }} - 16\frac{{A_4^{\left( m \right)} }}{{a^4 }}} \right]} \hfill \\ { + \overline G _i \left[ {6A_1^{\left( i \right)} a + \left( {2 + \frac{7}{{v_i }}} \right)A_2^{\left( i \right)} a^3 } \right] = 0} \hfill \\ \end{array} .$$

(34)

Further calculation of the effective shear modulus with help of Eq. 17 requires only three coefficients, $A_1^{\left( m \right)}$, $A_2^{\left( m \right)}$, and $A_3^{\left( m \right)}$. The solution of Eq. 34 gives:

$$\begin{aligned}&A_1^{\left( m \right)} = \left\{ {\overline G _i^2 \left( {7 + 5\overline v _i } \right)\left[ {2\left( {10v_m - 7} \right)\left( {10v_m - 8} \right)R^7 + 63a^5 R^2 - 25\left( {7 - 12v_m + 8v_m^2 } \right)a^7 } \right]} \right. \\&+ \overline G _i G_m \left[ {2\left( {10v_m - 7} \right)\left( {\left( {5v_m - 7} \right)\left( {7 + 5\overline v _i } \right) + 4\left( {10v_m - 8} \right)\left( {7 - 10\overline v _i } \right)} \right)R^7 } \right. \\&\left. { + 189\left( {7 - 15\overline v _i } \right)a^5 R^2 + 25\left( {49\overline v _i - 49 + 66\overline v _i v_m + 42v_m - 56v_m^2 + 20\overline v _i v_m^2 } \right)a^7 } \right] \\&\left. { + 2G_m^2 \left( {7 - 10\overline v _i } \right)\left[ {4\left( {10v_m - 7} \right)\left( {5v_m - 7} \right)R^7 - 126a_k^5 R_k^2 - 25v_m \left( {v_m^2 - 7} \right)a^7 } \right]} \right\}R^3 D^{\left( m \right)} \\ \end{aligned} $$

(35)

$$A_2^{\left( m \right)} = - 90\left[ {\overline G _i^2 \left( {7 + 5\overline v _i } \right) + 3\overline G _i G_m \left( {7 - 15\overline v _i } \right) - 4G_m^2 \left( {7 - 10\overline v _i } \right)} \right]\left( {R^2 - a^2 } \right)R^3 a^3 \overline v _i D^{\left( m \right)} $$

(36)

$$\begin{aligned}&A_3^{\left( m \right)} = - 5\left( {4v_m - 5} \right)\left\{ {\overline G _i^2 \left( {7 + 5\overline v _i } \right)\left( {10v_m - 7} \right)\left( {R^7 - a^7 } \right)} \right. \\&+ \overline G _i G_m \left[ {3\left( {10v_m - 7} \right)\left( {7 - 15\overline v _i } \right)R^7 + \left( {100\overline v _i v_m + 35\overline v _i + 35v_m - 98} \right)a^7 } \right] \\&\left. { + G_m^2 \left( {10\overline v _i - 7} \right)\left[ {4\left( {10v_m - 7} \right)R^7 - \left( {7 + 5v_m } \right)a^7 } \right]} \right\}R^3 a^3 D^{\left( m \right)} \\ \end{aligned} $$

(37)

where $D^{\left( m \right)}$ satisfies the following expression:

$$\begin{aligned}&D^{\left( m \right)} = \left\{ {\overline G _i^2 \left( {7 + 5\overline v _i } \right)\left[ {2\left( {10v_m - 7} \right)\left( {10v_m - 8} \right)R^{10} } \right.} \right. \\&+ 126\left( {7 + 5\overline v _i } \right)a^5 R^5 - 25\left( {7 - 12v_m + 8v_m^2 } \right)a^3 R^7 - 25\left( {7 - 12v_m + 8v_m^2 } \right)a^7 R^3 \\&\left. { + 4\left( {5v_m - 4} \right)\left( {10v_m - 7} \right)a^{10} } \right] \\&+ \overline G _i G_m \left[ { - 6\left( {10v_m - 7} \right)\left( {\left( {5v_m - 7} \right)\left( {7 + 5\overline v _i } \right) + 4\left( {10v_m - 8} \right)\left( {7 - 10\overline v _i } \right)} \right)R^{10} } \right. \\&- 75\left( {7 - 15\overline v _i } \right)\left( {7 - 12v_m + 8v_m^2 } \right)a^3 R^7 + 378\left( {7 - 15\overline v _i } \right)a^5 R^5 \\&+ 25\left( {49\overline v _i - 49 + 66\overline v _i v_m + 42v_m - 56v_m^2 + 20\overline v _i v_m^2 } \right)a^7 R^3 \\&\left. { - \left( {5v_m - 4} \right)\left( {100\overline v _i v_m + 35\overline v _i + 35v_m - 98} \right)a^{10} } \right] \\&+ 2G_m^2 \left( {7 - 10\overline v _i } \right)\left[ {4\left( {10v_m - 7} \right)\left( {5v_m - 7} \right)R^{10} + 50\left( {7 - 12v_m + 8v_m^2 } \right)a^3 R^7 } \right. \\&\left. { - 252a^5 R^5 - 25\left( {v_m^2 - 7} \right)a^7 R^3 - 2\left( {5v_m - 4} \right)\left( {7 + 5v_m } \right)a^{10} } \right\}^{ - 1} \\ \end{aligned} $$

(38)

Substituting Eqs. 35, 36, and 37 to Eq. 23 we arrive at Eq. 24 with the following notations:

$$N\left( {G_m ,\overline G _i ,v_m ,\overline v _i ,\Phi } \right) = (1 - \overline v _i )\left( {N_1 \Phi ^{{7 \mathord{\left/ {\vphantom {7 3}} \right. \kern-\nulldelimiterspace} 3}} + N_2 } \right)$$

(39)

$$N_1 = \left( {50\overline v _i v_m - 49} \right)\left( \,{\overline G _i - G_m } \right) + 70\left(\, {\overline G _i v_m - G_m \overline v _i } \right) + 35\left( {G_m v_m - \overline G _i \overline v _i } \right)$$

(40)

$$N_2 = \left( {7 - 10v_m } \right)\left[ \,{\overline G _i \left( {5\overline v _i + 7} \right) + 4G_m \left( {7 - 10\overline v _i } \right)} \right]$$

(41)

$$D\left( {G_m ,\bar G_k ,v_m ,\bar v_i ,\Phi } \right) = 4\left( {\bar G_k - G_m } \right)\left( {5v_m - 4} \right)N_1 \Phi ^{{{10} \mathord{\left/ {\vphantom {{10} 3}} \right. \kern-\nulldelimiterspace} 3}} + D_2 \Phi ^{{7 \mathord{\left/ {\vphantom {7 3}} \right. \kern-\nulldelimiterspace} 3}} + D_3 \Phi ^{{5 \mathord{\left/ {\vphantom {5 3}} \right. \kern-\nulldelimiterspace} 3}} + D_4 \Phi + D_5 $$

(42)

$$D_2 = 25\left\{ { - \left( {7 + 8v_m^2 - 12v_m } \right)\left( {7 + 5\bar v_i } \right)\bar G_i^2 + \left( {\left( {20v_m^2 + 66v_m - 49} \right)\bar v_i - 56v_m^2 + 42v_m - 49} \right)\bar G_i G_m + 2\left( {7 - v_m^2 } \right)\left( {7 - 10\bar v_i } \right)G_m^2 } \right.$$

(43)

$$D_3 = 126\left( {\bar G_i - G_m } \right)\left[ {\bar G_i \left( {5\bar v_i + 7} \right) + 4G_m \left( {7 - 10\bar v_i } \right)} \right]$$

(44)

$$D_4 = - 25\left( {7 - 12v_m + 8v_m^2 } \right)\left( {\bar G_i - G_m } \right)\left[ {\bar G_i \left( {5\bar v_i + 7} \right) + 4G_m \left( {7 - 10\bar v_i } \right)} \right]$$

(45)

$$D_5 = 2\left( {7 - 10v_m } \right)\left[ {2\bar G_i \left( {4 - 5v_m } \right) + G_m \left( {7 - 5v_m } \right)} \right]\left[ {\bar G_i \left( {7 + 5\bar v_i } \right) + 4G_m \left( {7 - 10\bar v_i } \right)} \right]$$

(46)