Establishment and simplification of micromechanical material model for viscoelastic woven fabric/hybrid composite

Abstract

The present research focuses on proposing a novel theoretical micromechanical model (TMM) designed to derive the frequency-dependent storage and loss moduli of woven fabric (WF)-matrix composites, as well as WF-particulate matrix (Hybrid) composites, based on their constituent properties. The TMM serves as a higher-order modulus operator, accounting for the composite woven fabric unit cell geometry and the effective modulus of both the fabric and matrix using equivalent modulus theory. This model also incorporates viscoelastic parameters obtained from literature and experiments for each constituent, namely woven glass fabric and SiC particles embedded in an epoxy matrix. The proposed TMM is validated by comparing its predictions of the frequency-dependent storage modulus and loss factor with experimental data acquired through dynamic mechanical analyzer tests on samples with varying fiber and particulate volume fractions. To address the inherent complexities of the higher-order modulus operator, the model is streamlined into a lower-order form expressed as a function of two separate variables: volume fraction and a differential time operator. This advancement enhances the applicability and usability of the model for predicting the mechanical behaviour of these complex composite materials. This novel mathematical model eliminates the cost and time for conducting the explicit experiments as well as can be applied to different range of similar hybrid composites considering the fact that the constituent properties are known.

Appendices

Appendix 1

$$E_{eq}^{WFC} \left( {} \right) = \frac{{p_{0}^{WFC} + p_{1}^{WFC} D + p_{2}^{WFC} D^{2} + p_{3}^{WFC} D^{3} + p_{4}^{WFC} D^{4} }}{{q_{0}^{WFC} + q_{1}^{WFC} D + q_{2}^{WFC} D^{2} + q_{3}^{WFC} D^{3} }}$$

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where,

$$p_{0}^{WFC} = c_{1} a_{0}^{l} b_{0}^{s} b_{0}^{m} + c_{2} a_{0}^{s} b_{0}^{l} b_{0}^{m} + c_{3} a_{0}^{m} b_{0}^{s} b_{0}^{l}$$

$$\begin{aligned} p_{1}^{WFC} = & \;\;c_{1} \left( {a_{1}^{l} b_{0}^{s} b_{0}^{m} + a_{0}^{l} b_{1}^{s} b_{0}^{m} + a_{0}^{l} b_{0}^{s} b_{1}^{m} } \right) \\ & \;\; + c_{2} \left( {a_{1}^{s} b_{0}^{l} b_{0}^{m} + a_{0}^{s} b_{1}^{l} b_{0}^{m} + a_{0}^{s} b_{0}^{l} b_{1}^{m} } \right) \\ & \;\; + c_{3} \left( {a_{1}^{m} b_{0}^{s} b_{0}^{l} + a_{0}^{m} b_{1}^{s} b_{0}^{l} + a_{0}^{m} b_{0}^{s} b_{1}^{l} } \right) \\ \end{aligned}$$

$$\begin{aligned} p_{2}^{WFC} = & \;\;c_{1} \left( {a_{2}^{l} b_{0}^{s} b_{0}^{m} + a_{1}^{l} b_{1}^{s} b_{0}^{m} + a_{1}^{l} b_{0}^{s} b_{1}^{m} + a_{0}^{l} b_{1}^{s} b_{1}^{m} } \right) \\ & \;\; + c_{2} \left( {a_{2}^{s} b_{0}^{l} b_{0}^{m} + a_{1}^{s} b_{1}^{l} b_{0}^{m} + a_{1}^{s} b_{0}^{l} b_{1}^{m} + a_{0}^{s} b_{1}^{l} b_{1}^{m} } \right) \\ & \;\; + c_{3} \left( {a_{2}^{m} b_{0}^{s} b_{0}^{l} + a_{1}^{m} b_{1}^{s} b_{0}^{l} + a_{1}^{m} b_{0}^{s} b_{1}^{l} + a_{0}^{m} b_{1}^{s} b_{1}^{l} } \right) \\ \end{aligned}$$

$$\begin{aligned} p_{3}^{WFC} = & \;\;c_{1} \left( {a_{1}^{l} b_{1}^{s} b_{1}^{m} + a_{2}^{l} b_{1}^{s} b_{0}^{m} + a_{2}^{l} b_{0}^{s} b_{1}^{m} } \right) \\ & \;\; + c_{2} \left( {a_{1}^{s} b_{1}^{l} b_{1}^{m} + a_{2}^{s} b_{1}^{l} b_{0}^{m} + a_{2}^{s} b_{0}^{l} b_{1}^{m} } \right) \\ & \;\; + c_{3} \left( {a_{1}^{m} b_{1}^{s} b_{1}^{l} + a_{2}^{m} b_{1}^{s} b_{0}^{l} + a_{2}^{m} b_{0}^{s} b_{1}^{l} } \right) \\ \end{aligned}$$

$$p_{4}^{WFC} = c_{1} a_{2}^{l} b_{1}^{s} b_{1}^{m} + c_{2} a_{2}^{s} b_{1}^{l} b_{1}^{m} + c_{3} a_{2}^{m} b_{1}^{s} b_{1}^{l}$$

$$p_{5}^{WFC} = b_{0}^{l} b_{0}^{s} b_{0}^{m}$$

$$q_{0}^{WFC} = b_{0}^{l} b_{0}^{s} b_{0}^{m}$$

$$q_{1}^{WFC} = b_{0}^{l} b_{0}^{s} b_{1}^{m} + b_{0}^{l} b_{1}^{s} b_{0}^{m} + b_{1}^{l} b_{0}^{s} b_{0}^{m}$$

$$q_{2}^{WFC} = b_{0}^{l} b_{1}^{s} b_{1}^{m} + b_{1}^{l} b_{0}^{s} b_{1}^{m} + b_{1}^{l} b_{1}^{s} b_{0}^{m}$$

$$q_{3}^{WFC} = b_{1}^{l} b_{1}^{s} b_{1}^{m}$$

$$c_{1} = \frac{{\left( {A_{w} + A_{f} } \right)\lambda_{1} }}{{A_{c} }}, \, c_{2} = \frac{{\left( {A_{w} + A_{f} } \right)\lambda_{2} }}{{A_{c} }}, \, c_{3} = \frac{{A_{m} }}{{A_{c} }}$$

Appendix 2

$$\begin{aligned} E_{eq}^{HyC} \left( {} \right) = & \;\;\frac{{\lambda_{1} \left( {A_{w} + A_{f} } \right)}}{A}\left( {\frac{{a_{0}^{l} + a_{1}^{l} D + a_{2}^{l} D^{2} }}{{b_{0}^{l} + b_{1}^{l} D}}} \right) \\ & \;\; + \frac{{\lambda_{2} \left( {A_{w} + A_{f} } \right)}}{A}\left( {\frac{{a_{0}^{s} + a_{1}^{s} D + a_{2}^{s} D^{2} }}{{b_{0}^{s} + b_{1}^{s} D}}} \right) \\ & \;\; + \frac{{A_{m} }}{A}\left( {\frac{{a_{{_{0} }}^{em} + a_{{_{1} }}^{em} D + a_{{_{2} }}^{em} D^{2} + a_{{_{3} }}^{em} D^{3} + a_{{_{4} }}^{em} D^{4} }}{{b_{{_{0} }}^{em} + b_{{_{1} }}^{em} D + b_{{_{2} }}^{em} D^{2} + b_{{_{3} }}^{em} D^{3} }}} \right) \\ \end{aligned}$$

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On solving and simplifying,

$$E_{eq}^{HyC} \left( {} \right) = \frac{{p_{0}^{HyC} + p_{1}^{HyC} D + p_{2}^{HyC} D^{2} + p_{3}^{HyC} D^{3} + p_{4}^{HyC} D^{4} + p_{5}^{HyC} D^{5} + p_{6}^{HyC} D^{6} + p_{7}^{HyC} D^{7} }}{{q_{0}^{HyC} + q_{1}^{HyC} D + q_{2}^{HyC} D^{2} + q_{3}^{HyC} D^{3} + q_{4}^{HyC} D^{4} + q_{5}^{HyC} D^{5} + q_{6}^{HyC} D^{6} }}$$

(41)

where,

$$p_{0}^{HyC} = c_{1} a_{0}^{l} b_{0}^{s} b_{{_{0} }}^{em} + c_{2} a_{0}^{s} b_{0}^{l} b_{{_{0} }}^{em} + c_{3} b_{0}^{s} b_{0}^{l} b_{{_{0} }}^{em}$$

$$\begin{aligned} p_{1}^{HyC} = & \;\;c_{1} \left( {a_{0}^{l} b_{0}^{s} b_{{_{1} }}^{em} + a_{1}^{l} b_{0}^{s} b_{{_{0} }}^{em} + a_{0}^{l} b_{1}^{s} b_{{_{0} }}^{em} } \right) \\ & \;\; + c_{2} \left( {a_{1}^{s} b_{0}^{l} b_{{_{0} }}^{em} + a_{0}^{s} b_{1}^{l} b_{{_{0} }}^{em} + a_{0}^{s} b_{0}^{l} b_{{_{0} }}^{em} } \right) \\ & \;\; + c_{3} \left( {b_{1}^{s} b_{0}^{l} a_{{_{0} }}^{em} + b_{0}^{s} b_{1}^{l} a_{{_{0} }}^{em} + b_{0}^{s} b_{0}^{l} a_{{_{1} }}^{em} } \right) \\ \end{aligned}$$

$$\begin{aligned} p_{2}^{HyC} = & \;\;c_{1} \left( {a_{0}^{l} b_{0}^{s} b_{2}^{em} + a_{1}^{l} b_{0}^{s} b_{1}^{em} + a_{0}^{l} b_{1}^{s} b_{1}^{em} + a_{1}^{l} b_{1}^{s} b_{0}^{em} + a_{2}^{l} b_{0}^{s} b_{0}^{em} } \right) \\ & \;\; + c_{2} \left( {a_{0}^{s} b_{0}^{l} b_{2}^{em} + a_{1}^{s} b_{0}^{l} b_{2}^{em} + a_{0}^{s} b_{1}^{l} b_{1}^{em} + a_{1}^{s} b_{1}^{l} b_{0}^{em} + a_{2}^{s} b_{0}^{l} b_{0}^{em} } \right) \\ & \;\; + c_{3} \left( {b_{1}^{s} b_{0}^{l} a_{1}^{em} + b_{0}^{s} b_{1}^{l} a_{1}^{em} + b_{0}^{s} b_{0}^{l} a_{2}^{em} + b_{1}^{s} b_{1}^{l} a_{0}^{em} } \right) \\ \end{aligned}$$

$$\begin{aligned} p_{3}^{HyC} = & \;\;c_{1} \left( {a_{0}^{l} b_{0}^{s} b_{3}^{em} + a_{1}^{l} b_{0}^{s} b_{2}^{em} + a_{0}^{l} b_{1}^{s} b_{2}^{em} + a_{1}^{l} b_{1}^{s} b_{1}^{em} + a_{2}^{l} b_{0}^{s} b_{1}^{em} } \right) \\ & \;\; + c_{2} \left( {a_{0}^{s} b_{0}^{l} b_{3}^{em} + a_{1}^{s} b_{0}^{l} b_{2}^{em} + a_{0}^{s} b_{1}^{l} b_{2}^{em} + a_{1}^{s} b_{1}^{l} b_{1}^{em} + a_{2}^{s} b_{0}^{l} b_{1}^{em} } \right) \\ & \;\; + c_{3} \left( {b_{1}^{s} b_{0}^{l} a_{2}^{em} + b_{0}^{s} b_{1}^{l} a_{2}^{em} + b_{0}^{s} b_{0}^{l} a_{3}^{em} + b_{1}^{s} b_{1}^{l} a_{1}^{em} } \right) \\ \end{aligned}$$

$$\begin{aligned} p_{4}^{HyC} = & \;\;c_{1} \left( {a_{0}^{l} b_{0}^{s} b_{4}^{em} + a_{1}^{l} b_{0}^{s} b_{3}^{em} + a_{0}^{l} b_{1}^{s} b_{3}^{em} + a_{1}^{l} b_{1}^{s} b_{2}^{em} + a_{2}^{l} b_{0}^{s} b_{2}^{em} + a_{2}^{l} b_{1}^{s} b_{1}^{em} } \right) \\ & \;\; + c_{2} \left( {a_{0}^{s} b_{0}^{l} b_{4}^{em} + a_{1}^{s} b_{0}^{l} b_{3}^{em} + a_{0}^{s} b_{1}^{l} b_{3}^{em} + a_{1}^{s} b_{1}^{l} b_{2}^{em} + a_{2}^{s} b_{0}^{l} b_{2}^{em} + a_{2}^{s} b_{1}^{l} b_{1}^{em} } \right) \\ & \;\; + c_{3} \left( {b_{0}^{s} b_{0}^{l} a_{4}^{em} + b_{1}^{s} b_{0}^{l} a_{3}^{em} + b_{0}^{s} b_{1}^{l} a_{3}^{em} + b_{1}^{s} b_{1}^{l} a_{2}^{em} } \right) \\ \end{aligned}$$

$$\begin{aligned} p_{5}^{HyC} = & \;\;c_{1} \left( {a_{1}^{l} b_{0}^{s} b_{4}^{em} + a_{0}^{l} b_{1}^{s} b_{4}^{em} + a_{1}^{l} b_{1}^{s} b_{3}^{em} + a_{2}^{l} b_{0}^{s} b_{3}^{em} + a_{2}^{l} b_{1}^{s} b_{2}^{em} } \right) \\ & \;\; + c_{2} \left( {a_{1}^{s} b_{0}^{l} b_{4}^{em} + a_{0}^{s} b_{1}^{l} b_{4}^{em} + a_{1}^{s} b_{1}^{l} b_{3}^{em} + a_{2}^{s} b_{0}^{l} b_{3}^{em} + a_{2}^{s} b_{1}^{l} b_{2}^{em} } \right) \\ & \;\; + c_{3} \left( {b_{0}^{s} b_{0}^{l} a_{5}^{em} + b_{1}^{s} b_{0}^{l} a_{4}^{em} + b_{0}^{s} b_{1}^{l} a_{4}^{em} + b_{1}^{s} b_{1}^{l} a_{3}^{em} } \right) \\ \end{aligned}$$

$$\begin{aligned} p_{6}^{HyC} = & \;\;c_{1} \left( {a_{1}^{l} b_{1}^{s} b_{4}^{em} + a_{2}^{l} b_{0}^{s} b_{4}^{em} + a_{2}^{l} b_{1}^{s} b_{3}^{em} } \right) \\ & \;\; + c_{2} \left( {a_{1}^{s} b_{1}^{l} b_{4}^{em} + a_{2}^{s} b_{0}^{l} b_{4}^{em} + a_{2}^{s} b_{1}^{l} b_{3}^{em} } \right) \\ & \;\; + c_{3} \left( {b_{1}^{s} b_{0}^{l} a_{5}^{em} + b_{0}^{s} b_{1}^{l} a_{5}^{em} + b_{1}^{s} b_{1}^{l} a_{4}^{em} } \right) \\ \end{aligned}$$

$$p_{7}^{HyC} = c_{1} a_{2}^{l} b_{1}^{s} b_{4}^{em} + c_{2} a_{2}^{s} b_{1}^{l} b_{4}^{em} + c_{3} b_{1}^{s} b_{1}^{l} a_{5}^{em}$$

$$q_{0}^{HyC} = b_{0}^{l} b_{0}^{s} b_{0}^{em}$$

$$q_{1}^{HyC} = b_{0}^{l} b_{0}^{s} b_{1}^{em} + b_{0}^{l} b_{1}^{s} b_{0}^{em} + b_{1}^{l} b_{0}^{s} b_{0}^{em}$$

$$q_{2}^{HyC} = b_{0}^{l} b_{0}^{s} b_{2}^{em} + b_{0}^{l} b_{1}^{s} b_{1}^{em} + b_{1}^{l} b_{0}^{s} b_{1}^{em} + b_{1}^{l} b_{1}^{s} b_{0}^{em}$$

$$q_{3}^{HyC} = b_{0}^{l} b_{0}^{s} b_{3}^{em} + b_{0}^{l} b_{1}^{s} b_{2}^{em} + b_{1}^{l} b_{0}^{s} b_{2}^{em} + b_{1}^{l} b_{1}^{s} b_{1}^{em}$$

$$q_{4}^{HyC} = b_{0}^{l} b_{0}^{s} b_{4}^{em} + b_{0}^{l} b_{1}^{s} b_{3}^{em} + b_{1}^{l} b_{0}^{s} b_{3}^{em} + b_{1}^{l} b_{1}^{s} b_{2}^{em}$$

$$q_{5}^{HyC} = b_{0}^{l} b_{1}^{s} b_{4}^{em} + b_{1}^{l} b_{0}^{s} b_{4}^{em} + b_{1}^{l} b_{1}^{s} b_{3}^{em}$$

$$q_{6}^{HyC} = b_{1}^{l} b_{1}^{s} b_{4}^{em}$$

Appendix 3

$$E_{{eq}}^{{WFC^{\prime }}} \left( \omega \right) = \frac{{\left( {p_{0} - p_{2} \omega ^{2} + p_{4} \omega ^{4} } \right)\left( {q_{0} - q_{2} \omega ^{2} } \right) - \left( {p_{1} \omega - p_{3} \omega ^{3} } \right)\left( {q_{1} \omega - q_{3} \omega ^{3} } \right)}}{{\left( {q_{0} - q_{2} \omega ^{2} } \right)^{2} + \left( {q_{1} \omega - q_{3} \omega ^{3} } \right)^{2} }}$$

(42)

$$E_{{eq}}^{{WFC^{\prime\prime}}} \left( \omega \right) = \frac{{\left( {p_{1} \omega - p_{3} \omega ^{3} } \right)\left( {q_{0} - q_{2} \omega ^{2} } \right) - \left( {p_{0} - p_{2} \omega ^{2} + p_{4} \omega ^{4} } \right)\left( {q_{1} \omega - q_{3} \omega ^{3} } \right)}}{{\left( {q_{0} - q_{2} \omega ^{2} } \right)^{2} + \left( {q_{1} \omega - q_{3} \omega ^{3} } \right)^{2} }}$$

(43)

$$\eta _{{eq}}^{{WFC}} = \frac{{E_{{eq}}^{{WFC^{\prime\prime }}} \left( \omega \right)}}{{E_{{eq}}^{{WFC^{\prime }}} \left( \omega \right)}}$$

(44)

$$E_{{eq}}^{{HyC^{\prime }}} \left( \omega \right) = \frac{{\left( {p_{0} - p_{2} \omega ^{2} + p_{4} \omega ^{4} - p_{6} \omega ^{6} } \right)\left( {q_{0} - q_{2} \omega ^{2} + q_{4} \omega ^{4} - q_{6} \omega ^{6} } \right) + \left( {p_{1} \omega - p_{3} \omega ^{3} + p_{5} \omega ^{5} - p_{7} \omega ^{7} } \right)\left( {q_{1} \omega - q_{3} \omega ^{3} + q_{5} \omega ^{5} } \right)}}{{\left( {q_{0} - q_{2} \omega ^{2} + q_{4} \omega ^{4} - q_{6} \omega ^{6} } \right)^{2} + \left( {q_{1} \omega - q_{3} \omega ^{3} + q_{5} \omega ^{5} } \right)^{2} }}$$

(45)

$$E_{{eq}}^{{HyC^{\prime\prime }}} \left( \omega \right) = \frac{{\left( {p_{1} \omega - p_{3} \omega ^{3} + p_{5} \omega ^{5} - p_{7} \omega ^{7} } \right)\left( {q_{0} - q_{2} \omega ^{2} + q_{4} \omega ^{4} - q_{6} \omega ^{6} } \right) - \left( {p_{0} - p_{2} \omega ^{2} + p_{4} \omega ^{4} - p_{6} \omega ^{6} } \right)\left( {q_{1} \omega - q_{3} \omega ^{3} + q_{5} \omega ^{5} } \right)}}{{\left( {q_{0} - q_{2} \omega ^{2} + q_{4} \omega ^{4} - q_{6} \omega ^{6} } \right)^{2} + \left( {q_{1} \omega - q_{3} \omega ^{3} + q_{5} \omega ^{5} } \right)^{2} }}$$

(46)

$$\eta _{{eq}}^{{HyC}} = \frac{{E_{{eq}}^{{HyC^{\prime\prime }}} \left( \omega \right)}}{{E_{{eq}}^{{HyC^{\prime }}} \left( \omega \right)}}$$

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