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A hybrid computational scheme on electromechanically coupled behaviors of aligned MWCNT/polymer nanocomposite sensors with strain-dependent tunneling effect

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Abstract

The sensitivity of randomly distributed multi-walled carbon nanotube (MWCNT)/polymer nanocomposite sensors has been extensively investigated, while that of aligned nanocomposite sensors is to be explored. In this paper, a novel hybrid computational scheme is presented for the transversely isotropic sensing behaviors of aligned MWCNT/polymer nanocomposite sensors combining micromechanics and finite element simulations. Specifically, the strain-dependent tunneling effect and multi-scale simulation of underlying heterogeneous microstructures are analyzed cooperatively to quantitatively illustrate the electromechanical coupling phenomenon of strain sensors. First, the effective elastic and electric properties of coated MWCNT with the weak interface connection are calculated by the Mori–Tanaka method on the microscopic scale. Then, the mesoscopic representative volume element (RVE) is established by coated MWCNTs as inclusions and polymer as the matrix. The tunneling effect and electric damage process are implemented with a proposed strain-dependent tunneling distance. Next, the macroscopic strain sensing behaviors of homogenized RVEs are evaluated with finite element simulations. The predicted resistance change ratio and sensitivity of aligned MWCNT/polymer nanocomposite sensors are both consistent with the experimental data over a wide loading range. This research has demonstrated the high sensing performance of aligned MWCNT/polymer nanocomposite sensors along the preferred direction.

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Acknowledgements

Funded by the Alexander von Humboldt Research Fellowship (Grant No. CHN 1210283 HFST-P) and the National Natural Science Foundation of China (Grant No. 12372158).

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  1. School of Civil Engineering, Central South University, Changsha, 410083, People’s Republic of China

    Xiaodong Xia

  2. Institute of Mechanics, University of Duisburg-Essen, 45141, Essen, Germany

    Xiaodong Xia, Rainer Niekamp, Dominik Brands & Jörg Schröder

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Appendices

Appendix A: The explicit expressions of \({\mathbf{S}}_{{\text{int}}}^{(\sigma )}\) and \({\mathbf{S}}_{{\text{int}}}^{(\chi )}\) in Eqs. (10) and (11)

The explicit expressions of \({\mathbf{S}}_{{\text{int}}}^{(\sigma )}\) are given as [57]

$$S_{1111}^{(\sigma )} = \frac{1}{{2(1 - \nu_{{\text{int}}}^{{}} )}}\left\{ {1 - 2\nu_{{\text{int}}}^{{}} + \frac{{3\alpha_{{{\text{CNT}}}}^{2} - 1}}{{\alpha_{{{\text{CNT}}}}^{2} - 1}} - \left[ {1 - 2\nu_{{\text{int}}}^{{}} + \frac{{3\alpha_{{{\text{CNT}}}}^{2} }}{{\alpha_{{{\text{CNT}}}}^{2} - 1}}} \right]g(\alpha_{{{\text{CNT}}}} )} \right\},$$
(A.1)
$$S_{2222}^{(\sigma )} = S_{3333}^{(\sigma )} = \frac{3}{{8(1 - \nu_{{\text{int}}}^{{}} )}}\frac{{\alpha_{{{\text{CNT}}}}^{2} }}{{\alpha_{{{\text{CNT}}}}^{2} - 1}} + \frac{1}{{4(1 - \nu_{{\text{int}}}^{{}} )}}\left[ {1 - 2\nu_{{\text{int}}}^{{}} - \frac{9}{{4(\alpha_{{{\text{CNT}}}}^{2} - 1)}}} \right]g(\alpha_{{{\text{CNT}}}}^{{}} ),$$
(A.2)
$$S_{2233}^{(\sigma )} = S_{3322}^{(\sigma )} = \frac{1}{{4(1 - \nu_{{\text{int}}}^{{}} )}}\left\{ {\frac{{\alpha_{{{\text{CNT}}}}^{2} }}{{2(\alpha_{{{\text{CNT}}}}^{2} - 1)}} - \left[ {1 - 2\nu_{{\text{int}}}^{{}} + \frac{3}{{4(\alpha_{{{\text{CNT}}}}^{2} - 1)}}} \right]g(\alpha_{{{\text{CNT}}}}^{{}} )} \right\},$$
(A.3)
$$S_{2211}^{(\sigma )} = S_{3311}^{(\sigma )} = - \frac{1}{{2(1 - \nu_{{\text{int}}}^{{}} )}}\frac{{\alpha_{{{\text{CNT}}}}^{2} }}{{\alpha_{{{\text{CNT}}}}^{2} - 1}} + \frac{1}{{4(1 - \nu_{{\text{int}}}^{{}} )}}\left\{ {\frac{{3\alpha_{{{\text{CNT}}}}^{2} }}{{\alpha_{{{\text{CNT}}}}^{2} - 1}} - (1 - 2\nu_{{\text{int}}}^{{}} )} \right\}g(\alpha_{{{\text{CNT}}}}^{{}} ),$$
(A.4)
$$S_{1122}^{(\sigma )} = S_{1133}^{(\sigma )} = - \frac{1}{{2(1 - \nu_{{\text{int}}}^{{}} )}}\left[ {1 - 2\nu_{{\text{int}}}^{{}} + \frac{1}{{\alpha_{{{\text{CNT}}}}^{2} - 1}}} \right] + \frac{1}{{2(1 - \nu_{{\text{int}}}^{{}} )}}\left[ {1 - 2\nu_{{\text{int}}}^{{}} + \frac{3}{{2(\alpha_{{{\text{CNT}}}}^{2} - 1)}}} \right]g(\alpha_{{{\text{CNT}}}}^{{}} ),$$
(A.5)
$$S_{2323}^{(\sigma )} = \frac{1}{{4(1 - \nu_{{\text{int}}}^{{}} )}}\left\{ {\frac{{\alpha_{{{\text{CNT}}}}^{2} }}{{2(\alpha_{{{\text{CNT}}}}^{2} - 1)}} + \left[ {1 - 2\nu_{{\text{int}}}^{{}} - \frac{3}{{4(\alpha_{{{\text{CNT}}}}^{2} - 1)}}} \right]g(\alpha_{{{\text{CNT}}}}^{{}} )} \right\},$$
(A.6)
$$S_{1212}^{(\sigma )} = S_{1313}^{(\sigma )} = \frac{1}{{4(1 - \nu_{{\text{int}}}^{{}} )}}\left\{ {1 - 2\nu_{{\text{int}}}^{{}} - \frac{{\alpha_{{{\text{CNT}}}}^{2} + 1}}{{\alpha_{{{\text{CNT}}}}^{2} - 1}} - \frac{1}{2}\left[ {1 - 2\nu_{{\text{int}}}^{{}} - \frac{{3(\alpha_{{{\text{CNT}}}}^{2} + 1)}}{{\alpha_{{{\text{CNT}}}}^{2} - 1}}} \right]g(\alpha_{{{\text{CNT}}}}^{{}} )} \right\}.$$
(A.7)

Here, \(g(\alpha_{{{\text{CNT}}}} ) = \left[ {\alpha_{{{\text{CNT}}}} /(\alpha_{{{\text{CNT}}}}^{2} - 1)^{3/2} } \right] \cdot \left[ {\alpha\,_{{{\text{CNT}}}} (\alpha_{{{\text{CNT}}}}^{2} - 1)^{1/2} - {\text{arccosh}}\,\alpha_{{{\text{CNT}}}} } \right]\) for MWCNT with \(\alpha_{{{\text{CNT}}}} > 1\),

In addition, the explicit expressions of \({\mathbf{S}}_{{\text{int}}}^{(\chi )} = {\text{diag}}\{ S_{11}^{(\chi )} ,S_{22}^{(\chi )} ,S_{33}^{(\chi )} \}\) are given as [58]

$$S_{22}^{(\chi )} = S_{33}^{(\chi )} = \frac{{\alpha_{{{\text{CNT}}}}^{{}} }}{{2(\alpha_{{{\text{CNT}}}}^{2} - 1)^{3/2} }}\left[ {\alpha_{{{\text{CNT}}}}^{{}} (\alpha\,_{{{\text{CNT}}}}^{2} - 1)^{1/2} - \text{arccosh}\, \alpha_{{{\text{CNT}}}}^{2} } \right],\quad S_{11}^{(\chi )} = 1 - 2S_{33}^{(\chi )} .$$
(A.8)

Appendix B: Some material and geometric parameters of RVEs listed in the tables

See Tables 1 and 2.

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Xia, X., Niekamp, R., Brands, D. et al. A hybrid computational scheme on electromechanically coupled behaviors of aligned MWCNT/polymer nanocomposite sensors with strain-dependent tunneling effect. Arch Appl Mech 93, 4305–4325 (2023). https://doi.org/10.1007/s00419-023-02493-2

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