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An electromechanical coupling isogeometric approach using zig-zag function for modeling and smart damping control of multilayer PFG-GPRC plates

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Abstract

In this article, a novel numerical approach based on electromechanical coupling isogeometric analysis employing a piecewise linear zig-zag function is proposed for modeling and analysis of smart constrained layer damping (SCLD) treatment in multilayer porous functionally graded graphene platelets-reinforced composite (PFG-GPRC) plates. The approach efficiently approximates the geometric, mechanical, and electric displacement fields by utilizing non-uniform rational B-splines (NURBS) basis functions. These basis functions are subsequently integrated with the zig-zag formulation to characterize the system dynamic and help handle both continuous/discontinuous material properties at all interfaces, as well as improve the effectiveness of global–local numerical solutions for the analysis of current structures. The multilayer PFG-GPRC plate model is designed to incorporate porous, uniformly, or non-uniformly distributed layers based on three different graphene platelet patterns. The analysis of the SCLD treatment encompasses an examination of the frequency response function of the damped structure under passive/hybrid mechanisms, taking into account viscoelastic behavior and the converse piezoelectric effect. Reliability in the current analysis is demonstrated through a validation study, and a comprehensive parametric investigation is undertaken to analyze the impact of various parameters related to graphene platelets (GPLs) and distribution types of porosity on the damping behavior of multilayer PFG-GPRC plates.

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Acknowledgements

This study was supported by Bualuang ASEAN Chair Professor Fund.

Funding

Bualuang ASEAN Chair Professor Fund (Bualuang ASEAN Chair Professor).

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Authors and Affiliations

  1. Laboratory for Applied and Industrial Mathematics, Institute for Computational Science and Artificial Intelligence, Van Lang University, Ho Chi Minh City, Vietnam

    T. Nguyen-Thoi

  2. Laboratory for Computational Mechanics, Institute for Computational Science and Artificial Intelligence, Van Lang University, Ho Chi Minh City, Vietnam

    Duy-Khuong Ly

  3. Faculty of Mechanical - Electrical and Computer Engineering, School of Technology, Van Lang University, Ho Chi Minh City, Vietnam

    T. Nguyen-Thoi

  4. Faculty of Civil Engineering, School of Technology, Van Lang University, Ho Chi Minh City, Vietnam

    Duy-Khuong Ly

  5. Bualuang ASEAN Chair Professor, Thammasat School of Engineering, Thammasat University, Pathumtani, 12120, Thailand

    T. Nguyen-Thoi

  6. Department of Mechanical Engineering, National Institute of Technology Karnataka, Surathkal Srinivasnagar PO, Mangalore, 575025, India

    S. Kattimani

  7. Thammasat University Research Unit in Structural and Foundation Engineering, Department of Civil Engineering, Thammasat School of Engineering, Thammasat University, Pathumtani, 12120, Thailand

    Chanachai Thongchom

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  1. T. Nguyen-Thoi
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  2. Duy-Khuong Ly
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  3. S. Kattimani
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Appendix

Appendix

1.1 Stiffness and rigidity matrices related to the PFG-GPRC/ACLD plate

The matrices appearing in Eqs. (39)–(40) are given by

The elemental elastic stiffness matrices:

$$\begin{aligned} \left[ K_{{ t t}}^{e}\right] =\left( \left[ K_{t b}^{e}\right] +\left[ K_{t s}^{e}\right] \right) +(\left[ K_{{ t b p}}^{e}\right] +\left[ K_{t s p}^{e}\right] )+\left[ K_{tsv}^{e}\right] \end{aligned}$$
(A.1)

in which

$$\begin{aligned} \left[ K_{t b}^{e}\right]= & {} \int _{\mathrm {\Omega }} \left[ B_{t b}\right] ^{\textrm{T}}\left[ D_{t b}\right] \left[ B_{t b}\right] \mathrm {d\Omega }; \quad \left[ D_{t b}\right] =\int _{h_{1}}^{h_{2}}\left[ {\bar{C}}_{b}^{{ FG}}(z)\right] \textrm{d} z;\nonumber \\ \left[ K_{{ t s}}^{e}\right]= & {} \int _{\mathrm {\Omega }} \left[ B_{{ t s}}\right] ^{\textrm{T}}\left[ D_{{ t s}}\right] \left[ B_{t s}\right] \mathrm {d\Omega }; \quad \left[ D_{t s}\right] =\int _{h_{1}}^{h_{2}}\left[ {\bar{C}}_{s}^{\textrm{FG}}(z)\right] \textrm{d} z;\nonumber \\ \left[ K_{{ t b p}}^{e}\right]= & {} \int _{\mathrm {\Omega }} \left( \left[ B_{t b}\right] ^{\textrm{T}}\left[ D_{t b}^{P}\right] \left[ B_{t b}\right] +\left[ B_{t b}\right] ^{\textrm{T}}\left[ D_{t b s}^{P}\right] \left[ B_{t s}\right] +\left[ B_{ts}\right] ^{\textrm{T}}\left[ D_{t b s}^{P}\right] \left[ B_{tb}\right] \right) \mathrm {d\Omega };\nonumber \\ \left[ D_{t b}^{P}\right]= & {} \int _{h_{3}}^{h_{4}}\left[ {\bar{C}}_{b}^{P}\right] \textrm{d} z; \quad \left[ D_{t b s}^{P}\right] =\int _{h_{3}}^{h_{4}}\left[ {\bar{C}}_{b s}^{P}\right] \textrm{d} z;\nonumber \\ \left[ K_{t s p}^{e}\right]= & {} \int _{\mathrm {\Omega }} \left[ B_{{ t s}}\right] ^{\textrm{T}}\left[ D_{{ t s}}^{P}\right] \left[ B_{t s}\right] \mathrm {d\Omega }; \quad \left[ D_{t s}^{P}\right] =\int _{h_{3}}^{h_{4}}\left[ {\bar{C}}_{s}^{P}\right] \textrm{d} z\nonumber \\ \left[ K_{t s v}^{e}\right]= & {} \int _{\mathrm {\Omega }} \left[ B_{{ t s}}\right] ^{\textrm{T}} \left[ {D}_{ts}^{Vis}\right] \left[ B_{t s}\right] \mathrm {d\Omega }; \quad \left[ D_{ts }^{Vis}\right] =\int _{h_{2}}^{h_{3}}\left[ {\bar{C}}_{s}^{Vis}\right] \textrm{d} z\nonumber \\ \left[ K_{tr}^{e}\right]= & {} \left( \left[ K_{trb}^{e}\right] +\left[ K_{trs}^{e}\right] \right) +\left( \left[ K_{trbp}^{e}\right] +\left[ K_{trsp}^{e}\right] \right) +\left[ K_{trsv}^{e}\right] ; \end{aligned}$$
(A.2)

in which

$$\begin{aligned} \left[ K_{t r b}^{e}\right]= & {} \int _{\mathrm {\Omega } } \left[ B_{t b}\right] ^{\textrm{T}}\left[ D_{t r b}\right] \left[ B_{r b}\right] \mathrm {d\Omega }; \quad \left[ D_{t r b}\right] =\int _{h_{1}}^{h_{2}}\left[ {\bar{C}}_{b}^{F G}(z)\right] \left[ Z_{1}\right] \textrm{d} z;\nonumber \\ \left[ K_{t r s}^{e}\right]= & {} \int _{\mathrm {\Omega }} \left[ B_{t s}\right] ^{\textrm{T}}\left[ D_{t r s}\right] \left[ B_{r s}\right] \mathrm {d\Omega }; \quad \left[ D_{t r s}\right] =\int _{h_{1}}^{h_{2}}\left[ {\bar{C}}_{s}^{F G}(z)\right] \left[ Z_{3}\right] \textrm{d} z;\nonumber \\ \left[ K_{t r b p}^{e}\right]= & {} \int _{\mathrm {\Omega }} \left( \left[ B_{t b}\right] ^{\textrm{T}}\left[ D_{t r b}^{P}\right] \left[ B_{r b}\right] + 1/2\left[ B_{t b}\right] ^{\textrm{T}}\left[ D_{t r b s}^{P}\right] \left[ B_{r s}\right] +1/2\left[ B_{t s}\right] ^{\textrm{T}}\left[ D_{r t b s}^{P}\right] ^{\textrm{T}}\left[ B_{r b}\right] \right. \nonumber \\{} & {} \left. +1/2\left[ \left[ B_{rb}\right] ^{\textrm{T}}\left[ D_{r t b s}^{P}\right] ^{\textrm{T}}\left[ B_{ts}\right] \right] ^\textrm{T} + 1/2\left[ \left[ B_{rs}\right] ^{\textrm{T}}\left[ D_{trb s}^{P}\right] ^{\textrm{T}}\left[ B_{tb}\right] \right] ^\textrm{T} \right) \mathrm {d\Omega }\nonumber \\ \left[ D_{t r b}^{P}\right]= & {} \int _{h_{3}}^{h_{4}}\left[ {\bar{C}}_{b}^{P}\right] \left[ Z_{2}\right] \textrm{d} z; \quad \left[ D_{t r b s}^{P}\right] =\int _{h_{3}}^{h_{4}}\left[ {\bar{C}}_{b s}^{p}\right] \left[ Z_{5}\right] \textrm{d} z; \quad \left[ D_{r t b s}^{P}\right] =\int _{h_{3}}^{h_{4}}\left[ Z_{2}\right] ^{\textrm{T}}\left[ {\bar{C}}_{b s}^{P}\right] \textrm{d} z;\nonumber \\ \left[ K_{t r s p}^{e}\right]= & {} \int _{\mathrm {\Omega }} \left[ B_{t s}\right] ^{\textrm{T}}\left[ D_{t r s}^{P}\right] \left[ B_{r s}\right] \mathrm {d\Omega }; \quad \left[ D_{t r s}^{P}\right] =\int _{h_{3}}^{h_{4}}\left[ {\bar{C}}_{s}^{P}\right] \left[ Z_{5}\right] \textrm{d} z;\nonumber \\ \left[ K_{{trsv }}^{e}\right]= & {} \int _{\mathrm {\Omega }}\left[ B_{t s}\right] ^{\textrm{T}}\left[ D_{{trs}}^{Vis}\right] \left[ B_{r s}\right] \mathrm {d\Omega }; \quad \left[ D_{{trs }}^{Vis}\right] =\int _{h_{2}}^{h_{3}}\left[ {\bar{C}}_{s}^{Vis}\right] \left[ Z_{4}\right] \textrm{d} z.\nonumber \\ \left[ K_{rr}^{e}\right]= & {} \left( \left[ K_{rrb}^{e}\right] +\left[ K_{rrs}^{e}\right] \right) +\left( \left[ K_{rrbp}^{e}\right] +\left[ K_{rrsp}^{e}\right] \right) +\left[ K_{rrsv}^{e}\right] ; \end{aligned}$$
(A.3)

in which

$$\begin{aligned} \left[ K_{r r b}^{e}\right]= & {} \int _{\mathrm {\Omega }}\left[ B_{r b}\right] ^{\textrm{T}}\left[ D_{r r b}\right] \left[ B_{r b}\right] \mathrm {d\Omega }; \quad [D_{r r b}]=\int _{h_{1}}^{h_{2}}\left[ Z_{1}\right] ^{\textrm{T}}\left[ {\bar{C}}_{b}^{F G}(z)\right] \left[ Z_{1}\right] \textrm{d} z;\nonumber \\ \left[ K_{r r s}^{e}\right]= & {} \int _{\mathrm {\Omega }} \left[ B_{r s}\right] ^{\textrm{T}}\left[ D_{r r s}\right] \left[ B_{r s}\right] \mathrm {d\Omega }; \quad \left[ D_{r r s}\right] =\int _{h_{1}}^{h_{2}}\left[ Z_{3}\right] ^{\textrm{T}}\left[ {\bar{C}}_{s}^{F G}(z)\right] \left[ Z_{3}\right] \textrm{d} z;\nonumber \\ \left[ K_{r r b p}^{e}\right]= & {} \int _{\mathrm {\Omega }} \left( \left[ B_{r b}\right] ^{\textrm{T}}\left[ D_{r r b}^{P}\right] \left[ B_{r b}\right] +\left[ B_{r b}\right] ^{\textrm{T}}\left[ D_{r r b s}^{P}\right] \left[ B_{r s}\right] +\left[ B_{r s}\right] ^{\textrm{T}}\left[ D_{r r b s}^{P}\right] \left[ B_{rb}\right] \right) \mathrm {d\Omega };\nonumber \\ \left[ D_{r r b}^{P}\right]= & {} \int _{h_{3}}^{h_{4}}\left[ Z_{2}\right] ^{\textrm{T}}\left[ {\bar{C}}_{b}^{P}\right] \left[ Z_{2}\right] \textrm{d} z; \quad \left[ D_{r r b s}^{P}\right] =\int _{h_{3}}^{h_{4}}\left[ Z_{2}\right] ^{\textrm{T}}\left[ {\bar{C}}_{b s}^{P}\right] \left[ Z_{5}\right] \textrm{d} z;\nonumber \\ \left[ K_{r r s p}^{e}\right]= & {} \int _{\mathrm {\Omega }} \left[ B_{r s}\right] ^{\textrm{T}}\left[ D_{r r s}^{P}\right] \left[ B_{r s}\right] \mathrm {d\Omega }; \quad \left[ D_{r r s}^{P}\right] =\int _{h_{3}}^{h_{4}}\left[ Z_{2}\right] ^{\textrm{T}}\left[ {\bar{C}}_{s}^{P}\right] \left[ Z_{5}\right] \textrm{d} z;\nonumber \\ \left[ K_{r r s v}^{e}\right]= & {} \int _{\mathrm {\Omega }} \left[ B_{r s}\right] ^{\textrm{T}}\left[ D_{r r s}^{Vis}\right] \left[ B_{r s}\right] \mathrm {d\Omega }; \quad \left[ D_{r r s}^{Vis}\right] =\int _{h_{2}}^{h_{3}}\left[ Z_{4}\right] ^{\textrm{T}}\left[ {\bar{C}}_{s}^{Vis}\right] \left[ Z_{4}\right] \textrm{d} z.\nonumber \\ \left\{ F^{e}\right\}= & {} \int _{\Omega }\left[ N_{t}\right] ^{\textrm{T}}\{f\} \textrm{d}\Omega \nonumber \\ \left\{ F_{t p}^{e}\right\}= & {} \left\{ F_{t p b}^{e}\right\} +\left\{ F_{t p s}^{e}\right\} \nonumber \\ \left\{ F_{r p}^{e}\right\}= & {} \left\{ F_{r p b}^{e}\right\} +\left\{ F_{r p s}^{e}\right\} \end{aligned}$$
(A.4)

in which

$$\begin{aligned} \left\{ F_{t p b}^{e}\right\}= & {} \int _{\mathrm {\Omega }} \left[ B_{t b}\right] ^{\textrm{T}}\left\{ D_{t p}^{b}\right\} \mathrm {d\Omega }; \quad \left\{ F_{t p s}^{e}\right\} =\int _{\mathrm {\Omega }}\left[ B_{t s}\right] ^{\textrm{T}}\left\{ D_{tp}^{s}\right\} \mathrm {d\Omega };\nonumber \\ \left\{ D_{t p}^{b}\right\}= & {} \int _{h_{3}}^{h_{4}} \frac{1}{h_{p}}\left\{ {\bar{e}}_{b}^{P}\right\} \textrm{d} z; \quad \left\{ D_{t p}^{s}\right\} =\int _{h_{3}}^{h_{4}} \frac{1}{h_{p}}\left\{ e_{s}^{P}\right\} \textrm{d} z;\nonumber \\ \left\{ F_{r p b}^{e}\right\}= & {} \int _{\mathrm {d\Omega }} \left[ B_{r b}\right] ^{\textrm{T}}\left\{ D_{r p}^{b}\right\} \mathrm {d\Omega }; \quad \left\{ F_{r p s}^{e}\right\} =\int _{\mathrm {\Omega }}\left[ B_{r s}\right] ^{\textrm{T}}\left\{ D_{r p}^{s}\right\} \mathrm {d\Omega }.\nonumber \\ \left\{ D_{r p}^{b}\right\}= & {} \int _{h_{3}}^{h_{4}} \frac{1}{h_{p}}\left[ Z_{2}\right] ^{\textrm{T}}\left\{ {\bar{e}}_{b}^{P}\right\} \textrm{d} z; \quad \left\{ D_{r p}^{s}\right\} =\int _{h_{3}}^{h_{4}} \frac{1}{h_{p}}\left[ Z_{5}\right] ^{\textrm{T}}\left\{ {\bar{e}}_{s}^{P}\right\} \textrm{d} z. \end{aligned}$$

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Nguyen-Thoi, T., Ly, DK., Kattimani, S. et al. An electromechanical coupling isogeometric approach using zig-zag function for modeling and smart damping control of multilayer PFG-GPRC plates. Acta Mech 235, 941–970 (2024). https://doi.org/10.1007/s00707-023-03785-y

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