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Nonlinear pyrocoupled deflection of viscoelastic sandwich shell with CNT reinforced magneto-electro-elastic facing subjected to electromagnetic loads in thermal environment

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Abstract

The current work puts forward a finite element (FE)-based numerical formulation to evaluate the nonlinear deflections of multifunctional sandwich composite (MSC) shells. These shells possess a viscoelastic core, and face sheets made of functionally graded carbon nanotube-reinforced magneto-electro-elastic (FG-CNTMEE) materials. The viscoelastic core is considered to be temperature-dependent and is modelled via the complex modulus approach. Two different forms of viscoelastic cores, such as Dyad 606 and EC 2216, are considered in this study. The shell kinematics is realized with the aid of the higher-order shear deformation theory (HSDT). Furthermore, Donnell's nonlinear strain displacement relations are incorporated to account for the nonlinear behaviour. The total potential energy principle is utilized to get the global equations of motion which are solved using direct iterative method. Predominant emphasis is also placed to assess the impact of pyroeffects, coupling fields and electromagnetic (EM) boundary restrictions on the nonlinear deflections of MSC shells working in the thermal environment and subjected to EM loads, which is first of its kind. Also, parametric studies dealing with the shell geometries, CNT distributions and volume fractions, core-to-face sheet thickness ratio, aspect ratio, curvature ratio has been discussed in detail. The results of this work are believed to be unique and serve as a guide for the design engineers towards developing sophisticated smart structures for various engineering applications.

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Abbreviations

a, b, h :

Length, width, and thickness of the composite sandwich shell

h 1 and h 2 :

z-Coordinate of the bottom and top surface of the bottom FG-CNTMEE face sheet

h 3 and h 4 :

z-Coordinate of the bottom and top surface of the top FG-CNTMEE face sheet

R 1 and R 2 :

Principal radii of curvature at the mid-plane of sandwich shell along x- and y- directions, respectively

\({V}_{CNT}^{*}\) :

Total volume fraction of CNT reinforcement

\({V}_{CNT}\) and\({V}_{m}\) :

The effective volume fractions of CNT and the matrix

[e], \(\left[{e}^{CNT}\right]\),\(\left[{e}^{m}\right]\) :

Piezoelectric coefficients of effective FG-CNTMEE composite, CNT fiber, matrix, respectively

[C],\(\left[{C}^{CNT}\right],\left[{C}^{m}\right]\) :

Elastic stiffness coefficients of effective FG-CNTMEE composite, CNT fiber, matrix, respectively

\(\left[{N}_{t}\right], \left[{N}_{r}\right], \left[{N}_{r*}\right],\) :

Shape function matrices related to translational displacement, rotational displacement, higher-order rotational displacement, respectively

\(\left[{N}_{\phi }\right] and \left[{N}_{\psi }\right]\) :

Shape function matrices related to electric potential and magnetic potential degrees of freedom, respectively

[q],\(\left[{q}^{CNT}\right], \left[{q}^{m}\right]\) :

Magnetostrictive coefficients of effective FG-CNTMEE composite, CNT fiber, matrix, respectively

[η],\(\left[{\eta }^{CNT}\right], \left[{\eta }^{m}\right]\) :

Dielectric coefficients of effective FG-CNTMEE composite, CNT fiber, matrix, respectively

[m], \(\left[{m}^{CNT}\right]\),\(\left[{m}^{m}\right]\) :

Electro-magnetic coefficients of effective FG-CNTMEE composite, CNT fiber, matrix, respectively

[μ],\(\left[{\mu }^{CNT}\right], \left[{\mu }^{m}\right]\) :

Magnetic permeability coefficients of effective FG-CNTMEE composite, CNT fiber, matrix, respectively

\(\left[\alpha \right], \left[{\alpha }^{CNT}\right],\left[{\alpha }^{m}\right]\) :

Thermal expansion coefficients of effective FG-CNTMEE composite, CNT fiber, matrix, respectively

\(\left[p\right],\left[\tau \right]\) :

Pyroelectric and pyromagnetic coefficients

\(\left\{\varepsilon \right\}\) :

Strain tensor

\(\left\{B\right\}\) :

Magnetic flux vector

\(\left\{E\right\}\) :

Electric field vector

\(\left\{\sigma \right\}\) :

Stress tensor

\(\left\{H\right\}\) :

Magnetic field vector

\(\left\{D\right\}\) :

Electric displacement vector

u 0, v 0, and w 0 :

Displacements at the midplane along the x-, y- and z-axes

\({\theta }_{x}, {\theta }_{y}\) :

Normal transverse rotation about x- and y-axes

\(\psi \) :

Magnetic potential

\(\phi \) :

Electric potential

\(\left\{{d}_{t}\right\}\) :

Linear displacement

\(\left\{{d}_{r}\right\}\) :

Rotational displacement

\(\left\{{d}_{r*}\right\}\) :

Higher-order rotational displacement

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  1. Department of Mechanical Engineering, National Institute of Technology, Silchar, Assam, 788010, India

    Vinyas Mahesh

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Appendix

Appendix

The relation between various stiffness matrices contributing towards equivalent linear stiffness matrix \(\left[{K}_{L\_eq}\right]\) appearing in Eq. (20a) is expressed as,

$$ \begin{aligned} \left[ {K_{1}^{e} } \right] & = \left[ {K_{{tb1}}^{e} } \right] + \left[ {K_{{ts1}}^{e} } \right];\quad \left[ {K_{2}^{e} } \right] = \left[ {K_{{rtb24}}^{e} } \right]^{T} + \left[ {K_{{rts13}}^{e} } \right]^{T} \hfill \\ \left[ {K_{3}^{e} } \right] & = \left[ {K_{{rtb4}}^{e} } \right]^{T} + \left[ {K_{{rts3}}^{e} } \right]^{T} ;\quad \left[ {K_{4}^{e} } \right] = \left[ {K_{{tb\phi 1}}^{e} } \right]^{T} + \left[ {K_{{ts\phi 1}}^{e} } \right]^{T} \hfill \\ \left[ {K_{5}^{e} } \right] &= \left[ {K_{{tb\psi 1}}^{e} } \right]^{T} + \left[ {K_{{ts\psi 1}}^{e} } \right]^{T} ;\left[ {K_{6}^{e} } \right] = \left[ {K_{{rtb24}}^{e} } \right] + \left[ {K_{{rts13}}^{e} } \right] \hfill \\ \left[ {K_{7}^{e} } \right] &= \left[ {K_{{rrb3557}}^{e} } \right] + \left[ {K_{{rrs3513}}^{e} } \right];\left[ {K_{8}^{e} } \right] = \left[ {K_{{rrb57}}^{e} } \right] + \left[ {K_{{rrs35}}^{e} } \right] \hfill \\ \left[ {K_{9}^{e} } \right] &= \left[ {K_{{rb\phi 24}}^{e} } \right]^{T} + \left[ {K_{{r\phi s13}}^{e} } \right]^{T} ;\left[ {K_{{10}}^{e} } \right] = \left[ {K_{{rb\psi 24}}^{e} } \right]^{T} + \left[ {K_{{r\psi s13}}^{e} } \right]^{T} \hfill \\ \left[ {K_{{11}}^{e} } \right] &= \left[ {K_{{rtb4}}^{e} } \right] + \left[ {K_{{rts3}}^{e} } \right];\left[ {K_{{12}}^{e} } \right] = \left[ {K_{{rrb57}}^{e} } \right] + \left[ {K_{{rrs35}}^{e} } \right] \hfill \\ \left[ {K_{{13}}^{e} } \right] &= \left[ {K_{{rrb7}}^{e} } \right] + \left[ {K_{{rrs5}}^{e} } \right];\left[ {K_{{14}}^{e} } \right] = \left[ {K_{{rb\phi 4}}^{e} } \right]^{T} + \left[ {K_{{r\phi s3}}^{e} } \right]^{T} \hfill \\ \left[ {K_{{15}}^{e} } \right] &= \left[ {K_{{rb\psi 4}}^{e} } \right]^{T} + \left[ {K_{{r\psi s3}}^{e} } \right]^{T} ;\left[ {K_{{16}}^{e} } \right] = \left[ {K_{{tb\phi 1}}^{e} } \right]^{T} + \left[ {K_{{ts\phi 1}}^{e} } \right]^{T} \hfill \\ \left[ {K_{{17}}^{e} } \right] &= \left[ {K_{{rb\phi 2}}^{e} } \right]^{T} + \left[ {K_{{rb\phi 4}}^{e} } \right]^{T} + \left[ {K_{{r\phi s1}}^{e} } \right]^{T} + \left[ {K_{{r\phi s3}}^{e} } \right]^{T} ;\left[ {K_{{18}}^{e} } \right] = \left[ {K_{{rb\phi 4}}^{e} } \right]^{T} + \left[ {K_{{r\phi s3}}^{e} } \right]^{T} ; \hfill \\ \left[ {K_{{19}}^{e} } \right] &= \left[ {K_{{tb\psi 1}}^{e} } \right]^{T} + \left[ {K_{{ts\psi 1}}^{e} } \right]^{T} \left[ {K_{{20}}^{e} } \right] = \left[ {K_{{rb\psi 2}}^{e} } \right]^{T} + \left[ {K_{{rb\psi 4}}^{e} } \right]^{T} + \left[ {K_{{r\psi s1}}^{e} } \right]^{T} + \left[ {K_{{r\psi s3}}^{e} } \right]^{T} ; \hfill \\ \left[ {K_{{21}}^{e} } \right] &= \left[ {K_{{rb\psi 4}}^{e} } \right]^{T} + \left[ {K_{{r\psi s3}}^{e} } \right]^{T} ;\left[ {K_{{22}}^{e} } \right] = \left[ {K_{{16}}^{e} } \right] - \left[ {K_{{\phi \psi }}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{19}}^{e} } \right]; \hfill \\ \left[ {K_{{23}}^{e} } \right] &= \left[ {K_{{17}}^{e} } \right] - \left[ {K_{{\phi \psi }}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{20}}^{e} } \right];\left[ {K_{{24}}^{e} } \right] = \left[ {K_{{18}}^{e} } \right] - \left[ {K_{{\phi \psi }}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{21}}^{e} } \right]; \hfill \\ \left[ {K_{{25}}^{e} } \right] & = \left[ {K_{{\phi \phi }}^{e} } \right] - \left[ {K_{{\phi \psi }}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{\phi \psi }}^{e} } \right]^{T} ;\left[ {K_{{26}}^{e} } \right] = \left[ {K_{{25}}^{e} } \right]^{{ - 1}} \left[ {K_{{22}}^{e} } \right]^{T} ;\hfill\\ \left[ {K_{{27}}^{e} } \right] & = \left[ {K_{{25}}^{e} } \right]^{{ - 1}} \left[ {K_{{23}}^{e} } \right]^{T} ;\left[ {K_{{28}}^{e} } \right] = \left[ {K_{{25}}^{e} } \right]^{{ - 1}} \left[ {K_{{24}}^{e} } \right]^{T} \hfill \\ \left[ {K_{{29}}^{e} } \right] &= \left[ {K_{{11}}^{e} } \right] - \left[ {K_{{15}}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{19}}^{e} } \right];\left[ {K_{{30}}^{e} } \right] = \left[ {K_{{12}}^{e} } \right] - \left[ {K_{{15}}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{20}}^{e} } \right]; \\ \left[ {K_{{31}}^{e} } \right] &= \left[ {K_{{13}}^{e} } \right] - \left[ {K_{{15}}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{21}}^{e} } \right];\left[ {K_{{32}}^{e} } \right] = \left[ {K_{{14}}^{e} } \right] - \left[ {K_{{15}}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{\phi \psi }}^{e} } \right]^{T} \hfill \\ \left[ {K_{{33}}^{e} } \right] & = \left[ {K_{{29}}^{e} } \right] - \left[ {K_{{32}}^{e} } \right]\left[ {K_{{26}}^{e} } \right];\left[ {K_{{34}}^{e} } \right] = \left[ {K_{{30}}^{e} } \right] - \left[ {K_{{32}}^{e} } \right]\left[ {K_{{27}}^{e} } \right]; \hfill \\ \left[ {K_{{35}}^{e} } \right] & = \left[ {K_{{31}}^{e} } \right] - \left[ {K_{{32}}^{e} } \right]\left[ {K_{{28}}^{e} } \right];\left[ {K_{{36}}^{e} } \right] = \left[ {K_{{35}}^{e} } \right]^{{ - 1}} \left[ {K_{{33}}^{e} } \right]^{T} ; \hfill \\ \left[ {K_{{37}}^{e} } \right] &= \left[ {K_{{35}}^{e} } \right]^{{ - 1}} \left[ {K_{{34}}^{e} } \right]^{T} ;\left[ {K_{{38}}^{e} } \right] = \left[ {K_{6}^{e} } \right] - \left[ {K_{{10}}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{19}}^{e} } \right]; \hfill \\ \left[ {K_{{39}}^{e} } \right] & = \left[ {K_{7}^{e} } \right] - \left[ {K_{{10}}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{20}}^{e} } \right];\left[ {K_{{40}}^{e} } \right] = \left[ {K_{8}^{e} } \right] - \left[ {K_{{10}}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{21}}^{e} } \right]; \hfill \\ \left[ {K_{{41}}^{e} } \right] & = \left[ {K_{9}^{e} } \right] - \left[ {K_{{10}}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{\phi \psi }}^{e} } \right]^{T} ;\left[ {K_{{42}}^{e} } \right] = \left[ {K_{{38}}^{e} } \right] - \left[ {K_{{41}}^{e} } \right]\left[ {K_{{26}}^{e} } \right] \hfill \\ \left[ {K_{{43}}^{e} } \right] & = \left[ {K_{{39}}^{e} } \right] - \left[ {K_{{41}}^{e} } \right]\left[ {K_{{27}}^{e} } \right];\left[ {K_{{44}}^{e} } \right] = \left[ {K_{{40}}^{e} } \right] - \left[ {K_{{41}}^{e} } \right]\left[ {K_{{28}}^{e} } \right]; \hfill \\ \left[ {K_{{45}}^{e} } \right] & = \left[ {K_{{42}}^{e} } \right] - \left[ {K_{{44}}^{e} } \right]\left[ {K_{{36}}^{e} } \right];\left[ {K_{{46}}^{e} } \right] = \left[ {K_{{43}}^{e} } \right] - \left[ {K_{{44}}^{e} } \right]\left[ {K_{{37}}^{e} } \right]; \hfill \\ \left[ {K_{{47}}^{e} } \right] &= \left[ {K_{{46}}^{e} } \right]^{{ - 1}} \left[ {K_{{45}}^{e} } \right];\left[ {K_{{48}}^{e} } \right] = \left[ {K_{1}^{e} } \right] - \left[ {K_{5}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{19}}^{e} } \right]; \hfill \\ \left[ {K_{{49}}^{e} } \right] & = \left[ {K_{2}^{e} } \right] - \left[ {K_{5}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{20}}^{e} } \right];\left[ {K_{{50}}^{e} } \right] = \left[ {K_{3}^{e} } \right] - \left[ {K_{5}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{21}}^{e} } \right] \hfill \\ \left[ {K_{{51}}^{e} } \right] & = \left[ {K_{4}^{e} } \right] - \left[ {K_{5}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{\phi \psi }}^{e} } \right]^{T} ;\left[ {K_{{52}}^{e} } \right] = \left[ {K_{{48}}^{e} } \right] - \left[ {K_{{51}}^{e} } \right]\left[ {K_{{26}}^{e} } \right] \hfill \\ \left[ {K_{{53}}^{e} } \right] & = \left[ {K_{{49}}^{e} } \right] - \left[ {K_{{51}}^{e} } \right]\left[ {K_{{27}}^{e} } \right];\left[ {K_{{54}}^{e} } \right] = \left[ {K_{{50}}^{e} } \right] - \left[ {K_{{51}}^{e} } \right]\left[ {K_{{28}}^{e} } \right] \hfill \\ \left[ {K_{{55}}^{e} } \right] & = \left[ {K_{{52}}^{e} } \right] - \left[ {K_{{54}}^{e} } \right]\left[ {K_{{36}}^{e} } \right];\left[ {K_{{56}}^{e} } \right] = \left[ {K_{{53}}^{e} } \right] - \left[ {K_{{54}}^{e} } \right]\left[ {K_{{37}}^{e} } \right] \hfill \\ \left[ {K_{{L\_eq}}^{e} } \right] & = \left[ {K_{{55}}^{e} } \right] - \left[ {K_{{56}}^{e} } \right]\left[ {K_{{47}}^{e} } \right] \hfill \\ \end{aligned} $$
(A1)

The equivalent nonlinear stiffness matrix [KNL_eq] can be condensed as follows:

$$ \begin{aligned} \left[ {K_{{NL\_1}}^{e} } \right] & = \left[ {K_{{tbNLbNL1\_tbtbNL1}}^{e} } \right];\left[ {K_{{NL\_2}}^{e} } \right] = \left[ {K_{{rbNL\_rtb24}}^{e} } \right]^{T} \hfill \\ \left[ {K_{{NL\_3}}^{e} } \right]& = \left[ {K_{{rbNL4}}^{e} } \right]^{T} ;\left[ {K_{{NL\_4}}^{e} } \right] = \left[ {K_{{bNL\phi 1}}^{e} } \right] \hfill \\ \left[ {K_{{NL\_5}}^{e} } \right] & = \left[ {K_{{bNL\psi 1}}^{e} } \right];\left[ {K_{{NL\_6}}^{e} } \right] = \left[ {K_{{rbNL24}}^{e} } \right]^{T} \hfill \\ \left[ {K_{{NL\_7}}^{e} } \right] & = \left[ {K_{{rbNL4}}^{e} } \right]^{T} ;\left[ {K_{{NL\_8}}^{e} } \right] = \left[ {K_{{bNL\phi 1}}^{e} } \right]^{T} ;\left[ {K_{{NL\_9}}^{e} } \right] = \left[ {K_{{bNL\psi 1}}^{e} } \right]^{T} \hfill \\ \left[ {K_{{NL\_10}}^{e} } \right]& = \left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{NL\_9}}^{e} } \right];\left[ {K_{{NL\_11}}^{e} } \right] = \left[ {K_{{\phi \psi }}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{NL\_9}}^{e} } \right] \hfill \\ \left[ {K_{{NL\_12}}^{e} } \right] & = \left[ {K_{{25}}^{e} } \right]^{{ - 1}} \left[ {K_{{NL\_11}}^{e} } \right];\left[ {K_{{NL\_13}}^{e} } \right] = \left[ {K_{{15}}^{e} } \right]^{{ - 1}} \left[ {K_{{NL\_10}}^{e} } \right]; \hfill \\ \left[ {K_{{NL\_14}}^{e} } \right] & = \left[ {K_{{32}}^{e} } \right]\left[ {K_{{NL\_12}}^{e} } \right];\left[ {K_{{NL\_15}}^{e} } \right] = \left[ {K_{{NL\_14}}^{e} } \right] - \left[ {K_{{NL\_13}}^{e} } \right]; \hfill \\ \left[ {K_{{NL\_16}}^{e} } \right] & = \left[ {K_{{35}}^{e} } \right]^{{ - 1}} \left[ {K_{{NL\_15}}^{e} } \right];\left[ {K_{{NL\_17}}^{e} } \right] = \left[ {K_{{10}}^{e} } \right]\left[ {K_{{NL\_10}}^{e} } \right]; \hfill \\ \left[ {K_{{NL\_18}}^{e} } \right] & = \left[ {K_{{NL\_6}}^{e} } \right] - \left[ {K_{{NL\_17}}^{e} } \right];\left[ {K_{{NL\_19}}^{e} } \right] = \left[ {K_{{41}}^{e} } \right]\left[ {K_{{NL\_12}}^{e} } \right]; \hfill \\ \left[ {K_{{NL\_20}}^{e} } \right] & = \left[ {K_{{NL\_19}}^{e} } \right] + \left[ {K_{{NL\_18}}^{e} } \right];\left[ {K_{{NL\_21}}^{e} } \right] = \left[ {K_{{44}}^{e} } \right]\left[ {K_{{NL\_16}}^{e} } \right]; \hfill \\ \left[ {K_{{NL\_22}}^{e} } \right] & = \left[ {K_{{NL\_20}}^{e} } \right] - \left[ {K_{{NL\_21}}^{e} } \right];\left[ {K_{{NL\_23}}^{e} } \right] = \left[ {K_{{46}}^{e} } \right]^{{ - 1}} \left[ {K_{{NL\_22}}^{e} } \right]; \hfill \\ \left[ {K_{{NL\_24}}^{e} } \right] & = \left[ {K_{{NL\_1}}^{e} } \right] - \left[ {K_{{NL\_5}}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{19}}^{e} } \right] - \left[ {K_{5}^{e} } \right]\left[ {K_{{NL\_10}}^{e} } \right] - \left[ {K_{{NL\_5}}^{e} } \right]\left[ {K_{{NL\_10}}^{e} } \right]; \hfill \\ \left[ {K_{{NL\_25}}^{e} } \right] & = \left[ {K_{{NL\_2}}^{e} } \right] - \left[ {K_{{NL\_5}}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{20}}^{e} } \right] \hfill \\ \left[ {K_{{NL\_26}}^{e} } \right] & = \left[ {K_{{NL\_3}}^{e} } \right] - \left[ {K_{{NL\_5}}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{21}}^{e} } \right] \hfill \\ \left[ {K_{{NL\_27}}^{e} } \right] & = \left[ {K_{{NL\_4}}^{e} } \right] - \left[ {K_{{NL\_5}}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \left[ {K_{{\psi \phi }}^{e} } \right] \hfill \\ \left[ {K_{{NL\_28}}^{e} } \right] & = \left[ {K_{{51}}^{e} } \right]\left[ {K_{{NL\_12}}^{e} } \right] \hfill \\ \left[ {K_{{NL\_29}}^{e} } \right] & = \left[ {K_{{NL\_28}}^{e} } \right] + \left[ {K_{{NL\_24}}^{e} } \right] - \left[ {K_{{NL\_27}}^{e} } \right]\left[ {K_{{26}}^{e} } \right] + \left[ {K_{{NL\_27}}^{e} } \right]\left[ {K_{{NL\_12}}^{e} } \right] \hfill \\ \left[ {K_{{NL\_30}}^{e} } \right] & = \left[ {K_{{NL\_25}}^{e} } \right] - \left[ {K_{{NL\_27}}^{e} } \right]\left[ {K_{{27}}^{e} } \right] \hfill \\ \left[ {K_{{NL\_31}}^{e} } \right] & = \left[ {K_{{NL\_26}}^{e} } \right] - \left[ {K_{{NL\_27}}^{e} } \right]\left[ {K_{{28}}^{e} } \right]; \hfill \\ \left[ {K_{{NL\_32}}^{e} } \right] & = \left[ {K_{{54}}^{e} } \right]\left[ {K_{{NL\_16}}^{e} } \right]; \hfill \\ \left[ {K_{{NL\_33}}^{e} } \right] & = \left[ {K_{{NL\_29}}^{e} } \right] - \left[ {K_{{NL\_32}}^{e} } \right] - \left[ {K_{{NL\_31}}^{e} } \right]\left[ {K_{{36}}^{e} } \right] - \left[ {K_{{NL\_31}}^{e} } \right]\left[ {K_{{NL\_16}}^{e} } \right] \hfill \\ \left[ {K_{{NL\_34}}^{e} } \right] & = \left[ {K_{{NL\_30}}^{e} } \right] - \left[ {K_{{NL\_31}}^{e} } \right]\left[ {K_{{37}}^{e} } \right] \hfill \\ \left[ {K_{{NL\_35}}^{e} } \right] & = \left[ {K_{{56}}^{e} } \right]\left[ {K_{{NL\_23}}^{e} } \right],\left[ {K_{{NL\_36}}^{e} } \right] = \left[ {K_{{NL\_34}}^{e} } \right]\left[ {K_{{47}}^{e} } \right] \hfill \\ \left[ {K_{{NL\_37}}^{e} } \right] & = \left[ {K_{{NL\_34}}^{e} } \right]\left[ {K_{{NL\_23}}^{e} } \right] \hfill \\ \left[ {K_{{NL\_eq}}^{e} } \right] & = \left[ {K_{{NL\_33}}^{e} } \right] - \left[ {K_{{NL\_35}}^{e} } \right] - \left[ {K_{{NL\_36}}^{e} } \right] - \left[ {K_{{NL\_37}}^{e} } \right] \hfill \\ \end{aligned} $$
(A2)

The equivalent force \(\{{F}_{eq}\}\) can be represented as follows:

$$ \left\{ {{F_{eq}}} \right\} = \left\{ {F_t} \right\} - \left[ {K_{F\phi }^e} \right]\left( {\left\{ {F_\phi } \right\} + \left\{ {{F_{t\phi }}} \right\}} \right) - \left[ {K_{F\psi }^e} \right]\left( {\left\{ {F_\psi } \right\} + \left\{ {{F_{t\psi }}} \right\}} \right)$$

where

$$ \begin{aligned} \left[ {K_{{F\phi }}^{e} } \right] & = \left[ {K_{{F31}}^{e} } \right] - \left[ {K_{{F35}}^{e} } \right] - \left[ {K_{{F33}}^{e} } \right]; \hfill \\ \left[ {K_{{F\psi }}^{e} } \right] & = \left[ {K_{{F32}}^{e} } \right] - \left[ {K_{{F36}}^{e} } \right] - \left[ {K_{{F34}}^{e} } \right] \hfill \\ \left[ {K_{{F1}}^{e} } \right] & = \left[ {K_{{25}}^{e} } \right]^{{ - 1}} \left[ {K_{{\phi \phi }}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} ;\left[ {K_{{F2}}^{e} } \right] = \left[ {K_{{15}}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \hfill \\ \left[ {K_{{F3}}^{e} } \right] & = \left[ {K_{{32}}^{e} } \right]\left[ {K_{{25}}^{e} } \right]^{{ - 1}} ;\left[ {K_{{F4}}^{e} } \right] = \left[ {K_{{32}}^{e} } \right]\left[ {K_{{F1}}^{e} } \right] \hfill \\ \left[ {K_{{F5}}^{e} } \right] & = \left[ {K_{{F2}}^{e} } \right] - \left[ {K_{{F4}}^{e} } \right];\left[ {K_{{F6}}^{e} } \right] = \left[ {K_{{35}}^{e} } \right]^{{ - 1}} \left[ {K_{{F5}}^{e} } \right] \hfill \\ \left[ {K_{{F7}}^{e} } \right] & = \left[ {K_{{35}}^{e} } \right]^{{ - 1}} \left[ {K_{{F3}}^{e} } \right];\left[ {K_{{F8}}^{e} } \right] = \left[ {K_{{10}}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \hfill \\ \left[ {K_{{F9}}^{e} } \right] & = \left[ {K_{{41}}^{e} } \right]\left[ {K_{{F1}}^{e} } \right];\left[ {K_{{F10}}^{e} } \right] = \left[ {K_{{41}}^{e} } \right]\left[ {K_{{25}}^{e} } \right]^{{ - 1}} ; \hfill \\ \left[ {K_{{F11}}^{e} } \right] & = \left[ {K_{{F8}}^{e} } \right] - \left[ {K_{{F9}}^{e} } \right];\left[ {K_{{F12}}^{e} } \right] = \left[ {K_{{44}}^{e} } \right]\left[ {K_{{F6}}^{e} } \right] \hfill \\ \left[ {K_{{F13}}^{e} } \right] & = \left[ {K_{{44}}^{e} } \right]\left[ {K_{{F7}}^{e} } \right];\left[ {K_{{F14}}^{e} } \right] = \left[ {K_{{F10}}^{e} } \right] - \left[ {K_{{F14}}^{e} } \right] \hfill \\ \left[ {K_{{F15}}^{e} } \right]& = \left[ {K_{{F11}}^{e} } \right] - \left[ {K_{{F12}}^{e} } \right];\left[ {K_{{F16}}^{e} } \right] = \left[ {K_{{46}}^{e} } \right]^{{ - 1}} \left[ {K_{{F14}}^{e} } \right]; \hfill \\ \left[ {K_{{F17}}^{e} } \right] & = \left[ {K_{{46}}^{e} } \right]^{{ - 1}} \left[ {K_{{F15}}^{e} } \right];\left[ {K_{{F18}}^{e} } \right] = \left[ {K_{5}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} \hfill \\ \left[ {K_{{F19}}^{e} } \right] & = \left[ {K_{{NL\_5}}^{e} } \right]\left[ {K_{{\psi \psi }}^{e} } \right]^{{ - 1}} ;\left[ {K_{{F20}}^{e} } \right] = \left[ {K_{{F18}}^{e} } \right] + \left[ {K_{{F19}}^{e} } \right] \hfill \\ \left[ {K_{{F21}}^{e} } \right] & = \left[ {K_{{51}}^{e} } \right]\left[ {K_{{F1}}^{e} } \right];\left[ {K_{{F22}}^{e} } \right] = \left[ {K_{{51}}^{e} } \right]\left[ {K_{{25}}^{e} } \right]^{{ - 1}} \hfill \\ \left[ {K_{{F23}}^{e} } \right] & = \left[ {K_{{NL\_27}}^{e} } \right]\left[ {K_{{F1}}^{e} } \right];\left[ {K_{{F24}}^{e} } \right] = \left[ {K_{{NL\_27}}^{e} } \right]\left[ {K_{{25}}^{e} } \right]^{{ - 1}} \hfill \\ \left[ {K_{{F25}}^{e} } \right] & = \left[ {K_{{F22}}^{e} } \right] + \left[ {K_{{F24}}^{e} } \right];\left[ {K_{{F26}}^{e} } \right] = \left[ {K_{{F20}}^{e} } \right] - \left[ {K_{{F21}}^{e} } \right] - \left[ {K_{{F23}}^{e} } \right] \hfill \\ \left[ {K_{{F27}}^{e} } \right] & = \left[ {K_{{54}}^{e} } \right]\left[ {K_{{F6}}^{e} } \right];\left[ {K_{{F28}}^{e} } \right] = \left[ {K_{{54}}^{e} } \right]\left[ {K_{{F7}}^{e} } \right] \hfill \\ \left[ {K_{{F29}}^{e} } \right] & = \left[ {K_{{NL\_31}}^{e} } \right]\left[ {K_{{F6}}^{e} } \right];\left[ {K_{{F30}}^{e} } \right] = \left[ {K_{{NL\_31}}^{e} } \right]\left[ {K_{{F7}}^{e} } \right] \hfill \\ \left[ {K_{{F31}}^{e} } \right] & = \left[ {K_{{F25}}^{e} } \right] - \left[ {K_{{F30}}^{e} } \right] - \left[ {K_{{F28}}^{e} } \right]; \hfill \\ \left[ {K_{{F32}}^{e} } \right] & = \left[ {K_{{F26}}^{e} } \right] - \left[ {K_{{F29}}^{e} } \right] - \left[ {K_{{F27}}^{e} } \right];\left[ {K_{{F33}}^{e} } \right] = \left[ {K_{{56}}^{e} } \right]\left[ {K_{{F16}}^{e} } \right];\left[ {K_{{F34}}^{e} } \right] = \left[ {K_{{56}}^{e} } \right]\left[ {K_{{F17}}^{e} } \right]; \hfill \\ \left[ {K_{{F35}}^{e} } \right] & = \left[ {K_{{NL\_34}}^{e} } \right]\left[ {K_{{F16}}^{e} } \right];\left[ {K_{{F36}}^{e} } \right] = \left[ {K_{{NL\_34}}^{e} } \right]\left[ {K_{{F17}}^{e} } \right] \hfill \\ \end{aligned} $$
(A3)

The different stiffness matrices appearing in the condensation process are as follows:

$$ \begin{gathered} \left[ {K_{{rrs35}}^{e} } \right] = \left[ {K_{{rrs3}}^{e} } \right] + \left[ {K_{{rrs5}}^{e} } \right],\left[ {K_{{rrs13}}^{e} } \right] = \left[ {K_{{rrs1}}^{e} } \right] + \left[ {K_{{rrs3}}^{e} } \right], \hfill \\ \left[ {K_{{rrs3513}}^{e} } \right] = \left[ {K_{{rrs35}}^{e} } \right] + \left[ {K_{{rrs13}}^{e} } \right],\left[ {K_{{rts13}}^{e} } \right] = \left[ {K_{{rts1}}^{e} } \right] + \left[ {K_{{rts3}}^{e} } \right], \hfill \\ \left[ {K_{{_{{r\psi s13}} }}^{e} } \right] = \left[ {K_{{_{{r\psi s1}} }}^{e} } \right] + \left[ {K_{{_{{r\psi s3}} }}^{e} } \right],\left[ {K_{{_{{r\phi s13}} }}^{e} } \right] = \left[ {K_{{_{{r\phi s1}} }}^{e} } \right] + \left[ {K_{{_{{r\phi s3}} }}^{e} } \right] \hfill \\ \left[ {K_{{tbtbNL1}}^{e} } \right] = \left[ {K_{{tb1}}^{e} } \right] + \left[ {K_{{tbNL1}}^{e} } \right],\left[ {K_{{rtb24}}^{e} } \right] = \left[ {K_{{rtb2}}^{e} } \right] + \left[ {K_{{rtb4}}^{e} } \right] \hfill \\ \left[ {K_{{tbNLbNL1\_tbtbNL1}}^{e} } \right] = \left[ {K_{{tbNLbNL1}}^{e} } \right] + \left[ {K_{{tbtbNL1}}^{e} } \right], \hfill \\ \left[ {K_{{rbNL\_rtb24}}^{e} } \right] = \left[ {K_{{rbNL24}}^{e} } \right] + \left[ {K_{{rtb24}}^{e} } \right],\left[ {K_{{rbNL\_rtb4}}^{e} } \right] = \left[ {K_{{rbNL4}}^{e} } \right] + \left[ {K_{{rtb4}}^{e} } \right], \hfill \\ \left[ {K_{{bNL\_tb\phi 1}}^{e} } \right] = \left[ {K_{{bNL\phi 1}}^{e} } \right] + \left[ {K_{{tb\phi 1}}^{e} } \right],\left[ {K_{{bNL\_tb\psi 1}}^{e} } \right] = \left[ {K_{{bNL\psi 1}}^{e} } \right] + \left[ {K_{{tb\psi 1}}^{e} } \right] \hfill \\ \left[ {K_{{rrb57}}^{e} } \right] = \left[ {K_{{rrb5}}^{e} } \right] + \left[ {K_{{rrb7}}^{e} } \right],\left[ {K_{{rtbrbNL4}}^{e} } \right] = \left[ {K_{{rtb4}}^{e} } \right] + \left[ {K_{{rbNL4}}^{e} } \right] \hfill \\ \left[ {K_{{rtbrbNL2}}^{e} } \right] = \left[ {K_{{rtb2}}^{e} } \right] + \left[ {K_{{rbNL2}}^{e} } \right],\left[ {K_{{rtbrbNL24}}^{e} } \right] = \left[ {K_{{rtbrbNL2}}^{e} } \right] + \left[ {K_{{rtbrbNL4}}^{e} } \right] \hfill \\ \left[ {K_{{rrb35}}^{e} } \right] = \left[ {K_{{rrb3}}^{e} } \right] + \left[ {K_{{rrb5}}^{e} } \right],\left[ {K_{{rrb5735}}^{e} } \right] = \left[ {K_{{rrb57}}^{e} } \right] + \left[ {K_{{rrb35}}^{e} } \right] \hfill \\ \left[ {K_{{rb\phi 24}}^{e} } \right] = \left[ {K_{{rb\phi 2}}^{e} } \right] + \left[ {K_{{rb\phi 4}}^{e} } \right],\left[ {K_{{rb\psi 24}}^{e} } \right] = \left[ {K_{{rb\psi 2}}^{e} } \right] + \left[ {K_{{rb\psi 4}}^{e} } \right] \hfill \\ \left[ {K_{{tbNLbNL1}}^{e} } \right] = \left[ {K_{{tbNL1}}^{e} } \right] + \left[ {K_{{bNL1}}^{e} } \right],\left[ {K_{{rbNL24}}^{e} } \right] = \left[ {K_{{rbNL2}}^{e} } \right] + \left[ {K_{{rbNL4}}^{e} } \right] \hfill \\ \end{gathered} $$
(A4)

The expressions for stiffness matrices and force vectors are as follows:

$$ \begin{aligned} \left[ {K_{{rtb4}}^{e} } \right] & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rb}} } \right]^{T} \left[ {D_{{b4}} } \right]} } \left[ {B_{{tb}} } \right]{\text{ }}dxdy;\left[ {K_{{rbNL4}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rb}} } \right]^{T} \left[ {D_{{bNL4}} } \right]} } \left[ {B_{1} } \right]{\text{ }}\left[ {B_{2} } \right]{\text{ }}dxdy \hfill \\ \left[ {K_{{rrb5}}^{e} } \right] & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rb}} } \right]^{T} \left[ {D_{{b5}} } \right]} } \left[ {B_{{rb}} } \right]{\text{ }}dxdy;\left[ {K_{{rrb7}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rb}} } \right]^{T} \left[ {D_{{b7}} } \right]} } \left[ {B_{{rb}} } \right]{\text{ }}dxdy \hfill \\ \left[ {K_{{rb\phi 4}}^{e} } \right] & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rb}} } \right]^{T} \left[ {D_{{b\phi 4}} } \right]} } \left[ {B_{\phi } } \right]{\text{ }}dxdy;\left[ {K_{{rb\psi 4}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rb}} } \right]^{T} \left[ {D_{{b\psi 4}} } \right]} } \left[ {B_{\psi } } \right]{\text{ }}dxdy \hfill \\ \left[ {K_{{rtb4}}^{e} } \right] & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rb}} } \right]^{T} \left[ {D_{{b4}} } \right]} } \left[ {B_{{tb}} } \right]{\text{ }}dxdy;\left[ {K_{{rtb2}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rb}} } \right]^{T} \left[ {D_{{b2}} } \right]} } \left[ {B_{{tb}} } \right]{\text{ }}dxdy \hfill \\ \left[ {K_{{rbNL2}}^{e} } \right] & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rb}} } \right]^{T} \left[ {D_{{bNL2}} } \right]} } \left[ {B_{1} } \right]{\text{ }}\left[ {B_{2} } \right]{\text{ }}dxdy;\left[ {K_{{rrb3}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rb}} } \right]^{T} \left[ {D_{{b3}} } \right]} } \left[ {B_{{rb}} } \right]{\text{ }}dxdy \hfill \\ \left[ {K_{{rrb5}}^{e} } \right] & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rb}} } \right]^{T} \left[ {D_{{b5}} } \right]} } \left[ {B_{{rb}} } \right]{\text{ }}dxdy;\left[ {K_{{rb\phi 2}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rb}} } \right]^{T} \left[ {D_{{b\phi 2}} } \right]} } \left[ {B_{\phi } } \right]{\text{ }}dxdy \hfill \\ \left[ {K_{{rb\psi 2}}^{e} } \right] & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rb}} } \right]^{T} \left[ {D_{{b\psi 2}} } \right]} } \left[ {B_{\psi } } \right]{\text{ }}dxdy;\left[ {K_{{tbNL1}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{tb}} } \right]^{T} \left[ {D_{{bNL1}} } \right]} } \left[ {B_{1} } \right]{\text{ }}\left[ {B_{2} } \right]{\text{ }}dxdy \hfill \\ \left[ {K_{{bNL1}}^{e} } \right] & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{2} } \right]{\text{ }}^{T} \left[ {B_{1} } \right]{\text{ }}^{T} \left[ {D_{{bbNL1}} } \right]} } \left[ {B_{1} } \right]{\text{ }}\left[ {B_{2} } \right]{\text{ }}dxdy; \hfill \\ \left[ {K_{{bNL\phi 1}}^{e} } \right] & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\phi } } \right]^{T} \left[ {D_{{bNL\phi 1}} } \right]} } \left[ {B_{1} } \right]{\text{ }}\left[ {B_{2} } \right]{\text{ }}dxdy;\hfill\\ \left[ {K_{{bNL\psi 1}}^{e} } \right] & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\psi } } \right]^{T} \left[ {D_{{bNL\psi 1}} } \right]} } \left[ {B_{1} } \right]{\text{ }}\left[ {B_{2} } \right]{\text{ }}dxdy \hfill \\ \left[ {K_{{tb1}}^{e} } \right] & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{tb}} } \right]^{T} \left[ {D_{{b1}} } \right]} } \left[ {B_{{tb}} } \right]{\text{ }}dxdy;\left[ {K_{{tb\phi 1}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{tb}} } \right]^{T} \left[ {D_{{b\phi 1}} } \right]} } \left[ {B_{\phi } } \right]{\text{ }}dxdy \hfill \\ \left[ {K_{{tb\psi 1}}^{e} } \right] & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{tb}} } \right]^{T} \left[ {D_{{b\psi 1}} } \right]} } \left[ {B_{\psi } } \right]{\text{ }}dxdy;\left[ {K_{{_{{rts3}} }}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rs}} } \right]^{T} \left[ {D_{{s3}} } \right]\left[ {B_{{ts}} } \right]} } {\text{ }}dxdy \hfill \\ \left[ {K_{{rrs3}} } \right] & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rs}} } \right]^{T} \left[ {D_{{s3}} } \right]\left[ {B_{{rs}} } \right]} } {\text{ }}dxdy;\left[ {K_{{_{{r\phi s3}} }}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rs}} } \right]^{T} \left[ {D_{{s\phi 3}} } \right]\left[ {B_{\phi } } \right]} } {\text{ }}dxdy \hfill \\ \left[ {K_{{ts\phi 1}}^{e} } \right] & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{ts}} } \right]^{T} \left[ {D_{{s\phi 1}} } \right]} } \left[ {B_{\phi } } \right]{\text{ }}dxdy;\left[ {K_{{ts\psi 1}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{ts}} } \right]^{T} \left[ {D_{{s\psi 1}} } \right]} } \left[ {B_{\psi } } \right]{\text{ }}dxdy \hfill \\ \left[ {K_{{_{{r\phi s1}} }}^{e} } \right] & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rs}} } \right]^{T} \left[ {D_{{s\phi 1}} } \right]\left[ {B_{\phi } } \right]} } {\text{ }}dxdy;\left[ {K_{{r\psi s1}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rs}} } \right]^{T} \left[ {D_{{s\psi 1}} } \right]\left[ {B_{\psi } } \right]} } {\text{ }}dxdy \hfill \\ \left[ {K_{{_{{rts1}} }}^{e} } \right] & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rs}} } \right]^{T} \left[ {D_{{s1}} } \right]\left[ {B_{{ts}} } \right]} } {\text{ }}dxdy;\left[ {K_{{_{{rrs1}} }}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rs}} } \right]^{T} \left[ {D_{{s1}} } \right]\left[ {B_{{rs}} } \right]} } {\text{ }}dxdy \hfill \\ \left[ {K_{{ts1}} } \right] & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{ts}} } \right]^{T} \left[ {D_{{s1}} } \right]\left[ {B_{{ts}} } \right]} } {\text{ }}dxdy;\left[ {K_{{_{{rts3}} }}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{rs}} } \right]^{T} \left[ {D_{{s3}} } \right]\left[ {B_{{ts}} } \right]} } {\text{ }}dxdy \hfill \\ \left\{ {F_{\psi } } \right\} & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {N_{\psi } } \right]^{T} Q^{\psi } } } {\text{ }}dxdy;\left\{ {F_{\phi } } \right\} = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {N_{\phi } } \right]^{T} Q^{\phi } } } {\text{ }}dxdy; \hfill \\ \left\{ {F_{t} } \right\}& = \int\limits_{{\Omega ^{N} }} {\left[ {C_{b} } \right]^{N} \left[ \alpha \right]^{N} \Delta T} d\Omega ^{N} ;\left\{ {F_{{t\phi }} } \right\} = \int\limits_{{\Omega ^{N} }} {\left[ p \right]\Delta T} d\Omega ^{N} ;\left\{ {F_{{t\psi }} } \right\} = \int\limits_{{\Omega ^{N} }} {\left[ \tau \right]\Delta T} d\Omega ^{N} ; \hfill \\ \end{aligned} $$
(A5)

The various rigidity matrices contributing to Eq. (A5) can be denoted as follows:

$$ \begin{aligned} \left[ {D_{{b1}} } \right] & = \sum\limits_{{N = 1}}^{3} {\int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\left[ {C_{b} } \right]^{N} } {\text{ }}dz;}\,\,\, \left[ {D_{{bbNL1}} } \right] = \frac{1}{4}\left[ {D_{{b1}} } \right];\left[ {D_{{bNL1}} } \right] = \frac{1}{2}\left[ {D_{{b1}} } \right]; \hfill \\ \left[ {D_{{b2}} } \right] & = \sum\limits_{{N = 1}}^{3} {\int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {z\left[ {C_{b} } \right]^{{\text{N}}} {\text{ }}} dz} ;\left[ {D_{{bNL2}} } \right] = \frac{1}{2}\left[ {D_{{b2}} } \right];\left[ {D_{{b3}} } \right] = \sum\limits_{{N = 1}}^{3} {\int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {z^{2} \left[ {C_{b} } \right]^{N} } dz;} \hfill \\ \left[ {D_{{b4}} } \right] & = \sum\limits_{{N = 1}}^{3} {\int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {c_{1} z^{3} \left[ {C_{b} } \right]^{N} } dz;}\,\,\, \left[ {D_{{bNL4}} } \right] = \frac{1}{2}\left[ {D_{{b4}} } \right];\left[ {D_{{b5}} } \right] = \sum\limits_{{N = 1}}^{3} {\int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {c_{1} z^{4} \left[ {C_{b} } \right]^{N} } dz;} \hfill \\ \left[ {D_{{b7}} } \right] & = \sum\limits_{{N = 1}}^{3} {\int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {c_{1} ^{2} z^{6} \left[ {C_{b} } \right]^{N} } dz;} \left[ {D_{{b\phi 1}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {\left[ {Z_{t} } \right]\left[ {e_{b} } \right]} dz + \int\limits_{{h_{1} }}^{{h_{2} }} {\left[ {Z_{b} } \right]\left[ {e_{b} } \right]} dz; \hfill \\ \left[ {D_{{bNL\phi 1}} } \right] & = \left[ {D_{{b\phi 1}} } \right];\left[ {D_{{b\phi 2}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {z\left[ {Z_{t} } \right]\left[ {e_{b} } \right]} dz + \int\limits_{{h_{1} }}^{{h_{2} }} {z\left[ {Z_{b} } \right]\left[ {e_{b} } \right]} dz; \hfill \\ \left[ {D_{{b\phi 4}} } \right] & = \int\limits_{{h_{3} }}^{{h_{4} }} {c_{1} z^{3} \left[ {Z_{t} } \right]\left[ {e_{b} } \right]} dz + \int\limits_{{h_{1} }}^{{h_{2} }} {c_{1} z^{3} \left[ {Z_{b} } \right]\left[ {e_{b} } \right]} dz;\left[ {D_{{s1}} } \right] = \sum\limits_{{N = 1}}^{3} {\int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\left[ {C_{s} } \right]^{N} } dz;} \hfill \\ \left[ {D_{{b\psi 1}} } \right] & = \int\limits_{{h_{3} }}^{{h_{4} }} {\left[ {Z_{t} } \right]\left[ {q_{b} } \right]} dz + \int\limits_{{h_{1} }}^{{h_{2} }} {\left[ {Z_{b} } \right]\left[ {q_{b} } \right]} dz;\left[ {D_{{bNL\psi 1}} } \right] = \left[ {D_{{b\psi 1}} } \right] \hfill \\ \left[ {D_{{b\psi 2}} } \right] & = \int\limits_{{h_{3} }}^{{h_{4} }} {z\left[ {Z_{t} } \right]\left[ {q_{b} } \right]} dz + \int\limits_{{h_{1} }}^{{h_{2} }} {z\left[ {Z_{b} } \right]\left[ {q_{b} } \right]} dz; \hfill \\ \left[ {D_{{b\psi 4}} } \right] & = \int\limits_{{h_{3} }}^{{h_{4} }} {c_{1} z^{3} \left[ {Z_{t} } \right]\left[ {q_{b} } \right]} dz + \int\limits_{{h_{1} }}^{{h_{2} }} {c_{1} z^{3} \left[ {Z_{b} } \right]\left[ {q_{b} } \right]} dz;\left[ {D_{{s3}} } \right] = \sum\limits_{{N = 1}}^{3} {\int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {c_{2} z^{2} \left[ {C_{s} } \right]^{N} } {\text{ }}dz{\text{ }}} \hfill \\ \left[ {D_{{s5}} } \right] & = \sum\limits_{{N = 1}}^{3} {\int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {c_{2} ^{2} z^{4} \left[ {C_{s} } \right]^{N} } dz;} \left[ {D_{{s\phi 1}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {\left[ {Z_{t} } \right]\left[ {e_{s} } \right]^{T} } dz + \int\limits_{{h_{1} }}^{{h_{2} }} {\left[ {Z_{b} } \right]\left[ {e_{s} } \right]^{T} } dz; \hfill \\ \left[ {D_{{s\phi 3}} } \right] & = \int\limits_{{h_{3} }}^{{h_{4} }} {c_{2} z^{2} \left[ {Z_{t} } \right]\left[ {e_{s} } \right]^{T} } dz + \int\limits_{{h_{1} }}^{{h_{2} }} {c_{2} z^{2} \left[ {Z_{b} } \right]\left[ {e_{s} } \right]^{T} } dz; \hfill \\ \left[ {D_{{s\psi 1}} } \right] & = \int\limits_{{h_{3} }}^{{h_{4} }} {\left[ {Z_{t} } \right]\left[ {q_{s} } \right]^{T} } dz + \int\limits_{{h_{1} }}^{{h_{2} }} {\left[ {Z_{b} } \right]\left[ {q_{s} } \right]^{T} } dz; \hfill \\ \left[ {D_{{s\psi 3}} } \right] & = \int\limits_{{h_{3} }}^{{h_{4} }} {c_{2} z^{2} \left[ {Z_{t} } \right]\left[ {q_{s} } \right]^{T} } dz + \int\limits_{{h_{1} }}^{{h_{2} }} {c_{2} z^{2} \left[ {Z_{b} } \right]\left[ {q_{s} } \right]^{T} } dz; \hfill \\ \left[ {D_{{\phi \phi }} } \right] & = \int\limits_{{h_{3} }}^{{h_{4} }} {\left[ {Z_{t} } \right]^{T} \left[ \eta \right]\left[ {Z_{t} } \right]} dz + \int\limits_{{h_{1} }}^{{h_{2} }} {\left[ {Z_{b} } \right]^{T} \left[ \eta \right]} \left[ {Z_{b} } \right]dz; \hfill \\ \left[ {D_{{\psi \psi }} } \right] & = \int\limits_{{h_{3} }}^{{h_{4} }} {\left[ {Z_{t} } \right]^{T} \left[ \mu \right]\left[ {Z_{t} } \right]} dz + \int\limits_{{h_{1} }}^{{h_{2} }} {\left[ {Z_{b} } \right]^{T} \left[ \mu \right]} \left[ {Z_{b} } \right]dz; \hfill \\ \left[ {D_{{\phi \psi }} } \right] & = \int\limits_{{h_{3} }}^{{h_{4} }} {\left[ {Z_{t} } \right]^{T} \left[ m \right]\left[ {Z_{t} } \right]} dz + \int\limits_{{h_{1} }}^{{h_{2} }} {\left[ {Z_{b} } \right]^{T} \left[ m \right]} \left[ {Z_{b} } \right]dz; \hfill \\ \end{aligned} $$
(A6)

The derivative of shape function matrices appearing in the FE formulation can be represented by

$$ \begin{aligned} \left[ {B_{{tb}} } \right] & = \left[ {\begin{array}{*{20}l} {N_{{i,x}} } \hfill & 0 \hfill & {\frac{1}{{R_{1} }}} \hfill \\ 0 \hfill & {N_{{i,y}} } \hfill & {\frac{1}{{R_{2} }}} \hfill \\ {N_{{i,y}} } \hfill & {N_{{i,x}} } \hfill & 0 \hfill \\ \end{array} } \right],\left[ {B_{{rb}} } \right] = \left[ {\begin{array}{*{20}l} {N_{{i,x}} } \hfill & 0 \hfill \\ 0 \hfill & {N_{{i,y}} } \hfill \\ {N_{{i,y}} } \hfill & {N_{{i,x}} } \hfill \\ \end{array} } \right],\left[ {B_{{ts}} } \right] = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & {N_{{i,x}} } \hfill \\ 0 \hfill & 0 \hfill & {N_{{i,y}} } \hfill \\ \end{array} } \right],\left[ {B_{{rs}} } \right] = \left[ {\begin{array}{*{20}l} 1 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill \\ \end{array} } \right] \hfill \\ \left[ {B_{1} } \right] &= \left[ {\begin{array}{*{20}l} {W_{{0,x}} } \hfill & 0 \hfill & {W_{{0,y}} } \hfill \\ 0 \hfill & {W_{{0,y}} } \hfill & {W_{{0,x}} } \hfill \\ \end{array} } \right],\left[ {B_{2} } \right] = \left[ {\begin{array}{*{20}l} {B_{{21}} } \hfill & {B_{{22}} } \hfill & \ldots \hfill & {B_{{28}} } \hfill \\ \end{array} } \right]; \hfill \\ \left[ {B_{\phi } } \right] & = \left[ {B_{\psi } } \right] = \left[ {\begin{array}{*{20}l} {N_{{i,x}} } \hfill \\ {N_{{i,y}} } \hfill \\ {N_{{i,z}} } \hfill \\ \end{array} } \right] \hfill \\ \end{aligned} $$
(A7)

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Mahesh, V. Nonlinear pyrocoupled deflection of viscoelastic sandwich shell with CNT reinforced magneto-electro-elastic facing subjected to electromagnetic loads in thermal environment. Eur. Phys. J. Plus 136, 796 (2021). https://doi.org/10.1140/epjp/s13360-021-01751-y

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