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Electro-magneto-mechanical formulation of a sandwich shell subjected to electro-magneto-mechanical considering thickness stretching

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Abstract

Thickness stretching included formulation of a multi-layered doubly curved shell in small scale is studied in the present work. Out-of-plane normal strain is accounted in our formulation based on a higher-order theory. Based on this theory, the total transverse deflection is divided into three portions named as bending, shear and stretching parts. Transient formulation of the nanoshell is derived using Hamilton’s principle and nonlocal formulation. The natural frequencies of the nanoshell are obtained in terms of main input parameters, such as initial electric and magnetic potentials, nonlocal parameters, aspect ratio, radii ratio and foundation parameters.

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Funding

This study was funded by X (4005279/025).

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Authors and Affiliations

  1. Department of Solid Mechanic, Faculty of Mechanical Engineering, University of Kashan, Kashan, 87317-51167, Iran

    Mohammad Arefi & Shayan Mannani

  2. Department of Engineering and Architecture, University of Parma, Parco Area delle Scienze 181/A, 43124, Parma, Italy

    Mohammad Arefi & L. Collini

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  1. Mohammad Arefi
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Correspondence to Mohammad Arefi.

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Appendix A

Appendix A

$$q_{out} = q\left[ {1 + \frac{h}{{2R_{1} }}} \right]\left[ {1 + \frac{h}{{2R_{2} }}} \right], q_{in} = R_{f} \left[ {1 + \frac{h}{{2R_{1} }}} \right]\left[ {1 + \frac{h}{{2R_{2} }}} \right]g_{{x_{3} = - \frac{h}{2}}}$$
(A.1)
$$\overline{{N_{{0x_{1} }} }} = \left( {N_{{0x_{1} }} + N_{{E0x_{1} }} + N_{{M0x_{1} }} } \right),\overline{{N_{{0x_{2} }} }} = \left( {N_{{0x_{2} }} + N_{{E0x_{2} }} + N_{{M0x_{2} }} } \right)$$
(A.2)
$$R_{f} = K_{w} w - K_{G} \nabla^{2} w$$
(A.3)
$$\left\{ {N_{{E0x_{2} }} ,N_{{M0x_{2} }} } \right\} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2} - h_{p} }}^{{ + \frac{{h_{c} }}{2} + h_{p} }} \frac{2}{h}Y_{2} \left\{ {e_{{32}} \Psi _{0} ,q_{{32}} {{\Phi }}_{0} } \right\}dx_{3}$$
(A.4)
$$\left\{ {N_{1} ,M_{1} ,S_{1} ,P_{1} } \right\} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2} - h_{p} }}^{{ + \frac{{h_{c} }}{2} + h_{p} }} \sigma_{1} Y_{1} \left\{ {1,x_{3} ,f\left( {x_{3} } \right),g\left( {x_{3} } \right)} \right\}dx_{3}$$
(A.5)
$$\left\{ {N_{2} ,M_{2} ,S_{2} ,P_{2} } \right\} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2} - h_{p} }}^{{ + \frac{{h_{c} }}{2} + h_{p} }} \sigma_{2} Y_{2} \left\{ {1,x_{3} ,f\left( {x_{3} } \right),g\left( {x_{3} } \right)} \right\}dz$$
(A.6)
$$\left\{ {N_{13} ,M_{13} ,S_{13} } \right\} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2} - h_{p} }}^{{ + \frac{{h_{c} }}{2} + h_{p} }} \sigma_{13} \left\{ {\frac{{Y_{1} }}{{R_{1} }},x_{3} \left\{ { - f^{\prime}\left( {x_{3} } \right) + 1} \right\},g\left( {x_{3} } \right)Y_{1} } \right\}dx_{3}$$
(A.7)
$$\left\{ {N_{23} ,M_{23} ,S_{23} } \right\} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2} - h_{p} }}^{{ + \frac{{h_{c} }}{2} + h_{p} }} \sigma_{23} \left\{ {\frac{{Y_{2} }}{{R_{2} }},x_{3} \left\{ { - f^{\prime}\left( {x_{3} } \right) + 1} \right\},g\left( {x_{3} } \right)Y_{2} } \right\}dx_{3}$$
(A.8)
$$\left\{ {N_{12} ,N_{21} ,M_{12} ,S_{12} } \right\} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2} - h_{p} }}^{{ + \frac{{h_{c} }}{2} + h_{p} }} \sigma_{12} \left\{ {Y_{1} ,Y_{2} ,x_{3} \left\{ {Y_{1} + Y_{2} } \right\},f\left( {x_{3} } \right)} \right\}dx_{3}$$
(A.9)
$$\left\{ {G_{3} } \right\} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2} - h_{p} }}^{{ + \frac{{h_{c} }}{2} + h_{p} }} \sigma_{3} g^{\prime}\left( {x_{3} } \right)dx_{3}$$
(A.10)
$$\left\{ {\overline{{D_{i} }} ,\overline{{B_{i} }} } \right\} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2} - h_{p} }}^{{ + \frac{{h_{c} }}{2} + h_{p} }} cos\frac{\pi z}{h}\left\{ {D_{i} Y_{i} ,B_{i} Y_{i} } \right\}dx_{3} ,i = 1,2$$
(A.11)
$$\left\{ {\overline{{D_{3} }} ,\overline{{B_{3} }} } \right\} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2} - h_{p} }}^{{ + \frac{{h_{c} }}{2} + h_{p} }} \frac{\pi }{h}sin\frac{{\pi x_{3} }}{h}\left\{ {D_{3} ,B_{3} } \right\}dx_{3} ,i = 1,2$$
(A.12)

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Arefi, M., Mannani, S. & Collini, L. Electro-magneto-mechanical formulation of a sandwich shell subjected to electro-magneto-mechanical considering thickness stretching. Archiv.Civ.Mech.Eng 22, 196 (2022). https://doi.org/10.1007/s43452-022-00514-5

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