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Electro-Magneto-Elastic Response of Laminated Composite Plate: A Finite Element Approach

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Abstract

The flexural behaviour of the laminated composite plate embedded with two different smart materials (piezoelectric and magnetostrictive) and subsequent deflection suppression have been investigated in this article. The mathematical model of the laminated composite plate embedded with and without smart materials are developed using the higher-order shear deformation kinematics in conjunction with finite element steps. The plate is assumed to be subjected to the combined effect of the mechanical load, electrical potential and the magnetic field induction. The desired responses are computed numerically with the help of a homemade computer code developed in MATLAB environment in association with the present finite element formulation. The convergence and the validity of the presently computed numerical responses have been established by comparing the responses with those available numerical and analytical results. Further, the desired responses of the laminated composite structure bonded with and without functional materials are computed using the commercial finite element package (ANSYS) and compared with the present numerical results. Finally, the present higher-order model is extended to examine the static responses of the laminated composite plate embedded with piezo and magnetostrictive material by solving the wide variety of numerical examples for different design parameters. The degree of deflection suppression capability due to the smart layers has been underlined and discussed in details.

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Author information

Authors and Affiliations

  1. School of Mechanical Engineering, KIIT University, Bhubaneswar, India

    Goutam Dutta & Trupti R. Mahapatra

  2. Department of Mechanical Engineering, National Institute of Technology, Rourkela, India

    Subrata K. Panda & Vijay K. Singh

Authors
  1. Goutam Dutta
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  2. Subrata K. Panda
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  3. Trupti R. Mahapatra
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  4. Vijay K. Singh
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Corresponding author

Correspondence to Subrata K. Panda.

Appendix

Appendix

Linear mid-plane strain terms

$$\begin{aligned} \varepsilon _{_{xx} } ^{0}=u,_x , \quad \varepsilon _{_{yy} } ^{0}=v,_{\mathrm{y}}, \!\quad \varepsilon _{_{zz} } ^{0}=w,_z , \quad \gamma _{_{yz} } ^{0}=v,_z +\, w,_y,\! \quad \gamma _{_{xz} } ^{0}=u,_z +\, w,_x , \!\quad \gamma _{_{xy} } ^{0}=u,_y +\, v,_x \end{aligned}$$

Individual terms of matrix [B]

$$\begin{aligned}&{[}\hbox {B}]_{ 1\mathrm{x}1}={\partial /\partial x}, [\hbox {B}]_{ 1\mathrm{x}3}=1/R_{\mathrm{x}}; [\hbox {B}]_{ 2\mathrm{x}2}={ \partial /\partial y}, [\hbox {B}]_{ 2\mathrm{x}3}= -1/R_{\mathrm{y}}; [\hbox {B}]_{ 3\mathrm{x}6}=1; \\&{[}\hbox {B}]_{ 4\mathrm{x}2= }-1/R_{\mathrm{y}}, [\hbox {B}]_{4\mathrm{x}3}={ \partial /\partial y}, [\hbox {B}]_{4\mathrm{x}5}=1; [\hbox {B}]_{ 5\mathrm{x}1}=-1/R_{\mathrm{x}}, [\hbox {B}]_{ 5\mathrm{x}3}={ \partial /\partial x},\\&{[}\hbox {B}]_{ 5\mathrm{x}4}=1; [\hbox {B}]_{ 6\mathrm{x}1}={ \partial /\partial y}, [\hbox {B}]_{ 6\mathrm{x}2}={ \partial /\partial x,} [\hbox {B}]_{ 6\mathrm{x}3}=2/R_{xy}; [\hbox {B}]_{ 7\mathrm{x}4}={ \partial /\partial x},\\&{[}\hbox {B}]_{ 7\mathrm{x}6}=1/R_{\mathrm{x}}; [\hbox {B}]_{ 8\mathrm{x}5}={ \partial /\partial y},[\hbox {B}]_{ 8\mathrm{x}6}=1/R_{\mathrm{y}}; [\hbox {B}]_{ 9\mathrm{x}5}=-1/R_{\mathrm{y}}, \\&{[}\hbox {B}]_{ 9\mathrm{x}6}={ \partial /\partial y}, [\hbox {B}]_{ 9\mathrm{x}8}=2; [\hbox {B}]_{ 10\mathrm{x}4}= -1/R_{\mathrm{x},} [\hbox {B}]_{ 10\mathrm{x}6}={ \partial /\partial x}, [\hbox {B}]_{ 10\mathrm{x}7}=2; \\&{[}\hbox {B}]_{ 11\mathrm{x}4}= { \partial /\partial y}, [\hbox {B}]_{11\mathrm{x}5}= { \partial /\partial x,} [\hbox {B}]_{ 11\mathrm{x}6}=2/R_{xy}; [\hbox {B}]_{ 12\mathrm{x}7}= { \partial /\partial x;}\\&{[}\hbox {B}]_{ 13\mathrm{x}8}={ \partial /\partial y}; [\hbox {B}]_{ 14\mathrm{x}8}= -1/R_{\mathrm{y},} [\hbox {B}]_{ 14\mathrm{x}10}=3; [\hbox {B}]_{15\mathrm{x}7}= -1/R_{\mathrm{x}}, [\hbox {B}]_{ 15\mathrm{x}9 }=3; \\&{[}\hbox {B}]_{16\mathrm{x}7}={ \partial /\partial y}, [\hbox {B}]_{ 16\mathrm{x}8}= { \partial /\partial x}; [\hbox {B}]_{ 17\mathrm{x}9}={ \partial /\partial x}; [\hbox {B}]_{ 18\mathrm{x}10}= { \partial /\partial y};\\&{[}\hbox {B}]_{18\mathrm{x}10}=-1/R_{\mathrm{y}}; [\hbox {B}]_{ 20\mathrm{x}9}= -1/R_{\mathrm{x}}; [\hbox {B}]_{ 21\mathrm{x}9}={ \partial /\partial y}, [\hbox {B}]_{ 21\mathrm{x}10}={ \partial /\partial x}; \end{aligned}$$

Thickness coordinate matrix

$$\begin{aligned} \left[ {T^{L}} \right] =\left[ {{\begin{array}{lllllllllllllllllllll} 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad z&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{2}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{3}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad z&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{2}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{3}}&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad z&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{2}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{3}}&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad z&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{2}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{3}}&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad z&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{2}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {z^{3}}\\ \end{array} }} \right] \end{aligned}$$

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Dutta, G., Panda, S.K., Mahapatra, T.R. et al. Electro-Magneto-Elastic Response of Laminated Composite Plate: A Finite Element Approach. Int. J. Appl. Comput. Math 3, 2573–2592 (2017). https://doi.org/10.1007/s40819-016-0256-6

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