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Nonlinear vibration of initially stressed hybrid composite plates on elastic foundations

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Mechanics of Composite Materials Aims and scope

Nonlinear vibration equations of motion based on the Mindlin plate theory are derived for a hybrid composite plate with an initial stress on elastic foundations. Using the governing equations derived, the nonlinear vibration behavior of an initially stressed hybrid composite plate on Pasternak and Winkler elastic foundations is studied. The initial stress is taken to be a combination of a pure bending stress and a tensile stress in the plane of the plate. The Galerkin method is employed to reduce the governing nonlinear partial differential equations to ordinary nonlinear differential equations, and the Runge–Kutta method is used to obtain the nonlinear frequencies. The linear frequency can be calculated by neglecting the nonlinear terms in the ordinary nonlinear differential equations. The results obtained reveal that the foundation stiffness and initial stresses lead to a drastic change in the nonlinear vibration behavior of the plate. The effects of various parameters on the nonlinear vibration of hybrid composite plates are investigated and discussed.

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Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Lunghwa University of Science and Technology, Guishan Shiang, 33306, Taiwan

    Ching-Long Wei & Chun-Sheng Chen

  2. Department of Mechanical Engineering, Hwa Hsia Institute of Technology, New Taipei City, 23568, Taiwan

    Chia-Sheng Shih

  3. Department of Civil Engineering, National Taipei University of Technology, Taipei, 10608, Taiwan

    Yi-Chen Chang

Authors
  1. Ching-Long Wei
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  2. Chun-Sheng Chen
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  3. Chia-Sheng Shih
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  4. Yi-Chen Chang
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Corresponding author

Correspondence to Chia-Sheng Shih.

Additional information

Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 48, No. 4, pp. 679-700, July-August, 2012.

Appendix

Appendix

$$ \begin{array}{*{20}{c}} {\left( {{u_{x,y}} + {u_{y,x}}} \right) + {B_{11}}{\varphi_{x,x}} + {B_{12}}\;{\varphi_{y,y}} + {B_{16}}\left( {{\varphi_{x,y}} + {\varphi_{y,x}}} \right),} \\ {{L_2} = {A_{12}}{u_{x,x}} + {A_{22}}\;{u_{y,y}} + {A_{26}}\left( {{u_{x,y}} + {u_{y,x}}} \right) + {B_{12}}{\varphi_{x,x}} + {B_{22}}\;{\varphi_{y,y}} + {B_{26}}\left( {{\varphi_{x,y}} + {\varphi_{y,x}}} \right),} \\ {{L_3} = {B_{11}}{u_{x,x}} + {B_{12}}\;{u_{y,y}} + {B_{16}}\left( {{u_{x,y}} + {u_{y,x}}} \right) + {D_{11}}{\varphi_{x,x}} + {D_{12}}\;{\varphi_{y,y}} + {D_{16}}\left( {{\varphi_{x,y}} + {\varphi_{y,x}}} \right),} \\ {{L_4} = {B_{12}}{u_{x,x}} + {B_{22}}\;{u_{y,y}} + {B_{26}}\left( {{u_{x,y}} + {u_{y,x}}} \right) + {D_{12}}{\varphi_{x,x}} + {D_{22}}\;{\varphi_{y,y}} + {D_{26}}\left( {{\varphi_{x,y}} + {\varphi_{y,x}}} \right),} \\ {{L_5} = {A_{16}}{u_{x,x}} + {A_{26}}{u_{y,y}} + {A_{66}}\left( {{u_{x,y}} + {u_{y,x}}} \right) + {B_{16}}{\varphi_{x,x}} + {B_{26}}{\varphi_{y,y}} + {B_{66}}\left( {{\varphi_{x,y}} + {\varphi_{y,x}}} \right),} \\ {{L_6} = {A_{45}}\left( {{w_{,y}} + {\varphi_y}} \right) + {A_{55}}\left( {{w_{,x}} + {\varphi_x}} \right),\;{L_7} = {A_{44}}\left( {{w_{,y}} + {\varphi_y}} \right) + {A_{45}}\left( {{w_{,x}} + {\varphi_x}} \right),} \\ {{L_8} = {B_{16}}{u_{x,x}} + {B_{26}}{u_{y,y}} + {B_{66}}\left( {{u_{x,y}} + {u_{y,x}}} \right) + {D_{16}}{\varphi_{x,x}} + {D_{26}}{\varphi_{y,y}} + {D_{66}}\left( {{\varphi_{x,y}} + {\varphi_{y,x}}} \right),} \\ \end{array} $$
$$ \begin{array}{*{20}{c}} {{N_1} = {A_{11}}{{{w_{,x}^2}} \left/ {{2 + {A_{12}}{{{w_{,y}^2}} \left/ {2} \right.} + {A_{16}}{w_{,x}}{w_{,y}},}} \right.}} \hfill \\ {{N_2} = {A_{12}}{{{w_{,x}^2}} \left/ {{2 + {A_{22}}{{{w_{,y}^2}} \left/ {2} \right.} + {A_{26}}{w_{,x}}{w_{,y}},}} \right.}} \hfill \\ {{N_3} = {A_{16}}{{{w_{,x}^2}} \left/ {{2 + {{{{A_{26}}w_{,y}^2}} \left/ {{2 + {A_{66}}{w_{,x}}{w_{,y}},\;{N_4} = \left( {{Q_1} + {L_1}} \right){w_{,x}},}} \right.}}} \right.}} \hfill \\ {{N_5} = \left( {{Q_5} + {L_3}} \right){w_{,x}},\;{N_6} = \left( {{Q_2} + {L_2}} \right){w_{,y}},\;{N_7} = \left( {{Q_5} + {L_3}} \right){w_{,y}},} \hfill \\ {{N_8} = {{{{B_{11}}w_{,x}^2}} \left/ {{2 + {B_{12}}{{{w_{,y}^2}} \left/ {{2 + {B_{16}}{w_{,x}}{w_{,y}},}} \right.}}} \right.}} \hfill \\ {{N_9} = {B_{12}}{{{w_{,x}^2}} \left/ {{2 + {{{{B_{22}}w_{,y}^2}} \left/ {{2 + {B_{26}}{w_{,x}}{w_{,y}},}} \right.}}} \right.}} \hfill \\ {{N_{10}} = {B_{16}}{{{w_{,x}^2}} \left/ {{2 + {{{{B_{26}}w_{,y}^2}} \left/ {{2 + {B_{66}}{w_{,x}}{w_{,y}},}} \right.}}} \right.}} \hfill \\ \end{array} $$
$$ \begin{array}{*{20}{c}} {{R_1} = {N_{xx}}{u_{x,x}} + {M_{xx}}{\varphi_{x,x}},\;{R_2} = {N_{xy}}{u_{x,y}} + {M_{xy}}{\varphi_{x,y}} + {N_{xz}}{u_{z,x}},} \hfill \\ {{R_3} = {N_{xx}}{u_{y,x}} + {M_{xx}}{\varphi_{y,x}} + {N_{xy}}{u_{y,y}} + {M_{xy}}{\varphi_{y,y}} + {N_{xz}}{u_{z,y}},\;{R_4} = {N_{xx}}{w_{,x}} + {N_{xy}}{w_{,y}},} \hfill \\ {{R_5} = {M_{xx}}{u_{x,x}},\;{R_6} = {M_{xy}}{u_{x,y}} + {M_{xz}}{\varphi_{z,x}},\;{R_7} = {M_{xx}}{u_{y,x}} + {M_{xy}}{u_{y,y}} + {M_{xz}}{u_{z,y}},} \hfill \\ {{R_8} = {N_{xz}}{u_{x,x}} + {M_{xz}}{\varphi_{x,x}} + {N_{zz}}{\varphi_x} + {N_{zy}}{u_{x,y}} + {M_{zy}}{\varphi_{x,y}},} \hfill \\ {{R_9} = {N_{xz}}{u_{y,x}} + {M_{xz}}{\varphi_{y,x}} + {N_{zz}}{\varphi_{yx}} + {N_{zy}}{u_{y,y}} + {M_{zy}}{\varphi_{y,y}},} \hfill \\ \end{array} $$
$$ \begin{array}{*{20}{c}} {{S_1} = {N_{yy}}{u_{x,y}} + {M_{yy}}{\varphi_{x,y}},\;{S_2} = {N_{xy}}{u_{x,x}} + {M_{xy}}{\varphi_{x,x}} + {N_{yz}}{u_{z,x}},} \\ {{S_3} = {N_{yy}}{u_{y,y}} + {M_{yy}}{\varphi_{y,y}} + {N_{xy}}{u_{y,x}} + {M_{xy}}{\varphi_{y,x}} + {N_{xz}}{u_{z,y}},\;{S_4} = {N_{xy}}{w_{,x}} + {N_{yy}}{w_{,y}},} \\ {{S_5} = {M_{yy}}{u_{x,y}},\;{S_6} = {M_{xy}}{u_{x,x}} + {M_{yz}}{u_{z,x}},\;{S_7} = {M_{yy}}{u_{y,y}} + {M_{xy}}{u_{y,x}} + {M_{xz}}{u_{z,y}},} \\ \end{array} $$
$$ \begin{array}{*{20}{c}} {{B_1} = {L_5} + {N_3} + {S_1},\;{B_2} = {L_2} + {N_2} + {S_2},\;{B_3} = {L_7} + {N_5} + {S_3},\quad {B_4} = {L_8} + {S_4},} \\ {{B_5} = {L_8} + {R_5},\;{B_6} = {L_1} + {N_1} + {R_1},\;{B_7} = {L_5} + {N_3} + {R_2},\;{B_8} = {L_6} + {N_4} + {R_3},} \\ {{B_9} = {L_8} + {S_4},\;{B_{10}} = {L_8} + {R_5},} \\ \end{array} $$
$$ \begin{array}{*{20}{c}} {{f_x} = \int\limits_{ - {{h} \left/ {2} \right.}}^{{{h} \left/ {2} \right.}} {\left( {{{\bar{X}}_x} + \varDelta {X_x}} \right)dz + \sigma_{zx}^{+} - \sigma_{zx}^{-}, } } \\ {{f_y} = \int\limits_{ - {{h} \left/ {2} \right.}}^{{{h} \left/ {2} \right.}} {\left( {{{\bar{X}}_y} + \varDelta {X_y}} \right)dz + \sigma_{zy}^{+} - \sigma_{zy}^{-}, } } \\ {{f_z} = \int\limits_{{{{ - h}} \left/ {2} \right.}}^{{{h} \left/ {2} \right.}} {\left( {{{\bar{X}}_z} + \varDelta {X_z}} \right)dz + \left( {\sigma_{zx}^{+} - \sigma_{zx}^{-} } \right){w_{,x}} + \left( {\sigma_{zy}^{+} - \sigma_{zy}^{-} } \right){w_{,y}} + \sigma_{zz}^{+} - \sigma_{zz}^{-}, } } \\ \end{array} $$
$$ \begin{array}{*{20}{c}} {{m_x} = \int\limits_{{{{ - h}} \left/ {2} \right.}}^{{{h} \left/ {2} \right.}} {\left( {{{\bar{X}}_x} + \varDelta {X_x}} \right)zdz + h{{{\left( {\sigma_{zx}^{+} - \sigma_{zx}^{-} } \right)}} \left/ {2} \right.},} } \\ {{m_y} = \int\limits_{{{{ - h}} \left/ {2} \right.}}^{{{h} \left/ {2} \right.}} {\left( {{{\bar{X}}_y} + \varDelta {X_y}} \right)zdz + h{{{\left( {\sigma_{zy}^{+} - \sigma_{zy}^{-} } \right)}} \left/ {2} \right.}.} } \\ \end{array} $$

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Wei, CL., Chen, CS., Shih, CS. et al. Nonlinear vibration of initially stressed hybrid composite plates on elastic foundations. Mech Compos Mater 48, 467–482 (2012). https://doi.org/10.1007/s11029-012-9291-5

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