Variational approach to dynamic analysis of third-order shear deformable plates within gradient elasticity

Abstract

A variational approach based on Hamilton’s principle is used to develop the governing equations for the dynamic analysis of plates using the Reddy third-order shear deformable plate theory with strain gradient and velocity gradient. The plate is made of homogeneous and isotropic elastic material. The stain energy, kinetic energy, and the external work are generalized to capture the gradient elasticity (i.e., size effect) in plates modeled using the third-order shear deformation theory. In this framework, both strain and velocity gradients are included in the strain energy and kinetic energy expressions, respectively. The equations of motion are derived, along with the consistent boundary equations. Finally, the resulting third-order shear deformation (strain and velocity) gradient plate theory is specialized to the first-order and classical strain gradient plate theories.

Appendices

Appendix 1

In 3D elasticity, the strain energy is [1]

$$ \delta U_{t} = - \int\limits_{V} {\sigma_{ij,j} \delta u_{i} dv} + \int\limits_{\delta V} {\left[ {n_{j} \sigma_{ij} - \left( {n_{k} \tau_{ijk} } \right)_{,j} + \left( {n_{k} n_{l} \tau_{ijk} } \right)_{,l} n_{j} } \right]\delta u_{i} da} + \int\limits_{\delta V} {n_{j} n_{k} \tau_{ijk} D\left( {\delta u_{i} } \right)da} + \oint\limits_{\varGamma } {[[m_{j} n_{k} \tau_{ijk} ]]\delta u_{i} dl}$$

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while D = n _l ∂ _l is the normal (directional) derivative operator. The external work [13]

$$ \delta W_{t} = \int\limits_{V} {f_{i} \delta u_{i} dv} + \int\limits_{\partial V} {t_{i} \delta u_{i} da} + \int\limits_{\partial V} {q_{i} D\left( {\delta u_{i} } \right)da} $$

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and the kinetic energy

$$ \delta K_{t} = \int\limits_{\varOmega } {\rho \left( {\dot{u}_{i} \delta \dot{u}_{i} + l^{2} \dot{u}_{i,j} \delta \dot{u}_{i,j} } \right)dv} $$

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can be used in the Hamilton’s principle. General stress components (σ _ij ) are defined as

$$ \sigma_{ij} \equiv \tau_{ij} - \tau_{ijk,k} = \tau_{ij} - l^{2} \tau_{ij,kk} $$

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Assuming that the initial (t = 0) and final (t = T) configurations of the plate are prescribed (initial and final conditions), substitution of (A1, A2 and A3) in Hamilton’s principle results in

$$ \int_{0}^{T} {\left\{ {\int\limits_{V} {\left( {\sigma_{ij,j} + f_{i} - \rho \textit{\"{v}}_{i} } \right)\delta u_{i} dv} + \int\limits_{\partial V} {\left[ {t_{i} - n_{j} \sigma_{ij} + \left( {n_{k} \tau_{ijk} } \right)_{,j} - \left( {n_{k} n_{l} \tau_{ijk} } \right)_{,l} n_{j} - n_{j} \rho l_{k}^{2} \textit{\"{u}}_{i,j} } \right]\delta u_{i} da} } \right.} \left. { + \int\limits_{\delta V} {\left[ {q_{i} - n_{j} n_{k} \tau_{ijk} } \right]D\left( {\delta u_{i} } \right)da} - \oint\limits_{\varGamma } {[[m_{j} n_{k} \tau_{ijk} ]]\delta u_{i} dl} } \right\}dt = 0 $$

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while v _i are the general stress and displacement components, i.e.

$$ v_{i} \equiv u_{i} - l_{k}^{2} u_{i,kk} $$

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In this approach, it is challenging to apply the dimension reduction to the integral over the boundaries.

Appendix 2

The non-zero stress components of first gradient elasticity for the TSDT are

$${\tau_{\alpha \beta } = \lambda^{\prime}\delta_{\alpha \beta } \left[ {u_{0\gamma ,\gamma } + z\phi_{\gamma ,\gamma } - cz^{3} \left( {\phi_{\gamma ,\gamma } + u_{0z,\gamma \gamma } } \right)} \right] + \mu \left[ {\left( {u_{0\alpha ,\beta } + u_{0\beta ,\alpha } } \right) + z\left( {\phi_{\alpha ,\beta } + \phi_{\beta ,\alpha } } \right) - cz^{3} \left( {\phi_{\alpha ,\beta } + \phi_{\beta ,\alpha } + 2u_{0z,\alpha \beta } } \right)} \right]},$$

$$ \tau_{\alpha z} = \mu \left( {1 - 3cz^{2} } \right)\left( {\phi_{\alpha } + u_{0z,\alpha } } \right) $$

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and

$${\tau_{\alpha \beta \delta } = l^{2} \lambda^{\prime}\delta_{\alpha \beta } \left[ {u_{0\gamma ,\gamma \delta } + z\phi_{\gamma ,\gamma \delta } - cz^{3} \left( {\phi_{\gamma ,\gamma \delta } + u_{0z,\gamma \gamma \delta } } \right)} \right]+ l^{2} \mu \left[ {\left( {u_{0\alpha ,\beta \delta } + u_{0\beta ,\alpha \delta } } \right) + z\left( {\phi_{\alpha ,\beta \delta } + \phi_{\beta ,\alpha \delta } } \right) - cz^{3} \left( {\phi_{\alpha ,\beta \delta } + \phi_{\beta ,\alpha \delta } + 2u_{0z,\alpha \beta \delta } } \right)} \right]},$$

$$ { \tau_{\alpha \beta z} = l^{2} \lambda^{\prime}\delta_{\alpha \beta } \left[ {\phi_{\gamma ,\gamma } - 3cz^{2} \left( {\phi_{\gamma ,\gamma } + u_{0z,\gamma \gamma } } \right)} \right] + l^{2} \mu \left[ { + \left( {\phi_{\alpha ,\beta } + \phi_{\beta ,\alpha } } \right) - 3cz^{2} \left( {\phi_{\alpha ,\beta } + \phi_{\beta ,\alpha } + 2u_{0z,\alpha \beta } } \right)} \right] }, $$

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$${\tau_{\alpha z\delta } = l^{2} \mu \left( {1 - 3cz^{2} } \right)\left( {\phi_{\alpha ,\delta } + u_{0z,\alpha \delta } } \right)},$$

$$ \tau_{\alpha zz} = - 6l^{2} cz\mu \left( {\phi_{\alpha } + u_{0z,\alpha } } \right)$$

while α, β and δ are x and y.

Appendix 3

The general bending moments in terms of generalized displacements are

$${\left\{ {N_{\alpha \beta } ,R_{\alpha \beta } } \right\} = \left\{ {g_{1} ,g_{3} } \right\}\left[ {\lambda^{\prime}\delta_{\alpha \beta } u_{0\gamma ,\gamma } + \mu \left( {u_{0\alpha ,\beta } + u_{0\beta ,\alpha } } \right)} \right]},$$

$$ { \left\{ {M_{\alpha \beta } ,P_{\alpha \beta } ,N_{\alpha \beta }^{z} ,R_{\alpha \beta }^{z} } \right\} = \lambda^{\prime}\delta_{\alpha \beta } \left[ {\left\{ {g_{3} ,g_{5} ,g_{1} ,g_{3} } \right\}\phi_{\gamma ,\gamma } - c\left\{ {g_{5} ,g_{7} ,3g_{3} ,3g_{5} } \right\}\left( {\phi_{\gamma ,\gamma } + u_{0z,\gamma \gamma } } \right)} \right] + \mu \left[ {\left\{ {g_{3} ,g_{5} ,g_{1} ,g_{3} } \right\}\left( {\phi_{\alpha ,\beta } + \phi_{\beta ,\alpha } } \right) - c\left\{ {g_{5} ,g_{7} ,3g_{3} ,3g_{5} } \right\}\left( {\phi_{\alpha ,\beta } + \phi_{\beta ,\alpha } + 2u_{0z,\alpha \beta } } \right)} \right]}, $$

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$${M_{\alpha z}^{z} = - 6cg_{3} \mu \left( {\phi_{\alpha } + u_{0z,\alpha } } \right)},$$

$$ \left\{ {N_{\alpha z} ,R_{\alpha z} } \right\} = \mu \left( {\left\{ {g_{1} ,g_{3} } \right\} - 3c\left\{ {g_{3} ,g_{5} } \right\}} \right)\left( {\phi_{\alpha } + u_{0z,\alpha } } \right)$$

while

$$ g_{n} = \frac{{h^{n} }}{{n \times 2^{n - 1} }} $$

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