, Volume 50, Issue 6, pp 1537–1550 | Cite as

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

  • S. M. Mousavi
  • J. Paavola
  • J. N. Reddy


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.


Strain gradient elasticity Velocity gradient Shear deformation Equations of motion Boundary conditions Third-order theory Variational approach 



The third author gratefully acknowledge the support of this work by the FidiPro grant.


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Civil and Structural EngineeringAalto UniversityAaltoFinland
  2. 2.Department of Mechanical EngineeringTexas A&M UniversityCollege StationUSA

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