Variational approach to dynamic analysis of third-order shear deformable plates within gradient elasticity
- 313 Downloads
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.
KeywordsStrain 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.
- 4.Koiter WT (1964) Couple-stresses in the theory of elasticity: I and II, Koninklijke Nederlandse Akademie van Wetenschappen (Royal Netherlands Academy of Arts and Sciences) B67:17–44Google Scholar
- 12.Reddy JN, Srinivasa AR (2014) Nonlinear theories of beams and plates accounting for moderate rotations and material length scales. Int J Non Linear Mech (In press)Google Scholar
- 19.Reddy JN (2007) Theory and analysis of elastic plates and shells. CRC Press, Taylor & Francis, PhiladelphiaGoogle Scholar
- 20.Reddy JN (2006) Mechanics of laminated composite plates and shells, 2nd edn. CRC Press, Boca RatonGoogle Scholar
- 30.Reddy JN (2013) An introduction to continuum mechanics, 2nd edn. Cambridge University Press, New YorkGoogle Scholar
- 38.Malvern LE (1969) Introduction to mechanics of a continuous medium. Prentice-Hall Inc, Inglewood CliffsGoogle Scholar
- 41.Neff P (2004) A geometrically exact Cosserat shell-model including size effects, avoiding degeneracy in the thin shell limit. Part I: formal dimensional reduction for elastic plates and existence of minimizers for positive Cosserat couple modulus. Contin Mech Thermodyn 16:577–628CrossRefADSzbMATHMathSciNetGoogle Scholar
- 42.Lazar M, Maugin GA, Aifantis EC (2005) On dislocations in a special class of generalized elasticity. Phys Stat Sol (b) 242:2365–2390Google Scholar
- 43.Tahaei Yaghoubi S, Mousavi SM, Paavola J (2015) Strain and velocity gradient theory for higher-order shear deformable beams. Arch Appl Mech (accepted)Google Scholar