A Constraint on the Consistence of Transverse Shear Strain Energy in the Higher Order Shear Deformation Theories of Elastic Plates

Abstract

This paper studies how to improve the thirdorder shear deformation theories of isotropic plates, which is a question raised by late Reissner in 1985 (ASME Appl. Mech. Rev., 38, pp.1453–1464). It is demonstrated in this paper that a proper displacement field with the higherorder shear deformations given by the method of displacement assumption should satisfy the constraint on the consistence of the transverse shear strain energy a priori in addition to the traction conditions on plate surfaces. This additional constraint on the assumed displacement fields with the higherorder shear deformations is in line with Love's criterion of the consistent first approximation to the strain energy wherein the transverse shear strain energy is included. The constraint on the consistence of the transverse shear strain energy is physically similar to the requirement for the use of the shear coefficients in the firstorder shear deformation plate theories proposed by Reissner and Mindlin, respectively. A procedure to improve the assumed displacement field with the thirdorder shear deformations is presented. The present study shows that the various displacement fields with the simple thirdorder shear deformations would be identical when the constraint on the consistence of the transverse shear strain energy is enforced.