A class of shear deformable isotropic elastic plates with parametrically variable warping shapes

A homogeneous shear deformable isotropic elastic plate model is addressed in which the normal transverse fibers are allowed to rotate and to warp in a physically consistent manner specified by a fixed value of a real non‐negative warping parameter ω. On letting ω vary continuously (at fixed load and boundary conditions), a continuous family of shear deformable plates P ω is generated, which spans from the Kirchhoff plate at the lower limit ω = 0 , to the Mindlin plate at the upper limit ω = ∞ ; for ω = 2 , P ω identifies with the third‐order Reddy plate. The boundary‐value problem for the generic plate P ω is addressed in the case of quasi‐static loads, for which a principle of minimum total potential energy is given. Taking the Kirchhoff deflection w K and the shear angle potentialsG , H as basic unknown fields, the problem turns out to be governed by three uncoupled fourth‐order PDEs, of which one coincides with the classical PDE for the Kirchhoff plate, the other two are PDEs of Helmholtz type. A closed‐form representation of the general solution for the plate family is given in terms of ( w K , G , H ) to within an arbitrary biharmonic function and an arbitrary pair of conjugate harmonic functions. These arbitrary functions are available to enforce the inherent boundary conditions. The connections between the present theory and other existing theories are pointed out. An application to a family of simply supported circular plates under uniform load is analytically worked out.