A dynamical model for multilayered plates with independent shear deformations
In this paper a dynamic model for an n layered plate is developed based upon the assumptions of Reissner Mindlin plate theory Each plate layer is assumed to be transversely isotropic transversely homogeneous and of a uniform thickness however no symmetry in the material properties or thicknesses of each plate is as sumed The layers are assumed to be perfectly bonded so that no slip occurs along the interface No additional a priori kinematic restrictions are imposed upon the motion of the plates The equations of motion are derived by the principle of virtual work Existence and uniqueness results are obtained In the case where the layers are symmetric we show that all solutions decouple into a bending solution with antisymmetric displacements about the mid plane and an in plane solution with symmetric displacements Introduction The rst model for a thin multilayered plate which includes the e ects of transverse shear deformation is probably due to Reissner where a static model is developed for a sandwich plate consisting of an inner core bonded to two symmetric face plates Since then numerous dynamic models for multilayered materials based upon similar ideas have been proposed see e g and the references therein In many of these models often referred to as constrained layer models the inner core is assumed to be much more exible than the face material so that bending stresses can be ignored within the core and the transverse shear deformations can be assumed negligible within the face plates Thus in the usual approach to modeling sandwich plates the outer layers are mod eled by plates which allow little or no shear while the inner layer is modeled as a material in which only shear stresses are considered relevant From an applications viewpoint a goal has been to identify e ective elastic modulii for the composite plate so that a low order system involving only the transverse displacement or Mathematics Subject Classi cation K possibly transverse displacement and e ective rotations can be obtained In order to accomplish this the number of degrees of freedom of a plate element are arti cially reduced by relating the the stresses and or strains in each layer with one another In this paper we develop a model for the motion of a multilayered plate which allows independent shear rotations in each layer Each plate layer is assumed to be transversely isotropic homogeneous in the transverse direction and of a uniform thickness however the material properties and thicknesses of each layer can be di erent Each layer is modeled under the assumptions of Reissner Mindlin plate theory i e within each layer the in plane displacements vary linearly with respect to the transverse coordinate and the transverse normal stress vanishes throughout the thickness The surfaces of each layer are perfectly bonded so that no slip occurs along the interfaces however a main point is that we do not impose additional a priori kinematic restrictions which couple stresses and or strains of each layer with one another The equations of motion are derived through the principle of virtual work and we discuss the modi cations needed to include strain rate damping The resulting system is shown to be well posed in an appropriate function space In the case where the layers are symmetric about the center sheet of the plate that is the thicknesses densities and elastic parameters are symmetric with re spect to the middle layer or middle interface as the case may be we show that the motions decouple in a bending solution with displacements which are antisymmet ric about the center sheet and a stretching solution with displacements that are symmetric about the center sheet This type of decoupling has been observed in three dimensional plate theories as well A similar model is obtained in where the equations for a symmetric three layer beam are derived under the assumption that the motion is antisymmetric about the centerline of the beam and Poisson e ects are negligible A fourth order beam model then is obtained when continuity of shear stresses across the interfaces is imposed and rotational inertia is ignored As mentioned in the symmetric case we obtain solutions with antisymmetric displacements about the center sheet however this is due to the decoupling and is not imposed a priori The assumption of shear stress continuity along the interfaces leads to a model in which these stresses are constant throughout the thickness and is not assumed here Basic assumptions Our plate consists of n plate layers which occupy the region h at equilibrium where is an open bounded domain in R with su ciently smooth say C boundary Let z z zn zn h hi zi zi i n We use the rectangular coordinates x fx x g to denote points in and x fx x g fx x x g to denote points in Q i Qi where Qi zi zi i n For x Q let U x fU U U g x denote the displacement vector of the point which when the plate in equilibrium has coordinates x fx x x g We suppress all time dependence where there is no possiblility of confusion In addition let us de ne u fui u g and u i n by