Сomputational model of axisymmetric bending of a round plate with accounting for its incompressibility | Mekhanika | kompozitsionnykh | materialov i konstruktsii
> Volume 25 > №1 / 2019 / Pages: 110-121

Сomputational model of axisymmetric bending of a round plate with accounting for its incompressibility


Incompressible isotropic elastic materials have a maximum Poisson ratio of 0.5. In the process of loading and deforming the structural elements of such materials their shape changes, while the volume of the structure remains unchanged. The property of incompressibility is a consequence of the physical relations of a linearly elastic material, in which the Poisson’s ratio is assumed to be 0.5, and the physical modulus characterizing the resistance of a material to a change in volume tends to infinity, as a result of which the physical relations of Hooke’s law turn into so-called «Neohooke» relations, in which normal stresses contain a common power function having the dimension of stresses. It replaces in physical relations the uncertainty with and where is the volume related modulus , and is the deformation of the change in volume. The unconditional fulfillment of the condition of the invariability of the volume, which relates linear deformations, substantially changes some classical models of mechanics of solid deformable body based on various hypotheses. The bending of thin plates with small deformations is described by the defining relations based on Kirchhoff’s hypotheses about the absence of shear deformations in the plane and transverse linear deformation in relation to a round axisymmetric plate. The fulfillment of the incompressibility condition leads to the necessity of abandoning these hypotheses, in particular, from the hypothesis of the absence of shear deformations. For plates with one boundary rigidly fixed contour or with two also rigidly fixed contours, there is no transverse linear deformation, which is a consequence of the invariability of the volume of the plate in the process of its deformation. In some problems in order to obtain simple and easily solvable equations for a round axisymmetric plate, the radial displacement can be specified as a linear function along the transverse coordinate. And at the same time, it is not necessary to go over to the integral characteristics of the stress state, which are the bending moment and the shear force. In the classical theory of plate bending, such a transition allows to eliminate the transverse coordinate; however, for some problems of bending plates of a material with an unchanged volume, this transition leads to serious errors.