Features of the bending of a thin rectangular plate of material with an unchanged volume | Mekhanika | kompozitsionnykh | materialov i konstruktsii
> Volume 24 > №3 / 2018 / Pages: 490-498

Features of the bending of a thin rectangular plate of material with an unchanged volume


The maximum value of the coefficient of transverse compression (Poisson) for isotropic linearly elastic materials is limited to 0.5. If we add linear deformations for a material with , we obtain as a consequence of physical relationships, which is valid for small deformation. This indicates that when a body from such a material is deformed, only its shape changes, and the volume remains unchanged. The deformation of the change of volume, therefore, is zero, and the modulus of elasticity characterizing the resistance of the medium to a change of volume tends to infinity. Therefore, in the physical relationships resolved with respect to normal stresses and containing the product of this module by the deformation of the volume change θ, instead of this product, which becomes indeterminate as a result of multiplying zero by infinity, a force function with a dimension of stress is introduced. Materials that have the property of constant volume during deformation are called incompressible. These include rubber, various types of rubbers and some others. These materials, which are not very common in technology, thanks to a unique property, allow us to test some of the classical problems of mechanics of a solid deformable body, based on certain hypotheses. One of these problems is the bending of thin plates. The classical problem of plate bending is based on Kirchhoff’s hypotheses: the absence of linear deformation in the direction perpendicular to the base of the plate, transverse shear deformations and normal stress in the transverse direction. The deflection of the plate from the action of forces acting in planes perpendicular to the bases of the plate due to one of the hypotheses is a two-dimensional function, which significantly simplifies the problem, in spite of the fact that the other desired functions of displacements and stresses linearly depend on the coordinate perpendicular to the base of the plate. The transition to integral characteristics of the state of stress by integrating the differential equations of equilibrium in stresses over the plate thickness allows to get rid of the linear function and finally formulate the bending problem as two-dimensional one. Static boundary conditions are also formulated with respect to the integral characteristics of the state of stress which leads to some errors in the solution near the boundaries of the plate according to the Saint-Venant principle. Some classical hypotheses of the theory of bending plates are inconsistent with respect to the incompressibility condition, therefore, it is necessary to refuse from classical hypotheses and build the problem using other assumptions. Also, for some problems, depending on the type of load and the boundary conditions, it is possible to obtain fairly simple solutions without going over to the integral characteristics of the state of stress.