Об осесимметричном нагружении пластины на локальной опоре в рамках теории Родионовой – Титаева – Черных

Abstract

The problem of an axisymmetric bending of a plate with constant thickness under its own weight is considered. The plate has a circular support. The problem is solved in an axisymmetric statement using a nonclassical shell theory of Rodionova-Titaev-Chernykh (RTCh), which takes plate compression in thickness into account. The solution for this theory is obtained using Godunov's orthogonal sweep method. This solution is compared with the solution obtained using the general three-dimensional theory of elasticity, implemented in an open-source package Code_Aster using axisymmetric finite elements. The motivation for this study is the description of a stress-strain state of some variants of primary mirrors of large optical telescopes under the action of gravity. The obtained results characterizing the optical quality of the mirror surface are: the peak value (PV) and the root-mean square (RMS) of its displacement. A parametric study was carried out, i.e., the thickness of the plate and the half-width of the support were varied. The two methods were compared. It is shown that, as the plate thickness increases or the half-width of the support decreases, non-physical behaviour of the mirror surface takes place within the limits of the nonclassical theory of RTCh. A criterion of its applicability is therefore proposed.