On Some Interior and Boundary Effects in Thin Plates Based on the Asymmetric Theory of Elasticity

Abstract

For plates with microstrueture, the classical theory of elasticity considering only force interactions between the material points results in some well-known errors. In this case, the plate or shell theories must be based on the asymmetric theory of elasticity (ATE) [3]. Below, several mechanical phenomena in the theory of mi-cro-polar elastic thin plates [4-10] are stated and discussed in connection with the use of the asymptotic method [1, 11]. Let us consider a plate of 2h constant thick-ness, bounded with the cylindric lateral surface as a three-dimensional problem of asymptotic theory of elasticity (ATE) with the independent fields of displacements and rotations [2], which will present in x k (k = 1,2,3) Cartesian coordinate System. The prineipal equations are: (1.1) % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHdp % WCdaWgaaWcbaGaamyAaiaadQgacaGGSaGaamOAaaqabaGccqGH9aqp % caaIWaGaaiilaaqaaiabeY7aTnaaBaaaleaacaWGPbGaamOAaiaacY % cacaWGQbaabeaakiabgUcaRiaadwgadaWgaaWcbaGaamyAaiaadQga % caWGRbaabeaakiabeo8aZnaaBaaaleaacaWGQbGaam4AaaqabaGccq % GH9aqpcaaIWaaaaaa!4DAD! $$ \begin{array}{l} {\sigma _{ij,j}} = 0, \\ {\mu _{ij,j}} + {e_{ijk}}{\sigma _{jk}} = 0 \\ \end{array}$$ .