AUTHOREA
Log in
Sign Up
Browse Preprints
LOG IN
SIGN UP
Essential Maintenance
: All Authorea-powered sites will be offline 9am-10am EDT Tuesday 28 May
and 11pm-1am EDT Tuesday 28-Wednesday 29 May. We apologise for any inconvenience.
Tengiz BUCHUKURI
Public Documents
1
Shell equations in terms of Günter’s derivatives, derived by the Г-convergence
Roland Duduchava
and 1 more
March 02, 2020
A mixed boundary value problem for the L\’ame equation in a thin layer $\Omega^h:\cC\times[-h,h]$ around a surface $\cC$ with the Lipshitz boundary is investigated. The main goal is to find out what happens when the thickness of the layer tends to zero $h\to0$. To this end we reformulate BVP into an equivalent variational problem and prove that the energy functional has the $\Gamma$-limit being the energy functional on the mid-surface $\cC$. The corresponding BVP on $\cC$, considered as the $\Gamma$-limit of the initial BVP, is written in terms of G\”unter’s tangential derivatives on $\cC$ and represents a new form of the shell equation. It is shown that the Neumann boundary condition from the initial BVP on the upper and lower surfaces transforms into a right-hand side term of the basic equation of the limit BVP.