On Solvability of Boundary Value Problems for Elastic Micropolar Shells with Rigid Inclusions

On solvability of initial boundary-value problems of micropolar elastic shells with rigid inclusions

“…Let us note that here equation (39) includes a part of the initial conditions for the velocities, whereas the initial data for translations and rotations are treated as the ones which should be formulated explicitly. Now for shortness, we introduce the notation as in Eremeev and Lebedev [34]: U = ((u, f), u 1 , . .…”

“…Now for shortness, we introduce the notation as in Eremeev and Lebedev [34]: U = ( ( u , ϕ ) , u 1 ,… , u n , ϑ 1 ,… , ϑ N ) and, respectively, δ U = ( ( δ u , δ ϕ ) , δ u 1 ,… , δ u n , δ ϑ 1 ,… , δ ϑ N ) . On the set of U where ( u , f ) are smooth functions satisfying boundary conditions (9) 3 and (9) 4 , we introduce an inner product defined by the energy terms for ( u , ϕ ) :…”

“…Clearly V is a Hilbert space. In Eremeyev and Lebedev [32,34], it is shown that if S is smooth, Cartesian components of ( u , ϕ ) pertain to space W 1 , 2 ( S ) and moreover the norm ‖ U ‖ V is equivalent to the norm of ( u , ϕ ) in Sobolev’s space ( W 1 , 2 ( S ) ) 3 × W 1 , 2 ( S ) ) 3 .…”

“…The mathematical analysis of boundary-value problems (BVPs) in the shell theory was performed in many works [30,31]. Existence of solutions for Cosserat and micropolar shells was studied in a less number of papers [9,[32][33][34] and the reference therein. In particular, in Eremeev and Lebedev [34], static BVPs for shells with rigid inclusions were studied, where only perfect contact conditions were considered.…”

“…Existence of solutions for Cosserat and micropolar shells was studied in a less number of papers [9,[32][33][34] and the reference therein. In particular, in Eremeev and Lebedev [34], static BVPs for shells with rigid inclusions were studied, where only perfect contact conditions were considered. Here, we extend the results [32,34] towards dynamic problems for linear micropolar shells with inclusions considering both perfect and non-perfect interface conditions.…”