Narrow Results

Donati compatibility conditions on a surface allow to reformulate the minimization problem for a linearly elastic shell through the intrinsic approach, i.e. as a quadratic minimization problem with the linearized change of metric and change of curvature tensors of the middle surface of the shell as the new unknowns. Such compatibility conditions typically take the form of variational equations with divergence-free tensor fields as test-functions. In a previous work, the first author and Oana Iosifescu have identified and justified Donati compatibility conditions for shells modeled by Koiter's equations. In this paper, Donati compatibility conditions are identified and justified for two specific classes of linearly elastic shells, the so-called elliptic membrane shells and flexural shells.

Korn’s inequalities on a surface constitute the keystone for establishing the existence and uniqueness of solutions to various linearly elastic shell problems. As a rule, they are, however, somewhat delicate to establish. After briefly reviewing how such Korn inequalities are classically established, we show that they can be given simpler and more direct proofs in some important special cases, without any recourse to J. L. Lions lemma; besides, some of these inequalities hold on open sets that are only assumed to be bounded. In particular, we establish a new “identity for vector fields defined on a surface”. This identity is then used for establishing new Korn’s inequalities on a surface, whose novelty is that only the trace of the linearized change of curvature tensor appears in their right-hand side.