A Constructive Proof of Korn’s Scaled Inequalities for Shells

Abstract

One of Korn’s scaled inequalities for shells asserts that the $H^1$-norm of a displacement field of a shell with thickness $2ε$ clamped on a portion of its lateral boundary, once scaled to a domain independent of $ε,$ is bounded above by the $L^2$-norm of the corresponding scaled infinitesimal strain tensor field multiplied by a constant of order $ε^{−1}.$ We give a constructive proof to this inequality, and to other two inequalities of this type, which is thus different from the original proof of Ciarlet et al. [Arch. Rational Mech. Anal. 136 (1996), 163–190].