Korn inequalities for elastic junctions of massive bodies, thin plates, and rods

Korn inequalities have been obtained for junctions of massive elastic bodies, thin plates, and rods in many different combinations. These inequalities are asymptotically sharp thanks to the introduction of various weight factors in the $L_2$-norms of the displacements and their derivatives. Since thin bodies display different reactions to stretching and bending, such Korn inequalities are necessarily anisotropic. Junctions of elastic bodies with contrasting stiffness are allowed, but the constants in the inequalities obtained are independent of both the relative thickness $h\in(0,1]$ and the relative rigidity $\mu\in(0,+\infty)$. The norms corresponding to rigidly clamped elements of a structure are essentially different from the norms corresponding to hard-movable or movable elements that are not fastened directly, but only by means of neighbouring elements; therefore, an adequate structure of the weighted anisotropic norms is determined by the geometry of the whole junction. Each variant of Korn inequality is supplied with an example confirming the optimal choice of the weight factors.