Korn's inequalities for junctions of spatial bodies and thin rods (Q2785659)

Korn's inequality in elasticity has been extended in several ways. Not only has the original proof been alternatively treated (as undertaken, for example, by Friedrichs and Duvaud-Lions), but methods have been devised for evaluating the constants appearing in the inequality. Interest has recently again been prompted by attempts to apply Korn's inequality to a body with complicated structure.NEWLINENEWLINENEWLINEThe present paper studies the extension of Korn's inequality to bodies containing an elastic matrix in which cylindrical long fibers, like the leg of a spider, are imbedded in another elastic material. The fibers are geometrically defined as cylindrical subsets of the entire body with a small cross section characterized by a small parameter \(\varepsilon\). Then Korn's inequality has the form \(|u|_{H^1(\Omega)}\leq C(\varepsilon)E(u)\), where \(u\) is a displacement vector and \(E(u)\) the strain energy. In addition, \(C(\varepsilon)\) is a constant which depends on \(\varepsilon\). The explicit value of an exponent \(\alpha\) such that \(C(\varepsilon)\leq c\varepsilon^{-\alpha}\) represents the effect of the fibers. If the fiber is considered as an isolated cylindrical body, Korn's inequality is a simple application of the general case and the constant \(C(\varepsilon)\) does not depend on \(\varepsilon\). If, conversely, the rods are immersed in the body, the function \(C(\varepsilon)\) depends on \(\varepsilon\) according to their geometrical arrangement, their rigidity, or their softness.NEWLINENEWLINENEWLINEThe paper is interesting, but, unfortunately, very hard to read even for people familiar with Korn's inequalities.