On Saint-Venant's principle in linear elastodynamics
The authors use the displacement characterization of linear elastodynamics for a nonhomogeneous anisotropic semi-infinite cylinder subject to homogeneous initial conditions and loaded on its plane end only (or for a long body whose boundary includes a loaded plane) in order to obtain space-time estimates for measure of energy of a portion of the cylinder beyond a distance \(z\) from the load region. Additionally, they establish an analogue of the Phragmén-Lindelöf principle for a cone-shaped semi-infinite homogeneous isotropic elastic body whose boundary includes a plane perpendicular to the cone-axis subject to a harmonically vibrating load. The reviewer remarks: (i) The authors are not aware of a dynamic pure-stress version of Saint-Venant principle formulated previously by the reviewer [cf. \([*]\) \textit{J. Ignaczak}, Bull. Acad. Polon. Sci., Sér. Sci. Tech., 22, 483-489 (1974; Zbl 0288.73010)]. In particular, the inequalities (2.18) in \([*]\) make a counterpart to the inequalities of theorem 2.1 by the authors. The inequality (2.16) by the authors with \(z_0= 0\), \(t_0>0\) can be obtained from the inequality (2.3) in \([*]\) by letting \(z= l\), \(t= t_0- z/c\), and by postulating a correspondence between the stress and displacement energies. (ii) The estimate (2.52) combined with (2.42) is not a genuine dynamic estimate, since the end load, by virtue of (2.51), belongs to a class of statically self-equilibrated tractions for every time \(t\geq 0\).
nonhomogeneous anisotropic semi-infinite cylinder
cone-shaped semi-infinite homogeneous isotropic elastic body
scientific article; zbMATH DE number 1022663
scientific article