On the general solution of linear-elastic problems in isotropic and anisotropic Cosserat continua

Abstract

The existence of couple stresses acting on the surfaces of elastic continua together with the usual force stresses had been object of numerous fundamental papers [1, 7–16]. For describing the complete deformation a geometrically independent rotation vector had been introduced besides the displacement vector. In this paper the basical equations of three-dimensional linear-anisotropic elasticity with all inertia effects are represented in general coordinates and their general solution is established by means of a six-functions-set. In this way all static and cinematic components can be derived directly. At isotropy the six-functions-set can be represented in analogy to the Papkovich-Netjbee set [2, 3, 4] of the classical theory of elasticity. The two-dimensional elastostatic case is equivalent to a set of H. Schäfer [12]. The theory also contains as limit case a set of Mindlin-Teuersten [14, 15], in which the curl of the displacement vector is used for the rotation vector. Some examples refer to the theory of stress concentration, where the existence of couple stresses leads to interesting aspects.