2.1 Basic kinematic relations

In this work, we consider a shell as a single director Cosserat surface (Naghdi, 1972, Simo and Fox, 1989; Ibrahimbegovi? , 1997b). This is a two-dimensional surface (typically chosen as the shell mid-surface) with a so-called director vector attributed to each point of the surface. The position vector for a particular point in a shell deformed configuration is assumed to be defined by the following expression

(1)

where defines the domain of the mid-surface parametrization and h ⁺ ? h ^? is the thickness of the shell. In equation (1), ? ¹ and ? ² are convected curvilinear coordinates and ? is through the thickness coordinate. Parameter t defines time with the interval of interest defined as t ? [ t ₀ = 0, T]. It is assumed that the director vector t remains a unit vector in any deformed configuration, i.e.

(2)

It follows from equation (1) that all deformed configurations of the shell are completely determined by pairs ( ?, t). In other words, the configuration space, denoted by , is then defined by

(3)

where S ² is a unit sphere (a space of all vectors of unit length), while ? _? and ? _t are parts of the boundary where the displacement and the director field are specified, respectively.

At each point of the mid-surface in...