A new algorithm is proposed for integrating the spin in the frame of large deformation analysis. The method is based on the integration of a matrix relation, obtained from an adapted decomposition of the deformation gradient, and directly written in convected co-ordinates. The input of this algorithm is either the Gram matrix at any time (if a kinematical method is used, for instance) or more generally, the incremental deformation gradient, and the output is the required rotation matrix on convected bases. The result takes a very simple form in the important case of classical shells.
KEY WORDS: co-rotational frame; convected bases; integration of the spin rotated frame: one is related to the spatial spin n, and may be denoted by Q, and the second R is simply generated from the classical polar decomposition of the total deformation gradient (see Dafalias' and Ladeveze,' for example).
To obtain these rotations, some computational effort is needed, and the Hughes and Winget4 algorithm may be used. Other related algorithms, like that of Flanagan and Taylor6 and more recently R a ~h i d , ~ are also available. The first two papers suppose a linear interpolation of the displacements over each time step, whereas in the last one, the deformation is assumed to vary over the time step in such a way that the stretching occurs along fixed principal directions and without rotation during each time increment. All the rotation is considered to occur at the end of the increment with no additional stretch. It is clear that all of these algorithms are objective. Nevertheless, they do suppose the introduction of orthonormal frames and are performed in Cartesian co-ordinate systems, which are natural for three-dimensional structures but not for shells, for instance. Therefore, for these authors, the required rotation is an orthogonal 3 x 3 matrix Q that satisfies the classical differential equation = SZ. Q. Furthermore, we can