Non-linear mapping and covariant strain approaches

Abstract

In Chapters 2 and 3 we recognized the need for a consistent definition of the strain-fields which may be constrained in the physical regime being simulated, e.g. a thin Timoshenko beam or a thin and deep curved arch undergoing inextensional bending. We also saw that in the cases examined so far, the consistency requirement could be met by using an ‘assumed strain’ approach, i.e. by using a strain interpolation that is derived from the originally constituted strain interpolation (using the strain gradient operators on the displacement interpolations) through an orthogonality condition that assures variational correctness. It was also useful to remember that in the cases examined so far, the mapping from the covariant natural coordinate system (in which the interpolating functions are defined) to the Cartesian system (in which the strains and stresses are derived and the energy functional is evaluated) was a linear one. In such a case, the representation of the Cartesian strains in a consistent form was an easy task once the shape functions and derivatives of these in the covariant system was consistently matched. This was also automatically achieved quite often by the use of simple reduced integration rules. However, difficulties loomed large when such a simple approach was extrapolated to quadrilateral plate and shell elements and general curved beam and shell elements, i.e. curved in Cartesian space. Simple reduced integration strategies failed when distortions from rectangular geometries became large. At this stage, it was found that an ‘assumed covariant base’ approach was effective [4.1–4.4] although no reasons were adduced as to why this step was necessary. In this chapter, we shall investigate the basis for such methods.