Ky. Yuan et Thh. Pian, THE REFERENCE COORDINATES AND DISTORTION MEASURES FOR QUADRILATERAL HYBRID STRESS MEMBRANE ELEMENT, Computational mechanics, 14(4), 1994, pp. 311-322
The skew Cartesian coordinate system determined by the Jacobian of the
isoparametric transformation evaluated at the origin can be shown to
be a geodesic coordinate system at the origin. By using a theory in di
fferential geometry, inverse relations of the isoparametric coordinate
transformation can be derived and expressed in terms of these geodesi
c coordinates. In the formulation of hybrid stress finite elements, it
is suggested as a new strategy for assumed stresses that such coordin
ates be used as the reference coordinates. The theory described is exe
mplified by its applications to the 4-node hybrid stress membrane elem
ents. A set of new distortion-measuring parameters for the quadrilater
al element are also proposed based on such theory.