An unlocked implicit G1 continuity multi patch B-spline interpolation for the analysis of 3D kirchhoff-love rod elements

We present a multi patch assumed strain formulation (with implicit G1-continuity at the ends of the element) for 3D space Kirchhoff-Love rod; rotations are introduced at the ends of the element as degree of freedom similarly to the Hermitian interpolation for Euler Bernoulli beam problem. In this way the G1 continuity is ensured. Due to the general curved geometry a strong coupling appears in the membrane-flexural-torsion (m-f-t) problem, so that a pure displacement formulation leads in general to a locked element (membrane, flexural and torsion locking phenomena can occur). The multi patch approach presented, based on G1 continuity (low degree of continuity), does not present locking in contrast to the B-Spline (high degree of continuity) element, in a pure displacement approach. However, both the approaches present spurious mode in the deformations, i.e. in the stress resultants. In order to avoid this pathology we adopt a standard assumed strain formulation (or B-bar) approach, projecting the tangent strain measures onto lower degree spaces, (by means of standard L2 projections). In particular, considering a polynomial degree interpolation (p) for the displacements, the membrane and torsional strain measures are projected on a (p-1) space, while the two flexural strain measures are projected on a (p-2) space. In this way a very easy definition of the B-bar operators is attained, since the integrations are performed numerically. The strategy is very appealing for the design of free-locking general curve rod elements, and it provides very accurate results for different polynomial degrees as it is shown by means of presented example.