Large deflection of 3D curved rods: An objective formulation with principal axes transformations

• Definition of the finite rotational transformation between configurations is novel.
• A general section triad and its relation with the Frenet–Serret frame is presented.
• Objectivity of stain measures is sustained throughout the element.
• Equilibrium equations are obtained with a fixed-point approach.
• Extraction of rotational parameters from a rotation matrix is avoided.

This study shows that the deformed configuration space of 3D rods can be spanned in an objective manner by means of a semi-configuration-dependent approach. We employ a partially dependent default rotational triad defined using the spatial reference curve, a custom director-unit-vector, and a twisting-like-angle interpolation to obtain the true orientations of the cross-sections. Numerical verification shows that, rotating the custom director-unit-vector in conjunction with the element’s first node, strain measure objectivity is ensured throughout the element domain. Employing the fundamental theorem of calculus, this simple but accurate finite-element implementation need not invoke the virtual work principle in the classical sense.