A weak form quadrature element formulation of geometrically exact beams with strain gradient elasticity

In this article, a numerical scheme for geometrically nonlinear analysis of elastic strain gradient beams is presented based on the weak form quadrature element formulation in combination with the geometrically exact beam model. Compared to standard geometrically exact beam elements, the present formulation incorporates the derivatives of displacements and rotations as additional degrees of freedom at element boundary nodes due to the C1 continuity requirements brought about by the introduction of strain gradients. A generalized differential quadrature scheme based on the Hermite interpolation functions is employed to approximate the derivatives of displacements and rotations. The quaternion-based scheme for spatial rotations and rotation derivatives is introduced to realize a total Lagrange formulation without singularities. This formulation is shown to be feasible to analyze strain gradient beam structures undergoing large displacements and rotations with the advantages of retaining strain objectivity and circumventing shear locking problems. Six typical examples are given to demonstrate the validity of the formulation. •A numerical scheme for geometrically exact strain gradient beam is established.•The weak form quadrature element method is employed.•C1 continuity requirements caused by strain gradients are met.•Quaternion parameterizations of rotations and their derivatives are introduced.•Benchmark examples are studied to investigate strain gradient effects.