An Arbitrary Lagrangian–Eulerian Formulation of Two-Dimensional Viscoelastic Beams Based on the Consistent Corotational Method

In this paper, an arbitrary Lagrangian–Eulerian (ALE) formulation based on the consistent corotational method is presented for the geometric nonlinear dynamic analysis of two-dimensional (2D) viscoelastic beams. In the ALE description, mesh nodes can be moved in some arbitrarily specified way, which is convenient for investigating problems with moving boundaries and loads. By introducing a corotational frame, the rigid-body motion of an element can be removed. Then, the pure deformation and the deformation rate of the element can be measured in the local frame. This method can avoid rigid-body motion damping. In addition, the elastic force vector, the inertia force vector, and the internal damping force vector are derived with the same shape functions to ensure the consistency and independence of the element. Therefore, different assumptions can be made to describe the local deformation of the element. In this paper, the interdependent interpolation element (IIE) and the Kelvin–Voigt model are introduced in the local frame to consider the shear deformation, rotary inertia, and viscoelasticity. Moreover, the presented method is capable of considering the arbitrary curved initial geometry of a beam. Numerical examples show that internal damping dampens only the pure elastic deformation of the beam but does not dampen the rigid-body motion. Three dynamic problems of a beam with a moving boundary or subjected to a moving load are investigated numerically by the presented formulation and the commercial software ansys to verify the validity, versatility, and computational efficiency of the presented formulation.