Equivalence of Lagrange's equations for non-material volume and the principle of virtual work in ALE formulation

The Arbitrary Lagrangian–Eulerian (ALE) formulation provides a high-efficiency approach to simulate the moving contact behaviors in mechanical systems using the relative movement between material points and mesh nodes. Two approaches have been widely adopted to obtain the governing equations of ALE elements, namely (i) Lagrange's equations for non-material volume and (ii) the principle of virtual work. Indeed, consistent numerical results should be obtained through these two approaches; however, their mathematical expressions are quite different at first glance. In this paper, the equivalence of the above-mentioned two approaches demonstrated analytically for a general case and a specific example of three-dimensional two-node ALE cable elements. Additionally, the existence and disappearance conditions for additional terms $L_1$ and $L_2$ of the generalized inertial forces, which distinguish the ALE formulation from the Lagrangian formulation, are explicitly presented. They are physically caused by the material flow through the boundary and mathematically caused by the dependence of the mass matrix $M$ on the material coordinates $p_1$ and $p_2$. When no mass is transported through the boundary surface or the kinetic energy of the material points flowing into or out of the control volume is zero, $L_1$ and $L_2$ automatically disappear, and the governing equations degenerate to the classical form of Lagrange's equations. Finally, an example of preloaded wire rope with variable length is used to verify the above conclusions.