Nowadays, micro-machining is an expanding field because it is often used in the manufacture of electronic components and miniaturized medical systems. Unlike conventional machining operations, micro-machining is very sensitive to the material microstructure because the size of the tool is comparable to that of grains. Moreover, at this scale, it is difficult to capture the physical fields involved in the cutting process experimentally. Thus, numerical simulation can be a helpful way to understand micro-machining. In recent years, some numerical simulations of the literature on the modeling of micromachining include a model of crystal plasticity to reproduce the behavior of the machined material. These models use a Lagrangian formulation and a damage model necessary to create the chip. Recently, a comparative study in the literature has shown that the Arbitrary Lagrangian (ALE) Eulerian and Coupled Eulerian-Lagrangian (CEL) approaches are interesting to simulate the machining with a continuous chip without including any damage model. Nevertheless, these approaches have never been combined with a model of crystalline plasticity. In this work, the ALE and CEL approaches have been compared to simulate micromachining. Both approaches already proposed in ABAQUS/EXPLICIT software are combined with a model of crystal plasticity which has been implemented in a VUMAT subroutine. The study reveals the advantages and drawbacks of these methods in micro-machining simulations