Using finite element method (FEM), the mechanical behaviors of 2D woven composites with different weave structures are investigated with considering the complex in-plane stress states. Periodic boundary conditions for arbitrary off-axis loading scenarios are established, and a progressive damage model based on Linde criterion is proposed, which is subsequently validated by the previous experiments. In addition, the failure envelopes predicted by classical failure criteria are compared to the results obtained by FEM. The effects of woven structures on elastic properties and failure behavior are also discussed, which demonstrates that the plain woven composites have inferior mechanical performance than twill and satin woven composites. Based on the assessment of the failure criteria, this study provides recommendations for selecting relevant failure criteria to anticipate mechanical properties under various loading conditions. The findings in this paper are expected to encourage the development of damage constitutive models and their engineering applications for 2D woven composites.