Dynamic 3D garment simulation has various applications in many domains. Recently, self-supervised learning for this task has been studied to reduce annotation costs. However, different material characteristics of garments have been rarely explored, limiting the generalization and flexibility of existing methods. Therefore, in this paper, a novel self-supervised deep learning architecture is proposed, namely Material-aware Self-supervised Network (MSN), as a material-aware approach for dynamically simulating garments with different materials. Specifically, a material-aware parameterized regressor is introduced based on the observation that material characteristics change continuously regarding the fabric parameters. As a result, MSN realises real-time garment simulation with various material properties without model re-training. Moreover, to simulate garments of different categories (e.g., t-shirts vs. dresses), a sampling-based linear skinning strategy is studied in MSN. Comprehensive experiments on the widely used AMASS dataset demonstrated the effectiveness of MSN both quantitatively and qualitatively.