Physics-informed neural networks for bulge test modeling of general anisotropic two-dimensional crystalline materials with decoupled elasticity

Two-dimensional (2D) crystalline materials have great potential for flexible electronics and strain engineering, but their mechanical characterization via bulge testing is challenging: commercial Finite Element Analysis (FEA) cannot fully capture decoupled in-plane and out-of-plane stiffnesses or complex constitutive behaviors, and analytical solutions are intractable for anisotropic crystals with irregular geometries. Here, we develop a physics-informed neural network (PINNs) framework for 2D material bulge testing, combining modified Föppl-von Kármán theory with energy-based loss functions to capture arbitrary symmetries and decoupled elasticity. Our approach achieves high accuracy while revealing symmetry-dependent behaviors: square materials (Mn₂S₂) demonstrate nearly isotropic deformation, rectangular materials (black phosphorene) show strong directional anisotropy, and oblique materials (PdCdCl₄) display asymmetric deformation from stretch–shear coupling. The framework accommodates both linear and nonlinear constitutive behaviors, with nonlinear effects in graphene enhancing bubble expansion due to negative higher-order elastic constants, and also adapts to various bubble geometries by configurable sampling and boundary conditions. This computationally efficient framework addresses the longstanding limitations of commercial software in 2D material modeling and lays a foundation for further studies of inverse analysis. All code and data are available at https://github.com/YanChen32/PINNs_bulge_tests.git.