Large deformation analysis of the inhomogeneous hyperelastic thick-walled sphere under internal/external pressure by Physics-Informed Neural Networks

Abstract Modeling the mechanical response of heterogeneous hyperelastic structural elements under complex loading conditions presents significant challenges, primarily due to nonlinear material behavior, geometric constraints, and spatially varying properties. This study introduces a novel framework based on Physics-Informed Neural Networks (PINNs) to predict the behavior of a heterogeneous neo-Hookean hyperelastic thick-walled sphere subjected simultaneously to internal and external pressures. In contrast to conventional finite element methods (FEM) which are often limited by challenges in mesh generation and high computational costs when addressing material heterogeneity the proposed PINNs approach incorporates the governing nonlinear partial differential equations, boundary conditions, and constitutive laws directly into the neural network’s loss function. Notably, the framework integrates spatial heterogeneity by embedding coordinate-dependent material parameters, such as the shear modulus, into the network architecture. Validation against analytical solutions confirms the method's high accuracy, with relative errors in displacement and stress predictions remaining below 1%. Further case studies demonstrate that radial and hoop stress distributions are significantly influenced by pressure gradients and material heterogeneity, thereby capturing localized mechanical effects. Moreover, the mesh-free nature of the PINNs method obviates the need for domain discretization, enhancing computational efficiency in parametric studies. This work bridges the gap between machine learning and continuum mechanics by providing a robust computational tool for the design of engineering systems that involve soft, heterogeneous materials; applications include biomedical implants, soft robotics, and pressurized energy storage devices. The versatility of the framework also paves the way for its extension to dynamic problems, multiphysics coupling, and the integration of experimental data.