Modelling fourth-order hyperelasticity in soft solids using physics informed neural networks without labelled data

Mild traumatic brain injury can result from shear shock wave formation in the brain in the event of a head impact like in contact sports, road traffic accidents, etc. These highly nonlinear deformations are modelled by a fourth-order Landau hyperelastic model, instead of the commonly used first or second order models like Neo-Hookean and Mooney-Rivlin models, respectively. The conventional finite element computational solvers produce robust and accurate estimates, yet they are not deployable for real-time prediction given the computational cost. The advent of physics-informed neural networks (PINNs) to solve partial differential equations (PDEs) has opened the possibility of real-time estimates of brain deformation. It involves developing a physics-informed neural network model that minimizes the residuals of the governing system of equations while ensuring boundary conditions are enforced. In this work, we propose a causal marching physics-informed neural network (CMPINN) model to capture the nonlinear mechanical response of higher-order hyperelastic materials. The CMPINN introduces a novel adaptive training scheme that incrementally updates the neural network weights. This approach incorporates several loss terms related to each material domain, boundary domain and internal domain that contributes to the total loss function, which is minimized during training. The proposed PINN framework is developed for a cube undergoing homogeneous isotropic incompressible canonical deformations: uniaxial tension/compression, simple shear, biaxial tension/compression, and pure shear. Three other tests for scenarios involving spatially varying material properties and inhomogeneous deformations are performed and benchmarked with analytical and numerical solutions.