Peridynamic formulation for hyperelastic Mooney–Rivlin materials employing a novel strain energy density approach

Peridynamics (PD), unlike classical continuum mechanics, formulates the governing equations using spatial integrals rather than relying on displacement derivatives. This paper introduces a new formulation of PD for a novel application in analyzing the behavior of hyperelastic Mooney-Rivlin membranes under uniaxial tensile loading, offering a fresh perspective on modeling complex material deformations. A new formulation is derived for the PD strain energy density function based on the Mooney–Rivlin model, incorporating the relationship between stress components and strain energy. The hydrostatic pressure term for incompressible isotropic hyperelastic materials is also explicitly derived, and PD parameters are calibrated by equating the PD strain energy to the classical strain energy. Finally, the PD equation of motion is completed by introducing a new formulation of PD force density, specifically for Mooney–Rivlin hyperelastic membranes under uniaxial tensile loading. The PD equation of motion, formulated as an integro-differential equation, is solved using advanced numerical techniques for both spatial and time integrations. The PD method (PDM) shows excellent consistency with the finite element method, demonstrating its high accuracy and reliability. The study also employs adaptive dynamics methods to handle static problems within a dynamic framework, highlighting the flexibility and efficiency of the PDM. These findings demonstrate that PDM is a robust and efficient alternative to traditional methods, particularly for applications involving large deformations and dynamic loading, marking a significant advancement in the analysis of hyperelastic membranes.