Closed-Form Analysis of Stress and Deformation in Functionally Graded Multi-Layer Hyperelastic Cylinders under Internal Pressure

This study develops a closed-form solution to predict pressure-driven stress and displacement fields in a thick-walled, functionally graded (FG), incompressible, multi-layer hyperelastic cylinder made from Polyvinyl Chloride (PVC), subjected to internal pressure. The exact solution ensures incompressibility, which finite element methods (FEM) may not guarantee. Properties vary smoothly through the thickness using a Mooney–Rivlin model. Two cases are examined: bi-layer and tri-layer cylinders, where the properties in the second layer of the bi-layer case are 50% lower than the first, and in the tri-layer case, the second- and third-layers’ properties are 30% and 60% lower, respectively. Two material grading conditions are considered: in the first, properties at the largest radius are 1.2 times those at the smallest radius, and in the second, they are 0.8 times. Gradation is modeled using an exponential-logarithmic function. The field equations reduce to a nonlinear scalar condition for the integration constant governing radius mapping, leading to explicit solutions for radial displacement and radial, tangential, and axial stresses under internal pressure. Both analytical and FEM solutions yield identical results, with errors under 1% in all cases. The analysis recovers homogeneous limits and provides conditions where continuous gradation reduces stress concentrations compared to discretely layered baselines.