Analytical and Finite Element Multi-Shell Modelling of an Intervertebral Disc

The objective of this study is to develop an analytical model of an intervertebral disc capable of predicting stresses and deformations in the anulus fibrosus. The anulus fibrosus is treated as a multi-shell structure for which membrane theory applies and the junction with the end plates assumed rigid is treated with beam on elastic foundation theory. The displacements and stresses of the anulus fibrosus, when the disc is subjected to a compressive force are investigated. The discontinuity stresses are superimposed to those of the membrane stresses to estimate the stress state of the anulus fibrosus. The analytical results are compared to those of a finite element model for validation. The analytical and the finite element models show the same general trend, but their relative difference in estimating the stresses is rather high. However, both models show that the highest circumferential stress is located on the innermost lamella at the transverse plane. Furthermore, the stresses decrease through the disc thickness and at the vicinity of the endplates. These results allow to indicate, analytically, the location for maximal principal stresses in the anulus fibrosus.