The use of a constant load to generate equivalent viscoelastic strain in finite element analysis of cemented prosthetic joints subjected to cyclic loading

Polymers such as polymethyl-methacrylate (PMMA) surgical cement undergo elastic and viscoelastic deformation (creep) in response to physiological cyclic loading. Theoretically, the effect of gradual creep deformation on the stresses, strains, and displacements of a prosthetic joint can be evaluated by running a finite element analysis (FEA) model through a large number of loading cycles. However, with complex (i.e. realistic) models, this approach may require extensive computational time, and may accumulate unacceptably large numerical errors over the many iterations of the model. The present study utilized a Fourier series to represent a periodic stress and incorporated it in the linear viscoelastic constitutive equation. It was demonstrated that, for a linear viscoelastic material, the time average (i.e. the constant in the Fourier series) of the cyclic stress determined the accumulated creep strain and the sinusoidal components of the stress produced the periodic creep strain with a zero average and negligible amplitude. For a geometrically linear FEA model, the solution based on a cyclic stress can be readily applied to an external cyclic load, that is, the creep strain is determined by the time average of the cyclic load. While femoral component models were considered as geometrically non-linear, an FEA model of a femur implanted with an Exeter hip prosthesis showed that there was only a minor difference between the profile of the applied sinusoidal load and that of the resulting displacement. In such cases, applying the time average of a cyclic load to calculate the resulting creep strain with a given duration of loading should expect to provide acceptable accuracy, with a marked reduction in the computational time.