Residual stress prediction

Although the evidence presented in section 3.3.2 suggests that residual stress potentially has a large, deleterious effect on the life of a THA [100], few authors have

achieved a satisfactory computational method for predicting the stress field within the cement mantle due to polymerisation of bone cement.

Nuno et al. used FEM to model a bone-cement-prosthesis system [88]. The stem was inserted into the cement mantle with a “press fit” (i.e. the stem was slightly larger than the cavity into which it was inserted). This method produced a residual stress field but does not take into account variations in polymerisation conditions within the cement mantle.

In order to predict the residual stress field it is necessary to relate the volumetric change to the degree of polymerisation. This is a complex process as the cement is thought to shrink at the same time as it heats up due to the exothermic polymerisation reaction. This heat causes an expansion due to thermal expansion coefficients. The volumetric change due to polymerisation must be coupled with mechanical property development and the thermal volumetric change, as the relative timing of the three separate events will have a great bearing on the residual stress field generated. If the polymerisation shrinkage happens when the properties of the cement are close to the initial fluid state then there will be little residual stress generation. If the polymerisation shrinkage happens when the cement is heating up, it may be masked by the thermal expansion of the cement.

Some attempts to measure and model the relative timing of these processes have been made. Lennon et al. produced a model using Ansys™ based on functions published by Baliga et al. and Starke et al., which generated a temperature profile during cure in a simplified hip replacement geometry [7, 59, 110]. The model used an assumption of stress locking at differing moments during polymerisation and then the residual stress generated due to thermal contraction after this point was calculated. This, however, does not explicitly account for the shrinkage due to polymerisation. The model was validated using a geometrically identical physical model. After full polymerisation this physical model exhibited pre-load cracks formed in the cement mantle due to shrinkage during polymerisation (Figure 3-23). The model used in this study is highly sensitive to the timing of stress locking, which was hypothesised as a single instant and assumed to occur at the same instant in all elements.

Li et al. used finite element modelling techniques to predict residual stresses in cement mantles [63]. A model based on an Arrhenius equation was used to predict the polymerisation as a function of time. Again complete stress locking was assumed to occur at one moment, at the point of maximum temperature. From this point the entire cement mantle began to cool to ambient temperature. During the period of cooling a thermal contraction led to net shrinkage of the cement. Two different stem temperatures were examined; one at ambient initial temperature and one at 45ºC initial temperature. The residual stresses in the cement were compared in the hoop, radial and longitudinal directions. The stresses seen were consistently ranked with the radial stress being the lowest and the hoop stress being the highest with the longitudinal stresses somewhere in the middle. Changing the initial stem temperature to 45ºC caused the peak stress to move away from the cement-implant interface.

Stress locking is a simplification employed by many models. This method assumes that there is one instant during the polymerisation at which a complete transition from a Newtonian fluid to an elastic solid is experienced within the material. This will therefore mean that there is no ability for the material to transfer load prior to transition and that immediately after transition any mechanical change will result in stresses being developed.

Stress locking assumes an instantaneous change in the mechanical properties of bone cement from a fluid to a solid. In reality this will take a finite amount of time however and so to state that there is one instant when the material changes from fluid to solid is something of a simplification. The peak temperature will occur before polymerisation is complete and so there must be some time during which the material is not in its fully solid final state where some thermal shrinkage is taking place. Also, there must be some time before the peak temperature when the material is stiff enough to transfer some load as the volume changes. These effects should be investigated and accounted for in any future modelling of residual stress generation during polymerisation of acrylic bone cement.

No papers could be found in the literature that reported the temperature as well as the viscosity parameters during polymerisation. This may be because rheometers

generally run isothermally to eliminate any effect on the viscosity due to a change in temperature.