Quasi-static validation

The validation of the quasi-static combined loading condition is a very challenging task, as shown by the first validation step performed in project F2028 (Mischke et al., 2007).

In Figure V-8 it can be seen that the compressive flexibility is highly dependent on the shear pre-load. Under a positive shear load (red curve), the compressive flexibility is more than two times higher compared to the loading condition without a shear load (blue curve). A negative shear load (green curve) increases the compressive flexibility even more. The displacement in the z-direction during pure shear loading shows that the functional spinal units expand in the z-direction. This can be seen in Figure V-9. Because of this expansion the lumbar spines are more flexible in the z-direction under combined loading than under single compression loading.

Figure V-8 Results of experiments ID 01, ID 02 and ID 03 for lumbar spine LWS 0032 (measurements from Huber et al., 2005)

0 50 100 150 200 250 300 350 400 450

Figure V-9 Displacement in x- and z-directions under pure shear load – ID 04, LWS 0032 (measurements from Huber et al., 2005)

Two main factors could be responsible for this phenomenon: the intervertebral disc and the facet joints. The influence of the intervertebral disc was discovered during the testing of a functional spinal unit without the posterior elements, and therefore without the influence of the facet joints.

The behaviour of the functional spinal unit shown in Figure V-9 is representative for all of the other tested lumbar spine segments in project F1899. In order to implement this behaviour in the submodel, a numerical model solely of the intervertebral disc was created. With the aid of this disc model, the following influencing factors were analysed:

• material laws and material parameters for the annulus

• spring elements in the annulus (modelling of the fibres)

• shape of the endplates.

2.3.1.1 Material laws and material parameters

In the reference submodel (Mischke et al., 2007), the annulus was modelled as hyperelastic isotropic material with spring elements (representing the fibres) connecting the adjacent endplates. However, this approach is unsuitable for describing the behaviour shown in Figure V-9.

Another approach was tested using a simple FE model of a brick first. There are some materials, mostly polymer foams, which have a negative Poisson’s ratio (-1 < < 0). Stretching these so-called ‘auxetic’ materials in one direction makes them thicker in perpendicular directions. This phenomenon is shown in Figure V-10

for an isotropic material. Under shear load (Figure V-10, lower line), a small undesired decrease in height can be observed.

Figure V-10 Behaviour of test model with auxetic material (isotropic)

Using orthotropic (instead of isotropic) auxetic materials and different material parameters in the three translational degrees of freedom would not have led to the desired results either because shear stress does not induce strain in either isotropic or orthotropic materials. Consequently, anisotropic material was used to define the annulus, as this is the only material description that implies a correlation between shear stress and strain.

2.3.1.2 Modelling of the fibres

The annulus fibrosus can be described as a fibre-reinforced matrix; fibres are arranged concentrically around the nucleus pulposus. The fibres criss-cross and connect the endplates of two adjacent vertebrae (Figure V-11).

Figure V-11 Interbred fibres connecting the endplates of two vertebrae (Kummer, 2005)

Figure V-12 displays the implementation of the above-mentioned fibre structure in the numerical model of the functional spinal unit.

Figure V-12 Modelling of the fibres (above: 3 layers; below: 1 layer)

The experiments showed that there is a pressure of 0.5 to 1.8 bar in the nucleus pulposus of the unstressed functional spinal units. Amongst other things, the variation of the pressure depends on health aspects and the posture of the donors after dying. The counterforce that compensates the resulting force of the pressure inside the nucleus pulposus is induced by the fibres. Consequently, the fibres are pre-stressed.

The application of the intervertebral disc model shows that the pre-stressed fibres play an important role in the above-mentioned behaviour of the functional spinal units under shear load (expansion in the z-direction).

2.3.1.3 Shape of the endplates

The third influencing factor, after the material laws/parameters and the modelling of the fibres, is the shape of the endplates. Up to now, the endplates were modelled as planar areas, which do not conform to human anatomy. For a more detailed description of the geometry, one additional geometric parameter was measured compared to the reference submodel (Mischke et al., 2007): the distance between the vertebrae in the middle of the endplates. This information allows the endplates to be modelled with a domed shape (Figure V-13).

Figure V-13 Intervertebral disc with domed shape (sagittal plane)

2.3.1.4 Conclusion of quasi-static validation

The attempt to implement the above-mentioned behaviour (increasing softness in the z-direction under the shear pre-load) in the numerical model did not satisfy expectations. Work on this specific problem will not continue because the new experiments (project F2069) did not show this behaviour to such extent as can be seen in Figure V-14.

One reason for the higher compressive flexibility of the FSUs under the shear pre-load is their expansion in the z-direction when force in the x-direction is applied. A higher expansion in the z-direction as a result of a pre-load is equivalent to a higher degree of compressive flexibility. The comparison shows the expansion in the z-direction under the shear pre-load. However, it can be seen that the expansion in the z-direction declines in the new experiments; comparison of the green curves: uz

project F1899 (old experiments) and uz project F2069 (new experiments).

Nevertheless, the described behaviour can still be recognised in the new experiments but cannot be implemented in the numerical model.

0 20 40 60 80 100 120 140 160 180 200

Figure V-14 Comparison of averaged results of experiments ID 04 between the two projects F1899 (solid lines) and F2069 (dashed lines)