To simulate the function of a spinal spacer implant, the spacer is initially loaded at room temperature 297 K. The spacer is compressed from the top by a rigid surface to a thickness of 3.375 mm. The compression is then removed, and the spacer undergoes elastic recovery. To remove the residual strain, the spacer is heated to 326 K and then cooled to body temperature 311 K.

Figure 40.2: Loading and Recovery of a Spinal Spacer Implant

A 3-D geometry of the spinal spacer is created in Unigraphics, using dimensions found in Petrini 2005.[2] The geometry is imported into Mechanical and meshed with linear tetrahedron elements. Because the spacer is symmetrical, the model can be reduced to include only one quarter of the spacer, as shown below.

Figure 40.3: Simulated model ¼ of Spinal Spacer

The spinal spacer analysis uses the following material properties:[2]

Symmetrical conditions are applied to the 1/4 model of the spinal spacer. A rigid surface contacts the top of the model, and a compressing displacement is applied to that surface. After the displacement is removed, a thermal load is applied to the whole model.

Figure 40.4: Displacement

Figure 40.5: Thermal Load

The nonlinear static analysis is performed with the Large Deflection property set to On in the Analysis Settings.

After the mechanical loading is applied, thermal loading is applied over three steps (4 - 6) for quicker convergence.

In step 4, the temperature is increased from 297 K to 311 K. Convergence is achieved quickly as this temperature is below T0.

In step 5, the temperature is again increased from 311 K to 324 K. The major phase transformation does not occur in this step, so convergence is again achieved quickly.

In step 6, the temperature is increased above 324 K, and the shape memory effect occurs, so convergence is slower.

The following figure illustrates the displacement of the central point A (shown in Figure 40.3: Simulated model ¼ of Spinal Spacer).

Figure 40.6: Displacements of Central Node A of the Spinal Spacer

The following figures show the deformation of the spacer at each step.

Figure 40.7: Spinal Spacer Deformation and Stress, t = 2 seconds

Figure 40.8: Spinal Spacer Deformation and Stress, t = 3 seconds

Figure 40.9: Spinal Spacer Deformation and Stress, t = 6 seconds

In step 2, the displacement is 4.04 mm and the stress is 594.2 MPa. After elastic recovery, the peak displacement decreases to 3.56 mm and the stress is 182.49 MPa. In the final step, displacement and stress approaches zero, indicating that the spacer has returned to its original shape.

The simulation accurately depicts the spacer under load (step 2), during elastic recovery (step 3), and at full recovery due to SMA thermal effects (step 6).

A vertical helical spring is simulated with shape memory effect using two different models, a BEAM188 element model and a SOLID185 element model.

The spring is loaded by a weight of 1830 N in the martensite state at a temperature of 250 K, then heated to 400 K. At the increased temperature, the spring lifts the weight. The spring is then cooled back to 250 K and stretches again. A repeatable, two-way motion occurs, as shown in the figure below.

Figure 40.10: Motion of a Vertical Helical Spring

The geometry of the spring actuator was created in DesignModeler with a wire diameter of 4 mm, a spring external diameter of 24 mm, a pitch size of 12 mm, with two coils, and an initial length of 28 mm, as shown in the following figure. See input files for links to download the .agdb files.

Figure 40.11: Finite Element Models of a Spring Actuator

The corresponding finite element model is created using BEAM188 elements. A 3-D model is generated by extruding the initial finite element model and meshing with SOLID185 elements.

The following material properties,[3] typical of nitinol, are used in the spring actuator simulation:

The top of the spring actuator is fixed, and the bottom is loaded with a weight of 1830 N. Displacements are constrained in the X and Y directions. After the spring is stretched by the weight at temperature 250 K, the temperature is raised to 400 K to lift the weight, and the is reduced back to 250 K to lower the weight. The following figures show the applied force and temperature histories.

Figure 40.12: Applied Force Steps

Figure 40.13: Applied Temperature Steps

The nonlinear static analysis is performed with the Large Deflection property set to On and the Newton-Raphson Option set to Unsymmetric in the Analysis Settings (see figure below).

Figure 40.14: Details of Analysis Settings

The spring actuator stretched by load W in step 1 is shown in the following figure.

Figure 40.15: Spring Actuator Deformation at Step 1

The maximum displacement is 43 mm, greater than the original length of 28 mm.

In step 2, after heating with the shape memory effect, the spring actuator recovers to a maximum displacement of 10 mm. The deformation is in the martensite state to support the weight, as shown below.

Figure 40.16: Spring Actuator Deformation at Step 2

In step 3, after cooling to 250 K, the spring actuator stretches back to its original length:

Figure 40.17: Spring Actuator Deformation at Step 3

Results from the BEAM188 and SOLID185 models are compared. The following figure shows the displacement history of the actuator.

Figure 40.18: Displacement History: Bottom of a Spring Actuator with Temperature

The displacement history indicates that the BEAM188 and SOLID185 models have similar results. The BEAM188 model is much more efficient, however, requiring about an hour to complete. In comparison, the SOLID185 model requires more than eight hours to complete.