*2.2. FE Analysis and Material Failure Model*

For providing a mechanical stability of porous bioceramics sufficient to address long-bone defects, a support structure is needed. This is offered either by providing a shell-like structure that mimics a complete artificial *corticalis* and/or by an outside, broad, accessible support structure that provides adequate strength. Both approaches are presented. However, the focus was on an outside accessible support. A simple column geometry was chosen as such support (Figure 1a, left). Its additive manufacturing by the VPP process should be uncomplicated and provide required mechanical support as well as load balance for inside lying foams. A FE (Finite Element) analysis was made (ANSYS v. 2020 R2, ANSYS Inc., Pittsburgh, PA, USA) to approximate mechanical loads appearing in the structure and, additionally, to identify possible locations of material failure. Therefore, a geometrical model was created with respect to the existing symmetry conditions (Figure 1a, right).

**Figure 1.** Simple column structure and boundary conditions: (**a**) column structure (left) for simulation using geometry regarding symmetry planes (right); (**b**) boundary conditions (left); meshed geometry (right).

Corresponding boundary conditions were applied to the model (Figure 1b, left), which implies fixation of translational DOF (Degree of Freedom) in the z-direction at the bottom face, symmetry conditions in the x- and y-direction and a given displacement at the top face of the structure. A displacement instead of a pressure load was used to match experimental conditions where a press specifies a defined displacement, and the resulting pressure (reaction force) was measured. Displacement values for the model were selected to achieve reaction forces of about 800 N for 1/4 of the geometry and 3200 N for the complete geometry. This load was well estimated because experimental

data indeed showed that the full column structure withstood average loads up to 3200 N (see Section 3.2 on mechanical characterization). Table 1 shows the displacements and reaction forces used for the simulation. Three load cases were chosen to simulate the complex deformation of the ceramic VPP structure. Because of the well-known brittleness of ceramic hydroxyapatite/TCP [30], the load was slowly increased, and the deformation was determined starting from a 780 N reaction force. Material failure should occur suddenly at a certain point, as shown by simulation results.

**Table 1.** The displacement, reaction forces and respective maximum principal stresses used in the structure.


The chosen material for the model geometry was *β*-TCP. Young's modulus can be assumed to be 5.37 GPa at a porosity of 18% (previously measured [31]). Previously measured porosities of sintered VPP column structures varied between 5 and 20%, because process conditions, not yet thoroughly optimized to the *β*-TCP suspension, led to micro delamination between layers and/or cavities within the bulk material. For simulations, a porosity of 5% was selected, representing a stable and load-bearing material. Young's modulus, depending on porosity, can be estimated using the following Equation [32]:

$$E(\phi) = E\_0(1 - 2\phi) \tag{1}$$

where *E*(0.05) = 7.552 GPa for porosity of 5% (*E* is Young's modulus and *φ* is porosity). There was no given data for Poisson's ratio. To be reasonable, 0.22 was chosen. The model geometry was discretized with a tetrahedral mesh of an appropriate density to get reliable results for a mechanical solution (Figure 1b, right). It was solved considering nonlinear mechanical behavior (large deflection effects) and a high-resolution load stepping for simulation runtime to capture complex deformation of the model. Whether those material failure studies fit the experimental compressive strength tests is further shown in Section 3.2.
