## 3 Results
### Finite Element Simulation  
The case under study chosen in this numerical analysis is the porous compact fabricated following the space holder technique, by using ammonium bicarbonate, NH\({}_{4}\)HCO\({}_{3}\), as a space holder and obtaining a 40 vol.% of total porosity. This is the porous compact that has shown the most interesting mechanical properties, with the best balance between stiffness and mechanical integrity and, consequently, the ideal candidate for the use in cortical bone replacement. Following the methodology described by [PERSON] et al. [6; 34], and making use of the main porosity characteristics extracted from experiments (total porosity, pore size distribution and elongation factor distribution), one finite element geometry has been randomly generated for the porous materials under study: 40 vol.% of total porosity. The simulated microstructure of the porous material is comprised of two different phases: a titanium matrix and a series of pores randomly distributed. The mechanical properties of cp Ti have been used to describe the behaviour of the titanium matrix: a Young's modulus \(E_{T\!I}=110\) GPa, a yield stress  
Figure 3: (a) Density vs. Relative Strength; (b) Interconnected Porosity and Relative Porosity vs. Relative Strength (porous material strength/full dense material strength).  
\(\sigma_{Ti}\) = 700 MPa and a [PERSON]'s ratio \(\
u\) = 0.33. In order to describe the hardening plasticity behaviour of titanium, an isotropic hardening with a very small tangent modulus \(E_{T}\) = 1 GPa has been used. The pore morphology (equivalent diameter, \(D_{\alpha\beta}\), and elongation factor, \(F_{e}\)) has randomly generated following a normal distribution from the experimental data. The following values have been extracted from experimentation to be used in the pore generation: \(D_{\alpha\beta}\) = 32 \(\upmu\)m and \(F_{e}\) = 0.65. In addition, the pore orientation has also been randomly generated following a uniform distribution.  
Making use of the FE models, two different compression tests have been simulated. First, with the aim of predicting the mechanical properties of the porous material, a compression test under displacement control up to 1% macroscopic strain has been simulated. Then, the predicted uniaxial stress-strain responses have been obtained. The results in the simulation are summarized in Table 4.  
From the presented results, it can be seen that very good agreement is achieved with the proposed FE model, for both [PERSON]'s modulus and yield stress.  
Second, in order to complete the numerical analysis and for a better understanding of the mechanical behaviour of the porous material, the stress distribution within the porous matrix has been analysed in detail. By using the proposed FE model, a compression test under displacement control was performed until the macroscopic yield is reached (\(\sigma_{y}\) = 170 MPa, in this case). The results of this virtual test can be seen in Figure 4, where the contour plot of the von Mises stress distribution at macroscopic yield is shown.  
In Figure 4, it can be observed that, even though the applied macroscopic stress is only 170 MPa, there is a considerable portion of the material with a high stress level, due to stress concentration around the pores, as it could be expected.