### Numerical procedure

**It is aimed to compute the macroscopic stresses** and strains using a homogenization procedure. The procedure consists on defining the deformation state at each integration point, in the FE model, as well as the current matrix average state that depends on the corresponding state at the previous step time. FE analysis provides an alternative approach for estimating material properties. A FE model for the heterogeneous test is constructed in which the material parameters to be determined are considered as variable. The simulation is performed and results are compared to data from a comparable experiment data set. The agreement between the model predictions and the data is quantified and judged to be adequate or not. If the agreement is not adequate, the parameter values are updated, a new FE model is created and run, and the process continues till obtaining satisfactory results.

### Boundary conditions and material properties

**The boundary condition is so set that** at the bottom border Ux, Uy, Uz = 0 and at the upper face of the specimen is imposed a negative displacement load Uy in y-direction to make via the rigid plate indenter. A static step with small step time is used to assess the gradual evolution of stresses and strains in the elements model, the equivalent reaction force is calculated and used to furnishes the maximum load, when the deformation at the contact zone riches comparable value with the considered limited experimental one.

**The mechanical HAP** bio composite material properties are derived from the particles inclusions properties. Hydroxyapatite bio composite ceramics reinforced with alumina – Al2O3, pure titanium and pure pulverised boron powder. The inclusion material is assumed to be linearly elastic with elastic modulus Ep =390GPa and Poisson’s ratio t>=0.33. Pure titanium with high purity fraction is selected to be the metallic matrix with elastic modulus Em =114GPa, Poisson’s ratio t>=0.34.

The FE solution used to simulate two-phase composites consisting of an elasto-plastic matrix reinforced by linear elastic inclusions. Uniaxial compressive loading are successively applied to the multi particle cells embedded in the ceramic matrix. The average of the macroscopic strain over a RVE computed at each time step provides the loading history for the corresponding FE models.

**Average equivalent stress in the** inclusions of two phase composite materials is determined for different volume fractions of the reinforcing phase. Multiparticles, FE predictions (FE with 25% volume fraction) correspond to a uniaxial compression test. For comparison, the predictions of the compressive results, by mean of deformation and maximum loading were presented for uniaxial compression (Figures 8b, c and 9 and also Table 1). The average volume fractions for particles were indicated in the same table. Only results due to compressive tests using a rigid plate are presented here.

**Fig. 8. a) Experimental setup scheme for modelling under compression with plate indenter and b, c) FEM**

**Fig. 9. a) Microstructure of the sample containing 20% Boron (b) details of sheet and plate indenter and c) FEM**

**Table 1. Physical and mechanical properties with different aspect ratio of the HAP composites**

## Conclusions

**This preliminary study revealed that** the addition of the pure metallic boron (atomic pulverised) and pure titanium powders improve the structure and mechanical behaviour of the samples produced by microwave sintering. The hardness and compressive strength were greatly achieved. Addition of the Paraffin gives very smooth porous structure. Compressive strength values of the higher boron containing samples were demonstrated the superiority of mechanical properties for all cases after microwave sintering.

**As for FEM,** the reaction of the inclusions in the HAP matrix can be predicted based on the solution of inclusions in a finite medium having the properties of the matrix. Here only a simple FEM allowing heterogeneous field was proposed to solve an equivalent inclusion problem. Macroscopic deformation histories corresponding to the non-monotonic uniaxial and the plane strain compression were consecutively considered. Here, a simple prediction has made by using a simple FEM Naturally, development of HAP biocomposites needs more investigations to attempt a high accuracy between experimental results and their equivalent FE predictions.