Influence of the Compaction Pressure and Sintering Temperature on the Mechanical Properties of Porous Titanium for Biomedical Applications

metals Article Influence of the Compaction Pressure and Sintering Temperature on the Mechanical Properties of Porous Titanium for Biomedical Applications Sandra M. Castillo, Sergio Muñoz * , Paloma Trueba, Eduardo Díaz and Yadir Torres Department of Engineering and Materials Science and Transportation, University of Seville, Av. Camino de los Descubrimientos s/n, 41092 Seville, Spain; [email protected] (S.M.C.); ptr[email protected] (P.T.); [email protected] (E.D.); ytorr[email protected] (Y.T.) *Correspondence: ser[email protected]; Tel.: +34-954482277 Received: 16 October 2019; Accepted: 20 November 2019; Published: 22 November 2019   Abstract: In the present work, the use of porous titanium is proposed as a solution to the difference in stiffness between the implant and bone tissue, avoiding the bone resorption. Conventional powder metallurgical technique is an industrially established route for fabrication of this type of material. The results are discussed in terms of the influence of compaction pressure and sintering temperature on the porosity (volumetric fraction, size, and morphology) and the quality of the sintering necks. A very good agreement between the predicted values obtained using a simple 2D finite element model, the experimental uniaxial compression behavior, and the analytical model proposed by Nielsen, has been found for both the Young’s modulus and the yield strength. The porous samples obtained by the loose sintering technique and using temperatures between 1000 ◦ C − 1100 ◦ C (about 40% of total porosity) are recommended for achieving a suitable biomechanical behavior for cortical bone partial replacement. Keywords: finite element method; porous titanium; powder metallurgy; loose sintering 1. Introduction Titanium and titanium alloys have been widely used as orthopedic implant materials due to their excellent mechanical and corrosion resistance, as well as their adequate biological behavior [ 1 – 8 ]. However, the stiffness of titanium is higher (100–110 GPa) than that of the cortical bone (20–25 GPa), which produces the stress-shielding phenomenon, promoting bone resorption surrounding the implant and compromising, in these cases, the implants reliability (clinical success). Despite the excellent advances, between 5% and 10% of bone implants can fail due to different causes during the five-year post-implantation period. Focusing on dental implants, after 10–15 years, survival rates over 89%, as well as a significant percentage of revision surgery (20%) to replace the failed implants (overload, peri-implantitis, etc.) have been reported [ 9 ]. Additionally, shorter recovery and loading times (e.g., immediate loading implants) are necessary to keep the quality of life of the patients. The design and manufacture of materials with a Young’s modulus close to that of cortical bone has been widely addressed in the scientific literature [ 10 – 13 ]. Among the possible solutions proposed are the use of metastable β -titanium alloys [ 14 – 18 ] and/or porous materials [ 19 – 27 ]. The bibliography includes up to 34 porous metal processing routes. All of them are summarized in the Doctoral Thesis of P. Trueba, and even a novel classification is proposed according to the state of the starting material [ 28 ]. In this context, some works show the limitations in controlling the quantity, size, distribution, and morphology of the pores by conventional techniques. Other works report the high cost and the difficulty in obtaining reproducible and versatile results using the new processing routes (laser sintering, spark plasma Metals 2019,9, 1249; doi:10.3390/met9121249 www.mdpi.com/journal/metals Metals 2019,9, 1249 2 of 15 sintering, electron beam melting or selective laser melting, electric current activated/assisted sintering, etc.) [ 29 – 31 ]. However, the production of porous materials via the conventional powder metallurgy (PM) route can be cost-effective, flexible, and lead to the desired design foams by controlling the temperature and sintering time, as well as the compaction pressure. Low temperatures and compaction pressures are employed to obtained higher porosities and lower Young’s modulus. The loose sintering (compaction pressure is not applied) is the extreme case (maximum porosity) of the PM route [ 32 – 34 ]. However, an excessive porosity and/or deficient sinterability (quality of the necks) of titanium powder at low temperatures, may compromise the mechanical behavior [35–39]. On the other hand, several researchers have tried to develop finite element (FE) models to predict the mechanical properties of porous materials. Some of them have used very simplistic approximations with two-dimensional (2D) FE models based on periodically distributed circular pores geometries [ 40 ], while others have develop complex three-dimensional (3D) FE models based on microstructural information of the pore morphology [ 41 , 42 ], whose complexity makes their extensive use unfeasible. In order to overcome this difficulty, a 2D finite element model was proposed by these authors in previous work [ 43 , 44 ]. In this current work, a very comprehensive investigation has been carried out with the aim of achieving a better understanding of the powder metallurgy process. In this scenario, the influence of the compaction pressure and sintering temperature on the global performance of the porous samples has been experimentally analyzed, particularly the microstructure (size, type, and morphology of the pores and the percentage of porosity) and the mechanical properties (Young’s modulus and yield strength). Furthermore, our results are compared to numerical simulations of the mechanical response using the finite element method (FEM). The proposed FE model has been based on geometries generated from the pore morphology information obtained from image analysis. 2. Experimental Methods 2.1. Sample Processing Titanium powder with an irregular morphology was obtained by a hydrogenation/dehydrogena tion process. The particle size distribution, according to the supplier SE-JONG Materials Co. Ltd. (Gojan-dong, South Korea) showed a size lower that 9.7 µ m (10 pct), 23.3 µ m (50 pct), and 48.4 µ m (90 pct), with a chemical composition equivalent to Grade 4 commercially pure titanium (c.p. Ti) ASTM F67-00. Powder metallurgy is a compaction process followed by sintering of the green compacts. Consequently, there are two essential processing variables to be evaluated: The pressure of compaction, P c , and the sintering temperature, T s . These variables have a great influence in the microstructure of the PM samples, for example their porosity (content, size, and morphology), the grain size and sinterability (quality of the necks between the metal particles), as well as their behavior in service (stiffness, yield stress, fatigue life and wear and corrosion resistance). The processing conditions used were chosen in order to obtain mechanical properties (Young’s modulus and yield strength) similar to those of the cortical bone (20–25 GPa and 150–180 MPa, respectively) [45–48]. The first series of samples was fabricated following the loose-sintering process [ 48 ]. First, the metal powder was poured into a mold and vibrated, with no compaction (Pc=0). Then, the samples were heated to the corresponding sintering temperatures (1000 ◦ C, 1100 ◦ C, 1200 ◦ C, and 1300 ◦ C) for 2 h in a furnace with a ceramic horizontal tube (CARBOLYTE STF 15/75/450, Derbyshire, UK) under high vacuum (<0.05 Pa =5×10−4mbar). Thesecondseriesofsampleswasobtainedfollowingthesamesinteringprocess, but withaprevious compaction of the powder at two compacting pressures (P c =13 and 38.5 MPa). The implementation of very low compacting pressures required control of the applied load rate and a very sensitive load cell. This step was carried out by using an INSTRON 5505 universal machine (Instron, HighWycombe, UK). The deformation rate was 6 kN/s, the dwell time was 2 min, and the unloading time was 15 s for decreasing loads down to 15 kgf. Two compaction dies (tool steel material) were used: One of 8 mm (uniaxial compression test) and one of 12 mm (ultrasound measurements). To lubricate the sleeve Metals 2019,9, 1249 3 of 15 walls, Ethlyne-Bis-Stearamide (EBS) wax was used during the compaction. The powder mass used to obtain the specimens with the suitable dimensions (height/diameter: 0.8 and 1.2) for compression and ultrasound tests were 5.14 and 13.21 g, respectively. These values were calculated taking into account the most unfavorable (no porosity) situation. At least three porous samples were manufactured for each condition. 2.2. Porosity Characterisation The density measurements were performed using the Archimedes’ method with distilled water impregnation (ASTM C373-88). The total and interconnected porosity were calculated from the density measurement by using well-known mathematical expressions [ 49 ]. In addition, the total porosity ( θ ), the equivalent diameter (D eq ), and the pore elongation factor (F e , ratio between major and minor axis of ellipse equivalent to pore) were evaluated by image analysis, using an optical microscope Nikon Epiphot (Nikon, Tokyo, Japan) coupled with a camera Jenoptik Progres C3 (Jenoptik, Jena, Germany), and the suitable analysis software (Image-ProPlus 6.2, Mediacibernetic, Bethesda, MD, USA). To perform this study, a conventional metallographic preparation of the specimen was necessary, including resin mounting, grinding, and mechano-chemical polishing with magnesium oxide (MgO) and hydrogen peroxide (H 2 O 2 ). Image analysis was performed using 5 pictures of 5x for each sample (one for each kind of material). 2.3. Mechanical Testing The mechanical properties of the different porous materials were measured using two different methodologies. Firstly, through the uniaxial compression testing, according to the general Standard ASTM E9-89A and, particularly, the Standard for Mechanical testing of metals-ductility testing-compression test for porous and cellular metals (ISO 13314: 2011), with a height/diameter aspect ratio of 0.8. The tests were performed using a universal electromechanical Instron machine 5505 (Instron, HighWycombe, UK) by applying a strain rate of 0.005 mm/mm · min. All tests were finished for a 50 pct strain and, afterwards, Young modulus (E c ) and yield strength ( σy ) were obtained. The Young’s modulus from the compression stress–strain curves was corrected with the testing machine stiffness (87.9 kN/mm). At least three different specimens were tested for each porous material under study. Furthermore, the value of dynamic Young’s modulus (E d ) was also evaluated through the ultrasound technique, using a Krautkramer USM 35 equipment (GE Measurement & Control Solutions, Minden, Germany). To evaluate longitudinal waves, a Panametrics S-NDT (Olympus, Tokyo, Japan) probe of 4 MHz and an ultrasonic couplant fluid (Sonotrace grade 30) was used, while for cross-sectional waves, a Panametric S-V153 probe of 1 MHz/.5 and a shear wave couplant (Panametric S-NDT, Olympus, Tokyo, Japan) was used. Finally, from the values obtained and by a proper mathematical expression, the dynamic Young’s modulus was calculated [50]. 2.4. Finite Element Modelling In the present work, a FE model previously proposed by the authors [ 43 , 44 ] was used. This numerical model combines the simplicity of a 2D periodic geometry with the complex information of the pore morphology extracted from experimentation. Thanks to the low computational cost of the FE model, a large number of simulations could be carried out in order to obtain more reliable results. The methodology to generate the 2D simulated microstructure is explained as follows (see Figure 1): The baseline configuration consisted of a square matrix with a side length a, with a series of nxn pores periodically distributed inside. The number of pores along any direction, n, had to be chosen to ensure a minimum representative volume element (RVE), which gives reliable results in the simulations. On the other hand, the ratio between the side length and the average pore equivalent diameter (a/D eq ) determined the total porosity of the simulated geometry. Once the defining parameters (D eq ,a, n) were determined, the pores are generated in a uniform spatial distribution, but with a pore morphology (D eq Metals 2019,9, 1249 4 of 15 and F e ) randomly generated following a normal distribution from the experimental data. In addition, the pore orientation was also randomly generated following a uniform distribution. Metals 2019, 9, x FOR PEER REVIEW 4 of 15 Figure 1. Geometry generation for finite element (FE) model. The baseline configuration consisted of a square matrix with a side length a, with a series of nxn pores periodically distributed inside. The number of pores along any direction, n, had to be chosen to ensure a minimum representative volume element (RVE), which gives reliable results in the simulations. On the other hand, the ratio between the side length and the average pore equivalent diameter (a/Deq) determined the total porosity of the simulated geometry. Once the defining parameters (Deq, a, n) were determined, the pores are generated in a uniform spatial distribution, but with a pore morphology (Deq and Fe) randomly generated following a normal distribution from the experimental data. In addition, the pore orientation was also randomly generated following a uniform distribution. The simulated microstructure was comprised of two different phases: Titanium matrix and pores. The mechanical properties of grade IV c.p. Ti were used to describe the behavior of the titanium matrix [42]: Young’s Modulus, Edense = 110 GPa, yield strength, σy-dense = 650 MPa, and Poisson’s ratio, ν = 0.33. In order to describe the hardening plasticity behavior of titanium, an isotropic hardening with a very small tangent modulus ET = 1 GPa was used. To describe the pores, a linear elastic material with very low Young’s modulus (E = 10−7 GPa) and Poisson’s ratio (ν = 0.33) was used in order to avoid possible computational difficulties. In the simulations, the boundary conditions were defined as follows: The bottom boundary is fixed in the y-direction; the left boundary is fixed in the x-direction; the right boundary with coupling constraint in the x-direction (all points of the right boundary have the same displacement in x-direction). A displacement boundary condition of up to 1% macroscopic strain was applied on the upper boundary in the y-direction. The use of displacement conditions allows an easy control of the calculation steps, being very helpful to obtain the stress/strain information. All 2D geometries had been meshed with eight-node biquadratic plane strain elements. All finite element simulations had been performed using the commercial finite element code ANSYS. Mesh convergence had been verified based on overall and local stress values. 4. Results and Discussion 4.1. Porosity Characterisation The porosity distribution obtained from image analysis of the different porous samples studied are shown in Figure 2. The contents (vol.%) and morphological parameters of the pores (equivalent diameter (Deq) and pore elongation factor (Fe)), and the total and interconnected porosity (θ and θi, respectively) are summarized in Table 1. The porosity analysis reveals a clear influence of the pressure of compaction and the sintering temperature. Higher values of Pc and Ts lead to lower values of porosity (higher density) and to smaller pores (lower equivalent diameters), mostly isolated and rounded (higher elongation factor), as could be expected. It should be pointed out that the porosity values measured by the Archimedes’ method was reasonably in agreement with the Figure 1. Geometry generation for finite element (FE) model. The simulated microstructure was comprised of two different phases: Titanium matrix and pores. The mechanical properties of grade IV c.p. Ti were used to describe the behavior of the titanium matrix [ 42 ]: Young’s Modulus, E dense =110 GPa, yield strength, σy-dense =650 MPa, and Poisson’s ratio, ν =0.33. In order to describe the hardening plasticity behavior of titanium, an isotropic hardening with a very small tangent modulus E T =1 GPa was used. To describe the pores, a linear elastic material with very low Young’s modulus (E=10 −7 GPa) and Poisson’s ratio ( ν =0.33) was used in order to avoid possible computational difficulties. In the simulations, the boundary conditions were defined as follows: The bottom boundary is fixed in the y-direction; the left boundary is fixed in the x-direction; the right boundary with coupling constraint in the x-direction (all points of the right boundary have the same displacement in x-direction). A displacement boundary condition of up to 1% macroscopic strain was applied on the upper boundary in the y-direction. The use of displacement conditions allows an easy control of the calculation steps, being very helpful to obtain the stress/strain information. All 2D geometries had been meshed with eight-node biquadratic plane strain elements. All finite element simulations had been performed using the commercial finite element code ANSYS. Mesh convergence had been verified based on overall and local stress values. 3. Results and Discussion 3.1. Porosity Characterisation The porosity distribution obtained from image analysis of the different porous samples studied are shown in Figure 2. The contents (vol.%) and morphological parameters of the pores (equivalent diameter (D eq ) and pore elongation factor (Fe)), and the total and interconnected porosity ( θ and θi , respectively) are summarized in Table 1. The porosity analysis reveals a clear influence of the pressure of compaction and the sintering temperature. Higher values of P c and T s lead to lower values of porosity (higher density) and to smaller pores (lower equivalent diameters), mostly isolated and rounded (higher elongation factor), as could be expected. It should be pointed out that the porosity values measured by the Archimedes’ method was reasonably in agreement with the porosity estimation done by image analysis. In addition, the interconnected porosity was always lower than the total porosity and showed a similar trend. In this context, a sintering temperature and a compacting pressure reduction produced the largest pores and more irregular shape because of the coalescence of the smaller ones. The pore sizes obtained by conventional PM routes at low pressures and loose sintering are Metals 2019,9, 1249 5 of 15 insufficient to ensure the bone ingrowth requirements (interconnected porosity and sizes larger than 100 µ m are required). However, previous studies have confirmed the role of surface roughness inside pores on cell differentiation and osteoblast adhesion [ 43 , 51 – 53 ]. The inner surfaces of pores obtained by loose sintering are less rough (adhesion of osteoblasts is poor) than those obtained by applying pressure during compaction. Inside the pores, osteoblast cells attached to the walls perceive higher load transfer and this local stress will promote cell differentiation and mineralization and, therefore, the potential implant osseointegration [54]. Metals 2019, 9, x FOR PEER REVIEW 5 of 15 porosity estimation done by image analysis. In addition, the interconnected porosity was always lower than the total porosity and showed a similar trend. In this context, a sintering temperature and a compacting pressure reduction produced the largest pores and more irregular shape because of the coalescence of the smaller ones. The pore sizes obtained by conventional PM routes at low pressures and loose sintering are insufficient to ensure the bone ingrowth requirements (interconnected porosity and sizes larger than 100 μm are required). However, previous studies have confirmed the role of surface roughness inside pores on cell differentiation and osteoblast adhesion [43,51–53]. The inner surfaces of pores obtained by loose sintering are less rough (adhesion of osteoblasts is poor) than those obtained by applying pressure during compaction. Inside the pores, osteoblast cells attached to the walls perceive higher load transfer and this local stress will promote cell differentiation and mineralization and, therefore, the potential implant osseointegration [54]. Figure 2. Optical images of the porous samples obtained with different compaction pressure and sintering temperature. Common scale bar. Table 1. Influence of processing conditions on the characteristics of the obtained porosity. T s (°C) P c (MPa) Density (g/cm 3 ) Archimedes’ method Images Analysis Total porosity (%) Interconnected porosity, θ i (%) θ (%) D eq (µm) Fe 1000 0 (LS) 2.47 ± 0.23 41.4 ± 1.4 39.1 ± 1.7 43.0 ± 1.2 16.9 ± 3.1 0.59 ± 0.17 13 3.17 ± 0.02 32.2 ± 0.7 28.27 ± 0.4 30.8 ± 1.6 13.8 ± 2.9 0.60 ± 0.17 38.5 3.31 ± 0.03 29.4 ± 0.8 25.08 ± 0.9 29.5 ± 3.3 14.9 ± 2.1 0.57 ± 0.17 1100 0 (LS) 2.79 ± 0.22 37.4 ± 1.5 36.2 ± 1.5 36.9 ± 4.0 18.5 ± 3.9 0.57 ± 0.17 13 3.37 ± 0.39 21.8 ± 0.66 22.2 ± 1.9 20.7 ± 2.1 11.8 ± 2.6 0.64 ± 0.18 Figure 2. Optical images of the porous samples obtained with different compaction pressure and sintering temperature. Common scale bar. Table 1. Influence of processing conditions on the characteristics of the obtained porosity. Ts(◦C) Pc(MPa) Density (g/cm3) Archimedes’ Method Images Analysis Total Porosity (%) Interconnected Porosity, θi(%) θ(%) Deq (µm) Fe 1000 0 (LS) 2.47 ±0.23 41.4 ±1.4 39.1 ±1.7 43.0 ±1.2 16.9 ±3.1 0.59 ±0.17 13 3.17 ±0.02 32.2 ±0.7 28.27 ±0.4 30.8 ±1.6 13.8 ±2.9 0.60 ±0.17 38.5 3.31 ±0.03 29.4 ±0.8 25.08 ±0.9 29.5 ±3.3 14.9 ±2.1 0.57 ±0.17 1100 0 (LS) 2.79 ±0.22 37.4 ±1.5 36.2 ±1.5 36.9 ±4.0 18.5 ±3.9 0.57 ±0.17 13 3.37 ±0.39 21.8 ±0.66 22.2 ±1.9 20.7 ±2.1 11.8 ±2.6 0.64 ±0.18 38.5 3.65 ±0.05 20.3 ±0.4 14.9 ±1.1 19.9 ±2.5 11.0 ±2.1 0.64 ±0.18 1200 0 (LS) 3.12 ±0.14 30.0 ±0.6 27.9 ±1.2 30.0 ±6.9 15.2 ±2.4 0.62 ±0.18 13 3.88 ±0.02 13.6 ±0.4 6.2 ±0.7 13.2 ±1.6 10.4 ±1.5 0.70 ±0.17 38.5 3.81 ±0.04 17.5 ±0.6 10.1 ±1.0 17.3 ±2.1 12.8 ±2.9 0.70 ±0.17 1300 0 (LS) 3.72 ±0.07 15.1 ±0.2 13.8 ±1.5 14.9 ±1.8 13.7 ±2.4 0.68 ±0.18 13 3.96 ±0.08 13.4 ±0.3 5.6 ±0.9 12.8 ±0.6 13.4 ±2.8 0.75 ±0.17 38.5 3.99 ±0.04 11.8 ±0.3 5.4 ±0.5 12.2 ±1.3 13.0 ±2.6 0.75 ±0.17 Metals 2019,9, 1249 6 of 15 3.2. Mechanical Behavior As it has been previously mentioned, the mechanical properties of the porous compacts have been measured using two different methodologies (see Table 2): The uniaxial compression test (Young’s modulus and yield strength) and the ultrasounds technique (dynamic Young’s module). As it has been pointed out by the authors in previous work [ 43 , 45 , 46 ], a discrepancy between the Young’s modulus obtained through both experimental approaches (compression test and ultrasounds) has been encountered. Greiner et al. (2005) [ 55 ] linked this to a super-elastic deformation within the linear-elastic range of NiTi materials; the stiffness measured by the ultrasonic technique decreases with increasing porosity, in agreement with elastic Eshelby-based theory for closed, spherical porosity. Torres et al. [ 46 ] reported a similar trend for others porous materials and related this difference to a stiffness testing machine effect in which the mechanical system and the sample were considered as two springs in series. Moreover, it must be remembered that the Ti matrix is different at each cross-section of the cylindrical sample during a compression test; the material collapse starts at the section with the lowest titanium content. However, in the ultrasonic technique, the Young’s modulus is estimated as a function of the wave velocity, which goes through all the sections of the sample. Therefore, a change in one of four parameters (time of transfer, attenuation, reflection, and frequency) associated with the high-frequency waves transition from side to side of the material, could be linked to variations of physical properties such as Young’s modulus, density, or homogeneity in the microstructure. In these works, the reliability and certainty of the ultrasound measurements were validated by comparing them to a well-known and accepted pore-elasticity model proposed by Nielsen [ 56 ], which includes porosity parameters experimentally determined. Consequently, the Young’s modulus obtained from the ultrasound technique seems to be more reliable than that obtained from compression tests. The porous samples obtained by loose sintering and 1100 ◦ C have the most suitable biomechanical behavior in terms of guaranteeing the requirements of the cortical bone tissue (E =20–25 GPa and σy=150–180 MPa, respectively). Table 2. Mechanical behavior: Young’s modulus and 0.2% offset yield strength. Ts(◦C) Pc(MPa) Ultrasound Test Compression Test Ed(GPa)Ec(GPa)σy(MPa) 1000 0 (LS) 29.3 ±2.2 13.4 ±4.1 123 ±10 13 49.5 ±0.8 40.8 ±3.6 342 ±8 38.5 54.6 ±0.3 53.6 ±5.2 358 ±7 1100 0 (LS) 34.9 ±0.9 27.9 ±3.2 189 ±11 13 61.2 ±7.0 42.3 ±4.4 477 ±12 38.5 63.8 ±3.8 34.5 ±2.8 452 ±9 1200 0 (LS) 46.4 ±5.8 40.2 ±3.7 422 ±8 13 76.3 ±2.7 46.4 ±3.2 544 ±8 38.5 72.1 ±0.3 55.2 ±4.2 567 ±10 1300 0 (LS) 69.9 ±2.2 33.2 ±3.2 467 ±8 13 81.3 ±6.2 56.2 ±5.2 544 ±7 38.5 80.7 ±0.7 68.3 ±5.4 567 ±8 Metals 2019,9, 1249 7 of 15 In order to clearly observe the influence of both processing variables (pressure of compaction, P c , and sintering temperature, T s ) on the mechanical properties of the processed compacts, the experimental results summarized in Table 2are plotted in Figure 3. With the aim of showing the influence of the total porosity, θ , on the mechanical properties, the values of this variable have also been included in the graphs. Metals 2019, 9, x FOR PEER REVIEW 7 of 15 13 76.3 ± 2.7 46.4 ± 3.2 544 ± 8 38.5 72.1 ± 0.3 55.2 ± 4.2 567 ± 10 1300 0 (LS) 69.9 ± 2.2 33.2 ± 3.2 467 ± 8 13 81.3 ± 6.2 56.2 ± 5.2 544 ± 7 38.5 80.7 ± 0.7 68.3 ± 5.4 567 ± 8 In order to clearly observe the influence of both processing variables (pressure of compaction, Pc, and sintering temperature, Ts) on the mechanical properties of the processed compacts, the experimental results summarized in Table 2 are plotted in Figure 3. With the aim of showing the influence of the total porosity, θ, on the mechanical properties, the values of this variable have also been included in the graphs. Figure 3. Experimental results of Young’s modulus (a) and yield strength (b), in terms of compaction pressure, for each sintering temperature. Values of total porosity, θ, are also included. In view of the results displayed in Figure 3, several conclusions can be drawn. On the one hand, the mechanical properties of the porous compacts depend on both processing variables (Pc and Ts), being the sintering temperature the more influential one. As it can be expected, both compression strength and Young’s modulus decrease as the porosity increases. It should be noted that, although the total porosity is the more influential porosity variable, there is also a noticeable influence of the other porosity variables related to the pore morphology (Deq and Fe) on the mechanical properties, especially on the yield strength. In this regard, the reduction in mechanical strength was more representative, as expected, for larger pores. This trend is related to the increasing titanium remaining in the matrix, as well as to softer pores contours (greater Ff), and to the better quality of the sintering necks, associated with a better cold welding of titanium powders and a decrease in surface energy (coalescence of titanium particles at higher sintering temperatures). The pores act like notches that produce stress concentrations (“notch strengthening”), besides reducing strength effective area. Within this context, it is expected that some changes from a close to an open porosity regime will generate a triaxial effect that can partially compensate for the strength loss associated with any increase of porous fraction. Therefore, under interconnected porosity conditions, yield strength will not be drastically reduced due to increments in porosity, which is the case when isolated porosity is increased. It can be concluded that, by varying the processing variables (Pc and Ts), the total porosity of the porous material can be carefully controlled, and thus, the desired mechanical properties. 4.3. Finite Element Method Results Following the methodology presented in Section 2.4 and making use of the main porosity characteristics shown in Table 1, several finite element geometries have been randomly generated for each porous material under study. Following this methodology, 60 different geometries (5 for each of the 12 porous materials under study) have been generated and used in simulations. One of Figure 3. Experimental results of Young’s modulus ( a ) and yield strength ( b ), in terms of compaction pressure, for each sintering temperature. Values of total porosity, θ, are also included. In view of the results displayed in Figure 3, several conclusions can be drawn. On the one hand, the mechanical properties of the porous compacts depend on both processing variables (P c and T s ), being the sintering temperature the more influential one. As it can be expected, both compression strength and Young’s modulus decrease as the porosity increases. It should be noted that, although the total porosity is the more influential porosity variable, there is also a noticeable influence of the other porosity variables related to the pore morphology (D eq and F e ) on the mechanical properties, especially on the yield strength. In this regard, the reduction in mechanical strength was more representative, as expected, for larger pores. This trend is related to the increasing titanium remaining in the matrix, as well as to softer pores contours (greater F f ), and to the better quality of the sintering necks, associated with a better cold welding of titanium powders and a decrease in surface energy (coalescence of titanium particles at higher sintering temperatures). The pores act like notches that produce stress concentrations (“notch strengthening”), besides reducing strength effective area. Within this context, it is expected that some changes from a close to an open porosity regime will generate a triaxial effect that can partially compensate for the strength loss associated with any increase of porous fraction. Therefore, under interconnected porosity conditions, yield strength will not be drastically reduced due to increments in porosity, which is the case when isolated porosity is increased. It can be concluded that, by varying the processing variables (P c and T s ), the total porosity of the porous material can be carefully controlled, and thus, the desired mechanical properties. 3.3. Finite Element Method Results Following the methodology presented in Section 2.4 and making use of the main porosity characteristics shown in Table 1, several finite element geometries have been randomly generated for each porous material under study. Following this methodology, 60 different geometries (5 for each of the 12 porous materials under study) have been generated and used in simulations. One of the five geometries for each porous material under study has been chosen as “representative” and presented in Figure 4. Note that, for each geometry, the values of the total porosity ( θ ) and side length (a) have been included in the figure. It is important to note that the porous geometries used in the numerical model (some of which have been presented in Figure 4) are not the same as the porous geometries extracted from the experiments Metals 2019,9, 1249 8 of 15 (some of which have been presented in Figure 2). In fact, they do not need to be the exactly the same, but “statistically similar”, as is explained as follows. The FE geometries have been randomly generated following the statistical information of the porosity in real compacts by image analysis. This way, both geometries (FE and experimental one) have the same porosity level and the same statistical distribution of the main pore morphology characteristics (elongation factor and diameter). Consequently, both geometries are similar and can be expected to behave similarly. Metals 2019, 9, x FOR PEER REVIEW 8 of 15 the five geometries for each porous material under study has been chosen as “representative” and presented in Figure 4. Note that, for each geometry, the values of the total porosity (θ) and side length (a) have been included in the figure. Figure 4. Representative geometries, for each porous material, with values of porosity (θ) and side length (a) included. It is important to note that the porous geometries used in the numerical model (some of which have been presented in Figure 4) are not the same as the porous geometries extracted from the experiments (some of which have been presented in Figure 2). In fact, they do not need to be the exactly the same, but “statistically similar”, as is explained as follows. The FE geometries have been randomly generated following the statistical information of the porosity in real compacts by image analysis. This way, both geometries (FE and experimental one) have the same porosity level and the same statistical distribution of the main pore morphology characteristics (elongation factor and diameter). Consequently, both geometries are similar and can be expected to behave similarly. Making use of the FE model, series of compression tests under displacement control up to 1% macroscopic strain have been simulated. From them, the predicted uniaxial stress–strain responses have been obtained and presented in Figure 5, for each of the 60 different simulations performed. Figure 4. Representative geometries, for each porous material, with values of porosity ( θ ) and side length (a) included. Making use of the FE model, series of compression tests under displacement control up to 1% macroscopic strain have been simulated. From them, the predicted uniaxial stress–strain responses have been obtained and presented in Figure 5, for each of the 60 different simulations performed. From the obtained stress–strain curves, the average response for each porous material is summarized in Table 3, where the predicted Young’s modulus and the 0.2% offset yield strength are listed in terms of overall average ± standard deviation. Notice that, in the results obtained from the simulations, the corresponding standard deviation for each case has been included. Thanks to the low computational cost of the proposed FE model, an important number of simulations can be carried out to obtain more feasible and representative predictions. As previously mentioned, five different geometries have been generated for each porous material under study. Consequently, from these five “virtual experiments”, we can obtain an average value and a standard deviation for each prediction. From the results summarized in Table 3, several conclusions can be drawn. Firstly, one expected trend: An increase in the compaction pressure and sintering temperature leads to an increase in both material properties, Young’s modulus, and yield strength. These results can be easily explained in view of the porous geometries presented in Figure 4. At lower values of P c and T s , the porosity level Metals 2019,9, 1249 9 of 15 increases and, therefore, the fraction of the matrix bearing stress also decreases. This is due to the fact that the stiffness of the porous material (Young’s modulus) decreases. Regarding the yield strength, the explanation is a bit more complex. At lower values of P c and T s , not only the level of porosity increases, but also the shape of the porous is sharper (lower values of elongation factor, see Table 1), which leads to higher level of stress concentration around pores. Both factors lead to greater probability of plastic deformation and, consequently, lower values of yield strength. Metals 2019, 9, x FOR PEER REVIEW 9 of 15 Figure 5. Macroscopic stress–strain response of porous titanium by FE model, for each sintering temperature: (a) 1000 °C; (b) 1100 °C; (c) 1200 °C; and (d) 1300 °C. From the obtained stress–strain curves, the average response for each porous material is summarized in Table 3, where the predicted Young’s modulus and the 0.2% offset yield strength are listed in terms of overall average ± standard deviation. Notice that, in the results obtained from the simulations, the corresponding standard deviation for each case has been included. Thanks to the low computational cost of the proposed FE model, an important number of simulations can be carried out to obtain more feasible and representative predictions. As previously mentioned, five different geometries have been generated for each porous material under study. Consequently, from these five “virtual experiments”, we can obtain an average value and a standard deviation for each prediction. Table 3. Estimated values for elastic modulus and 0.2% offset yield strength by FE model. T s (°C) P c (MPa) E (GPa) σ y (MPa) 1000 0 (LS) 29.9 ± 6.1 124.4 ± 31.5 13 44.3 ± 5.5 240.2 ± 56.0 38.5 49 .8 ± 4.3 275.1 ± 42.2 1100 0 (LS) 36.6 ± 6.0 201.3 ± 28.4 13 67.3 ± 3.7 434.4 ± 39.5 38.5 70.3 ± 3.6 481.7 ± 49.1 1200 0 (LS) 49.9 ± 4.1 271.1 ± 38.9 13 84.9 ± 2.2 609.2 ± 21.4 38.5 76.9 ± 2.6 526.5 ± 26.0 1300 0 (LS) 82.2 ± 1.4 575.8 ± 14.5 13 86.2 ± 1.6 615.1 ± 16.5 38.5 89.2 ± 1.4 636.3 ± 14.0 Figure 5. Macroscopic stress–strain response of porous titanium by FE model, for each sintering temperature: (a) 1000 ◦C; (b) 1100 ◦C; (c) 1200 ◦C; and (d) 1300 ◦C. Table 3. Estimated values for elastic modulus and 0.2% offset yield strength by FE model. Ts(◦C) Pc(MPa) E (GPa) σy(MPa) 1000 0 (LS) 29.9 ±6.1 124.4 ±31.5 13 44.3 ±5.5 240.2 ±56.0 38.5 49.8 ±4.3 275.1 ±42.2 1100 0 (LS) 36.6 ±6.0 201.3 ±28.4 13 67.3 ±3.7 434.4 ±39.5 38.5 70.3 ±3.6 481.7 ±49.1 1200 0 (LS) 49.9 ±4.1 271.1 ±38.9 13 84.9 ±2.2 609.2 ±21.4 38.5 76.9 ±2.6 526.5 ±26.0 1300 0 (LS) 82.2 ±1.4 575.8 ±14.5 13 86.2 ±1.6 615.1 ±16.5 38.5 89.2 ±1.4 636.3 ±14.0