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International Journal of Bioprinting b-Ti21S TPMS FGPs produced by laser powder bed fusion
and the yield strength s of the cellular structures have to optimize not only the mechanical performances but also
y
been correlated to relative density using Gibson–Ashby the bone regeneration seem to be one of the most promising
[41]
model , according to Equations I and II. solutions . By modifying the shape, the porosity and the
[19]
pore size of TPMS structures, it is possible to achieve a
E = C ρ n 1 (I) structure with great surface curvature and permeability to
E 0 1 ρ 0 promote bone regeneration [1,42] .
Going inside a human bone, it is possible to observe
σ =C ρ 2 n (II) a variable porosity of the trabecular structure depending
σ 0 2 ρ 0 on the position inside it . Creating a graded porosity
[43]
where inside the structures, namely functionally graded porous
structures (FGPSs), is of fundamental importance to
E = elastic modulus of the cellular structure attain mechanical and biological efficiency in terms
E = elastic modulus of the bulk alloy of high strength, low stiffness, and improved tissue
0 ingrowth. Indeed, a pore size in the range of 100–600 mm
r = density of the cellular structure is essentially on the side where osseointegration must be
r = density of the bulk alloy guaranteed [26,44-46] . Higher density level is desired in the
0 junction with the solid part or where wear resistance must
C , C , n and n are the Gibson–Ashby constants. n and n be high. Differently, maximum porosity level is desired
1
2
1
2
1
2
result equal to 2 and 3/2 for bending dominated behavior inside the implant to reduce even more the stiffness of
and equal to 1 and 1 in case of stretching-dominated the prosthetic device [37,40,47,48] . In the recent years, different
behavior, respectively. The cellular structures are divided authors have focused their attention on the design of
into two main groups, the strut-based lattices and the cellular structure with a gradient porosity, both in the case
triply periodic minimal surface (TPMS) structures . A of strut-based and TPMS structures [25,32,33,47,49-53] . The nature
[30]
main difference is the presence of struts and nodes in the of TPMS unit cells to be defined by implicit equations
strut-based lattices, while the TPMS shows a smoother and to be optimized to achieve optimal mechanical
transition at the connection point of the ligaments. The properties and promote bone ingrowth makes them a
sharp notches formed at the junction between struts act promising solution in the case of FGPSs. Indeed, they
as local stress concentrations and are deleterious in terms avoid the problem of discontinuity between layers that
of fatigue resistance. This does not occur in the TPMS occurs with trabecular unit cells [32,33,47,53-55] . The effect of
structures, where no sharp notches are present since the FGPS on the mechanical response was evaluated by de
they are characterized by mean surface curvature at each Galarreta et al. under a quasi-static compression load.
[56]
point equal to zero [23,31-37] . Thanks to these features, they They demonstrated a mixture rule dependency between
are characterized by a high compression fatigue resistance the elastic modulus and the porosity level in the case of
of around 60% of the yield stress in case of Ti-6Al-4V . radially graded porous structure. A different behavior is
[38]
TPMS structures are divided into two subgroups that are exhibited by longitudinally graded structures, where the
skeletal or sheet TPMS-based structures depending on the elastic modulus of the entire structure is dominated by the
way they are created, by thickening the minimal surface or collapse of the weakest layer of the FGPSs. The correlations
by solidifying the volumes between the minimal surfaces, between the overall elastic modulus of the FGPS and the
respectively . Skeletal TPMS structures are characterized single value associated with the different porosity levels are
[39]
by an interconnected porosity and a lower elastic modulus shown in Equations III and IV in the case of longitudinal
with respect to the sheet TPMS [26,32-35,37,40] . Al-Ketan et al. and radial/lateral porous graded structures, respectively .
[26]
[56]
compared strut-based cellular structures, namely Kelvin
n
and Octet-truss, with sheet and skeletal-TPMSs. They 1 = ∑ k 1 (III)
highlighted the lower elastic modulus of the skeletal-based E i= 1 i E i
structures. In detail, skeletal-based Schoen’s I-graph- n
wrapped package (IWP) results in the one with the lowest E = ∑ k E i (IV)
i
i=1
quasi-elastic modulus. Nevertheless, lower values of the n refers to the total number of layers, k the volume fraction
peak stress and the toughness are achieved in the skeletal- of the corresponding layer with respect to the total volume,
i
based TPMS. Comparing the diamond, IWP, and gyroid E the elastic modulus of the FGPSs, and E the elastic
geometries, skeletal-based gyroid TPMS offers the right modulus of the corresponding layer i. i
compromise between a low elastic modulus and a good
strength. Since cancellous bone has a TPMS-like structure, To the best of our knowledge, manufacturability and
application of TPMS structure in porous prosthetic devices mechanical properties of TPMS-FGPS structures made
Volume 9 Issue 4 (2023) 188 https://doi.org/10.18063/ijb.729
