Page 177 - v11i4

International Journal of Bioprinting Design of SLM-Ta artificial vertebral body

Find 1 2 ,..., n ; arrangement of identical unit cells. A strut is the basic
1 1 n p component of a strut-based unit cell and plays a crucial
minC UKU u k 0 i E u ; role in determining its mechanical properties. The elastic
T
T
s ii
i
2 2 i 1 (III) moduli of unit cells with straight struts can be obtained
V
st.. f using the Euler–Bernoulli and Timoshenko beam
V 0 theories. Based on the Euler–Bernoulli beam theory, the
51
0 min i 1 strut displacement along the z-axis under compression is
given by:
where ρ is the density of the element, C is compliance,
U is the global displacement vector, K is the global stiffness Fl cos 2 Flsin 2
3
matrix, n is the number of elements, u is the element 12 EI EA (IV)
z
i
displacement vector, k is the element stiffness matrix for s s
0
unit Young’s modulus, E is the stiffness of the material,
s
p is the penalization factor, V and V are the material where F is the force on the strut, θ is the angle between
0
ρ
volume and the design domain volume, respectively, f is the strut and the horizontal axis, A is the cross-sectional
the prescribed volume fraction, and ρ is the minimum area of the strut, and I is the moment of inertia, defined as
min
element density. I = πd /64.
4
Altair OptiStruct software (Altair Engineering Inc., In addition to the strut deformation caused by axial
TM
USA) was used for the topology optimization design of the loads and bending moments, the Timoshenko beam
thin-walled model. The mesh type was tetrahedral, and theory also considers the strut shear deformation. The strut
the mesh size was set to 0.5 mm. The material properties displacement along the z-axis can be calculated as follows:
of Ta were assigned to the mesh elements, with an elastic
modulus of 186 GPa, a Poisson’s ratio of 0.35, and a density
3
of 16.6 g/cm . The boundary conditions were defined such Fl cos 2 Flsin 2 Flcos 2 (V)
3
that the bottom surface of the thin-walled model was z 12 EI EA AG s
s
s
fully constrained and a pressure of 2 MPa was applied to
the top surface. The topology optimization parameters where κ is the shear coefficient factor, and G is the
were set to a minimum size of 0.5 and a maximum size shear modulus of the parent material. s
of 1. The prescribed volume fraction, penalization factor,
and minimum element density were set to 0.3, 3, and The displacement of the unit cell (δ UC,Z ) can be obtained
0.01, respectively. by accumulating strut displacements. Therefore, the strain
The key design parameters affecting the mechanical in the unit cell can be expressed as follows:
properties of thin-walled structures are wall thickness
and sidewall curvature. While it is evident that increasing UC Z,
wall thickness enhances load-bearing capacity, sidewall UC Z, h (VI)
curvature has more complex effects on the mechanical
properties of both the thin-walled structure and its internal where h is the height of the unit cell.
lattice. Furthermore, an excessively thick outer wall can
hinder the inward growth of bone tissue and reduce the According to Hooke’s law, the elastic modulus of a unit
fusion rate of the AVBs. This study focused on evaluating cell can be calculated as follows:
the effect of sidewall curvature on the mechanical properties
of AVBs with a wall thickness of 1 mm. As shown in
Figure 2, three topological thin-walled structures, named E UC Z , (VII)
TTS-1, TTS-2, and TTS-3, were designed with sidewall UC Z , UC Z ,
curvatures of 0.027, 0.014, and 0 mm , respectively.
−1
The interior of the topological thin-walled structure where σ UC,Z is the stress of the unit cell along the z-axis.
imitated the cancellous bone and was designed as According to the elastic–plastic cellular solid model
an interconnected porous architecture to enhance proposed by Gibson and Ashby, the plastic collapse
52
osseointegration and promote nutrient transport. The stress of the lattice structure under compression can be
lattice structure was constructed through a periodic
calculated as:

Volume 11 Issue 4 (2025) 169 doi: 10.36922/IJB025150133