Page 85 - MSAM-2-3

Materials Science in Additive Manufacturing Functional graded and hybrid TPMS lattices

.
where G(x, y, z) represents the radius of the cylinder that GC= 1212 (XIII)
is infilled with the first type of lattice. For the radial graded C
sheet-based gyroid lattices as shown in Figure 1D, the C 1212 . (XIV)
C
smooth grading can be implemented by controlling the c 1111 1122
value with Equations VII and VIII. To illustrate the effect brought by applying the
functional graded gyroid structures to mitigate the stress
3. Case study 1: Sheet-based graded gyroid shielding, finite element analysis was conducted using
lattices in orthopedic implant design commercial software ABAQUS/Explicit 2020. Based on
Simulation with the representative volume element (RVE) the conceptual implant design illustrated in Figure 2A, the
method was adopted to explore the elastic performance simulation focused on the exploration of the compressive
of the sheet-based gyroid structures with different behavior of the gyroid infillings with different density
relative densities. To implement the simulation, periodic gradients. According to the simulation set-up shown in
boundary conditions were applied to unit cells, which can Figure 2B, two rigid plates were utilized as the simplified
be expressed as follows: model in the simulation of the uniaxial compression
test. The displacement, which is equal to the 0.05 strain

u xL u x() 0 Lt x L,( ) t x( ) x e B, of the gyroid lattice, was applied on the top rigid plate.

i
i
i i
i
i
i
(IX) The bottom plate was fixed. General contact was applied
where u represents the displacement vector, x represents to simulate the contact behavior between the rigid plates
i
the point vector, and L is the characteristic length, ε is the and the sheet-based gyroid lattices. The friction coefficient
0
strain, and t represents the surface traction. The elastic for the tangential behavior was set to be 0.3. The normal
i
constitutive behaviors of the sheet-based gyroid lattice behavior was set with the hard contact formulation.
structure, which has cubic symmetry, can be expressed as The material property of titanium alloy was modeled in
[26]
the following equation: accordance with the Johnson-Cook model .
The dimension of the gyroid lattice was set to be

11
10 mm in width and length; the height of the lattice was
22 set to be 6 mm; and the size of the unit cell was set to be
2 mm. Three different sheet-based lattices were designed:
33 (i) Uniform gyroid lattice RD25 with a relative density of
12
25%; (ii) radial graded gyroid lattices RD05-45 with the
13 relative density 5% in the inner region and the relative

23 density 45% in the outer region; and (iii) radial graded
C 1111 C 1122 C 1122 11
A B
C 11222 C 1111 C 1122 22
C C C
1122 1122 1111 33 . (X)
C 1212 2 12
C 2
1212 13
2
C 1212 23
According to the periodic boundary conditions
proposed by Dong et al. , three independent components, C D
[25]
namely, C 1111 , C 1122 , and C 1212 , can effectively represent the
constitutive matrix of the sheet-based gyroid lattice. For the
base material, the properties of titanium alloy were applied.
From the constitutive matrix, elastic properties including
elastic modulus E, Poisson’s ratio ν, shear modulus G, and
bulk modulus K can be calculated as:
C C C 2 C
E 1111 1122 1111 1122 . (XI)
C 1111 C 1122 Figure 2. (A) Conceptual design of embedding sheet-based gyroid
1
K C 2 C . (XII) lattices in orthopedic implants. (B) Simulation set-up for the compression
test. (C) Full model of the gyroid lattices. (D) Quarter model of the gyroid
3 1111 1122 lattices.
Volume 2 Issue 3 (2023) 4 https://doi.org/10.36922/msam.1753