Page 116 - MSAM-3-2

Materials Science in Additive Manufacturing Hybrid lattice structures design with AI

is referred to as a TPMS. TPMS exhibits distinct geometric Equations IX and X define honeycomb-like structures
characteristics, notably smooth surfaces devoid of sharp based on TPMS Gyroid and Primitive. The relative density
corners. These surfaces are prevalent in numerous biological of each cell is approximately 0.25. Examples of unit
systems, including soap films, block copolymers, wings cells based on TPMS-Gyroid and TPMS-Primitive are
49
48
of butterflies, 50,51 and the skeleton of sea urchins. 52 illustrated in Figure 1A and B, respectively.
Various methods are utilized to model TPMS structures, 2.2. Hybrid lattice design
including parametric, implicit, and boundary functions.
The level-set approximation method is mainly employed As depicted in Figure 1C and D, G-Honeycomb and
in modeling TPMS-based honeycomb structures. TPMS in P-Honeycomb cells exhibit distinct stiffness, with the
3D space can be represented by a Fourier summation: P-Honeycomb cell being approximately four times stiffer

r F k cos2 kr 0 k (III) than the G-Honeycomb cell. By combining these hard and

soft cells within larger hybrid lattices, a broad spectrum
k
of tunable mechanical responses can be achieved. In
where k represents the reciprocal vector, α(k) denotes this study, a 10 by 10 lattice (100 mm by 100 mm) was
the phase shift, and F(k) represents the amplitude for designed, incorporating a total of 100 randomly distributed
vector k. Equation III can be simplified by truncating it to G-Honeycomb and P-Honeycomb cells, as shown in
53
a trigonometric function, φ, which satisfies: Figure 2A and B, respectively. Figure 2C illustrates an

c
xy z,, (IV) example design of the hybrid lattice, where cells are color-
coded: red representing P-Honeycomb (hard) and blue
where c is the iso-value controlling offset from the zero representing G-Honeycomb (soft). To streamline dataset
level-set. Then, lattice structures based on TPMS can be preparation and reduce complexity, the hybrid lattice is
created using: further simplified into a binary matrix representation.

2
2
xy z,, c (V) Figure 2D presents the simplified binary matrix, wherein
0 represents a hard cell (P-Honeycomb), and 1 represents a
where the intervals [−c,c] specify the fraction of
the solid region of the structure. The TPMS gyroid and soft cell (G-Honeycomb).
primitive surfaces can be modeled using the following 2.3. FE simulation
equations:
To validate the results obtained from the homogenization
x
f gyroid x yz,, cos wx sin wy cos method, FE analysis was conducted utilizing a pre-validated
y
34
y
x
wy sin wz cos wz s iin wx (VI) numerical model. ABAQUS/Explicit 2020 was employed
to predict compressive responses of the structure. To
z
z
f primitve x yz,, cos wx wy cos wz (VII) enhance computational efficiency, uniform G-Honeycomb,
cos
and P-Honeycomb structures were modeled under the
x
z
y
plane strain assumption, given their 2D surface-based
where x, y, and z represent coordinates in the 3D origins. The lattice structures were discretized using 2,437
Cartesian coordinate system. The variable w defines the CPE4R four-node bilinear plane strain elements with
periodicities of the TPMS function: reduced integration and hourglass control for each unit cell.
n The material behaviors of the base material were simplified
w 2 i fori xy z,, (VIII)
i
L i to be elastic and perfectly plastic, in line with testing data
obtained from 3D-printed tensile samples in literature, as
34
where n controls the number of unit cells along each summarized in Table 1. Two platens were modeled, with
i
direction. L defines the dimensions of the lattice along the bottom platen fixed, while displacement was applied
i
each direction. to the top platen to load the structure. Contact behaviors
To introduce new two-dimensional (2D) structures, the were characterized by a hard formulation along the normal
z periodicity in Equations VI and VII can be eliminated by direction and the penalty method with a friction coefficient
substituting z = 0 and then inserting them into Equation V: of 0.3 along the tangential direction. The elastic modulus
was determined as the slope of the stress-strain curve in the

x
sin
U GHoneycomb cos w x w y w x 2 c (IX) linear elastic region, while Poisson’s ratio was approximated
2
sin
x
y
as the ratio between the horizontal displacement of the

x
U PHoneycomb cos w x w y 2 c 2 (X) midpoint on the edge and the applied vertical displacement.
cos
In terms of the computational cost, the 2D FE model took
y
Volume 3 Issue 2 (2024) 3 doi: 10.36922/msam.3430