Page 80 - MSAM-3-3

Materials Science in Additive Manufacturing Gradient porous material design criteria

drop is named after W . Thus, W , W , and W can be where l and l ’ represent the height of layer 1 before
2
3
uρ
1
1
1
expressed as follows: and after the plastic deformation in the first densification
process, respectively; ρ represents the porosity of layer 1;
1
W = W u 1 ∫ ε ρ 1 σεd ρ 1 (5) ∆ρ is the porosity difference between layer 1 and layer
=
12
1
ρ
0
2; and ∆l is the displacement of layer 1 after the first
1
W = W u 2 ∫ ε ρ 2 σεd ρ 2 (6) densification process.
=
2
ρ
0
After that, densified layers 1 and 2 will deform together
W = W u 3 ∫ 0 ε ρ 3 σεd ρ 3 (7) and be in the state of plastic deformation, while layer 3
=
will still be in the state of elastic deformation due to the
3
ρ
lower porosity. During the second densification process,
where W , W , and W are the energy absorption the localized deformation of densified layers 1 and 2 will
uρ1
uρ3
uρ2
capacities for layers 1, 2, and 3, respectively. The ε , ε , and continue until their porosity approaches the porosity
ρ2
ρ1
ε are the strains at the first load drop that occurred in the of layer 3. Hence, the height and displacement can be
ρ3
SS curves of the relevant uniform porosity materials. expressed as Equations 10 and 11:
The second one is represented by Stage II, where the 1− ρ
1 (
’
l
’
materials are in the plateau and densification stage until l 12+ = ( 1− ρ 2 × l + ) (10)
2
the next layer takes over and dominates the deformation. 2 ) +∆ρ 23
In Stage II, the integral area can be expressed as W and
12
W . In the plateau stage, after breaking through the barrier ∆ = l |l ' − ( + l ' ) | = l ∆ ρ 23 × ( + l ' l
23
+
+
of maximum stress, the struts’ fracture would repeat 12 12 1 2 ( − ρ 2 ) +∆ ρ 23 1 2 ) (11)
1
continuously. The fractured struts could be stopped and
stuck by the struts nearby or the next layer; therefore, where l’ represents the height of densified layers 1 and
1+2
densification occurs. At this time, this layer still dominates 2 after the plastic deformation in the second densification
deformation and gradually causes an increase in stress process; l represents the height of layer 2 before the plastic
2
due to the decrease in porosity during the densification deformation; ρ represents the porosity of layer 2; ∆ρ is
23
2
process. Eventually, when the yield stress of the densified the porosity difference between layer 2 and layer 3; and
layer approaches the yield stress of the next layer, the ∆l is the total displacement of densified layers 1 and 2
1+2
next layer will dominate the deformation. For example, as during the second densification process.
shown in Figure 9, layer 1 will first be in the state of plastic
deformation, while the other two layers will still be in the As mentioned above, the strain can be estimated
state of elastic deformation due to the lower porosity. During because the displacement of each layer can be calculated,
the first densification process, the plastic deformation of the and the height of the sample is already known. Hence,
porous materials would be limited to layer 1, with higher if the stress at the beginning and the end of the plateau
porosity, since layer 2 and layer 3, with lower porosity, are and densification stage can be defined, the W and W
23
12
still under elastic deformation. Therefore, the localized can be estimated as well. As the fractured struts become
deformation of layer 1 will continue until the porosity of stuck by other struts, the stress decreases before the
layer 1 approaches the porosity of layer 2. Hence, the height plateau and densification stages are stopped. At that
of layer 1 before and after the deformation can be expressed time, the other struts are at high-stress levels and can
as Equation 8. It should be noted that the deformation of reach the yield stress and then begin the densification
layer 1 needs to occur until its porosity approaches the process quickly. Hence, the stress at the beginning of
porosity of layer 2, which could be different due to different the plateau and densification stage can be defined as the
unit cell designs. This will be corrected by introducing yield stress of the layer. On the other hand, when the
coefficients in the subsequent equations. Moreover, to ease porosity of the densified layer approaches the next layer,
the subsequent derivation, the displacement of layer 1 is the deformation will be dominated by the next layer. At
defined as Equation 9. that time, the stress of the densified layer will approach
the yield stress of the next layer, and thus, the stress at the
1− ρ end of the plateau and densification stage can be defined
l = ( 1− ρ 1 l × 1 (8) as the yield stress of the next layer. Moreover, the area
’
1
1 ) +∆ρ
12
of the elastic deformation stage should not be included
in this densification process. Therefore, the W and W
∆ ρ 12 23
| −l
l
∆= l ' | = 12 ×l (9) can be calculated and predicted by the estimated strain
1
1
1
1
( − ρ 1 ) +∆ ρ 12 1 and stress, as shown in Figure 11. According to Figure 11,
Volume 3 Issue 3 (2024) 9 doi: 10.36922/msam.4234