# Infill Microstructures for Additive Manufacturing

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Theoretical Background

- ${K}_{e}$ is the elements stiffness matrix;
- ${\rho}_{min}$ is the minimum penalty factor;
- N is the number of total elements in the design domain.

- (U
_{e}) is the displacement vector of element e; - (K
_{e}) is the stiffness of element e; - the vector {ρ} consists of the relative density of the ρ
_{e}elements.

#### 2.2. Homogenization

## 3. Design of Topologically Optimized Lattice Structures

## 4. Lattice Structures Fabrication

## 5. Results

#### 5.1. Theoretical Evaluation of Proposed Lattices

_{max}) and local minimum Young’s modulus (E

_{min}) is checked (Figure 13). According to Tancogne Dejean, a measure of the anisotropy of the lattice structure is derived from E

_{max}/E

_{min}[37].

#### 5.2. Zener Ratio Calculation

#### 5.3. Evaluation of Homogenied Microstructure with FE Analysis

#### 5.4. Exrerimental Results

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Intermediate shapes. Different iteration stages (

**b**–

**e**) from the initial design area (

**a**) of Topology Optimized ‘structure #1’: (

**a**) design area, (

**b**) step 5, (

**c**) step 10, (

**d**) step 18 and (

**e**) step 23.

**Figure 4.**Iterations steps (

**a**–

**e**) of the Topology Optimization study for ‘structure 2’: (

**a**) design area, (

**b**) step 10, (

**c**) step 20, (

**d**) step 25 and (

**e**) step 33.

**Figure 5.**(

**a**) Volume fraction per each step (

**b**), and the value of the volume fraction constraints on each iteration step of ‘structure #1’.

**Figure 6.**(

**a**) Volume fraction per each step (

**b**), and volume fraction constraints on each iteration step of ‘structure #2’.

**Figure 8.**Outline dimensions of unit cells and lattice models for the compression test of (

**a**) structure #1 with a volume fraction constraint = 0.4 and (

**b**) structure #2 with a volume fraction constraint = 0.1.

**Figure 9.**Fabricated ABS specimens without supports for both structures for the tensile strength test with a relative density of 0.4 for structure #1 and 0.1 for structure #2.

**Figure 10.**Outline dimensions of the tensile strength specimens (

**a**,

**b**), each lattice structure fabricated with 0°, 45° and 90° angles of single unit cell mapping.

**Figure 16.**The elastic moduli of both homogenized lattices in different relative densities ($\overline{\rho}$).

**Figure 17.**Displacement, strain and Von mises stress of homogenized structures #1 and #2 for different values of relative density.

**Figure 18.**Specimens’ behavior during the compression test: (

**a**) lattice structure #1 and (

**b**) lattice structure #2.

**Figure 19.**Samples behavior on the tensile strength test, and stress/strain curve of lattice structure #1 for 0°, 45° and 90° angles.

**Figure 20.**Strain at sB, Modulus of Elasticity, sB value and toughness of lattice structure #1 for different cell mapping angles.

**Figure 21.**Samples behavior on the tensile strength test, and stress/strain curve of lattice structure #2 for 0°, 45° and 90° angles.

**Figure 22.**Strain at sB, Modulus of Elasticity, sB value and toughness of lattice structure #2 for different cell mapping angles.

Young’s Modulus (MPa) | Density (kg/mm ^{3}) | Poisson’s Ratio |
---|---|---|

187,000 | 0.0079 | 0.33 |

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**MDPI and ACS Style**

Ntintakis, I.; Stavroulakis, G.E.
Infill Microstructures for Additive Manufacturing. *Appl. Sci.* **2022**, *12*, 7386.
https://doi.org/10.3390/app12157386

**AMA Style**

Ntintakis I, Stavroulakis GE.
Infill Microstructures for Additive Manufacturing. *Applied Sciences*. 2022; 12(15):7386.
https://doi.org/10.3390/app12157386

**Chicago/Turabian Style**

Ntintakis, Ioannis, and Georgios E. Stavroulakis.
2022. "Infill Microstructures for Additive Manufacturing" *Applied Sciences* 12, no. 15: 7386.
https://doi.org/10.3390/app12157386