# Design Data and Finite Element Analysis of 3D Printed Poly(ε-Caprolactone)-Based Lattice Scaffolds: Influence of Type of Unit Cell, Porosity, and Nozzle Diameter on the Mechanical Behavior

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. CAD-Based Lattice Structures Design

^{®}(Autodesk 2020, McInnis Parkway San Rafael, CA, USA), simulating the superimposition of different paths during the 3D printing procedure. The reticular, triangular, hexagonal, and wavy geometries were designed making the angle between different paths variables. In order to create the structures to be tested, each pattern was considered as a linear repetition of a unit cell size; five cell sizes were designed and tested (ranging from 1 to 5 mm). All designed scaffolds featured 5 × 5 cell units in the x and y directions, with 15 layers of thickness for triangular and hexagonal geometries and 16 layers of thickness for reticular and wavy ones. The porosity of the CAD models was obtained by measuring the occupied volume of each scaffold, and the designed porosity (

**P**

_{CAD}) was calculated as follows (Equation (1)):

_{f}is the occupied volume of the scaffold, and V

_{tot}is the bulk volume of the CAD model.

#### 2.3. Finite Element Method (FEM) Modeling

^{®}(Dassault Systemes Simulia Corporation., Johnston, RI, USA). Each CAD model was imported into the FE software as a continuous unique part. The PCL-based scaffolds properties were simulated by both linear and nonlinear elastic behavior. In the linear elastic condition, the elastic modulus and Poisson’s ratio of PCL were assumed to be 350 MPa and 0.3, respectively, as reported previously [30,31]. The elastic modulus was then used in the FE simulation with the assumption of linear elastic deformation and considering the PCL material as isotropic. In the STEP section of the model, the Nlgeom option was set to on in order to allow results with nonlinear behavior. An automatic incrementation was selected for the simulation, with a maximum number of increments equal to 100 and an increment size increasing from an initial size of 0.01 up to 0.1 to speed up the process. No particular problems in the iterations were registered. The minimum increment size was left to a default 1.0 × 10

^{−5}value. The output requested from the simulations contains stresses, strains, displacement, and Force/Reaction to allow us to calculate the related compressive modulus. These variables were calculated every n number of increments. Then, compression simulation was performed on the CAD models constraining the lower face of the scaffold, denying any possible movement or rotation with respect to the reference system, and 1.5% compressive strain was exerted to the top of the model. The compressive elastic modulus of each scaffold was obtained from Equation (2):

_{comp}is the equivalent maximum stress reached for the imposed displacement, and ε is the imposed strain. Since:

_{eq}is the resistant equivalent area of the scaffold (see Figure S1 for further details). The imposed strain was calculated from Equation (4) as follows:

_{x}and R

_{y}are the reaction forces in the x- and y-direction of the constrained portion of the tested scaffold, while A

_{eq}is the resistant equivalent area of the scaffold. The shear strain instead is calculated as follows:

_{el}is the elastic volume ratio, I

_{1}and I

_{2}are the first and second invariants of the deviatoric strain, μ, λ

_{m}, α, and β describe the deviatoric behavior of the material, and D introduces the compressibility [33].

_{0}(initial shear modulus) is equal to μ (shear behavior of the material in van der Waals), D value can be calculated as follow:

_{m}= 1.9, α = 6, β = 0, and D = 0.00659, as reported previously [31].

#### 2.4. Fused Deposition (FDM) Printing

^{−1}. Each object was modeled using Autodesk Inventor

^{®}software, then exported in stereolithography (.stl) format and sliced using Simplify3D

^{®}software to be printed. To ensure the adhesion of the scaffolds to the surface of the printing platform, a 3D glue stick (Magigoo™, Swieqi, Malta) was used.

#### 2.5. Spectroscopic Characterizations of 3D Printed Lattice Structures

^{−1}with transmission-type volume phase grating. The spectral resolution is 15–18 cm

^{−1}, and the detector is a thermoelectrically cooled indium gallium arsenide (InGaAs) [34]. The samples were analyzed with 60 cumulative scans with optimized laser power, aperture size, and duration (7 s) per exposure in order to achieve the best signal-to-noise ratio. All of the obtained spectra data were plotted using Origin™ 8 software (One Roundhouse Plaza, Northampton, MA, USA).

## 3. Results and Discussion

#### 3.1. CAD-Based Scaffolds Design and Printability

^{−1}, respectively. In the case of using Bowden drive extruder, with PCL as feedstock material, for obtaining the same accuracy in the deposition of filament (and therefore the same final lattice scaffolds), high values of retraction were needed to prevent unnecessary extrusion while the extruder travels between printed sections. Specifically, such settings must be of 6 mm and 4800 mm min

^{−1}for retraction distance and retraction speed, respectively (data not shown).

#### 3.2. FEM Analysis

^{5}. The elastic modulus was used in the FEM models with the assumption of linear elastic deformation, the FEM analysis was performed for each scaffold, and the mechanical behavior under compression was simulated. The distribution of von Mises stress for comparing the scaffolds with different geometries and nozzle diameters is shown in Figure 3. It is interesting to note that in the reticular, triangular, and wavy scaffolds, when subjected to stress, we have noticed the increase in the compressive elastic modulus due to the existence of thicker load columns. Conversely, the amount of maximum stress for the hexagonal scaffolds was lower than those of other scaffolds due to the absence of proper supporting columns. We assumed that the design parameters could influence the bonding strength between layers and filaments in the printed scaffolds, and the layer bonding of the hexagonal scaffold could be significantly weaker compared to other tested scaffolds because in every three layers, their contact points shifted, which could result in a decrease of the bonding strength.

#### 3.3. Spectroscopy Characterization of 3D Printed Scaffolds

^{−1}(in blue) associated with C-COO stretching, and three narrow bands at 1033 cm

^{−1}, 1066 cm

^{−1}, and 1110 cm

^{−1}(in light green) associated with C-C stretching. Some other bands are visible approximately at 1284 and 1301 cm

^{−1}(in magenta) due to the symmetric CH

_{2}stretching, and at 1419 cm

^{−1}, 1443 cm

^{−1}, and 1468 cm

^{−1}(in purple) ascribable to CH

_{2}bending; lastly, a peak was observed at 1722 cm

^{−1}(in orange) due to C=O stretching vibration.

#### 3.4. 3D Printed Wrist Brace Orthosis with Different Lattice Geometries

^{®}. The procedure to obtain the final device was divided into two different phases: after the measurement of the anatomic region of interest, all data are collected to draw the first representation of the orthosis. Then, via extrusion, the edges of the orthosis and the reinforced portions are created. Subsequently, linear repetition of lattice geometry was created and cut to fit the inner boundary of the drawn orthosis. Finally, combining the boundary with the selected pattern, it is possible to export it in stereolithography (STL) format and slice it using Simplify3D

^{®}software. The final 3D printed orthosis was 4 mm thick, weighed <20 g, and possessed breathability >80%.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**CAD models of designed lattice structures. (

**A**–

**D**) Top and 3D views of CAD design of four types of lattice geometries (

**A**) reticular, (

**B**) triangular, (

**C**) hexagonal, and (

**D**) wavy and related images of the 3D printed scaffolds.

**Figure 2.**Theoretical porosity of different lattice scaffolds. Porosity of each CAD model obtained by measuring the occupied volume of each scaffold.

**Figure 3.**FEM models under compression. Von Mises stress distribution of lattice scaffolds with different geometries and nozzle diameters. The stress value increases from blue to red colors. In orange, the top view magnification of stressed scaffold; in red and green, the magnification of load columns.

**Figure 4.**FEM-derived elastic modulus calculation. Elastic moduli of scaffolds with different microstructures and nozzle diameters obtained from FEM simulations.

**Figure 5.**FEM-derived shear moduli. The von Mises stress–strain distribution in the simulated shear test by FEM in the lateral x-direction and estimation of shear moduli of scaffolds with different microstructures.

**Figure 6.**FEM prediction of compressive behavior of PCL scaffolds with Van der Waals models. (

**A**–

**D**) Estimation of compressive moduli of scaffolds with different geometries using nonlinear vs. linear model; (

**E**) prediction of nonlinear compressive behavior of lattice structure using van der Waals hyperelastic model and its comparison with linear compression data. (

**F**) Raman spectroscopic analysis of lattice scaffold made of PCL printed with 0.4 mm (in blue) and 0.8 mm (in red) nozzle diameters. (A.U., arbitrary unit).

**Figure 7.**CAD design and wearability of 3D printed lattice-based orthosis. Representative images of shape memory 3D printed orthosis with reticular (upper images) and triangular (below images) lattice structures.

**Table 1.**Comparison of compressive elastic moduli (E

_{comp}) in the linear model of four types of scaffolds with different nozzle diameters and unit cell sizes.

Unit Cell Size (mm) | E_{comp} Reticular (MPa) | E_{comp} Triangular (MPa) | E_{comp} Hexagonal (MPa) | E_{c}_{omp} Wavy (MPa) |
---|---|---|---|---|

1 (nozzle 0.4) | - | - | 100.74 | 116.48 |

2 (nozzle 0.4) | 33.92 | 27.61 | 3.46 | 28.80 |

3 (nozzle 0.4) | 14.47 | 11.93 | 0.34 | 12.80 |

4 (nozzle 0.4) | 7.80 | 6.69 | - | - |

5 (nozzle 0.4) | - | - | - | - |

1 (nozzle 0.8) | - | - | - | |

2 (nozzle 0.8) | - | - | - | 88.00 |

3 (nozzle 0.8) | 40.25 | 33.05 | 2.25 | 38.40 |

4 (nozzle 0.8) | 22.04 | 18.43 | 0.28 | 21.40 |

5 (nozzle 0.8) | 14.52 | 11.70 | 0.07 | - |

**Table 2.**Compressive elastic moduli from the linear model in comparison with the Van der Waals nonlinear hyperelastic model.

E_{comp} | Linear | Nonlinear | Linear | Nonlinear | Linear | Nonlinear | Linear | Nonlinear |
---|---|---|---|---|---|---|---|---|

Unit Cell Size (mm) | Reticular (MPa) | Triangular (MPa) | Hexagonal (MPa) | Wavy (MPa) | ||||

1 (nozzle 0.4) | - | - | - | - | 100.74 | 89.86 | 116.48 | 101 |

2 (nozzle 0.4) | 33.92 | 29.85 | 27.61 | 24.07 | 3.46 | 3.39 | 28.80 | 25 |

3 (nozzle 0.4) | 14.47 | 11.98 | 11.93 | 10.28 | 0.34 | 0.34 | 12.80 | 11.11 |

4 (nozzle 0.4) | 7.80 | 6.38 | 6.69 | 5.74 | - | - | - | - |

5 (nozzle 0.4) | - | - | - | - | - | - | - | - |

1 (nozzle 0.8) | - | - | - | - | - | - | - | - |

2 (nozzle 0.8) | - | - | - | - | - | - | 88.00 | 77.5 |

3 (nozzle 0.8) | 40.25 | 35.18 | 33.05 | 28.63 | 2.25 | 2.18 | 38.40 | 33.56 |

4 (nozzle 0.8) | 22.04 | 19.18 | 18,43 | 15.9 | 0.28 | 0.29 | 21.4 | 18.73 |

5 (nozzle 0.8) | 14.52 | 12.28 | 11.7 | 10.14 | 0.07 | 0.07 | - | - |

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**MDPI and ACS Style**

Sala, R.; Regondi, S.; Pugliese, R.
Design Data and Finite Element Analysis of 3D Printed Poly(ε-Caprolactone)-Based Lattice Scaffolds: Influence of Type of Unit Cell, Porosity, and Nozzle Diameter on the Mechanical Behavior. *Eng* **2022**, *3*, 9-23.
https://doi.org/10.3390/eng3010002

**AMA Style**

Sala R, Regondi S, Pugliese R.
Design Data and Finite Element Analysis of 3D Printed Poly(ε-Caprolactone)-Based Lattice Scaffolds: Influence of Type of Unit Cell, Porosity, and Nozzle Diameter on the Mechanical Behavior. *Eng*. 2022; 3(1):9-23.
https://doi.org/10.3390/eng3010002

**Chicago/Turabian Style**

Sala, Riccardo, Stefano Regondi, and Raffaele Pugliese.
2022. "Design Data and Finite Element Analysis of 3D Printed Poly(ε-Caprolactone)-Based Lattice Scaffolds: Influence of Type of Unit Cell, Porosity, and Nozzle Diameter on the Mechanical Behavior" *Eng* 3, no. 1: 9-23.
https://doi.org/10.3390/eng3010002