# Development of an Elastic Material Model for BCC Lattice Cell Structures Using Finite Element Analysis and Neural Networks Approaches

^{*}

## Abstract

**:**

## 1. Introduction

_{e}), shear modulus (G

_{e}), and Poisson’s ratio (ν

_{e}) for several strut diameter/cell sizes. These results are then used to develop a neural networks NN model so that the equivalent solid properties of a BCC lattice cell can be predicted for any combination of strut diameters and cell sizes. The bulk material properties along with the cell configurations are used in the input dataset of NN and the equivalent solid mechanical properties are obtained from the output dataset.

## 2. Methodology Strategy

_{e}) and shear moduli (G

_{e}) and Poisson’s ratio (ν

_{e}).

## 3. Finite Element Modeling of Unit Cell

#### 3.1. Material and Physical Parameters

#### 3.2. Design and Mesh Generation

^{®}software is because it has more flexibility in selecting element types during mesh generation for a lattice structure. Most of the models are developed using Hexahedral mesh generation because of geometric complicacy imposed by branching and embedded structures. Although tetrahedral mesh generation can be easily automated, it gives inaccurate results as compared with hexahedral elements [16]. BCC unit cell and comprehensive configuration of cell connection of the lattice structure designed by using Micromechanics in Abaqus 2017 is illustrated in Figure 3.

#### 3.3. Applied Load and Boundary Conditions

#### 3.4. Material Properties

#### 3.5. Data Collection

^{2}. Strain is calculated from applied displacement divided by cell height L. Slopes of the stress-strain plots from the compression and shear models are elasticity modulus E

_{e}and shear modulus G

_{e}of the equivalent solid model, respectively. In addition, transverse displacement vs. longitudinal displacement is plotted from the compression model and Poisson’s ratio (ν

_{e}) is obtained from the slope of the curve. Variation of elasticity modulus E

_{e}, Poisson’s ratio ν

_{e}, and shear modulus G

_{e}, are shown in Figure 6, Figure 7 and Figure 8, respective, by solid line.

## 4. Neural Network for Equivalent Material Model

#### 4.1. The NN Model Used

#### 4.2. Training and Testing Patterns Used

## 5. Experimental Procedure

^{®}, i.e., software connected with the testing machine, and were plotted in Excel.

## 6. Finite Element Modeling of LCS and Equivalent Solid

## 7. Results and Discussion

_{e}, Poisson’s ratio ν

_{e}, and shear modulus G

_{e}, respectively. The figures include both the lattice cell FEA results (indicated by solid lines) and the NN output or random tasting data (indicated by discrete points). It is clear that the random tasting NN data are in satisfactory agreement with the FEA results for all cases. Furthermore, it is clear from Figure 6 that the elastic modulus of BCC configuration increases when the aspect ratio d/L. Figure 7 illustrated the NN for randomly choosing of the test for Poisson’s ratio that matched well with FEA results (solid). It is observed that the Poisson’s ratio decreases with an increase in aspect ratio. Finally, Figure 8 shows the intelligent model NN for a random testing data of Shear modulus at different d/L ratio are in good agreement with the FEA simulation (solid curves). Like elastic modulus, the shear modulus also increases when the aspect ratio increases. In all cases, the NN model is examined with 1 and 2 hidden layers with an increase in the number of nodes in each layer. The sensitivity of the neural network prediction increases for two hidden layers or more as a number of nodes for each layer increases. The outcomes demonstrate that the two hidden layer network accomplishes significantly better than the one hidden layer network. The optimum number of nodes in two hidden layers for NN that gives minimum mean square error (MSE) is 10:5 (10 nodes for the first hidden layer and 5 for the second hidden layer). As mentioned before that the complicacy of the problem lead to increase the number of nodes in each layer network. Studying various algorithms, the Resilient Backpropagation (trainrp) algorithm gives the best implementation as goal met MSE between the output of NN and the FEA results at least number of iterations. It is clear from statistical outcomes (R = 0.999) that the proposed neural network model accurately learned to map the relationship between the equivalent mechanical properties for quasi-isotropic material of BCC unit cell and varying parameters.

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A methodology strategy from finite element analysis (FEA) model to neural networks (NN) model to Equivalent solid model.

**Figure 4.**Mesh Sensitivity of BCC unit cell (5 × 5 × 5) mm and d = 1 mm for mesh size from 1.1 (Coarse) to 0.25 (Fine) mm.

**Figure 5.**Boundary conditions of BCC unit cell for FEA model (

**a**) Boundary condition for Shear modulus (

**b**) Boundary condition for Elastic modulus and Poisson’s ratio.

**Figure 7.**Comparison of equivalent Poisson’s ratio from FEA simulation and NN model for BCC pattern.

**Figure 11.**Experimental procedures, (

**a**) sample design in SolidWorks; (

**b**) sample fabrication using 3D printing; and (

**c**) compression test.

**Figure 13.**Comparison of the FEA simulation of LCS, experiment test and equivalent solid model results.

**Table 1.**Material properties of Acrylonitrile Butadiene Styrene (ABS) material [17].

Young’s Modulus $\left(\mathbf{M}\mathbf{P}\mathbf{a}\right)$ | Poisson’s Ratio | Density $(\mathbf{g}/\mathbf{m}{\mathbf{m}}^{3})$ | Yield Strength $\left(\mathbf{M}\mathbf{P}\mathbf{a}\right)$ | Ultimate Tensile Strength $\left(\mathbf{M}\mathbf{P}\mathbf{a}\right)$ | Plastic Strain $\left(\mathbf{m}\mathbf{m}/\mathbf{m}\mathbf{m}\right)$ |
---|---|---|---|---|---|

861.55 | 0.35 | 7.92 × 10^{−4} | 25.75 | 33.33 | 0.045 |

NO. | Algorithm (Acronym) | Detailing |
---|---|---|

1 | Trainbfg (BFG) | BFGS Quasi-Newton |

2 | Trainrp (RP) | Resilient Backpropagation |

3 | Trainscg (SCG) | Scaled Conjugate Gradient |

4 | Trainlm (LM) | Levenberg-Marquardt |

5 | Traincgb (CGB) | Conjugate Gradient Powell |

Input Parameters: Raw Material of ABS | Output Parameters: Equivalent BCC Lattice Properties |
---|---|

Elastic modulus ($E$), Poisson’s ratio $\left(\nu \right)$, Strut diameters ($d$), and Relative dimension $\left(d/L\right)$ | Elastic modulus (${E}_{e}$), Shear modulus (${G}_{e}$), and Poisson’s ratio $\left({\upsilon}_{e}\right)$ |

Data No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

$d$$\left(\mathrm{mm}\right)$ | 1 | 1 | 1 | 1.5 | 2.5 | 2 | 2.5 | 1 | 2.5 | 1.5 |

$d/L$ | 0.1 | 0.133 | 0.2 | 0.2 | 0.25 | 0.267 | 0.333 | 0.4 | 0.5 | 0.6 |

Data No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

E_{e} (MPa) | 1.223 | 2.349 | 3.282 | 3.282 | 13.25 | 16.113 | 33.696 | 64.837 | 145.92 | 243.15 |

G_{e} (MPa) | 4.809 | 8.506 | 19.248 | 19.248 | 19.248 | 30.534 | 34.942 | 56.056 | 82.197 | 129.66 |

ν_{e} | 0.4 | 0.39 | 0.36 | 0.36 | 0.35 | 0.34 | 0.32 | 0.3 | 0.28 | 0.25 |

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**MDPI and ACS Style**

Alwattar, T.A.; Mian, A.
Development of an Elastic Material Model for BCC Lattice Cell Structures Using Finite Element Analysis and Neural Networks Approaches. *J. Compos. Sci.* **2019**, *3*, 33.
https://doi.org/10.3390/jcs3020033

**AMA Style**

Alwattar TA, Mian A.
Development of an Elastic Material Model for BCC Lattice Cell Structures Using Finite Element Analysis and Neural Networks Approaches. *Journal of Composites Science*. 2019; 3(2):33.
https://doi.org/10.3390/jcs3020033

**Chicago/Turabian Style**

Alwattar, Tahseen A., and Ahsan Mian.
2019. "Development of an Elastic Material Model for BCC Lattice Cell Structures Using Finite Element Analysis and Neural Networks Approaches" *Journal of Composites Science* 3, no. 2: 33.
https://doi.org/10.3390/jcs3020033