# Derivation and Validation of Linear Elastic Orthotropic Material Properties for Short Fibre Reinforced FLM Parts

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. State of the Art and Objectives

^{®}ABS P400 material model by Matas et al. [24]. Bellini et al. [25] proposed a method to derive orthotropic material models for FLM parts with pre-defined infill rasters of [0° 90° +45° −45°] and Delaunay-Triangulation based “domain decomposition” infill. This method (which will be adapted and altered later in this contribution) is based on different build-up and infill orientations in six specimen geometries to derive the nine orthotropic material constants. Domingo-Espin et al. [26] also used Bellini’s orientation for a [+45° −45°] infill and validated the method successfully with a hook-like geometry made of polycarbonate (PC). The authors pointed out, however, that the quality of fit of the orthotropic model depends on build orientation and even proposed an isotropic material model for certain cases. Rodríguez et al. [27] proposed a unit-cell based mesoscale method to model FLM-printed ABS and validated their model with less than 10% deviation from experiment. Lee et al. [28] also used ABS to compare FLM, 3D printing (ink jet based) and “nano composite deposition system” (mechanical micro machining included) concerning raster orientation, air gap, bead width, color, and model temperature with focus on compression strength under different build directions (axial and transverse). Compressive strength for FLM was found to be 11.6% higher for axially than for transversely printed specimen. Other, more recent research on unreinforced FLM specimen encompasses Cantrell et al. [29], who included digital image correlation (DIC) in their tensile testing of different infill lay-ups ([+45 −45], [+30 −60] and [0 90]) of FLM ABS specimen. It was found that there is little effect of these stacking sequences on Young’s modulus and Poisson’s ratio; however, shear modulus and shear yield strength were largely affected with variation between results ranging up to 33%. Specimen made of PC behaved similarly; specimen printed upright (tension direction orthogonal to layers), on-edge (flat, but flipped 90° along the longest axis) and flat (tension direction parallel to layers) produced variations of similar magnitude [29]. A Finite Element (FE)-mesostructural model for ABS P400 was conceived and applied by Somireddy et al. [30]. The positive impact of a negative air gap was confirmed therein (“tightly packed”) and a Classical Laminate Theory (CLT)-model was derived to be applied on a 2D FE model of a component. Sheth et al. used a representative volume cell simulation approach with 4 by 4 roads (extrusion beads) and tested this model under varying angles from 0° in steps of 15° to 90°, which showed very good accordance.

#### 1.3. Objectives and Novelty of This Contribution

## 2. Materials and Methods

#### 2.1. Specimen Geometry, Slicing and Printing

- The tensile bar is printed flat on the printing platform with longitudinal infill (Flat 0°, F0° in the following), intended to primarily yield Young’s modulus in longitudinal direction ${E}_{x}$;
- Same orientation, but with perpendicular infill (F90°) for Young’s modulus in perpendicular direction in-plane ${E}_{y}$;
- Upright position with infill printed in the same direction as F90°, thus called Upright 90° (U90°), intended to explain interlayer modulus ${E}_{z}$;
- Flat printed position with 45° infill within plane (F45°) for shear modulus ${G}_{xy}$;
- Diagonally printed position (45° to plane) with parallel infill to walls, thus called U45°-90° for shear modulus ${G}_{yz}$;
- And diagonally printed position, yet with perpendicular infill to walls (like F0°), thus called U45°-0° for shear modulus ${G}_{xz}$.

#### 2.2. Simulation and Fitting of Orthotropic Material Parameters

- Average the experiment data to obtain one force-displacement, one longitudinal strain-displacement, and one transverse strain-displacement curve for each of the six samples (18 curves total).
- Find the linear parts of the curves (from displacement = 0 to the end of the linear part), for reference of the experimental data see the results section (Section 3). This is done to allow the linear elastic model to fit to the actual linear section of the curve. To obtain this linear limit, the following steps were taken and are proposed as a solution: (1) Smooth the average curve by a moving average of 10 measurement pairs; (2) Calculate the slope between each curve point and its predecessor $dy/dx$; (3) Calculate the curvature by calculating the “slope’s slope”, in turn; and (4) find the first occurrence where the percentage difference in curvature is smaller than a certain threshold (here, 0.75% were used arbitrarily). The threshold depends on the desired “strictness” of linearity; the smaller, the stricter. To avoid considering the initial, rather noisy data within the first part of the experiment, the linearity detection starts after 10% of the experiment curve data. (4) Finally, 10 equidistant displacement points are selected from the linear span of deformation $\left[{s}_{lin,min};{s}_{lin,max}\right]$. Using a mapping function, the closest (minimum-difference) data pairs from the averaged, but still discrete experiment data are selected.

#### 2.3. Validation

#### 2.4. Conduct and Evaluation of Experiments

## 3. Results

#### 3.1. Calibration

#### 3.1.1. Experiments

#### 3.1.2. Simulation and Parameter Fitting

#### 3.2. Validation

#### 3.2.1. Experiments

#### 3.2.2. Simulation and Material Model Validation

## 4. Discussion: Advantages, Disadvantages, and Open Research Questions of Simultaneous Parameter Fitting

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**FLM printing: (

**a**) Photo of short 15 wt.% carbon-fibre reinforced PETG-CF15 getting printed on a Raise3D Pro2 Plus (

**left**) and corresponding schematic of FLM hot-end (

**right**); (

**b**) Illustration of various current FLM material modeling approaches as described in the text.

**Figure 3.**Specimen geometries: (

**a**) Becker specimen, technical drawing; (

**b**) Specimen alignment and infill and their allocation to orthotropic constants, sliced by Raise3D ideaMaker software [38].

**Figure 4.**Building source (G-Code,

**left**) mapped to FE model (

**right**) for the first specimen (F0°) using the approach presented by the authors [39].

**Figure 5.**Building source (G-Code,

**left**) mapped to FE model (

**right**) for the fourth specimen (F45°) using the approach presented by the authors [39].

**Figure 6.**Fitting method in detail: F = flat, U = upright specimen as in Figure 3.

**Figure 9.**XX-rib specimen geometry: (

**a**) Specimen, technical drawing with main dimensions; (

**b**) 3D model of and FLM-printed XX-rib specimen; (

**c**) Bending load case for XX-rib specimen as applied in FE simulation. All contacts modeled using a coefficient of friction of $\mu =0.3$.

**Figure 11.**Results of experimental testing, averaged: (

**a**) Force-displacement curves; (

**b**) stress-strain curves.

**Figure 12.**Fitting of force-displacement, transverse strain-displacement, and longitudinal strain simulation results (grey lines) to experimental results (ten red points each, identified above).

**Figure 14.**Calibrated material model applied to upright-printed XX-rib specimen bending simulation compared to experimental result (force-displacement-curve above, average percentage error (APE) below; MAPE = 35%).

${\mathit{E}}_{\mathit{x}}$ | ${\mathit{E}}_{\mathit{y}}$ | $\mathit{z}$ | ${\mathit{\nu}}_{\mathit{x}\mathit{y}}$ | ${\mathit{\nu}}_{\mathit{y}\mathit{z}}$ | ${\mathit{\nu}}_{\mathit{x}\mathit{z}}$ | ${\mathit{G}}_{\mathit{x}\mathit{y}}$ | ${\mathit{G}}_{\mathit{y}\mathit{z}}$ | ${\mathit{G}}_{\mathit{x}\mathit{z}}$ |
---|---|---|---|---|---|---|---|---|

MPa | MPa | MPa | - | - | - | MPa | MPa | MPa |

8212 | 2615 | 1437 | 0.28 | 0.11 | 0.10 | 883 | 637 | 634 |

**Table 2.**Mean absolute percentage error (MAPE) for each fitting, rounded to full percent (larger values than 30% are highlighted in grey).

F0° | F90° | U90° | F45° | U45°-90° | U45°-0° | |
---|---|---|---|---|---|---|

$\mathrm{MAPE}(F-s$) | 14% | 9% | 25% | 13% | 49% | 17% |

$\mathrm{MAPE}({\epsilon}_{x}-s)$ | 10% | 8% | 35% | 19% | 6% | 42% |

$\mathrm{MAPE}({\epsilon}_{y}-s)$ | 20% | 21% | 14% | 14% | 21% | 12% |

**Table 3.**Material parameters determined from simulation-based fitting, rounded. Experimental start values from Table 1 are repeated in italics.

${\mathit{E}}_{\mathit{x}}$ | ${\mathit{E}}_{\mathit{y}}$ | ${\mathit{E}}_{\mathit{z}}$ | ${\mathit{\nu}}_{\mathit{x}\mathit{y}}$ | ${\mathit{\nu}}_{\mathit{y}\mathit{z}}$ | ${\mathit{\nu}}_{\mathit{x}\mathit{z}}$ | ${\mathit{G}}_{\mathit{x}\mathit{y}}$ | ${\mathit{G}}_{\mathit{y}\mathit{z}}$ | ${\mathit{G}}_{\mathit{x}\mathit{z}}$ |
---|---|---|---|---|---|---|---|---|

MPa | MPa | MPa | - | - | - | MPa | MPa | MPa |

8153 | 1949 | 1549 | 0.31 | 0.17 | 0.36 | 1096 | 642 | 1120 |

8212 | 2615 | 1437 | 0.28 | 0.11 | 0.10 | 883 | 637 | 634 |

**Table 4.**Linear correlation coefficients between input and output parameters. Blue: Correlation coefficient $>0.8$ or $<-0.8$; Orange: Correlation coefficient in [0.6; 0.8] or in [−0.8; −0.6].

${\mathit{E}}_{\mathit{x}}$ | ${\mathit{E}}_{\mathit{y}}$ | ${\mathit{E}}_{\mathit{z}}$ | ${\mathit{\nu}}_{\mathit{x}\mathit{y}}$ | ${\mathit{\nu}}_{\mathit{y}\mathit{z}}$ | ${\mathit{\nu}}_{\mathit{x}\mathit{z}}$ | ${\mathit{G}}_{\mathit{x}\mathit{y}}$ | ${\mathit{G}}_{\mathit{y}\mathit{z}}$ | ${\mathit{G}}_{\mathit{x}\mathit{z}}$ | ||
---|---|---|---|---|---|---|---|---|---|---|

F (reaction forces at boundary) | F0° | 0.99 | 0.40 | 0.32 | 0.23 | 0.23 | 0.38 | 0.43 | −0.08 | 0.15 |

F90° | 0.52 | 0.98 | 0.40 | 0.66 | 0.50 | 0.39 | 0.44 | 0.27 | 0.31 | |

U90° | 0.24 | 0.35 | 1.00 | 0.53 | 0.61 | 0.65 | 0.63 | 0.32 | 0.44 | |

F45° | 0.64 | 0.73 | 0.58 | 0.61 | 0.57 | 0.61 | 0.84 | 0.26 | 0.36 | |

U45°-90° | 0.10 | 0.46 | 0.77 | 0.58 | 0.66 | 0.67 | 0.62 | 0.81 | 0.69 | |

U45°-0° | 0.14 | 0.44 | 0.76 | 0.59 | 0.71 | 0.65 | 0.59 | 0.73 | 0.81 | |

${\u03f5}_{x}$ (transverse mean strain) | F0° | 0.05 | −0.63 | −0.45 | −0.92 | −0.63 | −0.46 | −0.41 | −0.38 | −0.35 |

F90° | 0.06 | −0.81 | −0.36 | −0.87 | −0.61 | −0.29 | −0.33 | −0.38 | −0.36 | |

U90° | 0.05 | −0.09 | −0.82 | −0.47 | −0.64 | −0.80 | −0.56 | −0.42 | −0.50 | |

F45° | −0.09 | −0.75 | 0.02 | −0.45 | −0.20 | 0.11 | 0.28 | −0.10 | −0.06 | |

U45°-90° | −0.43 | −0.01 | −0.59 | −0.10 | −0.12 | −0.27 | −0.23 | 0.56 | 0.19 | |

U45°-0° | −0.39 | −0.08 | −0.60 | −0.15 | −0.15 | −0.27 | −0.23 | 0.52 | 0.30 | |

${\u03f5}_{y}$ (longitudinal mean strain) | F0° | −0.56 | 0.25 | 0.30 | 0.50 | 0.49 | 0.35 | 0.36 | 0.79 | 0.51 |

F90° | −0.39 | 0.23 | 0.19 | 0.45 | 0.43 | 0.27 | 0.23 | 0.44 | 0.36 | |

U90° | −0.29 | −0.05 | −0.48 | 0.07 | 0.20 | 0.01 | −0.10 | 0.45 | 0.41 | |

F45° | −0.32 | 0.61 | 0.27 | 0.59 | 0.48 | 0.25 | 0.02 | 0.64 | 0.52 | |

U45°-90° | 0.55 | −0.08 | 0.05 | −0.11 | −0.12 | 0.01 | 0.08 | −0.85 | −0.40 | |

U45°-0° | 0.59 | 0.19 | 0.25 | 0.09 | 0.00 | 0.20 | 0.27 | −0.62 | −0.51 |

**Table 5.**Overview of the identified material parameters for 15 wt.% short carbon-fibre reinforced PETG filament (Formfuture CarbonFil).

${\mathit{E}}_{\mathit{x}}$ | ${\mathit{E}}_{\mathit{y}}$ | ${\mathit{E}}_{\mathit{z}}$ | ${\mathit{\nu}}_{\mathit{x}\mathit{y}}$ | ${\mathit{\nu}}_{\mathit{y}\mathit{z}}$ | ${\mathit{\nu}}_{\mathit{x}\mathit{z}}$ | ${\mathit{G}}_{\mathit{x}\mathit{y}}$ | ${\mathit{G}}_{\mathit{y}\mathit{z}}$ | ${\mathit{G}}_{\mathit{x}\mathit{z}}$ |
---|---|---|---|---|---|---|---|---|

MPa | MPa | MPa | - | - | - | MPa | MPa | MPa |

8153 | 1949 | 1549 | 0.31 | 0.17 | 0.36 | 1096 | 642 | 1120 |

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**MDPI and ACS Style**

Witzgall, C.; Völkl, H.; Wartzack, S. Derivation and Validation of Linear Elastic Orthotropic Material Properties for Short Fibre Reinforced FLM Parts. *J. Compos. Sci.* **2022**, *6*, 101.
https://doi.org/10.3390/jcs6040101

**AMA Style**

Witzgall C, Völkl H, Wartzack S. Derivation and Validation of Linear Elastic Orthotropic Material Properties for Short Fibre Reinforced FLM Parts. *Journal of Composites Science*. 2022; 6(4):101.
https://doi.org/10.3390/jcs6040101

**Chicago/Turabian Style**

Witzgall, Christian, Harald Völkl, and Sandro Wartzack. 2022. "Derivation and Validation of Linear Elastic Orthotropic Material Properties for Short Fibre Reinforced FLM Parts" *Journal of Composites Science* 6, no. 4: 101.
https://doi.org/10.3390/jcs6040101