# On the Numerical Modeling of Flax/PLA Bumper Beams

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

- Definition of the numerical model. It includes an intralaminar failure model, which considers nonlinear viscoplastic behavior and the influence of strain rate, and an interlaminar failure model based on cohesive interaction.
- Most of the material parameters were obtained through experimental characterization tests. However, some parameters included in the cohesive interaction are difficult to obtain experimentally. Thus, they were fitted using the bumper beam Type A results.
- Once the cohesive parameters were fitted using only the experimental results of the bumper beam Type A, the experimental results of the other bumper beam types (B, C, and D) were used to validate the prediction accuracy of the numerical model.
- Finally, the validated numerical models are used to analyze the failure modes and the influence of the cross section on the absorbed energy.

#### 2.1. Numerical Model

^{3}. According to the previous studies of flax/PLA, this biocomposite presents nonlinear elastic viscoplastic behavior. Therefore, the traditional mechanical behavior model for composites considering linear elastic behavior up to failure cannot be used to reproduce the behavior of green composites. In the present model, two regions were considered in the stress–strain curve: elastic and viscoplastic. As shown in Figure 1, the first elastic stage is assimilated to linear elastic with mechanical properties of E = 5 GPa and Poisson ratio = 0.3; the second region is viscoplastic and was defined using experimental strain-rate-dependent data. Tensile tests were carried out in the laboratory at different velocities to characterize the influence of strain rate. A constitutive viscoplastic model was defined in a previous work [21], which was implemented in an FEM model to reproduce the mechanical behavior of flax/PLA plates either under low-velocity impacts [10] or machining operations scenarios [22,23].

^{−1}, the ultimate strain was fixed as 0.056; otherwise, the ultimate strain obeys a linear function. These values are based on the results obtained in the experimental tensile tests; more detail can be found in [21]. Therefore, the subroutine calculates the ultimate strain as a function of the strain rate and deletes elements that exceed the ultimate strain.

#### 2.2. Fitting

_{nn}= K

_{ss}= K

_{tt}) were specified as the Young modulus of PLA for uncoupled traction–separation behavior (Table 2). These values were not modified during the fitting process.

_{Ic}), and few experimental tests can be found in the literature. To obtain the interlaminar fracture toughness energy in Mode I (G

_{Ic}), double cantilever beam (DCB) tests according to the ASTM D5528-13 standard [27] were carried out, Figure 5a). It should be noticed that this standard was developed for carbon-fiber-reinforced plastics. However, it was used in this study because no specific standard for natural fiber-reinforced composites was found. According to the ASTM D 5528 standard, G

_{ic}can be found using the area of the force–displacement curve, Figure 5b), using Equation (3):

_{Ic}value obtained in the experimental tests conducted on flax/PLA specimens was 2 mJ/mm

^{2}. However, no experimental tests were conducted on the flax/PLA specimens to determine the fracture toughness energy mode II (G

_{IIc}) and mode III (G

_{IIIc}). The reason is the lack of specific standards and the difficulty of applying standards developed for CFRPs because the stiffness of natural fibers is much lower than that of carbon fibers. Therefore, to fit G

_{IIc}and G

_{IIIc}, values between 2 and 3 mJ/mm

^{2}were checked. This range was selected because the fracture toughness energy modes II and III are equal to or higher than mode I in composites. After the fitting process, values of fracture toughness energies were fixed at G

_{Ic}= G

_{IIc}= G

_{IIIc}= 2 mJ/mm

^{2}.

#### 2.2.1. Contact Force History

#### 2.2.2. Absorbed Energy

#### 2.2.3. Delamination Damage Extension

## 3. Model Validation: Comparison with Experimental Data

## 4. Damage Analysis

## 5. Conclusions

- -
- Regarding the prediction of the absorbed energy of the bumper beam, the numerical results show excellent agreement with the experimental data. The model is able to predict the different behavior as a function of impact energy, from localized damage to complete failure.
- -
- Analysis of the force–displacement curves at impact energies of 30 J, 50 J, and 70 J shows that the permanent deformations of the bumper beam initiate at 30 J; the damage on the structure becomes significant and shows different severity for the four types at 50 J; and the complete failure of the beams is produced at 70 J.
- -
- Experimental and numerical results revealed that the threshold energy, where the maximum energy absorption capability is reached, for Type A is over 60 J; for Type B and C is around 60 J; and for Type D is at 50 J.
- -
- The damage evolution showed that the delamination manifests initially in the section corners and then spreads further. It implies that delaminations are more prone to propagate through the entire cross section and more prematurely for rounder types.
- -
- Adding the fact that the rounder section presents smaller peak force and threshold energy. Therefore, the squarest cross section, Type A, is the bumper beam with the highest energy-absorption capability. Conversely, the roundest one, Type D, presents the lowest energy-absorption capability. The lower the roundness of the cross section, the higher the energy-absorption capability.
- -
- Lastly, the numerical model predicted the same deformed shape of the four bumper beams under 60 J as the experimental scenario.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Stress-strain curves of flax/PLA biocomposite obtained in tensile tests at different strain rates.

**Figure 2.**The ultimate strain as a function of the strain rate. The element-deletion criterion is implemented in the VUSDFLD subroutine.

**Figure 3.**Numerical model of the bumper beam under low-velocity impact test. The overall setting and dimension.

**Figure 5.**Experimental tests conducted to determine the interlaminar fracture toughness energy in Mode I. (

**a**) Experimental setup; (

**b**) load–displacement curve; (

**c**) geometry of the specimens. Dimension not to scale.

**Figure 8.**Tomography of bumper beam type A, impact energy of 30 J; and comparison with numerical results. Observation of the interlaminar failure in the ends of the beam (sections A and C) and in the impacted section (section B). Red arrows point the appearance of failure in sections A and C in the numerical screenshot, red ovals point to failure at fillet radius in section B.

**Figure 9.**Tomography of bumper beam type A, impact energy of 50 J; comparison with numerical results. Observation of the interlaminar failure in the ends of the beam (sections A and C) and the center of impact (section B). The red arrows in the numerical screenshot point the appearance of failure in sections A and C.

**Figure 10.**Tomography of bumper beam type A, impact energy of 60 J; comparison with numerical results. Observation of the interlaminar failure in the ends of the beam (sections A and C) and the center of impact (section B). The red arrows in the numerical screenshot point the appearance of failure in sections A and C.

**Figure 11.**Absorbed energy as a function of impact energy. Experimental and numerical results comparison for all the bumper beam types. (

**a**) E

_{abs}-E

_{imp}for Type A; (

**b**) E

_{abs}-E

_{imp}for Type B; (

**c**) E

_{abs}-E

_{imp}for Type C; and (

**d**) E

_{abs}-E

_{imp}for Type D.

**Figure 12.**Force versus displacement curves of all the bumper beam types. Comparison between the experimental and the numerical results. (

**a**) Curves for Type A; (

**b**) Curves for Type B; (

**c**) Curves for Type C; and (

**d**) Curves for Type D.

**Figure 13.**Delamination initiation criterion at the middle cross section of bumper beam Type A. Impact energy: 30 J (

**a**), 40 J (

**b**), 50 J (

**c**), 60 J (

**d**), and 70 J (

**e**).

**Figure 14.**Delamination initiation criterion at the middle cross section of bumper beam Type B. Impact energy: 30 J (

**a**), 40 J (

**b**), 50 J (

**c**), 60 J (

**d**), and 70 J (

**e**).

**Figure 15.**Delamination initiation criterion at the middle cross section of bumper beam Type C. Impact energy: 30 J (

**a**), 40 J (

**b**), 50 J (

**c**), 60 J (

**d**), and 70 J (

**e**).

**Figure 16.**Delamination initiation criterion at the middle cross section of bumper beam Type D. Impact energy: 30 J (

**a**), 40 J (

**b**), 50 J (

**c**), 60 J (

**d**), and 70 J (

**e**).

**Figure 17.**Side view of the bumper beams impacted at 60 J. (

**a**) Photograph of Type B and numerical screenshots of Type A and B; (

**b**) photograph of Type C and numerical screenshots of Type C and D.

Bumper Beam | Radius (mm) | Height (mm) | Weight (g) |
---|---|---|---|

Type A | 10 | 40 | 97.4 |

Type B | 20 | 40 | 96.8 |

Type C | 24 | 40 | 95.7 |

Type D | 37 | 40 | 84.6 |

**Table 2.**Mechanical properties of PLA, data from [24].

Mechanical Properties | PLA |
---|---|

Tensile strength (MPa) | 54.27 |

Young modulus (MPa) | 3180 |

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**MDPI and ACS Style**

Jiao-Wang, L.; Loya, J.A.; Santiuste, C.
On the Numerical Modeling of Flax/PLA Bumper Beams. *Materials* **2022**, *15*, 5480.
https://doi.org/10.3390/ma15165480

**AMA Style**

Jiao-Wang L, Loya JA, Santiuste C.
On the Numerical Modeling of Flax/PLA Bumper Beams. *Materials*. 2022; 15(16):5480.
https://doi.org/10.3390/ma15165480

**Chicago/Turabian Style**

Jiao-Wang, Liu, José A. Loya, and Carlos Santiuste.
2022. "On the Numerical Modeling of Flax/PLA Bumper Beams" *Materials* 15, no. 16: 5480.
https://doi.org/10.3390/ma15165480