# Carbon Nano-Onions as Nanofillers for Enhancing the Damping Capacity of Titanium and Fiber-Reinforced Titanium: A Numerical Investigation

^{*}

## Abstract

**:**

## 1. Introduction

_{60}@C

_{240}carbon onion. Very recently, Pereira Júnior et al. [15] performed MD simulations based on the ReaxFF potential model to study the dynamical behavior and the structural transformations of CNOs under high-velocity impacts against rigid substrates.

## 2. Materials and Methods

#### 2.1. Numerical Treatment of Basic Phases

#### 2.1.1. Alphα Titanium Alloy

#### 2.1.2. Carbon Nano-Onions

#### 2.1.3. Carbon Fibers

#### 2.2. Numerical Treatment of α-Ti Filled with CNOs

_{f}of the CNO nanofillers in the α-Ti matrix was defined by:

_{CNO}and V

_{α-Ti}are the total volumes of the CNO nanofillers and the α-Ti matrix, respectively.

_{0}sin(2πft), where f is the frequency set to 100 Hz and t is the time, was applied to the front vertical plane x = l of the RVE. An appropriate frictionless support condition was applied to the behind boundary x = 0 of the RVE. To ensure that the periodic symmetry of the RVE was satisfied (zero shear stresses should appear on these surfaces at all times), the vertical planes y = 0, y = l, z = 0, and z = l were restricted to remain parallel to their original shape. Transient structural analysis was used to simulate the modeled MMNCs until the time instant t = 1.1T was reached, where T = 1/f = 0.01 s is the period of the external load.

#### 2.3. Numerical Treatment of CNOs/α-Ti Nanocomposite Reinforced with CFs

_{CF}and V

_{MMNC}are the total volumes of the CF reinforcements and the CNOs/α-Ti MMNC matrix, respectively.

_{fCF}= 0.05, 0.10). The FEM models of the developed RVEs, depicted in Figure 4b,d, were axially loaded with a sinusoidally time-varying normal stress along the effective x-axis of the CF-reinforced MMNCs to extract the dynamic properties governing the specific direction of the composite systems. For each investigated case, corresponding proper boundary conditions of periodic symmetry were imposed on the externally nonloaded RVE faces.

#### 2.4. Computation of Dynamic Properties

_{0}and ε

_{0}are the maximum of the input normal stress and the maximum (average nodal) of the output normal strain on the loaded boundary of the RVE models, respectively.

_{0}/ε

_{0}represents the elastic modulus or, equivalently, the stiffness of the MMNCs. The Poisson ratio of a given nanocomposite RVE may be calculated by the ratio of the average nodal normal strain on an externally nonloaded RVE face to the average nodal normal strain at the externally loaded RVE face.

## 3. Numerical Results and Discussion

_{f}. All computations were performed for an indicative load frequency of f = 100 Hz. The time variations in the average nodal normal strain on the loaded face of the RVEs are shown in Figure 7 for five different values of volume fractions V

_{f}. It may be observed that the higher the volume fraction V

_{f}of the CNOs within the pure metallic matrix, the lower the maximum output strain ε

_{0}and the longer the time lag Δτ, indicating a higher MMNC stiffness σ

_{0}/ε

_{0}and loss factor tanδ, respectively.

_{0}/ε

_{0}, Poisson’s ratio v, loss factor tanδ, and density ρ, respectively. For a better understanding of the CNOs’ influence, the axis of the independent variable in the material property diagrams was scaled not only according to the volume fraction V

_{f}but also to the mass fraction M

_{f}, which is defined as follows:

_{CNO}and ρ

_{α-Ti}are the densities of the CNO nanofillers and the α-Ti matrix, respectively, given in Table 1.

_{vM}, regarding two of the investigated RVEs at a volume fraction of 10% (Figure 10a) and 25% (Figure 10b) and at the time instant t = 0.8T = 0.08 s, i.e., near the peak of the observed stress concentrations. Note that the von Mises stress values σ

_{vM}at the scale bars are nondimensionalized by dividing them with the maximum applied stress σ

_{0}. It may be noticed that the maximum equivalent stresses were observed within the graphene-based nanoparticles relieving the matrix component. As the volume fraction was increased, the stresses were distributed to more nanoparticles. As a result, the maximum von Mises stress was lower for the case V

_{f}= 0.25.

_{f}= 0, 0.10, 0.15, 0.20, and 0.25. On the other hand, two different cases regarding the volume fraction of the CFs in the CNO/α-Ti matrix were investigated, i.e., V

_{fCF}= 0.05, 0.10. Figure 11a–d present the variation in the stiffness σ

_{0}/ε

_{0}, the loss factor tanδ, the storage modulus E′, and the loss modulus E″ of the CF-reinforced MMNCs, respectively, with regard to the CNO volume fraction V

_{f}and the CF volume faction V

_{fCF}.

_{f}rose. Nevertheless, the slope of the increase in the curves that correspond to the stiffness and storage modulus seems to reduce as V

_{f}reaches 25%. Instead, the loss factor and loss modulus curves have an almost linear ascending behavior. As in the case of simple MMNCs, and perhaps due to the impact of porosity, their slope shows a marginal increase as the V

_{f}increases. It becomes clear that the damping capacity, expressed by the loss factor, of the MMNC becomes lower with the addition of the CFs. In contrast, the higher the fiber volume fraction V

_{fCF}, the higher the stiffness of the CF-reinforced nanocomposite. In any case, the α-Ti offered better energy dissipation properties when reinforced with CNOs. However, most of the positive influence of the CNOs on the loss factor of the titanium matrix was not lost due to the addition of fiber reinforcements, which simultaneously provided a significant stiffness enhancement to the material system. An important observation is the fact that the addition of carbon fibers in an α-Ti matrix filled with CNO leads to an improvement in the storage modulus but to a reduction in the loss factor due to the inherent high effective elastic modulus and low loss factor of the carbon fibers, respectively. Some contour plots of the dimensionless equivalent von Mises stress σ

_{0}/σ

_{vM}for two CF-reinforced MMNC RVEs at a V

_{fCF}equal to 0.05 and 0.10 are illustrated in Figure 12a,b, respectively. The contours correspond to the highest tested CNO volume fraction V

_{f}= 0.25 and to the time point t = 0.0077 s around which stress peaks occur. As illustrated, for both the CF volume fraction cases, the maximum equivalent stress was detected in the fibers. In addition, the increase in the CF concentration led to a significant decrease in the level of von Mises stresses.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A CNO modeled as an isotropic hollow spherical nanoparticle: (

**a**) geometry and (

**b**) FE mesh.

**Figure 3.**A nanocomposite RVE made of α-Ti filled with CNOs: (

**a**) geometry for V

_{f}= 0.10, (

**b**) FE mesh for V

_{f}= 0.10, (

**c**) geometry for V

_{f}= 0.25, and (

**d**) FE mesh for V

_{f}= 0.25.

**Figure 4.**A nanocomposite RVE made of the CNOs/α-Ti reinforced with CFs: (

**a**) Geometry for V

_{fCF}= 0.05, (

**b**) FE mesh for V

_{fCF}= 0.05, (

**c**) geometry for V

_{fCF}= 0.10, and (

**d**) FE mesh for V

_{fCF}= 0.10.

**Figure 5.**The time lag between input stress and output strain, which was caused by the overall damping of the nanocomposite.

**Figure 6.**The two-level analysis proposed for the characterization of the mechanical and damping behavior of different types of MMNCs filled with CNOs.

**Figure 7.**Computed time variations in output strain of the CNOs/α-Ti RVEs at a variety of volume fractions.

**Figure 8.**Volume fraction variations in (

**a**) the stiffness σ

_{0}/ε

_{0}, (

**b**) Poisson’s ratio ν, (

**c**) the loss factor tanδ, and (

**d**) the density ρ of the CNOs/α-Ti nanocomposites.

**Figure 9.**Volume fraction variations in (

**a**) the storage modulus E′ and (

**b**) the loss modulus E″ of the CNOs/α-Ti nanocomposites.

**Figure 10.**Contours of the equivalent von Mises stress σ

_{vM}at time t = 0.8T for the CNOs/α-Ti and at a volume fraction V

_{f}of (

**a**) 10% and (

**b**) 25%.

**Figure 11.**The variation in (

**a**) the stiffness σ

_{0}/ε

_{0}, (

**b**) the loss factor tanδ, (

**c**) the storage modulus E′, and (

**d**) the loss modulus E″ of the CNOs/α-Ti nanocomposites reinforced with CFs versus the CNOs volume fraction and for two CF volume fractions.

**Figure 12.**Contours of the equivalent von Mises stress σ

_{vM}at time t = 0.77T for the CNOs/α-Ti reinforced with CFs at a CNOs volume fraction V

_{f}of 25% and a CF volume fraction V

_{fCF}of (

**a**) 5% and (

**b**) 10%.

Material/Phase | Behavior | Density (kg/m ^{3}) | Elastic Constant (GPa) | Poisson’s Ratio (−) | Loss Factor (−) | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

ρ | E_{xx} | E_{yy} | E_{zz} | G_{xy} | G_{yz} | G_{xz} | ν_{xy} | ν_{yz} | ν_{xz} | tanδ | ||

α-Ti | Isotropic | 4505 | 117 | 117 | 177 | 43.985 | 43.985 | 43.985 | 0.33 | 0.33 | 0.33 | 0.0278 |

CNOs | Isotropic | 2267 | 387 | 1050 | 1050 | 442.66 | 442.66 | 442.66 | 0.186 | 0.186 | 0.186 | 0.518 |

CFs | Orthotropic | 2150 | 17.24 | 17.24 | 17.24 | 7.1 | 7.1 | 7.1 | 0.214 | 0.214 | 0.214 | 0.000159 |

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**MDPI and ACS Style**

Giannopoulos, G.I.; Batsoulas, N.D.
Carbon Nano-Onions as Nanofillers for Enhancing the Damping Capacity of Titanium and Fiber-Reinforced Titanium: A Numerical Investigation. *Metals* **2023**, *13*, 1577.
https://doi.org/10.3390/met13091577

**AMA Style**

Giannopoulos GI, Batsoulas ND.
Carbon Nano-Onions as Nanofillers for Enhancing the Damping Capacity of Titanium and Fiber-Reinforced Titanium: A Numerical Investigation. *Metals*. 2023; 13(9):1577.
https://doi.org/10.3390/met13091577

**Chicago/Turabian Style**

Giannopoulos, Georgios I., and Nikolaos D. Batsoulas.
2023. "Carbon Nano-Onions as Nanofillers for Enhancing the Damping Capacity of Titanium and Fiber-Reinforced Titanium: A Numerical Investigation" *Metals* 13, no. 9: 1577.
https://doi.org/10.3390/met13091577