# Elastoplastic Deformation of Rotating Disk Made of Aluminum Dispersion-Hardened Alloys

^{1}

^{2}

^{*}

## Abstract

**:**

_{3}Er and TiB

_{2}particles are used as the initial materials. Tensile strength testing of the obtained alloys shows that the addition of Al

_{3}Er particles in the AA5056 alloy composition leads to an increase in its ultimate stress limit (USL) and plasticity from 170 to 204 MPa and from 14.7 to 21%, respectively, although the modifying effect is not observed during crystallization. The addition of TiB

_{2}particles to the A356 alloy composition also leads to a simultaneous increase in the yield strength, USL, and plasticity from 102 to 145 MPa, from 204 to 263 MPa, and from 2.3 to 2.8%, respectively. The study of the stress-strain state of the disk was carried out in the framework of deformed solid mechanics. The equilibrium equations were integrated analytically, taking into account the hardening conditions obtained from the experimental investigations. This made it possible to write the analytical relations for the radial and circumferential stresses and to determine the conditions of plastic deformation and loss of strength. The plastic resistance of a disk depends on the ratio between its outer and inner radii. The plastic resistance decreases with increasing disk width at a constant inner radius, which is associated with a stronger effect from the centrifugal force field. At a higher rotational rate of narrow disks, the tangential stresses are high and can exceed the USL value. A356 and A356–TiB

_{2}alloys are more brittle than the AA5056 and AA5056–Al

_{3}Er alloys. In the case of wide rotating disks, AA5056 and AA5056–Al

_{3}Er alloys are preferable.

## 1. Introduction

_{2}O

_{3}, TiO

_{2}), carbides, and borides for the grain structure modification of Al–Si alloys [10,11,12,13,14].

_{3}) nanoparticles with a negative thermal expansion coefficient. The alloy demonstrated a highly efficient use of ScF

_{3}-based modifiers and additional ultrasonic treatment. The mechanical mixing and vibration treatment used in [16] affected the structure and mechanical properties of the aluminum alloy A356–C consisting of ≤1 wt.% nanodiamonds. It was shown that the yield strength and ultimate stress limit (USL) of the A356 alloy increased without changing its plasticity. In [17], we conducted an integrated study of the alloy production process and the treatment effect on its physical and mechanical properties. It was found that the grain structure of the alloys consisting of TiB

_{2}particles was completely formed. The addition of TiB

_{2}particles increased the yield strength, USL, and plasticity of the alloys. The highest effect from the ground structure of the cast alloys was achieved by using a master alloy consisting of 1 µm TiB

_{2}particles.

## 2. Experimental Procedure and Results

#### 2.1. AA5056–Al3Er Alloy

_{3}Er master alloy was obtained by erbium (Er) hydrogenation in a chemical reactor with successive mixing with Al micropowder, compaction, and dehydrogenation in a vacuum furnace. A total of 1 kg of the AA5056 alloy was placed in a crucible and molted in a muffle furnace at 780 °C for 2 h. When the crucible was removed from the furnace, the Al–Al

_{3}Er master alloy was introduced into the melt via simultaneous ultrasonic treatment at a temperature of 730 °C. The ultrasonic treatment was performed using a magnetostriction water-cooling actuator at 4.1 kV voltage and 17.6 kHz frequency. After the complete dissolution of the master alloy, the melt was ultrasonically treated for 2 min. The melt was cast in a mold at 720 °C and underwent vibration treatment until complete crystallization. The initial AA5056 alloy was obtained under the same conditions but without the master alloy.

#### 2.2. A356–TiB_{2} Alloy

_{2}) particles were used to modify the alloy structure. The particle introduction and distribution in the A356 melt were performed by using master alloys with 60 wt.% Al and 40 wt.% TiB

_{2}obtained by gas-free burning of a mixture of aluminum, titanium, and boron powders [66]. The alloy was fabricated by melting 1 kg of the A356 alloy in a crucible at 750 °C for at least 4 h in a closed furnace. Master alloys were incorporated at 730 °C and mixed in a mechanical mixer [67] for 1 min in an open furnace. In order to reduce the porosity, obtain a homogeneous structure, and improve the distribution of alloying elements in the melt, vibration treatment was applied during 700 °C casting into the crucible. Vibration was conducted at a frequency of 60 Hz and 0.5 mm amplitude. An alloy without TiB

_{2}particles was created under the same conditions for comparison.

#### 2.3. Methods

_{2}particle diffusion decreases the average grain size from 180 to 140 µm due to the formation of new crystallization centers on the particle surface. Due to the structural proximity of TiB

_{2}and Al crystals, titanium diboride is an effective nucleation center in the Al melt, which, in turn, leads to the formation of the fine structure of ingots. Based on the data obtained, it can be assumed that the modified aggregative state (crystallization) of the introduced (nano- or submicron-sized) particle surface has a significant effect on the grain size of the fabricated alloys. This is associated with the low-stability state of the melt–particle system, which provides a weak thermal effect from the inoculator microparticles on the final state of the alloy, namely, its structure and physical/mechanical properties. After the thermal interaction between the particles and the melt, the temperature of which approaches crystallization, the alloy crystallizes on the particle surface.

_{3}Er particles into the melt improves the USL and plasticity from 170 to 204 MPa and from 14.7 to 21%, respectively, despite the absence of the modifying effect during crystallization. This effect is achieved through the introduction of Al

_{3}Er particles, which contribute to the deflection of the potential crack from the grain boundaries into its volume, as well as increase the role of the aluminum matrix in the deformation and fracture processes. The TiB

_{2}particle diffusion in the A356 alloy also leads to a simultaneous increase in the yield strength, USL, and plasticity from 102 to 145 MPa, from 204 to 263, and from 2.3 to 2.8%, respectively.

## 3. Mathematical Model

_{zz}becomes zero but so does its derivative $\frac{\partial {\sigma}_{\mathit{zz}}}{\partial z}$. On the strength of the symmetry conditions, in the center section, the axial shift and radial motion u

_{r}on the axial coordinate are both zero, i.e., ${u}_{z}=0$ and $\frac{\partial {u}_{r}}{\partial z}=0$, respectively. Therefore, at rather low disk thickness, σ

_{zz}and σ

_{rz}stresses are small. Thus, it can be considered that these stresses are zero throughout the disk, whereas the radial motion depends on the radial coordinate ${u}_{r}={u}_{r}\left(r\right)$ only.

_{rr}and σ

_{φφ}stress tensor components. The additional relation for elastic deformation can be obtained from the equation of strain compatibility, which in the plane state takes the form as follows:

- The whole disk is in the elastic state ($\omega <{\omega}_{\mathrm{el}}$).
- The disk is in the elastic state, but its inner surface is in the plastic state ($\omega ={\omega}_{\mathrm{el}}$).
- Inner disk layers $r\le {R}_{\mathrm{pl}}$ are in the plastic state, while outer layers $r>{R}_{\mathrm{pl}}$ are in the elastic state (${\omega}_{\mathrm{el}}<\omega <{\omega}_{\mathrm{pl}}$).
- The disk material is in the plastic state throughout its width ($\omega ={\omega}_{\mathrm{pl}}$).

_{zz}= 0, the following relations result from Hooke’s law:

_{rr}is as follows:

_{φφ}is derived from the balance equation (6):

_{el}, plastic deformation occurs on the inner disk surface. The rotational rate describes the elastic limit of the disk. In order to determine the rotational rate, at which plastic deformation occurs on the inner surface, we use the tangential stress from (16) at $r={R}_{\mathrm{in}}$. Note that the strain intensity relating to the elastic-plastic transition is ${\epsilon}_{\mathrm{pl}}={\sigma}_{0.2}/{G}_{\mathrm{eff}}$. Thus, we obtain the following:

_{rr}and tangential σ

_{φφ}stresses are tensile, they are positive, viz. σ

_{rr}> 0, σ

_{φφ}> 0. Moreover, we take into account that tangential stresses exceed radial (σ

_{φφ}> σ

_{rr}). Due to the plane state, the axial stress σ

_{zz}is absent (σ

_{zz}= 0). Hence, the conditions of the material hardening (9) and fracture (10) are calculated as follows:

_{zz}does not depend on the coordinate z and can be either the function of radial coordinate r or a constant. The absence of the shear stress (${\sigma}_{rz}=0$) implies that the axial strain is a constant value.

_{V}= 0) is written as follows:

_{zz}, let us use (12). It follows that the axial strain is as follows:

_{zz}is constant, (25) holds for the whole disk. Considering the plastic state of the whole disk, the radial stress on its outer boundary equals the yield strength:

_{1}according to (24) and (26):

_{φφ}for the case of nonlinear hardening:

_{2}is the integration constant and ${}_{2}F_{1}$ is the hypergeometric Gaussian function determined by the Gaussian series sum:

_{out}), radial stresses are assumed to be absent (${\sigma}_{rr}=0$). Hence, the integration constant C

_{2}is as follows:

## 4. Mathematical Simulation

_{ex}at the constant inner radius R

_{in}, i.e., h = R

_{ex}− R

_{in}, and increasing inner radius at a constant outer radius, i.e., with decreasing disk width h.

_{el}begins and the disk width h = R

_{ex}− R

_{in}. As can be seen in Figure 4, the circular disk resistance to plastic strain depends on the R

_{in}/R

_{ex}ratio. When the disk width expands at the constant R

_{in}, the resistance reduces due to the increase in the centrifugal force field. On the contrary, when the disk width expands at the constant R

_{ex}, the centrifugal force reduces and the plastic transfer occurs at a higher rotation frequency. Note that the width effect on the plastic strain ω

_{el}is more significant for a small R

_{in}/R

_{ex}ratio. The plastic strain slightly increases with the increasing R

_{in}/R

_{ex}ratio. When the inner and outer radii increase at the same disk width, the centrifugal force applied to the disk increases. This results in a reduction in the plastic resistance, which occurs at a lower plastic strain.

_{0}.

_{2}. Due to its growth, the plastic resistance increases, which requires a more intensive disk rotation. Thus, the plastic strain ω

_{el}becomes the highest for the A356–0.5%TiB

_{2}alloy, whereas, for the AA5056 alloy, it is the lowest.

_{ex}and r = R

_{in}, the radial stress is zero, viz. σ

_{rr}= 0. The maximum radial stress ${\sigma}_{\mathit{rr}}^{m}$ occurs at the ${R}_{m}=\sqrt{{R}_{\mathrm{in}}{R}_{\mathrm{ex}}}$ point, which is the geometric average of the inner and outer radii. The radial stress increases with an increasing disk width and rotational rate:

_{in}/R

_{ex}is the ratio between the inner and outer radii. The maximum radial stress ${\sigma}_{\mathit{rr}}^{m}$ monotonically decreases with increasing ξ. In the case of the solid disk (ξ = 0), this stress is 50% of the conventional yield strength, viz. ${\sigma}_{\mathit{rr}}^{m}=0.5{\sigma}_{0.2}$.

_{φφ}is presented in Figure 6. The tangential stress σ

_{φφ}is the highest on the inner disk radius and is calculated as follows:

_{φφ}appears on the outer disk boundary (r = R

_{ex}) and equals σ

_{0.2}. The highest tangential stress is observed on the inner disk boundary (r = R

_{in}) and is obtained from the following:

_{B}. At the disk rotational rate $\omega ={\omega}_{\mathrm{pl}}$ and with respect to (18) and (43), the condition of the disk strength retention can be defined as follows:

_{B}parameter values characterizing the disk strength at $\omega ={\omega}_{\mathrm{pl}}$ are given in Table 3.

_{pl}of rotation and the width h show the plastic zone, which occupies the whole disk with different radii R

_{in}. At the constant value of the inner radius R

_{in}, the centrifugal forces increase with an increasing width h. Therefore, the plastic strain of the disk occurs at a lower limit frequency ω

_{pl}. The plastic resistance increases with the decreasing inner radius ${R}_{\mathrm{in}}$, and total plastic deformation of the disk occurs at higher values of the limit frequency ω

_{pl}. In wide annular disks (${R}_{\mathrm{in}}\left(1/{\xi}_{B}-1\right)<h$), the strength loss occurs at the inner disk boundary at $\omega ={\omega}_{\mathrm{pl}}$, and the disk fracture begins. In Figure 7, one can see two zones for different geometrical parameters: A denotes no fracture, and B denotes a fracture at the inner disk boundary.

_{pl}must be significantly higher than the plastic strain ω

_{el}to gain total plastic deformation.

_{in}. In the narrow disks ($h\le 5$ mm), the parameter δ contributes only a few percent. At the increased width h and fixed R

_{in}, the δ parameter increases. This means that there is a wide range of rotational rates, which determines the transition of plastic strain on the inner boundary to the plastic strain of the whole disk. At a higher inner radius and constant width, the δ parameter decreases.

_{rr}and tangential σ

_{φφ}stresses is presented in Figure 9 and Figure 10. These curves are similar to those given in Figure 4 and Figure 5. The radial stress σ

_{rr}is characterized by the maximum, which increases with increasing R

_{in}/R

_{ex}ratio. In our case, this stress is zero on the outer and inner radii as they are not loaded with radial forces.

_{φφ}monotonically decreases with increasing radial coordinate r. The lowest tangential stress is observed on the outer disk boundary (r = R

_{ex}). At $\omega ={\omega}_{\mathrm{pl}}$, the tangential stress on the outer disk boundary equals the yield strength. When approaching the inner diameter, the tangential stress increases. It is worth noting that at the disk geometrical parameters in Figure 8 and Figure 9, its fracture does not occur. Additionally, with increasing R

_{in}/R

_{ex}ratio, the tangential stress decreases.

_{rr}and tangential σ

_{φφ}stresses in the disk material. The stress distribution over the disk width $\eta =r-{R}_{\mathrm{in}}$ calculated for different inner radii is identical.

## 5. Conclusions

_{3}Er and A356–TiB

_{2}were employed as initial materials. According to tensile strength testing of the obtained alloys, the diffusion of Al

_{3}Er particles improved the USL and plasticity of the AA5056 alloy from 170 to 204 MPa and from 14.7 to 21%, respectively, although a modifying effect was not observed during crystallization. The addition of TiB

_{2}particles to the A356 alloy composition also led to a simultaneous increase in its yield strength, USL, and plasticity from 102 to 145 MPa, from 204 to 263 MPa, and from 2.3 to 2.8%, respectively.

_{ex}, the centrifugal force was reduced, and the plastic transfer occurred at a higher rotation frequency.

_{φφ}monotonically decreased with increasing radial coordinate r. The lowest tangential stress was achieved at the outer disk boundary (r = R

_{ex}), whereas the highest tangential stresses were observed at the inner disk boundary (r = R

_{in}).

_{in}/R

_{ex}, the tangential stresses were high and could exceed the USL. The critical disk widths were $0.97{R}_{\mathrm{in}}$ and $0.58{R}_{\mathrm{in}}$ for A356 and A356–0.5TiB

_{2}alloys, respectively. Additionally, for AA5056 and AA5056–Al

_{3}Er alloys, the values were $3.37{R}_{\mathrm{in}}$ and $5.99{R}_{\mathrm{in}}$, respectively. Thus, A356 and A356–0.5TiB

_{2}alloys were more brittle than AA5056 and AA5056–Al

_{3}Er alloys. It was found that at $h<{R}_{\mathrm{in}}\left(1/{\xi}_{B}-1\right)$, the alloy sequence was A356–0.5TiB

_{2}, A356, AA5056, and AA5056–Al

_{3}Er for the disks to withstand the highest rotational rate without fracture. For wide disks (${R}_{\mathrm{in}}\left(1/{\xi}_{B}-1\right)<h$), AA5056 and AA5056–Al

_{3}Er alloys were preferable for fabrication.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Grain structure of different alloys. (

**a**) AA5056; (

**b**) AA5056–Al

_{3}Er; (

**c**) A356; (

**d**) A356–TiB

_{2}.

**Figure 4.**Dependences of disk width and rotational rate, at which plastic strain ω

_{el}begins in different alloys and different R

_{in}: (

**a**) A356; (

**b**) A356–0.5TiB

_{2}; (

**c**) AA5056; (

**d**) AA5056–Al

_{3}Er; 1—15 mm, 2—30 mm, 3—45 mm, 4—60 mm, 5—75 mm, 6—90 mm, 7—105 mm, 8—120 mm, and 9—135 mm.

**Figure 5.**Stress σ

_{rr}distribution over the disk radii: (

**a**) ${R}_{\mathrm{in}}=15$ mm, R

_{ex}= 20 mm; (

**b**) ${R}_{\mathrm{in}}=15$ mm, R

_{ex}= 30 mm; (

**c**) ${R}_{\mathrm{in}}=15$ mm, R

_{ex}= 75 mm; (

**d**) ${R}_{\mathrm{in}}=15$ mm, R

_{ex}= 150 mm; (

**e**) ${R}_{\mathrm{in}}=145$ mm, R

_{ex}= 150 mm; (

**f**) ${R}_{\mathrm{in}}=135$ mm, R

_{ex}= 150 mm; (

**g**) ${R}_{\mathrm{in}}=75$ mm, R

_{ex}= 150 mm; (

**h**) ${R}_{\mathrm{in}}=30$ mm, R

_{ex}= 150 mm; 1—A356, 2—A356–0.5 TiB

_{2}, 3—AA5056, and 4—AA5056–Al

_{3}Er.

**Figure 6.**Stress σ

_{φφ}distribution over the disk radii: (

**a**) ${R}_{\mathrm{in}}=15$ mm, R

_{ex}= 20 mm; (

**b**) ${R}_{\mathrm{in}}=15$ mm, R

_{ex}= 30 mm; (

**c**) ${R}_{\mathrm{in}}=15$ mm, R

_{ex}= 75 mm; (

**d**) ${R}_{\mathrm{in}}=15$ mm, R

_{ex}= 150 mm; (

**e**) ${R}_{\mathrm{in}}=145$ mm, R

_{ex}= 150 mm; (

**f**) ${R}_{\mathrm{in}}=135$ mm, R

_{ex}= 150 mm; (

**g**) ${R}_{\mathrm{in}}=75$ mm, R

_{ex}= 150 mm; (

**h**) ${R}_{\mathrm{in}}=30$ mm, R

_{ex}= 150 mm; 1—A356, 2—A356–0.5 TiB

_{2}, 3—AA5056, and 4—AA5056–Al

_{3}Er.

**Figure 7.**Dependences between the limit frequency ω

_{pl}of the rotation and disk width when the plastic zone occupies the whole disk made of different alloys: (

**a**) A356; (

**b**) A356–0.5TiB

_{2}; (

**c**) AA5056; (

**d**) AA5056–Al

_{3}Er; 1—R

_{in}= 15 mm, 2—R

_{in}= 30 mm, 3—R

_{in}= 45 mm, 4—R

_{in}= 60 mm, 5—R

_{in}= 75 mm, 6—R

_{in}= 90 mm, 7—R

_{in}= 105 mm, 8—R

_{in}= 120 mm, and 9—R

_{in}= 135 mm. A—no fracture, B—fracture on the inner disk boundary.

**Figure 8.**Parameter δ/width dependence for different alloys and radii: (

**a**) A356; (

**b**) A356–0.5TiB

_{2}; (

**c**) AA5056; (

**d**) AA5056–Al

_{3}Er; 1—R

_{in}= 30 mm, 2—R

_{in}= 60 mm, and 3—R

_{in}= 105 mm.

**Figure 9.**Stress σ

_{rr}distribution over the disk radii: (

**a**) ${R}_{\mathrm{in}}=15$ mm, ξ = 0.7; (

**b**) ${R}_{\mathrm{in}}=15$ mm, ξ = 0.8; (

**c**) ${R}_{\mathrm{in}}=15$ mm, ξ = 0.9; (

**d**) R

_{in}= 60 mm, ξ = 0.7; (

**e**) R

_{in}= 60 mm, ξ = 0.8; (

**f**) R

_{in}= 60 mm, ξ = 0.9; 1—A356, 2—A356–0.5TiB

_{2}, 3—AA5056, and 4—AA5056–Al

_{3}Er.

**Figure 10.**Stress σ

_{rr}distribution over the disk radii: (

**a**) ${R}_{\mathrm{in}}=15$ mm, ξ = 0.7; (

**b**) ${R}_{\mathrm{in}}=15$ mm, ξ = 0.8; (

**c**) ${R}_{\mathrm{in}}=15$ mm, ξ = 0.9; (

**d**) R

_{in}= 60 mm, ξ = 0.7; (

**e**) R

_{in}= 60 mm, ξ = 0.8; (

**f**) R

_{in}= 60 mm, ξ = 0.9; 1—A356, 2—A356–0.5TiB

_{2}, 3—AA5056, and 4—AA5056–Al

_{3}Er.

Materials | σ_{0.2}, MPa | σ_{B}, MPa | ɛ_{max},% | G, GPa | ρ, g/cm^{3} | ν |
---|---|---|---|---|---|---|

A356 | 102 | 204 | 2.3 | 27.2 [68] | 2.66 [69] | 0.33 [70] |

A356–0.5TiB_{2} | 145 | 263 | 2.8 | - | - | - |

TiB_{2} | - | - | - | 250 [71] | 4.52 [72] | 0.11 [73] |

AA5056 | 63 | 170 | 14.75 | 27 [74] | 2.65 [75] | 0.34 [76] |

AA5056–Al_{3}Er | 64 | 204 | 21.37 | - | - | - |

Al_{3}Er | - | - | 118 [77] | 5.55 [78] | 0.188 [79] |

Alloys | Γ_{eff} [MPa] | n |
---|---|---|

A356 | 1762 | 0.52 |

A356–0.5TiB_{2} | 2580 | 0.66 |

AA5056 | 343 | 0.34 |

AA5056–Al_{3}Er | 314 | 0.30 |

Alloys | ξ_{B} |
---|---|

A356 | 0.508 |

A356–0.5TiB_{2} | 0.631 |

AA5056 | 0.229 |

AA5056–Al_{3}Er | 0.143 |

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## Share and Cite

**MDPI and ACS Style**

Matvienko, O.; Daneyko, O.; Valikhov, V.; Platov, V.; Zhukov, I.; Vorozhtsov, A.
Elastoplastic Deformation of Rotating Disk Made of Aluminum Dispersion-Hardened Alloys. *Metals* **2023**, *13*, 1028.
https://doi.org/10.3390/met13061028

**AMA Style**

Matvienko O, Daneyko O, Valikhov V, Platov V, Zhukov I, Vorozhtsov A.
Elastoplastic Deformation of Rotating Disk Made of Aluminum Dispersion-Hardened Alloys. *Metals*. 2023; 13(6):1028.
https://doi.org/10.3390/met13061028

**Chicago/Turabian Style**

Matvienko, Oleg, Olga Daneyko, Vladimir Valikhov, Vladimir Platov, Ilya Zhukov, and Aleksandr Vorozhtsov.
2023. "Elastoplastic Deformation of Rotating Disk Made of Aluminum Dispersion-Hardened Alloys" *Metals* 13, no. 6: 1028.
https://doi.org/10.3390/met13061028