# Influence of θ′ Phase Cutting on Precipitate Hardening of Al–Cu Alloy during Prolonged Plastic Deformation: Molecular Dynamics and Continuum Modeling

^{1}

^{2}

^{*}

## Abstract

**:**

## Featured Application

**The proposed method for prediction of the strength of aluminum alloys can be used to determine the required size distribution of hardening particles to achieve the durability or maximum strength.**

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Statement of MD Task

^{3}, in the first case the system contains of about 2,000,000 atoms, and in the last one—about 5,000,000.

^{7}s

^{−1}. The considered strain rate falls within the range of experimentally available strain rates realized during the propagation of shock waves; today, in shock experiments on thin films, the strain rates reach 10

^{9}–10

^{10}s

^{−1}[71,72,73]. NVE fix is used for the determination of other atom positions except for layers moving at a given speed. The constant level of the temperature in the system is provided by the Berendsen thermostat [74]. Thus, the NVT ensemble is simulated in fact, which means the conservation of the number of particles, volume of system and temperature. Such an approach was previously implemented to study the dynamics of solitary dislocation and the dislocation–precipitate interactions in an Al–Cu system [34,57,58,69]. During MD calculations, about a hundred dislocation–precipitate interactions occur.

#### 2.2. Model of Dislocation–Precipitate Interaction

#### 2.3. Method of Discrete Dislocation Dynamics

^{6}s

^{−1}are calculated. This strain rate is chosen so as to be close to the rate considered in the MD one, as described in Section 2.1. In spite of its high value, the considered strain rate of 10

^{6}s

^{−1}corresponds to the transition to the quasi-stationary plastic response of alloy as shown in [57].

## 3. Results and Discussion

#### 3.1. Mechanism of Interaction of Dislocation with Inclusion in MD

#### 3.2. Influence of Multiple Interactions on the Structure of Inclusion and Stress State of the MD System

#### 3.3. Accumulation of Vacancies in the MD System during Prolonged Deformation

_{2}Cu intermetallic compound as voids. We chose two radii, 0.24 and 0.25 nm; the larger of them is not sensitive to the space inside θ′ phase, but underestimates the volume of vacancies in the system; the smaller one more accurately takes into account free volume inside the vacancies, but sometimes detects the volume in inclusion as free one. Figure 9a demonstrates the curves of the free volume fraction by dependence on time corresponding to two radii of 0.24 and 0.25 nm. Note that the difference between these two estimates at the end of MD calculations is 18%. The average curve for two obtained estimates of free volume is additionally given in Figure 9a; this average curve is presented in Figure 7b together with the vertical position of dislocation.

#### 3.4. Accounting for Precipitate Softening in the Equation of Dislocation Motion

#### 3.5. Strength of Alloys with Size-Distributed and Spatially Distributed Precipitates

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**Structure and form of θ′ precipitate. Panel (

**a**) is cross section of matrix-surrounded precipitate and (

**b**) is view of inclusion flat surface without aluminum matrix Aluminum atoms are colored in gray, copper atoms are red.

**Figure 2.**Dislocation and θ′ precipitate of 5 nm diameter in MD system after energy minimization. (

**a**)—view along $[11\overline{2}]$ direction, (

**b**)—view along $[111]$. Atoms with a centrosymmetry parameter more than 5 square angstroms are depicted. Aluminum atoms are colored in gray, copper atoms are red. Shockley partial dislocations are colored in green.

**Figure 3.**Stages of dislocation–precipitate interaction: (

**A**) dislocation free slip in front of θ′; (

**B**) dislocation looping around θ′; (

**C**) closing the Orowan loop; (

**D**) dislocation slip behind θ′.

**Figure 4.**Time dependences of the average stress in systems containing θ′ precipitate for various temperatures. Diameter of θ′ precipitate is 5 nm, thicknesses of systems, equal to the inter-precipitate distance, are 9.87 and 17.6 nm, shear velocity is 3 m/s.

**Figure 5.**The first dislocation–precipitate interaction. Atoms with a centrosymmetry parameter more than 5 square angstroms are depicted. Aluminum atoms are colored in gray, copper atoms are red. Shockley partial dislocations are colored in green.

**Figure 6.**View of θ′ precipitate after several interactions with dislocation; interaction number is indicated above each view together with the corresponding time. Diameter of θ′ phase is 5 nm; shear velocity is 5 m/s; and temperature is 300 K. Only copper atoms are shown.

**Figure 7.**Time dependences of averaged shear stresses (

**a**), volume fraction of vacancies (

**b**) and position along the y-axis of dislocation for the system of 17.6 nm thickness at a temperature of 300 K. Atomic distributions corresponding to the times indicated on the graph are shown. Copper atoms are shown in red; aluminum atoms are in gray (only atoms with the centrosymmetry parameter exceeding 5 square angstroms are shown); green lines are Shockley partial dislocation segments; blue lines are perfect dislocation segments; lilac lines are vertex dislocation segments.

**Figure 8.**Time dependence of averaged shear stress in a system containing θ′ phase of 18 nm in diameter. (

**a**) View of inclusion after multiple interactions with dislocation (

**b**) Thickness of system is 27 nm, displacement velocity is 3 m/s and temperature is 300 K.

**Figure 9.**Volume fraction α of voids by dependence on time: (

**a**) for a single system with two different radii of probe sphere of “Construct surface mesh” algorithm; and (

**b**) for different systems with additional linear approximation.

**Figure 10.**Time dependences of shear stresses averaged over system for precipitates with diameters d of 4.6 nm (

**a**) and 18 nm (

**b**) and different distances D between inclusions: 9.87 and 17.6 nm (

**a**) and 27 nm (

**b**). Comparison of results for multiple dislocation–precipitate interactions obtained by the model and the MD.

**Figure 11.**Results of 2D dislocation dynamics calculation for aluminum with θ′ precipitates for different typical sizes ((

**a**)—20 nm, (

**b**)—40 nm, (

**c**)—60 nm, (

**d**)—80 nm) and volume fractions: stress-strain curves calculated in approximation of constant size of precipitates (“uncuttable”) and accounting for their size evolution (“cuttable”).

**Figure 12.**Localization of plastic deformation in system in cases of (

**a**) “uncuttable” and (

**b**) “cuttable” precipitates.

**Figure 13.**Dependence of maximal shear strength on volume fraction and typical diameter of θ′ precipitates in Al–Cu alloy and region of strain softening due to precipitate cutting: Results of 2D dislocation dynamics accounting for precipitate evolution. Diagram is supplemented with experimental data on yield stress in Al–Cu alloy [37], given points corresponding to the experimental data on typical diameter and volume fraction of θ′ precipitates; additionally, maximal absolute errors in experimental data are indicated.

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**MDPI and ACS Style**

Krasnikov, V.S.; Mayer, A.E.; Pogorelko, V.V.; Gazizov, M.R.
Influence of θ′ Phase Cutting on Precipitate Hardening of Al–Cu Alloy during Prolonged Plastic Deformation: Molecular Dynamics and Continuum Modeling. *Appl. Sci.* **2021**, *11*, 4906.
https://doi.org/10.3390/app11114906

**AMA Style**

Krasnikov VS, Mayer AE, Pogorelko VV, Gazizov MR.
Influence of θ′ Phase Cutting on Precipitate Hardening of Al–Cu Alloy during Prolonged Plastic Deformation: Molecular Dynamics and Continuum Modeling. *Applied Sciences*. 2021; 11(11):4906.
https://doi.org/10.3390/app11114906

**Chicago/Turabian Style**

Krasnikov, Vasiliy S., Alexander E. Mayer, Victor V. Pogorelko, and Marat R. Gazizov.
2021. "Influence of θ′ Phase Cutting on Precipitate Hardening of Al–Cu Alloy during Prolonged Plastic Deformation: Molecular Dynamics and Continuum Modeling" *Applied Sciences* 11, no. 11: 4906.
https://doi.org/10.3390/app11114906