# A Unified Physical Model for Creep and Hot Working of Al-Mg Solid Solution Alloys

^{*}

## Abstract

**:**

## 1. Introduction

_{0}in Equation (1) is the pre-exponential factor in the Arrhenius form and Q is the activation energy for the relevant diffusional mechanism (self-diffusion or diffusion of Mg atoms in Al). In pure Aluminium the stress exponent n is close to 4.4–5; however, above a certain stress level, power-law breakdown occurs, and the slope of the curve describing the strain rate dependence on applied stress in double-log coordinates increases progressively with the stress level applied. This behaviour, which is associated to a stress exponent in the power-law regime close to 5 and an activation energy equivalent to that for vacancy diffusion, identifies “class M” (Metal) materials. The addition of Mg results in a more complex dependence of the secondary creep rate on applied stress. The stress exponent in the low stress regime is close to 4–5, it then becomes 3 in an intermediate stress range, and again, 4–5 before power-law breakdown [4,5,6]. This behaviour (“class A”) is generally interpreted by invoking the fact that, since glide and climb of dislocations occur in sequence during high-temperature deformation, the slower is rate controlling. In pure metals, glide is always faster than climb; therefore, the latter is, without exceptions, the rate controlling mechanism. In class A materials, glide is substantially slowed down by the formation of clouds of solute atoms around dislocations, and is, consequently, the rate controlling factor in the intermediate stress regime, leading to n = 3 and to an activation energy that is equivalent to the activation energy for diffusion of Mg atoms in Al. It is only when climb is very slow (in the low stress regime) or when the solute atoms no longer play any role, since dislocations have broken away from their atmospheres (in high stress regime), that the stress exponent is 4–5 and a class-M behaviour is apparent.

## 2. The Model

_{i}represents the strength of pure annealed metal, that is, the stress that is required to move a dislocation in the absence of other dislocations, while α is a constant. In solid solution alloys, the viscous drag of dislocations is thought to reduce dislocation mobility and control creep response in a wide interval of applied stresses. The term σ

_{ss}thus represents the stress required for dislocations to move by viscous drag in the presence of solute atom atmospheres.

^{2}), M is the dislocation mobility, and L is the dislocation mean free path, i.e., the distance travelled by a dislocation before it undergoes a reaction, customarily expressed as

_{L}being the strain-hardening constant. The first term on the right-hand side of Equation (3) represents the strain hardening effect due to dislocation multiplication, which is more rapid when L, and, consequently, C

_{L}assume low values and/or the dislocation density is high. The second and third terms on the right-hand side of Equation (3) describe the effect of dynamic recovery. Since, at high temperature, the last term of Equation (3), which includes an Arrhenius-type dependence on T, largely predominates on the second term, which is roughly a-thermal, and the main emphasis in this study is on describing high-temperature behaviour (T > 500 K), Equation (3) can be simplified to become

_{max}, which depends on material structure, is the flow stress that is required to plastically deform the material in the absence of thermal activation and Q

_{g}is the activation energy necessary to overcome the obstacle field. Recent studies [9,13] showed that Equation (7) works very well for pure Cu and Al with p = 2 and q = 1, with R

_{max}equivalent to the true stress corresponding to the ultimate tensile strength of the material considered, roughly quantified as 1.5R

_{uts}, where R

_{uts}is the ultimate tensile strength.

_{Mg}is the volume atomic misfit (details about Ω and δ

_{Mg}are given in [14]).

_{cg}= D

_{0cg}exp(−Q

_{cg}/RT) is the appropriate diffusion coefficient.

_{L}(C

_{L}= 86 in pure Al [13]) and of two of the terms in Equation (2), namely ${\sigma}_{i}$ and ${\sigma}_{ss}$.

_{i}, which is temperature and strain rate dependent, was based on the assumption that the dislocation density in annealed state (ρ

_{a}) and the ${\sigma}_{i}$ values account for the annealed yield strength of the pure metal [15]. The yield stress is thus given by Equation (2) (which is general, and holds for any single stage of the stress vs. strain curve) [15], where ρ

_{a}is virtually nihil. The traditional way of describing the temperature dependence of yield strength is to assume that, in the creep range, yield strength is proportional to creep strength, whereas below the creep range yield strength is proportional to shear modulus. On these bases, for pure Al, the following expression has been used in [13]

_{y}constant (A

_{y}= 4.2 × 10

^{−3}[13]) was determined so as to obtain a reliable estimate of the yield strength of high-purity Al at room temperature by Equation (2).

_{d}is the dislocation velocity, being c the Mg atomic concentration. The term D

_{Mg}= D

_{0Mg}exp(−Q

_{Mg}/RT) is the diffusivity of Mg in Al. The term I(z

_{0}) can be calculated by the numerical integration of

## 3. Description of High Purity Aluminium

_{L}= 86, U

_{ss}= 0, Q

_{cg}= Q

_{sd}= 122 kJ·mol

^{−1}, and D

_{cg}= D

_{0sd}= 8.34 × 10

^{−6}m

^{2}·s

^{−1}[13]. The original Figure in [13] was obtained by taking ω = 15; in this study, the value ω = 0 was used, which demonstrates that the simplified model of Equation (5) still works very well. The diffusion coefficient was recalculated in [13] by considering all the data reported in [17]. The basic model gives a very good description of the experimental data for the pure metal, without any need for data fitting, and, for this reason, it is an excellent basis for the implementation of the analysis of Al-Mg alloys.

## 4. Description of High Purity Aluminium-Magnesium Single Phase Alloys

#### 4.1. Diffusion Coefficient

_{0Mg}= 1.24 × 10

^{−4}m

^{2}·s

^{−1}, Q

_{sd}= 130.5 kJ·mol

^{−1}[18]. The accuracy of this estimate was challenged in a recent work [19], which reported a wide collection of literature results on the diffusivity of Mg in Al, which is illustrated in Figure 2. A good fitting is obtained with Q

_{sd}= 119 kJ·mol

^{−1}and D

_{0sd}= 1.9 × 10

^{−5}m

^{2}·s

^{−1}, which actually give a curve very close to the one from [18].

#### 4.2. Drag Stress Calculation and Experimental Datasets on Dislocation Density and Strain Rate

_{cg}can be considered to be equivalent to the self-diffusion coefficient of Al. Once the value of the ultimate tensile strength of the different Al-Mg alloys is quantified (with the above-mentioned assumption, R

_{max}= 1.5R

_{UTS}), all of the parameters in Equations (10), (12), and (19) are known. The model curves presented in Figure 3 were thus calculated by Equations (15) and (18). Since the agreement between the curves and the experimental data is excellent, a direct and independent confirmation that the estimation of the drag stress is sufficiently reliable to be used in the model for the steady-state creep rate dependence on applied stress is obtained.

_{c}, the rate-controlling mechanism should be viscous glide (n = 3). It is only when the applied stress exceeds a limiting value (${\sigma}_{ba}^{s}$, frequently identified with the break-away stress, ${\sigma}_{ba}$) that dislocations are able to break-away from solute atom atmospheres and creep should again become climb-controlled (n = 4–5). The data from [4] are thus well suited to assess the accuracy of the model in describing the effect of different amounts of Mg in solid solution.

#### 4.3. Viscous-Glide Controlled Creep: Strain Rate Dependence on Stress and Temperature at T ≥ 523 K

#### 4.4. Creep Above the Transition for Break-Away of Dislocations from Solute Atom Atmospheres

_{ss}being the drag stress calculated by Equation (18) A suitable phenomenological form for F is

_{a}the difference in volume between solute and solvent atoms (that is, with the formalism used in Equation (8), W

_{m}is directly related to β). In Friedel’s original formulation A

_{ba}= 1, giving, for example, a break-away stresses above 400 MPa for 2%Mg at 750 K, a level so high that it is hardly conceivable for this mechanism to play any role in creep. These very high values were considered by some authors [4,24,25] as incompatible with experimental evidence. Their analyses rather suggested that ${\sigma}_{ba}$ is one order of magnitude lower, that is, A

_{ba}≅ 0.1 [24,25] or even A

_{ba}≅ 0.065 [26,27]: this reduced value of the break-away stress will be here, denoted as ${\sigma}_{ba}^{*}$. Thus, Equation (23) was here provisionally used with A

_{ba}= 0.065, to obtain the values of the reduced drag stress ${\sigma}_{ss}^{*}$. The curves obtained by replacing ${\sigma}_{ss}$ with ${\sigma}_{ss}^{*}$ into Equation (11) are presented in Figure 6.

#### 4.5. Hot Working as an Extension of Creep: The Model in the High Strain Rate Plasticity Regime

## 5. Conclusions

_{L}, which, in combination with free dislocation density, determines the dislocation mean free path. Once the effect of break-away of dislocations from solute atmospheres has been described by a specific relationship, the model proposed does not require any variation in the constitutive equations to describe the whole experimental range of steady-state creep rate, nor data-fitting, which is a notable advancement over other phenomenological equations.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Steady-state creep rate dependence on applied stress for Al 99.999% [16]; the curves were calculated by Equations (10) and (11) with C

_{L}= 86, ω = σ

_{ss}= U

_{ss}= 0.

**Figure 2.**Diffusion coefficient in Al-Mg (set of literature data from [19]). The Figure illustrates the best fitting of the data as well as the curve representing the diffusion coefficient used in the majority of previous studies.

**Figure 3.**Experimental values of the dislocation density as a function of stress for Al-3%Mg [20,21,22], Al-4%Mg [21], Al-5%Mg [21], and Al-6.9%Mg [20]. The shaded area, which illustrates the scatter band for Al-Mg alloys, was reported in [22]. The Figure also plots the model curve for pure Al [13] and for Al-Mg alloys (3.24% and 5.5%Mg at 600 and 623 K, respectively).

**Figure 5.**Description of the experimental data for T > 523 K from [16] by Equations (10) and (11), where the drag stress σ

_{ss}is obtained from Equations (19) and (20).

**Figure 7.**Variation of ${\sigma}^{*}{}_{ss}$ as a function of temperature and applied stress for Al-2%Mg.

**Figure 8.**Plots for ${\sigma}^{*}{}_{ss}$, c* and ${\sigma}_{ss}$ as a function of modulus compensated stress for 3.24%Mg at 600 K.

b | Burgers vector | 2.86 × 10^{−10} m |

c | concentration of Mg in solid solution | [at %] |

C_{L} | work hardening constant | 86 |

D_{0sd} | pre-exponential factor in equation for self-diffusion | 8.34 × 10^{−6} m^{2}·s^{−1} |

D_{0Mg} | pre-exponential factor in equation for diffusion of Mg in Al | 1.9 × 10^{−5} m^{2}·s^{−1} |

G | shear modulus at the testing temperature | (3.022 × 10^{10}–1.6 × 10^{7} T) Pa |

k | Boltzmann constant | 1.38 × 10^{−23} J·K^{−1} |

L | mean dislocation free path | [m] |

m | Taylor factor | 3.06 |

M_{c} | climb mobility of dislocations | [m^{2}·N^{−1}·s^{−1}] |

M_{cg} | climb and glide mobility of dislocations | [m^{2}·N^{−1}·s^{−1}] |

n | stress exponent in power-law equation | |

Q_{sd} | activation energy for vacancy diffusion (self-diffusion) | 122 × 10^{3} J·mol^{−1} |

Q_{dMg} | activation energy for diffusion of Mg in Al | 119 × 10^{3} J·mol^{−1} |

R | universal gas constant | [J·mol^{−1}·K^{−1}] |

R_{max} | maximum back stress | [Pa] |

R_{UTS} | ultimate tensile strength | [Pa] |

T | absolute temperature | [K] |

v_{d} | velocity of dislocations | [m·s^{−1}] |

α | material constant in Taylor equations | 0.3 |

δ_{Mg} | volume atomic misfit | |

$\epsilon $ | strain | |

$\dot{\epsilon}$ | strain rate | [s^{−1}] |

ν | Poisson’s ratio | 0.3 |

ρ | free dislocation density | [m^{−2}] |

ρ_{a} | free dislocation density in annealed state | [m^{−2}] |

σ | stress (creep or constant strain rate experiments) | [Pa] |

σ_{ba} | break-away stress | [Pa] |

σ_{i} | internal stress | [Pa] |

σ_{ss} | solid solution strengthening stress | [Pa] |

σ^{*}_{ss} | reduced solid solution strengthening stress | [Pa] |

σ_{y} | yield strength | [Pa] |

${\tau}_{l}$ | dislocation line tension | [N] |

ω | recovery constant | [Pa] |

Ω | atomic volume of the host atom (Al) | 1.66 × 10^{−29} m^{3} |

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**MDPI and ACS Style**

Spigarelli, S.; Paoletti, C.
A Unified Physical Model for Creep and Hot Working of Al-Mg Solid Solution Alloys. *Metals* **2018**, *8*, 9.
https://doi.org/10.3390/met8010009

**AMA Style**

Spigarelli S, Paoletti C.
A Unified Physical Model for Creep and Hot Working of Al-Mg Solid Solution Alloys. *Metals*. 2018; 8(1):9.
https://doi.org/10.3390/met8010009

**Chicago/Turabian Style**

Spigarelli, Stefano, and Chiara Paoletti.
2018. "A Unified Physical Model for Creep and Hot Working of Al-Mg Solid Solution Alloys" *Metals* 8, no. 1: 9.
https://doi.org/10.3390/met8010009