A Combined Precipitation, Yield Strength, and Work Hardening Model for Al-Mg-Si Alloys

Abstract

In the present article, a new two-internal-variable model for the work hardening behavior of commercial Al-Mg-Si alloys at room temperature is presented, which is linked to the previously developed precipitation and yield strength models for the same class of alloys. As a starting point, the total dislocation density is taken equal to the sum of the statistically stored and the geometrically necessary dislocations, using the latter parameters as the independent internal variables of the system. Classic dislocation theory is then used to capture the overall stress-strain response. In a calibrated form, the work hardening model relies solely on outputs from the precipitation model and thus exhibits a high degree of predictive power. In addition to the solute content, which determines the rate of dynamic recovery, the two other microstructure parameters that control the work hardening behavior are the geometric slip distance and the corresponding volume fraction of nonshearable Orowan particles in the base material. Both parameters are extracted from the predicted particle size distribution. The applicability of the combined model is illustrated by means of novel process diagrams, which show the interplay between the different variables that contribute to work hardening in commercial Al-Mg-Si alloys.

Appendix

1.1 Symbols and units

A ₀ :

parameter related to the energy barrier for nucleation (J/mol)

b :

magnitude of the Burgers vector (m)

$\overline{C}$ :

mean solute concentration in matrix (wt pct)

C ₀ :

nominal solute concentration in matrix (wt pct)

C _e :

equilibrium solute concentration at the particle/matrix interface (wt pct)

$C_{\text{eff}}^{\text{Si}}$ :

effective silicon content in alloy (wt pct)

C _i :

solute concentration at the particle/matrix interface (wt pct)

C _Mg :

solute concentration of magnesium in matrix (wt pct)

$\hat{C}_{\text{Mg}}$ :

equivalent magnesium concentration (wt pct)

$\hat{C}_{\text{Mg}}^{\text{ref}}$ :

value of $\hat{C}_{\text{Mg}}$ in reference alloy (wt pct)

C _p :

concentration of alloying element inside the particle (wt pct)

D :

diffusion coefficient (m²/s)

f :

particle volume fraction

f _o :

volume fraction of nonshearable Orowan particles

$f_{o}^{\text{ref}}$ :

value of f _o in reference alloy

G :

shear modulus (N/m²)

j :

nucleation rate (#/m³s)

j ₀ :

pre-exponential term in expression for j (#/m³s)

k ₁ :

parameter related to statistical storage of dislocations (m⁻¹)

k ₂ :

parameter related to dynamic recovery of dislocations

$k_{2}^{\text{ref}}$ :

value of k ₂ in reference alloy

k ₃ :

parameter determining the solute dependence of k ₂ (N/m² wt pct^3/4)

ℓ:

mean planar particle spacing along the bending dislocation (m)

M :

Taylor factor

N _i :

number of particles per unit volume within the size class r _i (#/m³)

Q _d :

activation energy for diffusion (J/mol)

R :

universal gas constant (8.314 J/Kmol)

r :

particle radius (m)

r _i :

particle radius within size class i (m)

r _c :

critical particle radius for the transition from shearing to bypassing (m)

r _o :

particle radius of nonshearable Orowan particles (m)

t :

time (s)

T :

temperature (K or °C)

T _p :

peak temperature (K or °C)

α :

constant in expression for Δσ _d

Δσ _d :

net contribution from dislocation hardening to flow stress (N/m²)

Δσ _d,s :

value of Δσ _d at saturation (N/m²)

$\Updelta \sigma_{d,s}^{ref}$ :

value of Δσ _d,s in reference alloy (N/m²)

Δ :

plate thickness (m)

ɛ :

tensile strain

ɛ _c :

critical strain at which ρ _g saturates

$\varepsilon_{c}^{\text{ref}}$ :

value of ɛ _c in reference alloy

ɛ _p :

plastic tensile strain

ɛ _u :

true uniform plastic strain

γ :

shear strain

γ ^ref :

value of γ in reference alloy

λ _g :

geometric slip distance (m)

λ _g,o :

geometric slip distance of Orowan particles (m)

$\lambda_{g,o}^{ref}$ :

value of λ _g,o in reference alloy (m)

λ _s :

slip distance for statistical storage of dislocations (m)

ρ _g :

number density of geometrically necessary dislocations (m⁻²)

$\rho_{g}^{\text{ref}}$ :

value of ρ _g in reference alloy (m⁻²)

ρ _g,s :

value of ρ _g at saturation (m⁻²)

$\rho_{g,s}^{\text{ref}}$ :

value of ρ _g,s in reference alloy (m⁻²)

ρ _s :

number density of the statistically stored dislocations (m⁻²)

ρ _s,s :

value of ρ _s at saturation (m⁻²)

ρ _t :

total dislocation density (m⁻²)

ρ _t,s :

value of ρ _t at saturation (m⁻²)

σ :

flow stress (N/m²)

σ _i :

intrinsic yield strength of pure aluminum (N/m²)

σ _p :

contribution from hardening precipitates to the overall macroscopic yield strength (N/m²)

σ _ss :

contribution from alloying elements in solid solution to the overall macroscopic yield strength (N/m²)

σ _sat :

value of σ at saturation (N/m²)

σ _y :

overall macroscopic yield strength (N/m²)