Precipitation Strengthening Model

The precipitation hardening arises from the interactions between dislocations and precipitates. Specifically, the dispersed precipitate particles act as pinning points or obstacles and impede the movement of dislocations through the lattice, and therefore strengthen the material. In general, either a dislocation would pass the obstacles by cutting through the small and weak particles (the shearing mechanism) or it would have to by-pass the strong impenetrable precipitates (the by-passing mechanism). The shearing mechanism is believed to predominate in lightly aged alloys with fine coherent precipitates or zones, while the by-passing mechanism is more characteristic of over-aged alloys with coarser precipitates.

Following an usual approach to determine the critical resolved shear stress at which a dislocation overcomes the obstacles in the slip plane, the response equation can be derived and written as [1979Ger,1998Des],

(1)

where M is the Taylor factor, the mean obstacle strength, b the Burgers vector, and L the average particle spacing on the dislocation line. It has been shown that the Friedel statistics gives fairly good results in the calculation of the average particle spacing [1998Des]. Accordingly, this statistics is adopted and Eq. 1 becomes,

(2)

where G is the shear modulus, b a constant close to 0.5, V_f the particle volume fraction and the mean particle size.

The calculation of the mean obstacle strength in Eq. 2 is determined by the obstacle size distribution and the obstacle strength:

(3)

where N_i is the obstacle number density of the size class R_i and F_i is the corresponding obstacle strength, which is related to the size of the obstacles and how the obstacles are overcome (particle shearing or particle by-passing).

In the case of particle shearing for weak obstacles, a rigorous expressions of the obstacle strength is quite complex and depends on different strengthening mechanisms (e.g., chemical strengthening, modulus hardening, coherency strengthening, ordering strengthening and so on). Therefore, we will not consider the detailed mechanisms involved. Instead, a more general model proposed by Gerold [1979Ger] is used in the present study and the obstacle strength of a precipitate of radius R is given by,

(4)

where k is a constant and G is the shear modulus.

On the other hand, the obstacle strength is constant and independent of particle radius in the case of particle by-passing [1979Ger],

(5)

where b is a constant close to 0.5. From Eq. 4 and Eq. 5, one can find that the critical radius for the transition of the shearing and the by-passing mechanism is . By treating R_C an adjustable parameter as suggested by Myhr et al. [2001Myh], the equations Eq. 4 and Eq. 5 can be written as,

(6)

Combining equations Eq. 2, Eq. 3 and Eq. 6 gives the yield strength contributed by a particle of radius R_i,

(7)

where . This leads to the expression for the yield strength arising from precipitation hardening in the general case where both shearable (weak) and non-shearable (strong) particles are present,

(8)

Eq. 7 and Eq. 1 yield the relationships and , respectively, in the two extreme case of pure shearing (i.e., all particles are small and shearable) and pure by-passing (i.e., all particles are bigger than the critical radius and non-shearable). The two relationships are consistent with the usual expressions derived from the classical models.

Besides the precipitate hardening σ_P , the other two major contributions should be considered in the calculation of the overall yield: 1) σ₀, the baseline contribution including lattice resistance σ_i, work-hardening σ_WH and grain boundaries hardening σ_GB; 2) σ_SS, the solid solution strengthening. If all three contributions can be estimated individually, the overall yield strength of the material can thus be obtained according to the rule of additions. Several types of these rules have been proposed to account for contributions from different sources [1975Koc, 1985Ard]. A general form is written as [1985Ard],

(9)

when q = 1, it is a linear addition rule and it becomes the Pythagorean superposition rule when q = 2. The value of q can also be adjusted between 1 and 2 in terms of experimental data. When applying to aluminum alloys, the linear addition rule has been shown to be the appropriate one [1985Ard, 1998Des, 2001Myh, 2003Esm] and therefore the overall yield strength is given by the following equation,

(10)

where ; it does not change during precipitation process. σ_SS is the solid solution strengthening term, which depends on the mean solute concentration of each alloying element [1964Fri, 2001Myh],

(11)

where W_j is the weight percentage of the j^th alloying element in the solid solution matrix phase and a_j is the corresponding scaling factor. σ_P is the precipitation hardening term defined by Eq. 1. By applying a regression formula [1997Gro, 2001Myh], the yield strength σ in MPa can be converted to hardness HV in VPN as,

(12)

where A and B are treated as fitting parameters in terms of experimental data.