Evolution of internal dislocation at transient

The transient of dislocation evolution can be calculated in terms of dislocation density storage from working hardening and dislocation density decrease by recovery. During plastic deformation, internal ‘random’ dislocations are created by work hardening and annihilated by dynamic recovery. The plastic strain increase and the dislocation density storage can be related by Orowan equation:

= (5.35)

where is the magnitude of Burgers vector, is the Taylor factor and is the mean distance travelled by the dislocation before it is stopped. Here ∝ / is assumed.

At the same time, dislocation density decrease due to recovery may be described by (Nix et al. 1985):

= −2 ̅ (5.36)

where is distance between the sites of cross-slip or climb events, is the length of dislocations annihilated. For aluminium alloys / ≈ is acceptable. For Al-Mg alloys the mean velocity of mobile dislocation:

̅ = / (5.37)

where is the diffusion coefficient and is the drag force, depending on the solute concentration and misfit in atomic size, and is the friction stress, effectively driving dislocation motion.

The total internal ‘random’ dislocation density is the balance between the increment due to strain hardening and annihilation due to recovery:

= + = + −2 ̅ = + −2 _∙ = + −2 _∙ (− / ) = + −2 _∙ (− / ) = + −2 _{∙ =} ̇ + −2 _∙ = −2 _∙ = −2 _∙ (5.38) where = and = are constants.

Two points to be noted about the equation:

1) At steady state deformation = + = 0 , therefore there is the relationship = _/

;

2) The ratio can be assumed to be constant because of ̇ ∝ (Raj and Pharr 1986).

5.2.7.1.2.1 Evolution of geometrically necessary internal dislocation

Equation (5.38) can only be used to describe the evolution of ‘random’ internal dislocation density. To predict the total internal dislocation density, the ‘geometrically necessary’ dislocation density, , which relates to the lattice curvature, must be taken into account. Baxter et al. (1999) observed that the subgrain structure was in the form of microbands with low misorientation subgrain boundaries within them and they concluded that the higher misorientation boundaries in the bands are geometrically necessary boundaries to accommodate local lattice curvatures. Assumed that the local lattice curvature arises both from the misorientation across these boundaries and from excess density of dislocations of the same Burgers vector within subgrains (geometrically necessary dislocations), Baxter et al. (1999) developed the model for the calculation of shown as in Figure 5.2.

Figure 5.2 Relationships between local lattice curvature and (a) excess dislocations of a given Burgers vector and (b) subgrain boundary (microband)

spacing and misorientation (Baxter et al. 1999)

In Figure 5.2, an area A of matrix in which there are NA intersections per unit area of excess dislocation of a given Burgers vector b leads to a curvature of radius (see Figure 5.2(a)) and there is a relation

(b) (a)

∆ = ∆ ∅ =

where = ∅ ∆

hence

1

=

Then considering a subgrain boundary between two subgrains shown in Figure 5.2(b)

=_{ℎ =}

where ℎ = 1 ∕ is the spacing of dislocations in the subgrain boundary, is the number of intersections per unit length along the subgrain boundary.

And from Figure 5.2(b)

_{2 ≅2 =} 1 2 hence 1 = ₌

1 =

1

+ 1 = +

which can be rewritten as 1 =

1

+ 1 = +

where is the dislocation density in the microband boundary. Finally according to experimental measurements, Baxter et al. (1999) modified the equation to relate misorientation, subgrain size, the geometrically necessary dislocation density and the local lattice curvature as below

1

= + ̅

̅ (5.39)

Although Sellars and Zhu (2000) claimed that using equation (5.39), ‘the calculated data for the total internal dislocation density including and in reasonable agreement with experimental values for both constant and changing strain rate deformation’, doubt is cast about its application in extrusion. First because this equation largely based on the microband observed in rolling and plane strain compression at relatively small strains less than 2 while at large strains microband doesn’t appear to be encouraged; second substituting the data in Figure 5.3 and Figure 5.4 into equation (5.39) the geometrically necessary dislocation density = − / ≅ _× − _{× . ×} . × / = −2.5 × 10 / and a negative value for the dislocation density is impossible. The reason for this could be the model for is over-simplified or the experimental data is wrong or maybe ̅ can take a negative value when the curvatures are in opposite directions, which probably is a subject that needs more research. Since so far there is no reasonable explanation for this, it would not be meaningful to use this method to calculate the

internal dislocation density.

Figure 5.3 Local lattice curvature of Al-1%Mg deformed at 385ºC (Sellars and Zhu 2000)

Figure 5.4 Comparison of calculated and experimental data of (a) internal dislocation density; (b) subgrain size and (c) misorientation between subgrains

5.2.7.1.2.2 Incremental equations for the evolution of subgrain structure

Zhu and Sellars (1996) noted that the observed microstructure was well approximated by an exponential evolution with strain and a general equation of the microstructure evolution was proposed as below

= + ( − ) 1 − −

,

/ _{, 1/ ,}

(5.40)

The form of equation (5.40) can also be changed as follows (Zhu 1994)

= + ( − ) −

,

/ _{, 1/ ,}

(5.41)

where is a characteristic strain which controls the strain over which steady- state is reached, and are the values of the microstructure state variable at steady state before and after a change of deformation conditions.

In spite of a lack of theoretical analysis of the evolution of subgrain structure the semi-empirical equations (5.40) and (5.41) have been proved to be successful in modelling evolution of subgrain size and misorientation between subgrains during hot deformation at constant strain rate and temperature.

Substitute with into equation (5.41), that is = + ( − ) − differentiate it with respect to ,

Likewise, if is replaced by , after differentiating, a similar form of equation (5.42) will be obtained: 1 ₌ 1 1 − 1 − 1 = 1 − − 1 = 1 − Finally it becomes: = ( − ) (5.43)

Equations (5.38), (5.42) and (5.43) can be also written in integral form:

= / − ∙ _(5.44)

= 1 ( − ) (5.45)

=

( − ) (5.46)

Their integration with FEM and the determination of the parameters will be discussed in following relevant sections.