3D DISCRETE DISLOCATION DYNAMICS APPLIED TO ... - NUMODIS

112 Dislocation-precipitate interactions
4.2.4 Increment in hardening stress
Although a difference in shear modulus has a little effect on the flow stress, its effect is much stronger
on the hardening stress. In this section, the effects of ∆G on the hardening stress are discussed.
The stress required to force a dislocation to glide between particles which have remaining Orowan
loops, are plotted in figure 4.13 both for the case of Gp = 4Gm (filled symbols) and for no image
stress (∆G = 0 : open symbols). The change in the shear modulus of the particles results in an
increased work hardening rate and the effect of ∆G increases with the particle radius. The image
stress fields are the sum of the interactions of the particle and each dislocation present around the
particles. Thus the dislocation lines have to overcome the additional image stress field coming from
the interaction of the residual loops and the particles. This additional stress is directly related to the
number of Orowan loops stored around the particles. As a result, compared to the case of no image
stress, a higher shear stress is required to bypass the particles and this effect is more pronounced as
more dislocation lines are passing, which leads to a higher material hardening. Considering that the
range and the magnitude of the image stress increases as the radius Rp, it can be understood that the
hardening rate is increased as Rp. Fisher et al ([Fisher et al. 53]) have investigated the hardening
of metal crystals induced by precipitate particles. They computed the back stress resulting from
Orowan loops and calculated the effective critical stresses of the Frank-Read sources. The argument
is that the hardening stress (τh) is related to the number of loops (N) by τh = γN, where γ is a
function of the particle radius Rp and the inter-particle distance L. They obtained
γ = 0.65cbGm
1 −
ν
2(1 − ν)
R 2 p
(L + Rp) 3
, where c is the parameter describing the closest distance between a source and a particle. We
obtained the slopes of each graph in figure 4.13 by linear fitting and the dependence of these slopes
on b(Rp) 2 /(L + Rp) 3 as shown in figure 4.14. It is found that the argument of back stress proposed
by Fisher et al ([Fisher et al. 53]) still holds in the case of moving dislocation line through two
particles. The parameter c is around 3.45 for the case of no image stress and 4.43 when Gp = 4Gm,
which means, based on their argument, that the effective distance of a stress source is shorter or
the back stress is higher if the image stress is included.
It is observed experimentally that all the dislocations left around the particles by the gliding dis-
locations are not rigorously confined to a single glide plane, but are rather of the prismatic form
([Humphreys & Martin 67]). If cross-slip is easy, a dislocation may overcome an obstacle in its
glide plane by slipping on another slip plane, with the formation of long jogs. The simulations pre-
(4.6)