Dipole formation and yielding in a two-dimensional continuum dislocation model

While a Taylor-type yield stress, proportional to the square-root of the dislocation density, may appear at a macroscopic scale, it can be shown with discrete dislocation simulations that it does not accurately describe dislocation motion on the scale of individual dislocations. In this article, we first demonstrate that the Taylor term fails to capture a number of features of dislocation dynamics by comparing the results of a continuum formulation using this yield stress term to discrete dislocation dynamics simulations. We then present an alternate model, based on a mean free path formulation, and demonstrate that this model effectively reproduces the results of the discrete simulation. This mean free path model is proposed as an extension to an existing continuum theory, making use of the key fact that the velocity of dislocations in a realistic system is not single valued, but distributed over several values. The velocity distribution may also change with time. It is demonstrated that this formulation predicts features of both pileups and homogenous distributions which are in agreement with discrete dislocation simulations, but are not reproducible by traditional statistical continuum theories. Finally, possible extensions to this model are discussed, which may enhance the ability to reproduce key features of dislocation yielding.

Figure 1

Dipole formation as two bundles of oppositely oriented dislocations pass through one another due to an external shear stress. It is expected that, regardless of the number of dislocations, some finite fraction will be able to move through the system, with the exact number depending on the density and applied stress.

Figure 2

Converged state for a bundle of eight SSDs under the influence of an external stress (${\tau }_{\mathrm{ext}}=7.35\text{MPa}$) for DDD (left), and CDD using a dislocation density dependent yield stress [Eq. (7)] for several values of the fitting parameter $\alpha$ (right). The critical density of dislocations for a given $\alpha$ is also listed. The system is embedded in an infinite elastic medium, and dislocations are free to flow through the left and right boundaries. In the discrete system, some dislocations migrate out of the system and some remain, regardless of the local density. In the continuous system, it can seen that all dislocations will migrate if the local density is under ${\rho }_{\mathrm{crit}}$, and all will remain if the local density is above it. The yield stress, as normally formulated, cannot reproduce the discrete results given on the left.

Figure 3

Plastic slip (proportional to the integral of the GND density) in a double-pileup for both DDD and CDD using the dislocation density dependent yield stress for various values of the free parameter $\alpha$. An external stress of ${\tau }_{\mathrm{ext}}=7.35\text{MPa}$ is applied to the system. Boundaries are impenetrable to dislocations, causing them to pile up on either side. Regardless of the value of $\alpha$ chosen, the continuum model fails to accurately predict the distribution of dislocations found in discrete simulation.

Figure 4

Logarithmic plot of the transmission coefficient for a single dislocation moving through a distribution of $N$ oppositely oriented dislocations under the influence of an external stress $\tau$ in units of $\frac{Gb}{2\pi (1-\nu )}$, as a function of $\frac{N}{\tau }$. The solid line shows the function $T=e^{-0.4\frac{N}{\tau }}$.

Figure 5

Density of SSDs distributed according to a Gaussian distribution and allowed to relax. The red curve shows the converged immobile density, ${\rho }_{\mathrm{im}}$ predicted using the mean free path model presented in Sec. 5. In this model some fraction of dislocations are capable of moving regardless of the local density (contrast with Fig. 2). The applied external stress is ${\tau }_{\mathrm{ext}}=7.35\text{MPa}$.

Figure 6

Integral of the GND density (proportional to the plastic slip) of the double-pileup system for several different numbers of dislocations $N$ and applied external stresses ${\tau }_{\mathrm{ext}}$, calculated using both DDD and CDD using the mean free path method. Without any fitting parameter, good agreement can be seen between the two methods. The number of dislocations and external stress for the DDD results are the same as the CDD curves they lie on and ${\tau }_0=29.4\mathrm{MPa}$.