Origin of Plasticity Length-Scale Effects in Fracture

Fracture in metals is controlled by material behavior around the crack tip where size-dependent plasticity, now widely demonstrated at the micron scale, should play a key role. Here, a physical origin of the controlling length scales in fracture is identified using discrete-dislocation plasticity simulations. Results clearly demonstrate that the spacing between obstacles to dislocation motion controls fracture toughness. The simulations support a continuum strain-gradient plasticity model and provide a physical interpretation for that model’s phenomenological length scale. Analysis of a dislocation pileup under a stress gradient predicts the yield stress to increase with increasing obstacle spacing, physically rationalizing the simulations.

Figure 1

Schematic of the top half of the symmetric DD[CZ] fracture simulation geometry. A remote stress intensity loading is applied to a material that deforms via dislocation plasticity and that contains an initial crack and cohesive zone along the line ahead of the crack. The nanoscale material model consists of dislocations, dislocation slip planes, dislocation sources (solid circles), and dislocation obstacles (open circles). Lengths are in microns.

Figure 2

Normalized fracture energy ${\Gamma }_c/{\Gamma }_0$ versus dislocation obstacle spacing $L_{\mathrm{obs}}$, for various values of the tensile yield stress. Error bars show variations using three statistically different realizations of dislocation obstacles and sources in the simulation. Dashed lines: predictions of SGP theory [17] with gradient length scaled to match simulations (upper scale). Inset: growth of fracture toughness with crack growth, due to evolution of dislocation plasticity, for one particular sample ($L_{\mathrm{obs}}=80\mathrm{nm}$, ${\sigma }_Y=700\mathrm{MPa}$).

Figure 3

Contours of averaged local plastic strain from DD simulations and crack opening profile (curves, displacement $\times 10$) for materials with (a) $L_{\mathrm{obs}}=80\mathrm{nm}$ and (b) $L_{\mathrm{obs}}=250\mathrm{nm}$, at the same load $\Gamma /{\Gamma }_0=5.25$ and tensile yield stress ${\sigma }_Y=700\mathrm{MPa}$. Actual dislocations are shown as symbols, and the dislocation densities in the region shown are (a) $2\times {10}^{14}/{\mathrm{m}}^2$ and (b) $1\times {10}^{14}/{\mathrm{m}}^2$. All lengths are in microns.

Figure 4

Fracture toughness versus ${\sigma }_{\mathrm{coh}}/{\sigma }_Y^{\prime }$ where ${\sigma }_Y^{\prime }$ is the near-crack-tip yield stress [see Eq. (3)], which approximately collapses data from Fig. 2 onto a single curve (dashed line to guide the eye).