Size effect on micro-metal strength simulated by discrete dislocation and dislocation density-function dynamics

Abstract

The pronounced smaller-being-stronger size effect of nano- and micron-metals has been extensively explored by experimental approaches and simulations in recent decades. In this study, 2D dislocation dynamics (2D DD) was used to simulate the tensile deformation of micron-sized polycrystalline fcc metals. When the grain size is constant, changing the specimen thickness increases the yield strength either when the thickness-to-grain size ratio (t-to-d ratio) is <1 regardless of the initial dislocation density (and dislocation source density), or when the initial dislocation and dislocation source density are both low regardless of the t-to-d ratio. The above is the “smaller being stronger” phenomenon in polycrystals. “Smaller being weaker”, on the other hand, occurs when 1<t-to-d ratio<̃ 3 and the initial dislocation and source density are high. In addition, when the specimen thickness is kept constant, the Hall-Petch relationship is not strictly obeyed when the thickness is approaching the smallest grain size studied.

Apart from the size effect in poly-crystals, single crystals are also known to exhibit pronounced size effect. The size effect of single crystals can be expressed in a power-law σ~D-m, in which the exponent m ranges from ~0.3 to ~1.0. Dislocation-density function dynamics was used in this study to explore the relationship between the size dependence of the yield strength and the dislocation microstructure in different specimen sizes. The “post-mortem” dislocation structure was analyzed to work out a length scale governing the size effect. It was found that the initial dislocation structure has a significant effect on the yield strength of single crystals. For the dislocation microstructure studied, specimens of sizes ~4000 b and ~8000 b are more significantly affected by the stochasticity of the initial dislocation micron structure, whereas the initial dislocation structure could be easily disentangled in even smaller specimens and thus the strength depend less on the initial microstructure. For larger specimen (~16000 b), there is less stochasticity in the initial microstructure, since the specimen size is larger than the characteristic length describing the dislocation mean free path of the specimen.

DDFD is also used to study the tensile deformation of tri-crystals. The grain size effect on the strength was studied. The simulated relationship between the 0.2% proof stress and the inverse square root obeys the Hall-Petch relation. In addition, the internal length scale was computed by applying a strain gradient plasticity theory (SG) to the 1D strain profile derived from DDFD results. To shed light on the identity of the internal length scale (l), l is compared with two characteristic lengths that describe the tri-crystal microstructure: the mean dislocation spacing 1/√p and the dislocation pile-up length. The internal length scale in general approximates to the pile-up length. Exception is found for the largest grain size L=500 nm when the strain is large in which l approximates to the mean dislocation spacing. This suggest that in general plasticity is governed by a length scale related to the dislocation-pileup length, whereas for the exceptional case the bowing out of dislocations through forest dislocations is the governing factor for plasticity.