Analysis of particle induced dislocation structures using three-dimensional dislocation dynamics and strain gradient plasticity

doi:10.1016/j.commatsci.2011.02.014

Computational Materials Science

Volume 52, Issue 1, February 2012, Pages 33-39

https://doi.org/10.1016/j.commatsci.2011.02.014 Get rights and content

Abstract

Investigations of precipitation hardening are performed in term of analysis of distributions of geometrically necessary dislocations (GND) surrounding particles. The dislocation microstructures are computed from three dimensional discrete dislocation dynamics (DDD) and strain gradient plasticity (SGP) models. DDD simulations of spherical particle embedded in a single crystal matrix undergoing single slip provide the GND structures and the associated work-hardening. A 3D periodic arrangement of particles with cubic symmetry is considered. It is found that a network of slip and kink deformation bands develops, which is strongly dependent on the crystal lattice orientation of the matrix with respect to the particle array. For some relative orientations, the strain hardening is increased by the distributions of GND which act as additional barrier against slip. Some of these features are also captured with the SGP model in contrast to conventional continuum crystal plasticity.

Research highlights

3D dislocation dynamics simulations are performed for a metal matrix composite. The deformation structure around a periodic array of elastic particles is predicted. The dislocation structure in the single crystal matrix is made of slip and kink bands. The microcurl crystal plasticity model predicts the relative intensity of the bands.

Introduction

In crystal plasticity, precipitation hardening is commonly associated to a yield strength increase. The hardening can be explained by the formation of typical dislocation structures around particles, made of geometrically necessary dislocations (GND). Such dislocation microstructures have also been evidenced experimentally [1], [2]. The aim of this work is to describe the relationship between the GND structure and the matrix crystal lattice orientation using computational analysis.

During the last few years, many numerical models have been developed with the aim to investigate GND structures around particles [3]. Some of them are based on generalized continuum mechanics. As an example, the strain gradient plasticity (SGP) models are based on the dislocation density tensor directly related to GND densities [4]. Some relationships have then been established between the SGP constitutive equations and specific dislocation structures like pile-up formation at interfaces [5]. Most of these studies are however restricted to two-dimensional computations so that no relationship have been derived yet within a full 3D framework.

Other models, such as discrete dislocation dynamics (DDD) are explicitly dealing with dislocation lines. Obviously, DDD give access to the dislocation microstructure and consequently the evolution of the GND quantities as a result of collective interactions. 2D and 3D DDD simulations are expected to provide a physics-based description of crystal plasticity. Therefore, it is widely believed that DDD simulations can bridge the gap from the physical origin of plasticity to continuum models. This technique has already been applied in 3D to the question of precipitation hardening but without any detailed description of the dislocation structures around particles [6].

In this paper, GND distributions are investigated in 3D using DDD [7] and SGP [8] simulations of periodic sets of spherical particles embedded in a single crystal matrix loaded in pure shear. Only single slip is considered as a first step for the understanding of 3D particle hardening. DDD simulations show that the dislocation microstructure develops as wall-like structures which strongly depends on the slip system orientation with respect to the array of particles. The physical meaning and origin of the dislocation spreading are analyzed and discussed. They are interpreted as a network of slip and kink bands. A simple three-dimensional SGP model based on the micromorphic approach is then used to predict the plastic strain distribution around the particle under the same conditions but within a continuum framework. SGP results are compared to the DDD predictions.

Section snippets

Geometry of the composite and DDD model

Fig. 1 depicts the unit cell of the simulation volume used in DDD. Because all of six faces of the simulation volume are considered as periodic boundaries, the unit cell corresponds to a 3D cubic arrangement of particles embedded in a single crystal matrix. The axes X1, X2 and X3 in the figure represent the geometrical (global) coordinate system attached to the spatial distribution of particles. A second crystal frame is attached to the crystal lattice of the matrix material. Accordingly, the

Dislocation structures from DDD

DDD results obtained in the case of the simple configuration are given in Fig. 4. The red and pink points represent positive and negative edge dislocation segments whereas yellow² and green points represent positive and negative screw dislocations, respectively. All results are given after 0.5% average applied shear strain ɛ₁₂. For this special slip orientation, only a few pile-ups are observed

Conclusion

The dislocation microstructures formed around particles have been investigated in the case of a periodic arrangements of particles for two slip system orientations using two numerical tools: three-dimensional DDD and SGP model. In DDD simulations, remarkable slip and kink band structures around the particles have been found. It is shown that kink bands, associated with high GND densities enhance the hardening rate. The GND content close to particles is essentially linked to the formation of

Acknowledgments

This research was carried out under Project ANR-07-MAPR-0023-04 CAT-SIZE Matériaux et Procédés. Financial support is gratefully acknowledged. The authors want to thank Nicolas Cordero for help in programming the 3D version of the microcurl model.

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Analysis of particle induced dislocation structures using three-dimensional dislocation dynamics and strain gradient plasticity

Abstract

Research highlights

Introduction

Section snippets

Geometry of the composite and DDD model

Dislocation structures from DDD

Conclusion

Acknowledgments

Acta Metallurgica

Materials Science Engineering A

Acta Materialia

Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids

International Journal of Solids Structures

Journal of the Mechanics and Physics of Solids