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Research Papers

Climb-Enabled Discrete Dislocation Plasticity Analysis of the Deformation of a Particle Reinforced Composite

[+] Author and Article Information
C. Ayas

Structural Optimization and Mechanics,
Delft University of Technology,
Mekelweg 2,
Delft 2628 CD, The Netherlands

L. C. P. Dautzenberg, M. G. D. Geers

Mechanical Engineering Department,
Eindhoven University of Technology,
P.O. Box 513,
Eindhoven 5600 MB, The Netherlands

V. S. Deshpande

Department of Engineering,
Cambridge University,
Trumpington Street,
Cambridge CB2 1PZ, UK
Mechanical Engineering Department,
Eindhoven University of Technology,
P.O. Box 513,
Eindhoven 5600 MB, The Netherlands,
e-mail: vsd@eng.cam.ac.uk

Diffusional deformation of the elastic particles is also a potential mechanism in the deformation of superalloys. This mechanism is not included in this study.

Oscillatory motion of dislocations in DDP are common resulting from the finite time steps used in the explicit time integration schemes.

In the context of both annihilation and pinning at an obstacle, we interpret Le as radius measured from the dislocation over which core effects dominate. Opposite signed dislocations on the same slip system passing by each other within a proximity of Le are captured to annihilate and similarly dislocations passing a point obstacle within this radius is captured by the obstacle and pinned.

We performed spot checks on the periodicity of the total stress fields to confirm that the 86 × 50 mesh is sufficient to enforce periodicity for all the unit cell sizes considered here.

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received November 29, 2014; final manuscript received January 28, 2015; published online June 3, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(7), 071007 (Jul 01, 2015) (13 pages) Paper No: JAM-14-1547; doi: 10.1115/1.4030319 History: Received November 29, 2014; Revised January 28, 2015; Online June 03, 2015

The shear deformation of a composite comprising elastic particles in a single crystal elastic–plastic matrix is analyzed using a discrete dislocation plasticity (DDP) framework wherein dislocation motion occurs via climb-assisted glide. The topology of the reinforcement is such that dislocations cannot continuously transverse the matrix by glide-only without encountering the particles that are impenetrable to dislocations. When dislocation motion is via glide-only, the shear stress versus strain response is strongly strain hardening with the hardening rate increasing with decreasing particle size for a fixed volume fraction of particles. This is due to the formation of dislocation pile-ups at the particle/matrix interfaces. The back stresses associated with these pile-ups result in a size effect and a strong Bauschinger effect. By contrast, when dislocation climb is permitted, the dislocation pile-ups break up by forming lower energy dislocation wall structures at the particle/matrix interfaces. This results in a significantly reduced size effect and reduced strain hardening. In fact, with increasing climb mobility an “inverse size” effect is also predicted where the strength decreases with decreasing particle size. Mass transport along the matrix/particle interface by dislocation climb causes this change in the response and also results in a reduction in the lattice rotations and density of geometrically necessary dislocations (GNDs) compared to the case where dislocation motion is by glide-only.

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Figures

Grahic Jump Location
Fig. 1

Sketch of the combined glide and climb of an edge dislocation during its motion from location A to location C. In (a), the dislocation moves from A to C by first gliding to the intermediate location B and then climbing to C while in (b) the motion involves first climbing to the intermediate location B and then gliding to the final position C. The global co-ordinate system x1-x2 and the local axes attached to the dislocation x'1-x'2 are both illustrated.

Grahic Jump Location
Fig. 2

Sketch of the doubly periodic composite subjected to shear. The elastic particles are shaded dark and the three slip systems oriented at φ(α) are indicated along with the convention used to denote positive dislocations on each of the slip systems. The global co-ordinate system x1-x2 is also illustrated.

Grahic Jump Location
Fig. 3

The shear stress τ¯ versus shear strain γ¯ responses of composites oriented for single slip with φ(1)=0 deg. The response for glide-only motion of the dislocations (Bg/Bc=0) is shown in (a) while the climb-assisted glide case with Bg/Bc=10-4 is shown in (b). In each case, results are shown for four values of the unit cell size h. Note the scale difference for τ¯ in (a) and (b).

Grahic Jump Location
Fig. 4

The evolution of the dislocation density ρd with imposed shear strain γ¯ for composites oriented for single slip with φ(1)=0 deg. Results for (a) glide-only (Bg/Bc=0) and (b) climb assisted glide (Bg/Bc=10-4) are shown for four values of the unit cell size h.

Grahic Jump Location
Fig. 5

The dislocation structures and the associated σ12 stress distribution at an applied strain of γ¯=0.04 in the composite oriented for single slip with φ(1)=0 deg. Results are shown for the h=0.5 μm unit cell case with (a) glide-only (Bg/Bc=0) and (b) climb-assisted glide (Bg/Bc=10-4) dislocation motion as well as the h=2 μm unit cell with (c) Bg/Bc=0 and (d) Bg/Bc=10-4.

Grahic Jump Location
Fig. 6

The (a) mean shear stress 〈τ¯〉 and corresponding (b) mean dislocation density 〈ρd〉 given as a function of the unit cell size h for composites oriented for single slip with φ(1)=0 deg. Results are shown for selected values of the climb to glide mobility ratio Bg/Bc.

Grahic Jump Location
Fig. 7

The deformed configurations of the h=0.5 μm composite oriented for single slip (φ(1)=0 deg) at an applied strain of γ¯=0.04 for the cases of (a) glide-only dislocation motion (Bg/Bc=0) and (b) climb-assisted glide (Bg/Bc=10-3). The deformations are illustrated on the deformed FE meshes with the deformations magnified by a factor of 50.

Grahic Jump Location
Fig. 8

Distributions of the lattice rotation ω21 in the composite oriented for single slip (φ(1)=0 deg) at an applied strain of γ¯=0.04. Results are shown for (a) the h=0.5 μm unit cell with glide-only motion of the dislocations (Bg/Bc=0) and for the climb-assisted glide cases (Bg/Bc=10-3) with unit cell sizes (b) h=0.5 μm and (c) h=2 μm.

Grahic Jump Location
Fig. 9

(a) Contours of vacancy density Nve in the h=0.5  μm composite (Bg/Bc=10-3) oriented for single slip (φ(1)=0 deg) at an applied strain of γ¯=0.04 and (b) sketch of the climb motion of dislocations around the particles as inferred from the contours in (a). In (b) the crosses denote the absorption of vacancies and the circles the emission of vacancies.

Grahic Jump Location
Fig. 10

(a) The shear stress τ¯ versus shear strain γ¯ and (b) dislocation density ρd versus shear strain γ¯ responses of composites oriented for single slip with φ(1)=0 deg and climb-assisted glide motion of dislocations with Bg/Bc=10-4. Results are shown for loading up to γ¯=0.04 and unloading back to γ¯=0 for three values of the unit cell size h. The instants at which the dislocation structures are shown in Fig. 11 are marked by circles on the curves.

Grahic Jump Location
Fig. 11

The dislocation structures in the composites oriented for single slip with φ(1)=0 deg and climb-assisted glide motion of the dislocations with Bg/Bc=10-4. Results are shown for the composites with (a) h=1 μm and (b) h=0.25 μm unit cells at the four instants indicated in Fig. 10.

Grahic Jump Location
Fig. 12

The mean shear stress 〈τ¯〉 for composites with three active slip systems. Results are shown for selected values of the climb to glide mobility ratio Bg/Bc. The corresponding predictions for the composite oriented for single slip with φ(1)=0 deg from Fig. 6(a) (error bars omitted for the sake of clarity) are also included.

Grahic Jump Location
Fig. 13

The dislocation structures and the associated σ12 stress distribution at an applied strain of γ¯=0.04 in the composite with three active slip systems. Results are shown for the h=0.5 μm unit cell case with (a) glide-only (Bg/Bc=0) and (b) climb-assisted glide (Bg/Bc=10-4) dislocation motion, as well as the h=2 μm unit cell with (c) Bg/Bc=0 and (d) Bg/Bc=10-4.

Grahic Jump Location
Fig. 14

The shear stress τ¯ versus shear strain γ¯ responses of the matrix material with three active slip systems. The response for glide-only motion of the dislocations (Bg/Bc=0) is shown in (a) while the climb-assisted glide case with Bg/Bc=10-4 is included in (b). In each case, results are shown for four values of the unit cell size h used in the computations.

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