Dislocation transport parameters

In this work both the internal characteristics of the material domain and the applied external con- ditions are altered to investigate how this affects the trends in deformation behaviour. A control configuration of the system was required in order to base the comparisons in the following FDM results sections. This configuration is created using the control parameters. For all simulations that are presented, any model parameter that is not explicitly defined in the text or figure caption is assumed to take the control value.

The control parameters are given in Table 5.5.1. Some parameters, including Burgers vector mag- nitude, jog height, activation energy and anti-phase boundary energy have been kept constant universally, while others such as grain size and temperature are varied in later results sections. Most results are presented for a variety of jog spacings between 100 - 1000b (25.4 - 254nm), as ex- perimental observations of jog spacing are seen to vary significantly depending on the work history of an alloy [145] [146]. A spacing between dislocation sources of 200nm was used as standard as this was found to produce a good plastic response.

To maintain accuracy within these simulations, the Finite Difference timestep within the model is required to be less than 2 ns (see Section 5.4.1): a limitation due to the high glide velocity over a short domain. Because this velocity scales with the local stress, it does not change significantly for a lower applied strain rate, and so the small FD timestep must be maintained in all simulations. For low strain rates, when the timescale for the deformation increases, more timesteps are required to reach a given strain and so the simulation becomes more computationally expensive (e.g taking over 3 weeks to produce the 0.01 s−1 curve). To reduce this computational expense this work uses a standard strain rate of 100s−1.

A value of 2.56x103 Pa−1s−1 has been used for the dislocation mobility parameter M . This was based upon both the simulations of Dongsheng Li et al [115] and the experimental work of Urabe and Weertman [116], which were considered in Section 4.2.2 . This value is half of that predicted by eqn (4.2.4), but is far closer to the measured values for this type of FCC alloy, and produces velocities within the expected range [25].

Table 5.5.1: Details of the control parameters (used in all simulations unless stated otherwise)

Parameter Value Reference

b (m) 2.54×10−10 [113] ν 0.33 [147] av (m3) 1.64×10−29 Calculated as 4₃π ₂b
3 Ds,0 (m2s−1) 10−4 [148] Q (kJ ) 310 [113] EAP B (J ) 0.2 [113] hjog (m) 2.07×10−10 Calculated as √ 2 √ 3b λjog (m) 2.54×10−8 [145] [146] T (K) 973 - µ [at 973K] (GP a) 54.69 [138] Material density % (gcm−3) 7.98 [143] vsound (ms−1) 2.61×103 Calculated as q_µ % M (Pa−1s−1) 2.56×103 [115] Grain size d (m) 5×10−6 - LF R 250b ↔ 350b [149] Source spacing (m) 2×10−7 - Number of sources 23 -

Generation density ρsource (m−2) 1014 Section 5.4.4

Chapter 6

Full-field Simulations I: Pure Matrix

Assumed geometry for this chapter

Within this chapter the numerical results from Abaqus FE simple shear tests are presented for the case of a single crystal domain containing only γ phase. The model geometry is in plane-strain as depicted in Fig. 5.2.2, with the XLength/YLengthedge dimensions set to 5 µm (except in Section 6.6).

The control parameters for the simulations are established in the previous Section 5.5.2. These parameters are taken as the default while individual parameters are varied in each section. The default applied strain rate rate was 100s−1 and the temperature was 973K. As this setup describes a crystal of only the bulk matrix phase, these simulations will be referred to as either Pure-Matrix or Open-Matrix interchangeably.

Results analysis

Results will be presented largely in the form of domain-averaged (macro) flow stress curves. If the microstress and microstrains at position X are π(X) and (X) respectively, then the macro-value counterparts can be given by the volume average over the domain Ω

σ = 1 VΩ Z Z Z Ne π∞(X) d3X ε = 1 VΩ Z Z Z Ne (X) d3X (6.0.1) where VΩ is the volume of the domain and Neis the collection of elements that compose it. As this

model is applied within a plane-strain setup then the area average over the 2D simulation domain is used to calculate the macrostress. (Shear stress τ ≡ σ12 is plotted against applied shear strain).

6.1

Penetrable and Impenetrable Grain Boundaries

The effect of imposing closed and open α flux boundary conditions on the flow stress at 973K and 100s−1 is shown in Fig. 6.1.1 for a 5µm square grain. For both cases yield occurs after 0.36 % applied strain with a shear stress of 170 MPa. The open boundary grain has a post-yield behaviour that is almost ideally plastic - running on to 3% strain with only a small stress increase of ∼ 25 MPa. The closed boundary grain shows considerable hardening post-yield, with the stress increasing by ∼ 400 MPa. The hardening rate in latter case is linear. The contrast between the two simulations is stark as the dislocation content flows unobstructed out of the open boundary domain but is caused to pile-up on the grain wall of the closed boundary domain. In this instance the stress field from the large static dislocation content produces two mechanisms for hardening within the grain. Firstly, the motive force upon a dislocation approaching the pile-up is reduced due to the repulsive stress from the pile-up, lowering the velocity of the mobile dislocation. Secondly, the dislocation backstress will also lower the effective stress at the dislocation sources behind the pile-up, causing fewer generation events within the grain. A combination of slower moving dislocations and less dislocation content available within the grain significantly lowers dislocation activity and so also the plastic shear rate.

Fig. 6.1.2 shows the total dislocation source activity within the grain during these simulations. In both instances initial generation events are seen to occur at ∼ 0.1% strain, followed by a pe- riod of inactivity until yield. Activation of the dislocation sources occurs at the generation stress threshold (see Section 5.4.4), thereafter dislocations are emitted and propagate at a jog-controlled velocity. For this simulation, the jog-controlled velocity after initial generation is low and the dislocation stress fields render the sources inactive until the density field can be advected away. Once the stress has built to ∼170 MPa the jog-controlled velocity of the emitted field ahead of the sources is large enough and the density begins to move across the domain more effectively. At this point the sources can resume operation and generation events begin to occur at a steady rate in both simulations. As expected the rate of generation is lower for a grain with an impenetrable boundary, due to the back stress acting on sources as dislocation pile-ups develop.

Figure 6.1.1: Flow stress response for identical grains with open or closed grain-wall boundary conditions (λjog = 25nm).

Figure 6.1.2: Number of dislocation generation events during deformation for identical grains with open or closed grain-wall boundary conditions (λjog= 25nm).