4.3 Gradient plasticity simulations
4.3.2 Gradient plasticity model with Voce-hardening
Model parameters
The GP model parameters are calibrated such that the overall mechani-cal response matches the averaged stress-strain curves of the DDD sim-ulations. This calibration is, in general, not sufficient by its own since distributions of the plastic strain, e.g., along a line segment of the whole volume, are not necessarily predicted correctly. Therefore, the local dis-tribution of plastic strain has to be taken into account in the calibration, too. While, in principle, the determination of the GP model parameters in the fitting procedure is not unique, some guidelines, therefore, are shown in Table 4.2 which should help in the calibration. Regarding the elastic constants, the calibration yields a uniform Young’s modu-lus of E = 65 GPa for all GP simulations (except for one special case, cf. Section 4.3.4). The value of the Young’s modulus is slightly lower, compared to the value of 72.7 GPa used in the DDD model. This is due to the bow-out of dislocations from the very beginning of the loading (Bayerschen et al., 2015). The Poisson’s ratio is kept identical to the DDD simulations (ν = 0.347) and the resulting initial defect energy density is W0= 0.5G = 12.06GPa.
Table 4.2:Fitting procedure for gradient plasticity model with Voce-hardening, reprinted from Bayerschen et al. (2015).
1. A least-squares fit (LSF) of the DDD stress-strain curves is obtained.
2. The cross-section averaged plastic strain profiles along the loading axis are obtained. This is performed in order to ensure comparability with the (averaged) plastic strain profiles from the DDD simulations.
The occurring differences within the cross-section distributions of GP results are relatively small and not as pronounced as they are in the DDD simulations.
3. The Young’s modulus of the GP model is calibrated in order to match the elastic stiffness obtained from the LSF of the DDD data.
4. The initial yield stress τ0Cof the GP bulk model and the initial yield strength ΞC0 of the GBs as well as the initial yield strength ΞC0,∂B of the boundary (planes at x = 0 and x = xmax) are calibrated using the DDD-LSF and the averaged plastic strain profiles. Therefore, the plastic strain profiles along the loading axis of GP results and DDD results are compared. On that account, plastic strain profiles are obtained at three, representatively chosen, fixed overall plastic strain values in the well established plastic regime. The values used are εp∈ {0.001, 0.002, 0.003}.
5. In case of Voce hardening: The initial hardening modulus Θ and the saturation stress τ∞C are adjusted to the hardening behavior of the DDD-LSF.
4.3 Gradient plasticity simulations
It is remarked that the defect energy normalization constant g0, al-though in principle introducing an internal length scale into the model, mainly controls the elastic-plastic transition behavior if the GP model with GB yielding is used (see Section 3.5.1). Consequently, a value of g0= 16.95/µm is chosen such that the GP simulations show similar stress-strain results as the DDD simulations in the elastic-plastic transi-tion regime. Hardening resulting from the defect energy is negligible compared to the hardening relations investigated in the following (cf.
also the parameter study in Fig. G.1). The internal length scale in the GP model can be obtained from the normalization constant g0
by lint=p
2W0/(g02E) ≈ 36 nm, considering a Young’s modulus of E = 65GPa. This result is, remarkably, of the same order of magnitude as the mean dislocation spacing in the pile-ups at the GBs of the DDD simulations (cf. Section 4.2.3). However, it is not expected that g0
(and thus, the internal length scale) is a constant, in general. For all simulations, a reference shear rate of ˙γ0= 10−3/s, a rate sensitivity exponent of p = 20, and a drag stress of τD= 1MPa are considered. The used penalty parameter is Hχ= 108MPa. At first, two cases of different misorientation are investigated. They are summarized in Table 4.3. In the case NLC35V, the misorientation of the central grain is 35◦and, in the case NLC5V, the misorientation is 5°. The listed model parameters are obtained by the fitting procedure outlined in Table 4.2.
Table 4.3:Setup and model parameters of GP simulations for comparison to DDD results.
The abbreviation NLC indicates that lateral contraction is prevented on the boundary planes at x = 0 and x = xmax. The gradient hardening contribution is negligible in the investigated NLC cases, see Fig. G.1 in the appendix. Data reprinted from Bayerschen et al. (2015).
Name Ang- Harde- ΞC0,Γ ΞC0,∂B τ0C τ∞C Θ le ϕ ning (N/m) (N/m) (MPa) (MPa) (MPa)
NLC5V 5◦ Voce 3.5 25 30.0 108.51 1075
NLC35V 35◦ Voce 3.5 25 30.0 108.51 1075
Numerical results
At first, the framework presented in Chapter 3 is used taking into ac-count the parameter calibration guidelines described above. A misori-entation of 35° of the central grain is considered. This case exhibits weak interaction of dislocations across the GBs in the discrete simulations due to the substantially differing slip system orientations of the adjacent grains. The plastic strain profiles of the DDD simulations are evalu-ated and averaged at three constant overall plastic strains (cf. Fig. 4.2a).
Subsequently, the cross-section averaged plastic strain profiles of the GP simulations are compared to the DDD profiles, see Fig. 4.2b. The model parameters for this case can be found in Table 4.3. When calibrating the GP parameters, it can be seen, e.g., in Fig. 4.2b, that the plastic strain profiles for εp= 0.001are in good agreement. The subsequent evolution of plastic strain close to the GBs, however, cannot be accounted for by the GP simulations using Voce hardening. Significant deviations occur (see exemplary red arrow indicators in Fig. 4.2b) which are caused by an obvious limitation of this approach to account for the accumulation of dislocations at the GBs observed in the discrete simulations. Next, for the case of a small misorientation of the central grain (5°), the stress-strain response of the GP model is fitted to the DDD results (Fig. 4.2c).
The corresponding GP plastic strain profiles are in better agreement with the DDD simulation results within the grains (Fig. 4.2d), compared to the case of 35° misorientation (Fig. 4.2b). For small misorientations, the interaction of dislocations in the discrete simulations is higher, lead-ing to a more homogeneous distribution of plastic strain over all three grains. However, similar deviations occur between GP and DDD results close to the GBs (Fig. 4.2d) as the evolution of the plastic strain and its gradients cannot be accounted for sufficiently by the used model.
Therefore, an additional hardening relation for the GBs is investigated in the following and, after a calibration of the new hardening parameter, the GP results are compared to the DDD results, again.
4.3 Gradient plasticity simulations
0 40 80 120
0 0.001 0.002 0.003 0.004 0.005
NominalstressσinMPa
Nominal strain ε NLC35V
DDD Simulations DDD εpxx-evaluation
GP εpxx-evaluation GP Simulation
0 0.001 0.002 0.003 0.004 0.005
NominalstressσinMPa
Figure 4.2:(a) Stress-strain curves of DDD and GP simulations for 35° misorientation.
Blue error bars indicate that the DDD results are averaged at the corresponding overall plastic strains. (b) Distribution of cross-section averaged plastic strain along loading direction of DDD and GP simulations for 35° misorientation, obtained at the fixed overall plastic strain values from (a). (c,d) GP and DDD simulation results for 5° misorientation.
Red arrows indicate deviations at the GBs. GP simulations obtained with quadratic defect energy. Central grain rotated by misorientation angle around loading axis. DDD data (a-d) and GP data (a,b) from Bayerschen et al. (2015).