4.2 Microstructure model
4.3.1 Basic formulation versus the augmented Lagrangian for-
We will start with a benchmark plane strain example of 3D polycrystalline mi-crostructure simulated using the crystal visco-plasticity fast Fourier transform method (elastic response is neglected). The single-grid strategy is adopted. The crystal visco-plastic constitutive model proposed in [7] along with a Voce type hardening model described in [97] are implemented in the basic framework highlighted earlier in this paper as well as in the augmented Lagrangian for-mulation [68, 97]. Through this particular constitutive law, the stress-strain rate relation formulated in Eq.(4.18) is specified as follows:
ε˙p(x)= Mps(σ(x)) : σ(x), (4.46)
where the secant plastic compliance Mps is taken as:
Mps(σ(x)) = ˙γ0 Ns
∑
α
m(α)(x)⊗ m(α)(x) κ(α)(x)
m(α)(x) :σ(x) κ(α)(x)
(1/m−1). (4.47)
In the equation above, ˙γ0 is a reference rate of shearing, m characterizes the material rate sensitivity,κ(α)(x)is the slip resistance of systemα, Nsis the number of active slip systems, and m(α) denotes the symmetric Schmid tensor of slip systemα:
m(α) = sym(S(α))= 1 2
(s(α)⊗ n(α)+ n(α)⊗ s(α))
, (4.48)
where s(α) and n(α) are the slip direction and slip plane normal of the systemα, respectively.
The volume average of the local secant plastic modulus is taken to be the modulus of the reference medium defined by Eq. (4.21) (Cp0 = ⟨Mps−1⟩h). The
corresponding tangent plastic compliance in Eq. (4.29) at this specific case is Mtp= m1Msp. The plastic compliance in Eq. (4.36) for the integrated method high-lighted in Section 4.1.5 is Mp = Mps.
FCC aluminum is considered. A cubic polycrystalline microstructure com-posed of 64 grains is generated using the Voronoi tessellation scheme [106] in an 1mm3domain. The microstructure is discretized by 16× 16 × 16 equally spaced voxels. The macroscale velocity gradient is
L= ∇V =
0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 −1.0
× 10−3(s−1). (4.49)
The rate-dependent flow rule as given in [97] is used with ˙γ0 = 1s−1 and m= 0.1. The parameters in the Voce type hardening law are selected according to [97]: κ0 = 47.0MPa, κ1 = 86.0MPa, θ0 = 550.0Mpa, and θ1 = 16.0MPa. An ar-bitrary random texture is assigned to the microstructure. The initial microstruc-ture configuration and pole figures showing the random orientation distribu-tion are shown in Fig. 4.4.
This example is aiming at validating the implementation of the basic FFT framework by providing a comparison with the alternative augmented La-grangian implementation. After the thickness of the microstructure reduces by 50%, the deformed microstructure and its stress and strain fields are plotted in Fig. 4.5. For both implementations, the stress distribution over the microstruc-ture is consistent with the grain geometry. The local mechanical responses of the two simulations are very close. Both intergranular and intragranular het-erogeneities of the stress and strain rate fields are captured.
X1
Figure 4.4: (a) The image representation of a 3D polycrystalline crostructure containing 64 grains. (b) Pole figures of the mi-crostructure with randomly assigned orientations.
The homogenized (i.e. volume-averaged) effective stress-strain responses of the entire microstructure computed by the two different algorithms are shown in Fig. 4.6(a). The two curves almost overlap. Pole figures showing the orien-tation distribution of deformed microstructure are depicted in Fig. 4.6(b), from which we observe the typical plane strain deformation texture pattern for both cases. From the above comparison of the local and effective mechanical re-sponses, we conclude that consistent results are obtained from the two algo-rithms.
An error analysis is also conducted to reveal the performance of the two for-mulations. The convergence error, as a function of iteration steps, of the basic formulation (defined by Eq. (4.33)) and of the augmented Lagrangian formula-tion (the larger value of the strain error err(ε) and stress error err(σ) as defined in [62]) are depicted in Fig. 4.7(a), while the equilibrium error (Eq. (4.32)) of both algorithms is shown in Fig. 4.7(b). The error is captured at the first time step with∆t = 0.1. It was observed that the convergence rate of the basic formu-lation is comparable with the augmented Lagrangian formuformu-lation for the
cur-Figure 4.5: Contour plots of plane strain deformed microstructures evalu-ated by different algorithms. The top layer is the equivalent (plastic) strain field, and the bottom layer is the equivalent stress field. (a) Crystal visco-plasticity fast Fourier transform approach implemented in the basic formulation (b) Crystal visco-plasticity fast Fourier transform approach implemented in the augmented Lagrangian formulation.
rent example. The equilibrium error of the two formulations approaches a very close value as the number of iterations increases, although the fluctuation of the basic formulation is larger than that of the augmented Lagrangian case. The basic formulation performances sufficiently well for the current polycrytalline plasticity problem. The main reason is that the contrast between grains with different orientations is mild. For problems with high contrast, the augmented Lagrangian is expected to offer better convergence, while the basic formulation may even fail to converge [84, 85, 120]. Considering the simple structure of the basic formulation, we are employing it for the CEPFFT implementation.
It is also worth mentioning that the equilibrium error is mostly determined by the resolution of the microstructure. For high resolution, the equilibrium
Equivalentstress(MPa)
Figure 4.6: (a) The homogenized effective stress-strain responses com-puted by the basic and augmented Lagrangian crystal visco-plasticity FFT algorithms. (b) Pole figures of the deformed mi-crostructure texture.
Figure 4.7: (a) Evolution of the convergence error as a function of the num-ber of iterations of the augmented Lagrangian formulation in comparison with the convergence error of the basic formula-tion. The error axis uses logarithmic scale. (b) Evolution of the absolute (equilibrium) error as a function of the number of it-erations of the two formulations. The error axis uses normal scale.
condition is fulfilled with smaller error (see Fig. 4.8). When the number of pixels per side is doubled, the equilibrium error is approximately halved.
Pixel number per side
Equilibrium error
8 4.292371e-02
16 2.195340e-02 32 1.116329e-02 64 5.609229e-03
Equilibriumerror
0.01 0.02 0.03 0.04 0.05
Pixel number per side
0 10 20 30 40 50 60 70
0 0.01
Figure 4.8: Equilibrium error as a function of resolution (number of pixels per side) computed by the basic formulation. The equilibrium error is evaluated when the convergence error reaches below 10−7.