Geometry Definition
The hexahedral volume elements used in this study have an edge length of \(250\,\mu \hbox {m}\) and are discretized into \(100\times 100\times 100\) points of which 2.7% pertain to the void. To investigate the influence of crystallographic orientation, three different orientations are assigned to the matrix material: cube, rotated cube, and P. In Bunge Euler notation (\(\varphi _1\), \(\varPhi \), \(\varphi _2\)) these orientations read in degree (0.0, 0.0, 0.0), (45.0, 0.0, 0.0), and (54.7, 45.0, 0.0).
Realistic Pore The shape of the investigated pore is obtained using a Zeiss Xradia 520 Versa x-ray \(\mu \)-CT. With a resolution of up to \(1\,\mu \hbox {m}\), this \(\mu \)-CT enables an accurate reconstruction of internal defects in the investigated Fe-rich Al-Si-Cu alloy based on a frontal scanning of the rotating sample. Details on material processing, chemical composition, and experimental procedure are published elsewhere.13 After reconstruction of the initial microstructure (Fig. 1a), the following post-processing steps have been applied using the Python package scikit-image:31 (1) denoising using a non-local means filter, Fig. 1b; (2) placing markers based on a threshold, Fig. 1c; (3) segmenting with a random walker algorithm,32 Fig. 1d; and (4) removing all features consisting of less than 5000 voxel.
Finally, a \(100\times 100\times 100\) voxel subvolume containing one complex-shaped pore comprised out of 26,969 points (Fig. 2a) was extracted for the following simulations using a realistic pore shape. The complex shape of the pore, with many ramified branches consisting of alternating convex and concave regions, is representative for shrinkage pores found in the investigated material class.13,14 As a measure inspired by the surface-to-volume ratio that allows quantifying the complexity of a given shape, we calculated the ratio of total voxels in the pore to voxels located at the surface of the pore as 0.675.
Idealized Pore To investigate the influence of the pore geometry, a second volume element containing a pore with idealized shape is created. To this end, a spherical inclusion with diameter of 37 points (26,745 points in the volume) is placed in the center of the volume element (Fig. 2b). The shape of this pore resembles convex-shaped gas pores that result from polymer degassing.14 The ratio of total voxels in this sphere-shaped pore to voxels located at its surface is 0.243, i.e. significantly smaller than in the case of the realistic experimentally measured pore.
Model Parameters
For both, the crystal plasticity as well as the damage model, parameters need to be selected. Although we aim at obtaining qualitative results, where the exact values of the parameters for the constitutive models are of minor interest, their origin is briefly given in the following paragraphs for the sake of completeness.
Crystal Plasticity Constitutive Model The parameters used for the crystal plasticity constitutive model are based on an existing dataset that reproduces the behavior of a soft aluminum alloy.22 Since obtaining suitable parameters for the employed phenomenological model33,34 or the development of a physics-based model35,36,37 are challenging tasks by themselves, the existing parameters (except for the elastic constants38) are manually adjusted to increase yield point and hardening. The parameters used are given in Table I and the resulting stress–strain curve (including damage) can
be seen in Fig. 3b.
Table I Values for the elastic and plastic response of the aluminum matrix
Isotropic elasto-plastic behavior (without additional damage mechanisms) is assumed inside the pore as it cannot be taken out of the computation domain as in, e.g., finite element simulations. Low elastic stiffness (\(C_{11}={10.0}\hbox { MPa}\), \(C_{12}={6.7}\hbox { MPa}\)) and initial and final yield strength of 10.0 MPa and 63.0 MPa, respectively, ensure that the residual mechanical response of the pore does not influence the simulation results.Footnote 2
Phase Field Fracture Model The damage model is parameterized in terms of the fracture surface energy, \(g_0\), characteristic length scale, \(l_0\), and mobility M. The characteristic length scale \(l_0\) was set to the length of two voxels and the residual stiffness for \(\varphi =0\) was set to 0.01% of the elastic stiffness tensor in the undamaged bulk material. A mobility of \(M = 0.01\) is used, which is reasonable for the strain rates used in the present work. The fracture surface energy was adjusted such that the macroscopic stress–strain curve of a polycrystalline material shows the expected behavior of failure at the later stage of the plastic deformation. To this end a simple grain structure of 60 grains with randomly selected orientations created by a standard Voronoi tessellation approach was created. The stress–strain curve for uniaxial loading in x-direction and the corresponding crack surface for the finally selected value of \(g_0/l_0 = {5.0}\,\hbox {MN}\,\hbox {m}^{-1}\) are shown in Fig. 3.
Loading
Uniaxial tension along x-direction is applied at a rate of \(1 \times 10^{-3} \hbox { s}^{-1}\) with a time step of 0.0025 s. For the cube orientation, this corresponds to loading along the [1 0 0] direction, for the rotated cube orientation to loading along the [1 1 0] direction and for the P orientation along the [1 1 1] direction.