In this work, porous single crystals are idealized as two-phase materials with vacuous inclusions (phase 2) embedded in the single-crystal matrix (phase 1). The single- crystal matrix is assumed to have a crystal lattice defined by means of three linearly
independent crystallographic axes l1, l2 and l3 (see Fig. 3.1). The lattice can be
completely general, so that l1, l2 and l3 are not necessarily mutually orthogonal or have the same length. There are two main deformation mechanisms for the single- crystal matrix: (i) the elastic distortion of the atomic lattice and (ii) the plastic deformation through the motion of dislocations. In this work, however, we mainly focus on the response of porous single crystals under large plastic deformations and, for this reason, the elastic strains, which are typically very small (of the order 10−3), will be neglected. However, lattice rotation will be accounted for. Thus, the crystal matrix is assumed to deform by dislocation glide alongKwell-defined crystallographic slip systems, and the local constitutive behavior of the single crystal is taken to be viscoplastic, and can be characterized by
D= ∂u (1)(σ) ∂σ , u (1)(σ) =∑K k=1 φ(k)(τ(k)), (3.1)
where u(1) denotes the stress potential for the crystal matrix,σ is the Cauchy stress
and D is the Eulerian strain rate. The convex functions φ(k) (k = 1, ..., K) are the
slip potentials characterizing the response of the K slip systems, and depend on the resolved shear (or Schmid) stresses
τ(k)=σ⋅µ(k), where µ(k)=
1
Here the µ(k) are the second-order Schmid tensors obtained from the symmetrized dyadic product ofn(k)andm(k), withn(k)andm(k)denoting the unit vectors normal
to the slip plane and along the slip direction of the kth slip system, respectively. Al- though more general constitutive response could be considered for the crystal matrix, for simplicity, the slip potentials φ(k) are assumed to be of the usual power-law form
φ(k)(τ) = ˙ γ0(τ0)(k) n+1 ∣ τ (τ0)(k) ∣ n+1 , k=1, ..., K, (3.3)
where ˙γ0 denotes the reference strain rate,(τ0)(k)>0 is the reference flow stress of the
kth slip system andn is the creep exponent (the inverse of the strain rate sensitivity
m = 1/n). This class of slip potentials is known to be particularly appropriate for
exploring the effect of nonlinearity and crystallographic anisotropy for a wide range of material behaviors. Note that the (τ0)(k) can be very different for different slip
systems, which may lead to strongly anisotropic behavior for the single crystal. Also note that the creep exponentncould be taken to be different for different slip systems, but here, for simplicity, it will be taken to be identical for all slip systems, such that the stress potentialu(1)for the crystal matrix is a homogeneous function of degreen+1
inσ. In particular, the two limiting cases asntends to 1 and∞are of special interest, since they respectively describe linearly viscous and rigid ideally plastic behavior for the single crystal.
As shown in Fig. 3.1, the voids are assumed—on average—to be ellipsoidal in shape, and to be aligned in a given direction, but distributed withrandompositions in the surrounding single-crystal matrix, as described by two-point probability functions (for their centers) with “ellipsoidal symmetry” (Willis, 1977; Ponte Casta˜neda and Willis, 1995). In general, the ellipsoid characterizing the void distribution can be different from the ellipsoid characterizing the voids (Ponte Casta˜neda and Willis,
1995;Agoras and Ponte Casta˜neda,2013). However, the effect of the void distribution on the macroscopic behavior of the porous single crystals is only of second order in the volume fraction of the voids (porosity) and becomes less important at low to moderate porosities. For this reason, it is further assumed that the ellipsoidal shape
Zoom in 1 l α β γ Slip system 1 n 3 n 1 a 2 a 3 a 3 e 1 e 2 e Geometrical Features of an Ellipsoid 2 l 3 l 2 n 1 l
Figure 3.1: Schematic representation of a porous single crystal consisting of aligned, ellipsoidal voids (solid lines) that are distributed with the same ellipsoidal symmetry (dotted lines) in a single-crystal matrix.
and orientation of the distribution function are identical to the ellipsoidal shape and orientation of the voids (Agoras and Ponte Casta˜neda, 2014; Song et al., 2015). In view of these hypotheses, the microstructure of the porous single crystal under consideration can be completely described by the set of microstructural variables
s≡ {l1,l2,l3, f, w1, w2,n1,n2,n3}, (3.4)
wherel1,l2 andl3 characterize the lattice vectors of the crystal matrix,f denotes the volume fraction of the voids (or porosity), w1 =a3/a1, w2 =a3/a2 are the two aspect ratios of the representative ellipsoids characterizing the shape and distribution of the voids (a1, a2 and a3 are the lengths of the three semi-axes of the ellipsoid), and n1,
n2 and n3 are unit vectors along the three principal directions of the representative ellipsoid (see Fig. 3.1). It is remarked that, among the above-defined microstructural variables (3.4),l1,l2 andl3 describe the underlying anisotropy of the crystal matrix, or the “crystallographic” anisotropy, while the others (f, w1, w2,n1,n2,n3) characterize the “morphological” anisotropy of the porous single crystal.
For a given fixed state of the microstructure, as described by the microstructural variables (3.4), the instantaneous effective viscoplastic response of the porous single crystal, characterizing the relation between the average strain rate ⟨D⟩ = D and the
1998) D=∂ ̃ u(σ) ∂σ , ̃u(σ) = (1−f)σ∈S(σ)min ⟨u(x,σ)⟩ (1). (3.5)
Here ̃u is the effective stress potential for the porous single crystal, S(σ) is the set
of statically admissible stress fields, including all σ fields that are divergence free, lead to zero traction on the void surfaces, and satisfy the condition ⟨σ⟩ = σ. The
triangular brackets ⟨⋅⟩ denote volume averages over a representative volume element
(RVE) of the porous material, while ⟨⋅⟩(r) denotes volume averages over phase r in
the RVE.
In summary, the instantaneous macroscopic response of the porous single crystals considered in this work can be completely determined by the effective stress potential
̃
u defined in (3.5). However, given the nonlinear constitutive relations of the crystal matrix and the complexity of the random microstructure, the determination of the exact values of ̃u is impossible in practice, since it requires solving sets of nonlinear
partial differential equations with randomly oscillating coefficients. In this work, ap- proximate estimates for the effective potential will be obtained by means of a novel iterative homogenization scheme, making use of the recently developed fully opti- mized second-order (FOSO) variational homogenization method (Ponte Casta˜neda,
2015). In the next section, for completeness, we first recall the main features of the FOSO method in some detail. The FOSO method is then used in an iterative fashion, following the work of Agoras and Ponte Casta˜neda (2013), to obtain new estimates for the instantaneous effective behavior of porous single crystals. Finally, consistent homogenization estimates for the average strain-rate and vorticity fields in the phases of porous single crystals are used to develop complementary evolution laws for the microstructural variables (3.4), characterizing the evolution of both the “crystallographic” and “morphological” anisotropy of the porous single crystals at finite deformations.