The FOSO variational method of Ponte Casta˜neda (2015) makes use of the effective behavior of an appropriately chosen porous “linear comparison composite” (LCC)—
with uniform matrix properties and microstructure identical to that of the nonlinear
porous single crystal of interest—to estimate the effective behavior of the nonlinear composite. To set the stage for the FOSO estimates, we first describe the porous LCC involved in the FOSO procedure.
For the class of porous single crystals defined in section6.2, consider a porous LCC with the same microstructure as the nonlinear porous single crystal (described by the microstructural variables (3.4)), but with a crystal matrix (phase 1) characterized by the quadratic stress potential
u(L1)(σ) = 1
2σ⋅M
(1)σ+η(1)⋅σ, (3.6)
where M(1) and η(1) are uniform, anisotropic fourth- and second-order tensors, re-
spectively, corresponding to the viscous compliance tensor and eigenstrain-rate tensor of the crystal matrix, which are defined by
M(1)= K ∑ k=1 1 2µ(k) µ(k)⊗µ(k), and η(1)= K ∑ k=1 η(k) µ(k). (3.7)
Here the scalarsµ(k)andη(k)(k=1, ..., K) are respectively the positive slip viscosities
and slip eigenstrain rates, which are unknown a priori but will be specified later. Dif- ferentiation of the stress potential (3.6) with respect toσ shows that the constitutive relation of the LCC matrix is linear, i.e.,
D=M(1)σ+η(1). (3.8)
Note that the constitutive relation (3.8) for the matrix of the LCC, corresponding to a linearly viscous material with prescribed eigenstrain rates, is mathematically
analogous to that for a ‘thermoelastic’ material. On the other hand, the stress poten- tial for the vacuous inclusion (phase 2) is also assumed to be of the form (3.6), but with a viscous compliance tensor M(2) → ∞ and an eigenstrain-rate tensor η(2)=0.
Then, for any prescribed macroscopic stress σ, the effective stress potential ũL for
the porous LCC may be estimated by means of the estimates ofPonte Casta˜neda and Willis (1995) (to be referred to here as PCW estimates), and is given by
̃ uL(σ) = 1 2σ⋅ ̃ Mσ+η̃⋅σ. (3.9)
In the above expression,M̃ and̃ηare the effective compliance tensor and the effective
eigenstrain-rate tensor of the porous LCC, respectively. They are given by
̃
M=M(1)+ f
1−fQ
−1, and ̃η=η(1). (3.10)
Here Q is a fourth-order microstructural tensor related to the Eshelby tensor, de- pending on the matrix propertyM(1)and the shape and orientation of the voids, and
is given by Q= 1 4πw1w2 ∫ ∣ζ∣=1 (M(1))−1−(M(1))−1H(ζ) (M(1))−1 ∣Z−1ζ∣3 dS, (3.11)
where ζ is a unit vector and Hijkl = Kik−1ζjζl∣(ij)(kl) (the parentheses denote sym-
metrization with respect to the corresponding indices), with Kik = (M(1))
−1
imknζmζn
denoting the acoustic tensor. The symmetric second-order tensor Z serves to char- acterize the shape and orientation of the voids (and their distribution), and can be written in the form
Z=w1n1⊗n1+w2n2⊗n2+n3⊗n3. (3.12)
Correspondingly, the first and second moments of the stress field over the matrix of the porous LCC, which are required by the FOSO method, are determined by
σ(1)= 1 1−fσ, and ⟨σ⊗σ⟩ (1)= 2 1−f ∂̃uL ∂M(1). (3.13)
Following the work of Ponte Casta˜neda (2015), the FOSO estimates for the effective stress potential ˜u(as defined by (3.5)) of the nonlinear porous single crystals are given by ̃ uSO(σ) = (1−f) K ∑ k=1 [αφ(k)(τˇ(k))+(1−α)φ(k)(τˆ(k))], (3.14)
where α is an appropriately chosen constant weight factor between 0 and 1, the ˇ
τ(k) and ˆτ(k) (k=1, ..., K) are stress variables depending on both the first and second
moments of the stress field over the matrix of the porous LCC (see expression (3.13)). They satisfy the relations (Ponte Casta˜neda,2015)
αˇτ(k)+(1−α)τˆ(k)=σ(1)⋅µ(k)=τ(k), (3.15)
and
α(τˇ(k))2+(1−α) (τˆ(k))2=µ
(k)⋅⟨σ⊗σ⟩(1)µ(k)=τ(k), (3.16)
whereτ(k)and τ(k) correspond to the first and second moments of the resolved shear
stresses over the kth slip system of the LCC matrix. More specifically, ˇτ(k) and ˆτ(k)
are chosen to be such that ˇτ(k)≤τˆ(k) and can be easily obtained by solving the set of
quadratic equations (3.15) and (3.16), so that
ˇ τ(k)=τ(k)− √ 1−α α √ τ(k)−τ2(k)=τ(k)− √ 1−α α SD (1)(τ (k)) (3.17) and ˆ τ(k)=τ(k)+ √ α 1−α √ τ(k)−τ2(k)=τ(k)+ √ α 1−αSD (1)(τ (k)), (3.18) with SD(1)(τ(k)) = √
τ(k)−τ2(k)denoting the standard deviation of the resolved shear
stresses over slip system k in the LCC matrix. In this connection, it should be emphasized that the quantities ˇτ(k) and ˆτ(k)—depending on both the first and second
moments of the stress field over the LCC matrix—are functions of the properties of the LCC, as determined by the variables µ(k) and η(k) in (3.7).
viscosities µ(k) and the slip eigenstrain rates η(k) satisfy the linearization conditions φ′(k)(τˆ(k))− 1 2µ(k) ˆ τ(k)=η(k)=φ′(k)(τˇ(k))− 1 2µ(k) ˇ τ(k). (3.19)
Note that these two conditions imply that 1 2µ(k) = φ′ (k)(ˆτ(k))−φ′(k)(τˇ(k)) ˆ τ(k)−τˇ(k) , (3.20)
which identifies the slip viscosities µ(k) of the LCC matrix with ‘generalized secant’
linearizations of the nonlinear slip potentials for the viscoplastic single-crystal matrix, accounting for both the first and second moments of the stress field in the crystal matrix. (Note further that this expression reduces to the ‘tangent’ linearization when there are no field fluctuations in the phase and the ˆτ(k)→τˇ(k).) Expressions (3.17)-
(3.19) provide a system of 4K nonlinear algebraic equations for the variables ˇτ(k),
ˆ
τ(k),µ(k) and η(k) (k=1, .., K), which can be easily solved numerically.
As already mentioned, the macroscopic constitutive behavior and the correspond- ing field statistics (e.g., the first and second moments of the stress and strain rate fields) of the nonlinear porous single crystals can be obtained directly from those of the porous LCC (Ponte Casta˜neda,2015). (This follows from the full stationarity of the FOSO estimates in the properties of the LCC, together with the results of Idiart and Ponte Casta˜neda (2007c), and is independent of the choice of the weights α.) In particular, the FOSO estimate for the macroscopic strain rate D of the nonlinear porous single crystal under the applied loading σ is given by
D= ∂ ̃ uSO ∂σ (σ) = ∂̃uL ∂σ (σ) = ̃Mσ+η̃, (3.21)
where it is recalled thatM̃ and ̃ηare the effective compliance tensor and the effective
eigenstrain-rate tensor of the porous LCC, respectively, which are given by (3.10). We should emphasize that M̃ and ̃η depend nonlinearly on σ and, therefore, the
macroscopic constitutive relation (3.21) is also nonlinear, as expected.
discussed in this section are valid for any choice of the weights α. Unfortunately, at the present time there is no mathematically or physically motivated prescription available to select these weights in ‘optimal’ fashion. One alternative possibility would be to use these weights to try to fit our theoretical predictions to the results of numerical simulations or experimental results, but—in this first application of the method for porous single crystals—we choose not to pursue this option, preferring instead to make use of the simplest choice α=1/2 (see below).
On the other hand, the average strain rate and spin fields in the voids, which are useful for establishing the complementary equations for the evolution of the mi- crostructure, can be consistently obtained from the PCW estimates for the corre- sponding fields in the LCC. In particular, the average strain rate in the voids D(2)
may be expressed in terms of the macroscopic strain rate D as
D(2)=A(2)D+a(2), (3.22)
where A(2) and a(2) are the associated strain-rate concentration tensors given by
A(2)= [fI+(1−f)M(1)Q]−1, (3.23)
and
a(2)=−(1−f)A(2)(I−M(1)Q)η(1), (3.24)
with Idenoting the fully symmetric fourth-order identity tensor. Similarly, the aver- age spin in the voids can be written in terms of the macroscopic strain rate D and the macroscopic spin W as
W(2)=W−C(2)D−β(2), (3.25)
where C(2) and β(2)
are the associated spin-concentration tensors provided by
and
β(2)=−(1−f)Π(a(2)−η(1)). (3.27)
HereΠis the fourth-order Eshelby rotation tensor determining the spin of an isolated void in an infinite linearly viscous matrix, and is given by
Π= 1 4πw1w2 ∫ ∣ζ∣=1 ˆ H(ζ) (M(1))−1 ∣Z−1ζ∣3 dS, (3.28)
with ˆHijkl =Kik−1ζjζl∣[ij](kl) (the square bracket denotes the skew symmetric part of
the first two indices, while the round bracket denotes the symmetric part of the last two indices). We recall that Kik = (M(1))
−1
imknζmζn is the acoustic tensor, and Z as
given by (3.12) is a symmetric second-order tensor describing the instantaneous shape and orientation of the voids (and their distribution).
Next, it is necessary to determine the average slip rates γ(k) (k = 1, ..., K) over
different slip systems in the single-crystal matrix. Letting D(1) denote the average strain rate in the crystal matrix, they will be required here to satisfy the relation
D(1)= K ∑ k=1
γ(k)µ(k). (3.29)
Note that D(1) can be directly estimated from the constitutive relation (3.8) for the LCC matrix, i.e., D(1)=M(1)σ(1)+η(1)= K ∑ k=1 ( 1 2µ(k) τ(k)+η(k))µ(k), (3.30)
or, using (3.7), (3.15) and (3.19), from the expression
D(1)= K ∑ k=1 [αφ′ (k)(τˇ(k))+(1−α)φ′(k)(τˆ(k))]µ(k). (3.31)
'generalized secant' linearization
nonlinear constitutive relation
'secant' linearization
Figure 3.2: The ‘generalized secant’ linearization (3.20) and the ‘secant’ linearization (3.36) of the nonlinear constitutive response of the viscoplastic single crystals. The evaluation of the corresponding average slip rate γ(k) is also shown in the figure. expressed in the equivalent forms
γ(k)=
1 2µ(k)
τ(k)+η(k)=αφ′(k)(τˇ(k))+(1−α)φ′(k)(τˆ(k)). (3.32)
Note that γ(k) ≠2D(1)⋅µ(k), except when the Schmid tensors µ(k) are orthogonal
to each other for the crystal matrix. Thus, the γ(k) (k =1, ..., K) can be estimated
directly from the LCC, whose properties are determined by (3.19) (or, equivalently, the ‘generalized secant’ condition (3.20)). For visualization purposes, the ‘generalized secant’ condition (3.20) and the evaluation of the corresponding γ(k) from expression (3.32) is depicted graphically in Fig. 3.2. In particular, it can be seen that the ‘generalized secant’ condition provides a linear interpolation between the slip rates ˇ
γ(k) = φ′(k)(τˇ(k)) and ˆγ(k) = φ′(k)(τˆ(k)), associated with the resolved shear stresses
ˇ
τ(k) and ˆτ(k), respectively. More importantly, the average slip rates γ(k) are seen
in Fig. 3.2 to be related to the average resolved shear stresses τ(k)—lying between
ˇ
constitutive relation, i.e.,
γ(k)≠φ′(k)(τ(k)). (3.33)
Note further that, due to the nonlinear stress-strain rate response of the single-crystal matrix, the inequality (3.33) is to be expected. This is because the average of a nonlinear function is generally different from the function of the average. For this reason, the fact that the FOSO estimates (3.32) for γ(k) are entirely consistent with (3.33) is a distinguishing feature of the FOSO method in comparison to the earlier second-order estimates of Liu and Ponte Casta˜neda (2004a), which involves the use of an “affine” approximation of the average slip rates, i.e., γ(k)=φ′(k)(τ(k)), violating
the general expectation of relation (3.33). Finally, note that the weight factor α will be chosen to be 1/2, which is the most symmetric choice, although there may be other
better choices for the value ofα. This is a point that will require further investigation in future works.
In this context, we should mention that the FOSO estimate (3.14) for ̃u is a
generalization of the variational homogenization (VH) estimate ofdeBotton and Ponte Casta˜neda(1995), which may be recovered from the FOSO estimate (3.14) by formally setting the eigenstrain-rate tensorη(1)=0. In its final form, the VH estimate for the
effective stress potential ̃u of the nonlinear porous single crystal can be written as
̃ uVH(σ) = (1−f) K ∑ k=1 φ(k)(τˆ(k)). (3.34)
Here the stress variables ˆτ(k) depend only on the second moments of the resolved
shear stresses over the LCC matrix, which are given by
ˆ τ(k)= √ τ(k)= √ µ(k)⋅⟨σ⊗σ⟩(1)µ (k), (3.35)
where the second moment⟨σ⊗σ⟩(1)may be obtained from the PCW estimates(3.13)2
with the η(1) = 0. In turn, the slip viscosities µ
(k) in the porous LCC are given by
the conditions 1 2µ(k) = φ′ (k)(τˆ(k)) ˆ τ(k) . (3.36)
Note that (3.36) identifies the slip viscosities µ(k) of the LCC matrix with ‘secant’
linearizations of the nonlinear constitutive response for the corresponding slip systems in the viscoplastic single-crystal matrix (see the dashed straight line in Fig. 3.2), accounting for the second moment of the stress field in the crystal matrix. Also note that the VH estimates for the average slip rates γ(k) over different slip systems can be obtained directly from the LCC, i.e., γ(k)=τ(k)/ (2µ(k)) ≠φ′(k)(τ(k)).
The FOSO estimates (3.14) are known to be exact to second order in the hetero- geneity contrast (Ponte Casta˜neda, 2015), when used in combination with estimates for the LCC that are exact to second order in the heterogeneity contrast (e.g., the PCW estimates used in this work). For this reason, they are more accurate then the VH estimates (3.34), which are only exact to first order in the heterogeneity contrast. In fact, the VH estimate can be shown to be a rigorous lower bound for all other estimates for the effective stress potential ̃u of the porous single crystals (deBotton and Ponte Casta˜neda,1995). Although the FOSO estimates (3.14) provide fairly ac- curate estimates in most cases, they become less accurate for low porosity and high nonlinearity, especially at high stress triaxialities. As already noted in the context of the VH method (Agoras and Ponte Casta˜neda,2013), this drawback is attributed to the assumption, employed in their derivation, that the matrix property in the LCC is uniform. For this reason, in the next section we develop improved estimates by incorporating non-uniform properties of the matrix phase in the LCC.