Clustering-based Reduced-Order Modeling of Heterogeneous Materials
4.3 Clustered Lippmann-Schwinger integral equation
4.3.2 Infinitesimal strains
Continuous Lippmann-Schwinger integral equation
Liu and coworkers (Liu et al., 2016) proposed the original SCA’s formulation under infinitesimal strains by postulating the continuous Lippmann-Schwinger integral equation as
εµ(Y)= − Z
Ωµ,0
Φ0(Y−Y0) :σ∗µ(Y0)dv0+ε0µ, ∀Y∈Ωµ,0, (4.58)
8In the particular case of the SCA’s formulation, the mechanical behavior similarity between different domain points is based on the fourth-order local elastic strain concentration tensor (see Appendix C).
that, attending to the definition of the eigenstress field (see Equation (4.24)), can be conveniently expanded as
εµ(Y)= − Z
Ωµ,0
Φ0(Y−Y0) :³
σµ(Y0)−De,0:εµ(Y0)´
dv0+ε0µ, ∀Y∈Ωµ,0. (4.59)
Several comments are in order to complement the original publication:
• Homogeneous field strain. In comparison with Equation (4.28), the homogeneous far-field strain,ε0µ, has been introduced as a replacement for the macro-scale strain tensor, ε(X), and assumed as an additional unknown of the equilibrium problem. Such tensor is here interpreted as a convenient numerical artifact to enforce general macro-scale strain and/or stress tensors formulated through homogenization-based constraints as
1 vµ
Z
Ωµ,0εµ(Y)dv=ε(X), ∀Y∈Ωµ,0, (4.60) and
1 vµ
Z
Ωµ,0
σµ(Y)dv=σ(X), ∀Y ∈Ωµ,0. (4.61)
It is numerically observed that the homogeneous far-field strain tensor,ε0µ, recovers the macro-scale strain tensor, ε(X), in the most common case of a pure strain homogenization-based constraint. This observation contrasts with the original paper (Liu et al., 2016), where such recovery is presented as a consequence of adopting a self-consistent scheme to set the reference material Lamé parameters. In the more general case of mixed strain and stress constraints, the prescribed macro-scale strain tensor components are numerically recovered as well;
• Green operator singularity. It should be noticed that the explicit form of the Green operator derived in Equation (4.21) has a singularity at ζ =0. Such singularity is circumvented in the FFT-based homogenization basic scheme (Moulinec and Suquet, 1994) (see Appendix B) because the macro-scale strain tensor is directly enforced at the zero frequency of the micro-scale strain field, i.e., ˘εµ(0)=ε(X) (see Equation (4.27)).
Given that in the SCA’s formulation the macro-scale constraints are explicitly enforced in the spatial domain by Equations (4.60) and (4.61), the zero frequency of the Green operator should be set to zero, i.e., ˘Φ0(ζ=0)=0. Such enforcement seems to numerically guarantee that the homogeneous far-field strain tensor, ε0µ, recovers the macro-scale strain tensor,ε(X), as expected from the Lippmann-Schwinger integral equation derived in Equation (4.28);
• Extension to constitutive nonlinearity. The derivation of the Lippmann-Schwinger integral equation described in Section 4.2.1 assumed linear elastic phases. The extension to the case where the material phases have a general nonlinear behavior calls for a suitable time discretization of the Lippmann-Schwinger integral equilibrium equation and can be achieved by means of an implicit scheme based on approximated incremental constitutive functions (see Section 2.8). The resulting incremental Lippmann-Schwinger integral equation is established in the following section.
Incremental Lippmann-Schwinger integral equation
Considering the general (pseudo-)time increment £
tm,tm+1¤
, the incremental Lippmann-Schwinger integral equilibrium equation can be written as
εµ,m+1(Y)= − Z
Ωµ,0
Φ0(Y−Y0) :³
σˆµ,m+1(Y0)−De,0:εµ,m+1(Y0)´
dv0+ε0µ,m+1, ∀Y ∈Ωµ,0, (4.62) as well as the macro-scale constraints,
1 vµ
Z
Ωµ,0εµ,m+1(Y)dv=εm+1(X), ∀Y∈Ωµ,0, (4.63) and
1 vµ
Z
Ωµ,0
σˆµ,m+1(Y)dv=σm+1(X), ∀Y∈Ωµ,0, (4.64)
where ˆσµdenotes the incremental constitutive function for the Cauchy stress tensor such that σµ,m+1=σˆµ(εµ,m+1,βm)9.
Furthermore, given the additive nature of the infinitesimal strain tensor,
∆εµ,m+1(Y)=εµ,m+1(Y)−εµ,m(Y), (4.65) and of the Cauchy stress tensor,
∆σµ,m+1(Y)=σµ,m+1(Y)−σµ,m(Y), (4.66) it is possible to rewrite the equilibrium formulation having the incremental infinitesimal strain tensor as the primary unknown. Substitution of the right-hand side terms of Equation (4.65) by the corresponding (pseudo-)time Lippmann-Schwinger integral equilibrium equation (Equation (4.62)) and simple algebraic operations10yield
∆εµ,m+1(Y)=
− Z
Ωµ,0
Φ0(Y−Y0) :³
∆σˆµ,m+1(Y0)−De,0:∆εµ,m+1(Y0)´
dv0+∆ε0µ,m+1, ∀Y∈Ωµ,0, (4.67) with the macro-scale constraints formulated accordingly as
1 vµ
Z
Ωµ,0
∆εµ,m+1(Y)dv=∆εm+1(X), ∀Y ∈Ωµ,0, (4.68)
and
9The explicit dependence of the Cauchy stress tensor incremental constitutive function, ˆσµ, on the infinitesimal strain tensor,εµ,m+1, and on the set of internal variables,βm, is often omitted hereafter for the sake of compactness.
10Due to the fact that the Lippmann-Schwinger integral equilibrium equation is formulated in the reference configuration, the volume integrals stemming fromtmandtm+1can be conveniently added.
1 vµ
Z
Ωµ,0
∆σˆµ,m+1(Y)dv=∆σm+1(X), ∀Y∈Ωµ,0. (4.69)
Clustered Lippmann-Schwinger integral equation
In order to formulate the reduced equilibrium problem, the incremental Lippmann-Schwinger integral equation (see Equation (4.62)) can now be averaged over each material clusterIof the CRVE as
1 vµ(I)
Z
Ω(I)µ,0εµ,m+1(Y)dv= − 1 v(I)µ
Z
Ω(I)µ,0
"Z
Ωµ,0Φ0(Y−Y0) :
³σˆµ,m+1(Y0)−De,0:εµ,m+1(Y0)´ dv0
# dv+ 1
vµ(I) Z
Ω(I)µ,0ε0µ,m+1dv, I=1,2,...,nc. (4.70) Attending to the relation established in Equation (4.57), the previous equation can be rewritten as
1 f(I)vµ
Z
Ωµ,0
χ(I)(Y)εµ,m+1(Y)dv= − 1 f(I)vµ
Z
Ωµ,0
"Z
Ωµ,0
χ(I)(Y)Φ0(Y−Y0) :
³σˆµ,m+1(Y0)−De,0:εµ,m+1(Y0)´ dv0
#
dv+ε0µ,m+1, I=1,2,...,nc, (4.71)
wheref(I)denotes the volume fraction of theIth cluster,
f(I)=v(I)µ
vµ
. (4.72)
In addition, the cluster piecewise uniform assumption established in Equation (4.56) yields εµ,m+1(Y0)=
nc
X
J=1
χ(J)(Y0)ε(J)µ,m+1, (4.73)
σµ,m+1(Y0)=
nc
X
J=1
χ(J)(Y0)σ(J)µ,m+1, (4.74)
which after substitution in Equation (4.71) renders 1
f(I)vµ
Z
Ωµ,0
χ(I)(Y)εµ,m+1(Y)dv= − 1 f(I)vµ
Z
Ωµ,0
"Z
Ωµ,0
χ(I)(Y)Φ0(Y−Y0) :
"n Xc J=1
χ(J)(Y0)³
σˆ(J)µ,m+1−De,0:ε(J)µ,m+1´#
dv0
#
dv+ε0µ,m+1, I=1,2,...,nc. (4.75) Finally, by performing some rearrangements and noticing that the first term is actually the homogeneous incremental strain in theIth cluster,
1 f(I)vµ
Z
Ωµ,0χ(I)(Y)εµ,m+1(Y)dv=ε(I)µ,m+1, I=1,2,...,nc, (4.76)
the clustered incremental Lippmann-Schwinger integral equilibrium equation can be written as
ε(I)µ,m+1= −
nc
X
J=1
à 1 f(I)vµ
Z
Ωµ,0
Z
Ωµ,0χ(I)(Y)χ(J)(Y0)Φ0(Y−Y0)dv0dv
! :
³σˆ(J)µ,m+1−De,0:ε(J)µ,m+1´
+ε0µ,m+1, I=1,2,...,nc, (4.77)
or in a more compact way as
ε(I)µ,m+1= −
nc
X
J=1
T(I)(J):³
σˆ(J)µ,m+1−De,0:ε(J)µ,m+1´
+ε0µ,m+1, I=1,2,...,nc, (4.78)
where the so-called cluster interaction tensors are defined as
T(I)(J)= 1 f(I)vµ
Z
Ωµ,0
Z
Ωµ,0
χ(I)(Y)χ(J)(Y0)Φ0(Y−Y0)dv0dv, I,J=1,2,...,nc. (4.79)
Recalling the physical meaning of the reference homogeneous material Green operator introduced in Section 4.2.1, it transpires from the previous definition that the fourth-order interaction tensorT(I)(J)physically represents the influence of the stress in theJth cluster on the strain in theIth cluster, i.e., describes the strain-stress interaction between clustersIandJ.
Concerning the discretization of the macro-scale constraints (Equations (4.63) and (4.64)), substitution of Equations (4.73) and (4.74), and the use of the cluster piecewise uniform assumption (see Equation (4.57)) yields
nc
X
I=1
f(I)ε(I)µ,m+1=εm+1(X), (4.80)
and
nc
X
I=1
f(I)σˆ(I)µ,m+1=σm+1(X). (4.81)
In summary, collecting Equations (4.78), (4.80) and (4.81), the reduced micro-scale equilibrium problem consists of the solution of a system of nonlinear equations composed of (i)ncclustered Lippmann-Schwinger integral equilibrium equations and (ii) macro-scale strain and/or stress constraints. The clustered Lippmann-Schwinger system of equilibrium equations can then be established as
ε(1)µ,m+1= −
nc
X
J=1
T(1)(J): µ
σˆ(J)µ,m+1³
ε(J)µ,m+1,βm
´−De,0:ε(J)µ,m+1
¶
+ε0µ,m+1,
ε(2)µ,m+1= −
nc
X
J=1
T(2)(J): µ
σˆ(J)µ,m+1³
ε(J)µ,m+1,βm
´−De,0:ε(J)µ,m+1
¶
+ε0µ,m+1, ...
ε(nµ,mc)+1= −
nc
X
J=1
T(nc)(J): µ
σˆ(J)µ,m+1³
ε(J)µ,m+1,βm´
−De,0:ε(J)µ,m+1
¶
+ε0µ,n+1,
nc
X
I=1
f(I)ε(I)µ,m+1=εm+1(X) ; Xnc
I=1
f(I)σˆ(I)µ,m+1³
ε(I)µ,m+1,βm´
=σm+1(X),
(4.82)
and must be solved for the unknowns εµ,m+1=n
ε(1)µ,m+1,ε(2)µ,m+1, ...,ε(nµ,m+1c) ,ε0µ,m+1o
. (4.83)
By performing the same steps over the additive formulation established in Equations (4.67), (4.68) and (4.69), the clustered incremental Lippmann-Schwinger integral equilibrium equation and the macro-scale constraints come
∆ε(I)µ,m+1= −
nc
X
J=1
T(I)(J):³
∆σˆ(J)µ,m+1−De,0:∆ε(J)µ,m+1´
+∆ε0µ,m+1, I=1,2,...,nc, (4.84)
nc
X
I=1
f(I)∆ε(I)µ,m+1=∆εm+1(X), (4.85)
and
nc
X
I=1
f(I)∆σˆ(I)µ,m+1=∆σm+1(X). (4.86)
For the sake of completeness, the corresponding clustered Lippmann-Schwinger system of equilibrium equations can then be alternatively established as
∆ε(1)µ,m+1= −
nc
X
J=1
T(1)(J): µ
∆σˆ(J)µ,m+1³
∆ε(J)µ,m+1,βm
´−De,0:∆ε(J)µ,m+1
¶
+∆ε0µ,m+1,
∆ε(2)µ,m+1= −
nc
X
J=1
T(2)(J): µ
∆σˆ(J)µ,m+1³
∆ε(J)µ,m+1,βm
´−De,0:∆ε(J)µ,m+1
¶
+∆ε0µ,m+1, ...
∆ε(nµ,mc)+1= −
nc
X
J=1
T(nc)(J): µ
∆σˆ(J)µ,m+1³
∆ε(J)µ,m+1,βm´
−De,0:∆ε(J)µ,m+1
¶
+∆ε0µ,n+1,
nc
X
I=1
f(I)∆ε(I)µ,m+1=∆εm+1(X) ; Xnc
I=1
f(I)∆σˆ(I)µ,m+1³
∆ε(I)µ,m+1,βm
´=∆σm+1(X),
(4.87)
and must be then solved for the unknowns
∆εµ,m+1=n
∆ε(1)µ,m+1,∆ε(2)µ,m+1, ...,∆ε(nµ,m+1c) ,∆ε0µ,m+1o
. (4.88)
Incremental reference homogeneous material properties
In order to derive the Lippmann-Schwinger integral equation in Section 4.2.1, a reference (fictitious) homogeneous linear elastic material is conveniently introduced into the constitutive formulation. In addition, this reference material is assumed isotropic (see Equation (4.18)) so that a closed-form of the Green operator can be derived in the frequency domain. However, it should be noticed that so far, (i) the reference material Lamé parameters, (λ0,µ0), have been assumed constant throughout the deformation path and (ii) a suitable choice of such parameters has not been discussed yet.
As pointed out by Liu and coworkers (Liu et al., 2016) and recently addressed by Schneider (Schneider, 2019), the solution of the continuous Lippmann-Schwinger integral equilibrium equation does not depend on the choice of the reference material Lamé parameters.
Nonetheless, it should be remarked that such a choice plays a major role in the convergence rate of several FFT-based homogenization methods and has been a topic of extensive research over the last years. In contrast, and most importantly, the solution of the clustered Lippmann-Schwinger integral equilibrium equation does actually depend on the reference material’s properties (Liu et al. (2016), Schneider (2019)). Despite the recent progress in understanding polarization schemes popular in FFT-based homogenization methods, the mathematical quantification of such dependency still remains an open and crucial challenge to the best of the author’s knowledge.
Although not addressing this mathematical quantification, Liu and coworkers (Liu et al., 2016) propose a well-known self-consistent micromechanical approach (see Appendix A) to find an ‘optimal’ choice of the reference material Lamé parameters (see details in Appendix C).
As in the self-consistent method of classical micromechanics, the idea consists of setting, at each instanttm, the homogeneous tangent modulus of the reference material,De,0m , that best approximates the effective tangent modulus of the CRVE, Dm. Rather than discussing the effectiveness of such a strategy at this point, the aim here is instead to evaluate the impact of an incremental update of the reference homogeneous material properties in the previous derivations.
In the first place, it is important to clearly identify the actual mathematical sources of influence of the reference material’s properties on the incremental Lippmann-Schwinger integral equilibrium equation (see Equation (4.62)): (1) the Green operator (see Equation (4.21)); and (2) the polarization stress, σ∗µ, by means of the reference material homogeneous tangent modulus,De,0 (see Equation (4.18)). Note that these sources are later transferred to the clustered Lippmann-Schwinger integral equilibrium equation (see Equation (4.78)) by means of the cluster interaction tensors,T(I)(J), and the cluster polarization stress, (σ∗µ)(J), respectively.
With this in mind, the incremental Lippmann-Schwinger integral equilibrium equation can now be written as
εµ,m+1(Y)= − Z
Ωµ,0
Φ0m+1(Y−Y0) :³
σˆµ,m+1(Y0)−De,0m+1:εµ,m+1(Y0)´
dv0+ε0µ,m+1,
∀Y ∈Ωµ,0, (4.89) where the subscript (•)m+1has been included in the identified sources to denote the incremental
nature of the reference material’s properties, and its clustered counterpart as ε(I)µ,m+1= −
nc
X
J=1
T(I)(J)m+1:³
σˆ(J)µ,m+1−De,0m+1:ε(J)µ,m+1´
+ε0µ,m+1, I=1,2,...,nc. (4.90) Given the additive nature of the infinitesimal strain tensor and the Cauchy stress tensor (see Equations (4.65) and (4.66)), the additive incremental Lippmann-Schwinger integral equilibrium equation comes
∆εµ,m+1(Y)= − Z
Ωµ,0
Φ0m+1(Y−Y0) :³
∆σˆµ,m+1(Y0)−De,0m+1:∆εµ,m+1(Y0)´
dv0+∆ε0µ,m+1
− Z
Ωµ,0
³Φ0m+1(Y−Y0)−Φ0m(Y−Y0)´
: ˆσµ,m(Y0)
−³
Φ0m+1(Y−Y0) :De,0m+1−Φ0m(Y−Y0) :De,0m´:εµ,m(Y0)dv0, ∀Y∈Ωµ,0. (4.91) The comparison between this result and the previously derived Equation (4.67) reveals an additional term intrinsically related to the incremental update of the reference material’s properties. In the particular case of a constant reference material throughout the deformation path, i.e.,Φ0=Φ0m+1=Φ0mandDe,0=De,0m+1=De,0m , Equation (4.67) is recovered as expected.
Moreover, by performing the same cluster averaging procedures that led to Equation (4.78), the clustered incremental Lippmann-Schwinger integral equilibrium equation comes
∆ε(I)µ,m+1= −
nc
X
J=1
T(I)(J)m+1:³
∆σˆ(J)µ,m+1−De,0m+1:∆ε(J)µ,m+1´
+∆ε0µ,m+1
+
nc
X
J=1
T(I)(J)m+1:³
σˆ(J)µ,m−De,0m+1:ε(J)µ,m´
−
nc
X
J=1
T(I)(J)m :³
σˆ(J)µ,m−De,0m :ε(J)µ,m´
, I=1,2,...,nc.
(4.92)
Remark.
Despite adopting a self-consistent scheme to perform the incremental update of the reference material Lamé parameters, Liu and coworkers (Liu et al., 2016) neglected such additional terms and formulated the clustered Lippmann-Schwinger system of equilibrium equations given by Equations (4.84)-(4.88) (see Appendix C). Nonetheless, provided a smooth evolution of the reference material’s properties and a sufficiently refined incrementation, it is expected that the magnitude of the additional terms is small in comparison with the remaining terms.