 Research Article
 Open Access
 Published:
On the variationally consistent computational homogenization of elasticity in the incompressible limit
Advanced Modeling and Simulation in Engineering Sciences volume 2, Article number: 1 (2015)
Abstract
Background
Computational homogenization is a wellestablished approach in material modeling with the purpose to account for strong microheterogeneity in an approximate fashion without excessive computational cost. However, the case of macroscopically incompressible response is still unresolved.
Methods
The computational framework for Variationally Consistent Homogenization (VCH) of (near) incompressible solids is discussed. A canonical formulation of the subscale problem, pertinent to a Representative Volume Element (RVE), is established, whereby complete macroscale incompressibility is obtained as the limit situation when all constituents are incompressible.
Results
Numerical results for single RVEs demonstrate the seamless character of the computational algorithm at the fully incompressible limit.
Conclusions
The suggested framework can seamlessly handle the transition from (macroscopically) compressible to incompressible response. The framework allows for the classical boundary conditions on the RVE as well as the generalized situation of weakly periodic boundary conditions.
Background
Computational homogenization is a wellestablished approach in material modeling with the purpose to account for strong microheterogeneity in an approximate fashion without excessive computational cost. Such an approach can be applied to the situation when the intrinsic material properties are linear, leading to direct “upscaling”. It can also be applied to the more complex situation when the subscale properties are nonlinear and/or the subscale problem is inherently transient, whereby it is necessary to resort to nested macrosubscale computation (FE^{2}). Since the literature on classical as well as computational homogenization is abundant, it is neither necessary nor possible to give a comprehensive account. Selected references are [13] addressing different aspects on homogenization and multiscale modeling. An important issue is the choice of boundary conditions (and data) for the subscale RVEproblem. Selected references are [48]. In particular, [8] presents a quite general framework based on “weak periodicity”. How to accommodate the coalescence of microcracks presents a special challenge, e.g. [9]. How to incorporate interfaces and thin membranes are discussed in [1012].
Despite the extensive developments, there are still fundamental issues that need to be further addressed. Among these issues we note, (i) variational consistency and the macrohomogeneity condition in the context of selective homogenization (in particular, for multifield problems) (cf. [13,14]), (ii) how to establish bounds on the effective properties within a given confidence interval, and (iii) high contrast material properties of the constituents (e.g. rigid inclusions or pores) for which the classical Dirichlet and Neumann conditions give poor results.
The particular aspect considered in this paper, which represents an unresolved issue even in the simplest case of elastic response, is that of Variationally Consistent Homogenization (VCH) in the limit of macroscopic incompressibility. This situation is encountered when the microconstituents of a composite are intrinsically incompressible (or nearly incompressible), which infers macroscale incompressibility as well. An example is when a composite of metallic particles (or fibers) embedded in an elastomer matrix is subjected to stresses that are sufficiently large to cause significant plastic deformations in the particles. Another class of problems is characterized by an initially compressible macroscale response, which may become incompressible as the result of the deformation process. An important example is the evolving porous microstructure of a PMproduct during the process of sintering, whereby the homogenized response is compressible until the porosity vanishes inferring incompressible macroscale response, cf. [15]. This process is thus characterized by a transition from the compressible to incompressible regimes.
Applying traditional deformation controlled boundary conditions to macroscopically incompressible RVEs is technically possible, see e.g. Moran et al. [16] and Öhman et al. [14]. However, in such cases, special care must then be taken in applying compatible deformations (i.e. no scaling). This leads to a singular, but solvable, RVEproblem. This approach, however, raises three significant issues: (i) FE^{2} is only possible with approximate penalty methods, (ii), the situations of macroscopically compressible and incompressible response require different solution strategies, (iii) the average pressure is not coupled between the scales. One attempt to adress these issues based on a mixed variational framework, was presented by Öhman et al. [17]. In this paper, an alternative formulation is presented, where the mixed variational format on the macroscale is derived directly via homogenization. As compared to [17], extension is also made to weakly periodic boundary condition for the RVE.
The paper is outlined as follows: The appropriate variational setting of the finescale elasticity problem in a mixed format is given. Next, the corresponding VCH framework and macroscale problem are outlined. The canonical form of the RVEproblem, based on weakly periodic boundary conditions is established, followed by the Dirichlet and Neumann type boundary conditions resulting in upper and lower bounds on the RVE response. Numerical examples of RVEproblems are then shown, followed by conclusions.
Methods
Subscale modeling of isotropic elasticity allowing for the incompressible limit
A mixed displacementpressure weak format
We consider a generic microheterogeneous, i.e. polycrystalline, material in a given body whose macroscopic configuration occupies the region Ω in space with (presumed smooth) boundary Γ. We are then lead to defining a Representative Volume Element (RVE), that represents the topology of the microheterogeneous microstructure, as shown in Figure 1. The total domain occupied by the cubic RVE is denoted with external boundary .
We consider a model material as follows: The stress is decomposed in terms of deviator and pressure as σ=σ _{d}−p I where \(p \overset {\text {def}}{=} \frac 13\text {tr}(\boldsymbol {\sigma }\!)\), and the strain is decomposed in terms of deviator and dilation as \(\boldsymbol {\epsilon } = \boldsymbol {\epsilon }_{\mathrm {d}} + \frac {1}{3} e\boldsymbol {I}\) where \(e\overset {\text {def}}{=} \text {tr}(\boldsymbol {\epsilon }) = \boldsymbol {u} \cdot \boldsymbol {\nabla }\). With the kinematic definition \(\boldsymbol {\epsilon }_{\mathrm {d}}[\!\boldsymbol {u}]\overset {\text {def}}{=} \,[\!\boldsymbol {u}\otimes \boldsymbol {\nabla }]^{\text {sym}}\frac {1}{3}[\!\boldsymbol {u}\cdot \boldsymbol {\nabla }]\boldsymbol {I}\), we introduce the constitutive relations
Hence, \(\hat {\boldsymbol {\sigma }}_{\mathrm {d}}(\bullet)\) and \(\hat {e}(\bullet)\) denote suitable constitutive functions. Obviously, in the simplest case of linear isotropic elasticity, we have \(\hat {\boldsymbol {\sigma }}_{\mathrm {d}}(\boldsymbol {\epsilon }_{\mathrm {d}})=2G\boldsymbol {\epsilon }_{\mathrm {d}}\) and \(\hat {e}(p)=\frac {1}{K}p\), where G(x),K(x) for x∈Ω are the standard elastic moduli that fluctuate strongly. Moreover, intrinsic incompressibility is defined as \(\hat {e}(p)=0\) for any value of p. We are now in the position to formulate the strong format of the finescale problem under standard quasistatic conditions and small strain kinematics:
The corresponding weak format is: Find \(\boldsymbol {u}\in \mathbb {U}, p\in \mathbb {P}\) s.t.
where
The solution space and the test space \(\mathbb {U}^{0}\) are defined in standard fashion. In particular, all \(\boldsymbol {v}\in \mathbb {U}\) are characterized by v=u _{pre} on Γ ^{D}, whereas all \(\boldsymbol {v}\in \mathbb {U}^{0}\) satisfy v=0 on Γ ^{D}. The pressure space does not satisfy any boundary conditions.
It is illuminating (although not necessary from an operational point of view) to invoke the potential Π(u,p)
where ψ _{ u }(ε _{d}) and \(\psi _{p}^{*}(p)\) are constitutive energy densities^{a} such that
The stationarity conditions of Π(u,p) are
which are identical to the weak form in (3).
Variationally Consistent Homogenization
VMSansatz and scale separation
The appropriate variational setting of the homogenized problem is obtained upon replacing the integrands in the weak forms in (4)–(7) by running averages of the type
representing a smoothing approximation on a RVE. In practice, the RVE’s are finitesized and occupies the subscale region with boundary . The typical dimension of an RVE is . The RVE is centered at the macroscale position for any given \(\bar {\boldsymbol {x}}\in \Omega \). Boundary integrals can be homogenized in similar fashion, by considering Representative Surface Elements Γ _{ # }
The weak forms in (4)–(7) are thus approximated as
where the RVEfunctionals in (13)–(16) are defined as
Likewise, we homogenize the volumespecific energy potential Λ(u,p):
where the RVEfunctional is given as
In the spirit of the Variational MultiScale method (VMS) [18], we introduce the ansatz that the fields \(\boldsymbol {u}\in \mathbb {U}\) and \(p\in \mathbb {P}\) can be decomposed into macroscale (smooth) and subscale (fluctuating) parts inside each RVE via the unique hierarchical split \(\mathbb {U} = \mathbb {U}^{\mathrm {M}} \oplus \mathbb {U}^{\mathrm {s}}\) and \(\mathbb {P} = \mathbb {P}^{\mathrm {M}} \oplus \mathbb {P}^{\mathrm {s}}\). As a result, we may assume that it is possible to solve for the fluctuation fields \(\boldsymbol {u}^{\mathrm {s}}\in \mathbb {U}^{\mathrm {s}}\) and \(p^{\mathrm {s}}\in \mathbb {P}^{\mathrm {s}}\) as “local approximations” on each RVE for given macroscale solutions \(\boldsymbol {u}^{\mathrm {M}}\in \mathbb {U}^{\mathrm {M}}\) and \(p^{\mathrm {M}}\in \mathbb {P}^{\mathrm {M}}\), i.e. we construct the complete solution on each RVE as^{b}.
On the boundary of the macroscale domain, Γ, we assume smooth variation of u defined by the explicit relations u=u ^{M}, p=p ^{M} on Γ _{ # }.
In addition, the test function \(\delta \boldsymbol {u}\in \mathbb {U}^{0}\) in (3a) is replaced by \(\delta \boldsymbol {u}^{\mathrm {M}}\in \mathbb {U}^{{\mathrm {M}},0}\), whereas \(\delta p\in \mathbb {P}\) in (3b) is replaced by \(\delta p^{\mathrm {M}}\in \mathbb {P}^{\mathrm {M}}\). Altogether, these assumptions infer that \(\boldsymbol {u}^{\mathrm {M}}\in \mathbb {U}^{\mathrm {M}}\) and \(p^{\mathrm {M}}\in \mathbb {P}^{\mathrm {M}}\) can be solved from the homogenized problem
Explicit format of macroscale (homogenized) problem
In practice, the scales are linked by expressing \(\boldsymbol {u}^{\mathrm {M}}(\bar {\boldsymbol {x}},{\boldsymbol {x}})\) ^{c} and \(p^{\mathrm {M}}(\bar {\boldsymbol {x}},\boldsymbol {x})\) using Taylor series expansions of suitable order for \(\bar {\boldsymbol {x}}\in \Omega \) and in terms of the macroscale solution \(\bar {\boldsymbol {u}}(\bar {\boldsymbol {x}})\) and \(\bar {p}(\bar {\boldsymbol {x}})\) respectively. We thus introduce the macroscale fields \((\bar {\boldsymbol {u}},\bar {p})\in \bar {\mathbb {U}}\times \bar {\mathbb {P}}\) such that the macroscale solutions u ^{M},p ^{M} inside each RVE are expanded as follows:
Hence, u ^{M} is assumed to have linear variation in pertinent to standard “first order homogenization”, whereas p ^{M} is constant in . Now, we require that
which leads to the constraints
As a result, the hierarchical split (\(\mathbb {U} = \mathbb {U}^{\mathrm {M}} \oplus \mathbb {U}^{\mathrm {s}}\) and \(\mathbb {P} = \mathbb {P}^{\mathrm {M}} \oplus \mathbb {P}^{\mathrm {s}}\)) is guaranteed.
We can thus establish at the outset, before any further analysis, that the displacement and pressure fields within each RVE are implicit functions of the values \(\bar {\boldsymbol {u}}\), \(\bar {\boldsymbol {h}}\), \(\bar {p}\), such that \(\boldsymbol {u} = \tilde {\boldsymbol {u}}\{\bar {\boldsymbol {u}}, \bar {\boldsymbol {h}}, \bar {p}\}\) and \(p = \tilde {p}\{\bar {\boldsymbol {u}}, \bar {\boldsymbol {h}}, \bar {p}\}\).
With the representations in (25) and the constraints in (26) and (27), we are in the position to compute the homogenized quantities that enter the system (24):
The applied macroscale loads \(\bar {\boldsymbol {f}}\), \(\bar {\boldsymbol {t}}_{\text {pre}}\) and “moments” \(\,\bar {\bar {\boldsymbol {f}}}\), \(\bar {\bar {\boldsymbol {t}}}_{\text {pre}}\) are defined as
Remark.
Henceforth, we restrict to the situation when and Γ _{ # }→0; hence \(\bar {\bar {\boldsymbol {f}}}\) and \(\bar {\bar {\boldsymbol {t}}}_{\text {pre}}\) will vanish. We will also focus on the homogenization of \(\bar {\boldsymbol {\sigma }}_{\mathrm {d}}\) and \(\bar {e}\), so \(\bar {\boldsymbol {f}}\) and \(\bar {\boldsymbol {t}}_{\text {pre}}\) are considered as given macroscopic quantities.
The variationally consistent macroscale “flux” variables \(\bar {\boldsymbol {\sigma }}_{\mathrm {d}}\) and \(\bar {e}\) are obtained from homogenization of the constitutive functions as follows:
By combining (23) with (25) we note that \(\bar {\boldsymbol {\sigma }}_{\mathrm {d}}\) and \(\bar {e}\) are indeed implicit functions of the values of the macroscale variables \(\bar {\boldsymbol {u}}\), \(\bar {\boldsymbol {h}}\) and \(\bar {p}\), pertinent to the considered RVE.
Finally, upon inserting (28)–(33) into the system (24), we obtain the macroscale problem: Find \((\bar {\boldsymbol {u}},\bar {p})\in \bar {\mathbb {U}}\times \bar {\mathbb {P}}\) that solve
where
If we consider the macroscale fields, \(\bar {\boldsymbol {\sigma }}_{\mathrm {d}}\) and \(\bar {e}\), we conclude that they are implicit functions of the fields \(\bar {\boldsymbol {u}}\) and \(\bar {p}\). The macroscale spaces \(\bar {\mathbb {U}}\) and \(\bar {\mathbb {P}}\) are chosen as the standard ones for the finescale problem.
Canonical formulation of RVEproblem
Preliminaries – Concept of weak periodicity of fluctuation displacement
To avoid unnecessary technical complexity, we henceforth consider the situation without volume load, i.e. f=0. As a preliminary for establishing the proper variational format of the RVEproblem, we consider the most general weak form of the quasistatic momentum balance by introducing the boundary integral with boundary tractions:
or, more explicitly,
which is supposed to hold true for all possible δ u,δ p in suitable function spaces (as discussed below). However, this problem is not solvable without further specification of the solution fields u,p,t. In this paper we adopt a recently proposed variational framework allowing for weak satisfaction of microperiodicity, cf. Larsson et al. [8], and this framework will be briefly summarized in what follows. We then assume that the subscale fluctuation field u ^{s} is periodic across the RVE boundaries w.r.t. the chosen local coordinate axes. This model assumption, which may be termed “microperiodicity”, is a key ingredient (and frequently adopted) in the literature on mathematical homogenization and can be viewed as an approximation between the stiffer Dirichlet and the weaker Neumann boundary conditions. Indeed, both these cases can be obtained as special cases of the most general variational format of periodicity (as will be discussed further below).
In order to formulate the conditions on microperiodicity, we consider the RVE in Figure 2, where the boundary has been split into two parts: . Here, is the image boundary (later chosen as the computational domain for boundary integration), whereas is the mirror boundary. We shall now introduce the proper mapping such that any point is mirrored in a selfsimilar fashion to the corresponding point ; hence, x ^{−}=φ _{per}(x ^{+}).
In particular, we express microperiodicity of the displacement fluctuation field as
or, equivalently, in terms of the “jump” between the fluctuation fields on the image and mirror parts of the boundary as follows:
Subsequently, we shall not enforce the condition (45) strongly as the point of departure; rather it is done weakly. To this end, we assume that the boundary tractions \(\boldsymbol {t}\overset {\text {def}}{=}\boldsymbol {\sigma }\cdot \boldsymbol {n}\) satisfy the following antisymmetry condition for any regular mirror point
as depicted in Figure 2. We now evaluate, upon using (46), the boundary term in (42a) and (43a), as follows:
A weak statement of the microperiodicity constraint, given in strong form in (46), is
where the space of test functions that “live” only on the image boundary is given as .
RVEproblem – Original “variationally consistent” weak format
In order to establish the most straightforward formulation of the RVEproblem, based on microperiodicity, we first use the constraints in (27) and introduce the following spaces for the fluctuation fields:
It is then obvious that, for given macroscale values \(\bar {\boldsymbol {u}}\), \(\bar {\boldsymbol {h}}\), and \(\bar {p}\), we can introduce the unique decompositions
Next, we aim for a unique decomposition of the tractions (which are antiperiodic by assumption) in a fashion that is similar to (51). To this end, we first associate each traction field t along with the average stress \(\bar {\boldsymbol {\tau }}[\!\boldsymbol {t}] \in \mathbb {R}^{3\times 3}\), defined as
This definition for the average stress is chosen so that for any stress field τ in equilibrium and such that t=τ·n on we obtain . We also conclude that
As a direct consequence of (52), we may introduce the unique split
where is the space of the traction fluctuations that are selfequilibrating and thus defined as
The proof of uniqueness of the split in (54) follows from the identity
and the fact that
Hence, . □
As preliminaries for establishing the RVEproblem, we establish two identities: Firstly, from (53) and (54) follows that
Secondly, it follows from (25a) and the properties of that
whereby it is noted that the macroscale part of u is “filtered out”.
We are now in the position to establish the subscale problem: For given values \(\bar {\boldsymbol {u}}\), \(\bar {\boldsymbol {h}}\), and \(\bar {p}\), that represent the macroscale fields (and which solve the macroscale problem), find the subscale fluctuations that solve the system
where the RVEfunctionals were introduced in (17)–(19) and (48).
By inspecting the system in (60), we note that it is not the entire \(\bar {\boldsymbol {h}}\) that is used as input to the RVEproblem. In fact, it is readily concluded that it is only the deviatoric part \(\bar {\boldsymbol {\epsilon }}_{\mathrm {d}}=\bar {\boldsymbol {h}}^{\text {sym}}_{\mathrm {d}}\) that enters as “data” to the RVEproblem. In other words, neither \(\bar {\boldsymbol {u}}\), the volumetric part \(\bar {h}_{\mathrm {v}}\overset {\text {def}}{=}\bar {\boldsymbol {h}}:\boldsymbol {I}\), nor the skewsymmetric part \(\bar {\boldsymbol {h}}^{\text {skw}}=\frac {1}{2}\!\left [\bar {\boldsymbol {h}}\bar {\boldsymbol {h}}^{\mathrm {T}}\right ]\) will affect the RVEsolution. In conclusion, (\(\bar {\boldsymbol {\epsilon }}_{\mathrm {d}},\bar {p})\) are the macroscale variables that are used as data for the RVEproblem; hence, the solution of (60) is parameterized as \(\boldsymbol {u}^{\mathrm {s}}=\boldsymbol {u}^{\mathrm {s}}\{{\bar {\boldsymbol {\epsilon }}}_{\mathrm {d}}, \bar {p}\}\), \(p^{\mathrm {s}}=p^{\mathrm {s}}\{{\bar {\boldsymbol {\epsilon }}}_{\mathrm {d}}, \bar {p}\}\), and \(\boldsymbol {t}^{\mathrm {s}}=\boldsymbol {t}^{\mathrm {s}}\{{\bar {\boldsymbol {\epsilon }}}_{\mathrm {d}}, \bar {p}\}\). In a postprocessing step the homogenized “fluxes” can be represented as
Note that \(\bar {e}\{{\bar {\boldsymbol {\epsilon }}}_{\mathrm {d}}, \bar {p}\}\neq \bar {h}_{\mathrm {v}}\) in general!
From the aforesaid, it appears that there is a problem associated with the presented “consistent” format in the sense that the strong form of the continuity equation does not (necessarily) represent that one of the fine scale. The reason is that the formulation (60b) “filters out” any imposed constant volumetric strain \(\bar {\lambda }\); hence, the strong format generally reads
Clearly, testing (62) with a constant pressure, we obtain .
Remark.
In the special case of intrinsically incompressible material response, defined by \(\hat {e}(p)=0\) in , for any p, then we have . However, from (60b), (60c) it is concluded that the solution of the RVEproblem is , whereby we obtain \(\bar {\lambda }=\bar {h}_{\mathrm {v}} \neq 0\). It is only when the macroscale solution represents incompressible response, i.e. when \(\bar {h}_{\mathrm {v}} = 0\), that we obtain \(\bar {\lambda } = 0\). In conclusion, the RVEproblem is able to provide an “incompressible solution” even in the case that the macroscale solution represents compressible response. This anomaly is the main motivation for establishing an alternative format of the problem, which is discussed next.
Macrohomogeneity condition (VCMC)
The Variationally Consistent Macrohomogeneity Condition (VCMC) (or generalized HillMandel condition) is reviewed in Appendix ‘Variationally Consistent Homogenization (VCH)’. In order to establish its localized form for the present problem, we first identify the tangent spaces
whereby and represent sensitivity fields (or directional derivatives) for given changes and of the macroscale fields within each RVE.
The macrohomogeneity condition is satisfied if, for any given state u ^{M}, p ^{M} localized to the considered RVE, the following relations hold:
In order to show that this condition is, indeed, satisfied automatically by the solution of the RVEproblem as defined in (60), it suffices to consider (60c). Upon differentiating this relation w.r.t. u ^{M} and p ^{M}, we obtain
for any (by definition of (u ^{s})^{′} {u ^{M},p ^{M};du ^{M},dp ^{M}}). Now choosing δ t ^{s}=t ^{s} in (65), we obtain
Finally, upon choosing in (60a) and in (60b) while noting the identity in (66) we recover (64). In conclusion the VCMC is satisfied.
Remark.
In the present case there is, obviously, no need to compute the sensitivities du ^{s}=(u ^{s})^{′} and dp ^{s}=(p ^{s})^{′} explicitly. However, it is always possible to compute the sensitivities of all the fluctuation fields, u ^{s}, p ^{s}, and t ^{s} from the (linear) system obtained by linearizing (60). This system is closely related to the sensitivity problem that must be established as part of computing the pertinent macroscale tangent tensors exploited in Newton iterations on the macroscale problem. The explicit format of that sensitivity problem is discussed in Appendix ‘Sensitivity problem’ for the Canonical format of the RVEproblem that is introduced subsequently.
RVEproblem – Canonical weak format
A generalized formulation of the RVEproblem that does not contain the abovementioned inconsistency with the strong format is considered next. Firstly, we introduce the following spaces for the total (macroscale and fluctuation) fields:
Secondly, the finescale fields within an RVE are decomposed as
where \(\boldsymbol {x}_{\mathrm {m}}\overset {\text {def}}{=}\frac {1}{3}[\!\boldsymbol {x}\bar {\boldsymbol {x}}]\), and where \(\bar {e}\) is an additional scalar quantity which, at the outset, does not depend on the macroscale field(s).
Remark.
As a consequence of the ansatz in (70a), the rigid body motion is removed; \(\bar {\boldsymbol {u}} = \boldsymbol {0}\) and \(\bar {\boldsymbol {h}}^{\text {skw}} = \boldsymbol {0}\). Hence, the ansatz u is not identical to the solution u=u ^{M}+u ^{s} as expressed in (51a).
We now propose the alternative, subsequently denoted canonical, formulation of the RVEproblem as follows: For given values \(\bar {\boldsymbol {\epsilon }}_{\mathrm {d}}\), \( \bar {p}\), that represent the macroscale fields (which solve the macroscale problem), find the subscale fields that solve the system
If follows from (72) that the new variable \(\bar {e}\) correctly obtains the value in accordance with (61b). To show this result, we first use the divergence theorem to obtain the identities
Using these identities together with (71b), (71c), we deduce
The solution of the “canonical” problem (71) contains, in fact, the solution of the original “variationally consistent” problem in (60); however, without the apparent drawback that is associated with that format. The fluctuations
are identical to that solve the system (60). This important result can be shown by introducing the unique split of the unknowns
Firstly, inserting these relations into (71c) and testing with \(\delta \check {\boldsymbol {\tau }}\) we obtain
which gives
Next, testing (71a) with \(\delta \check {\boldsymbol {h}}\) we obtain
which gives
Lastly, from (71d) we obtain
With (81), this result implies that \(\check {p} = \bar {p}\).
Now, testing (71a)–(71c) with the fluctuations δ u ^{s}, δ p ^{s}, and δ t ^{s} respectively, we obtain exactly the original system in (60). Hence, we compute the identical quantities \(\boldsymbol {u}^{\mathrm {s}} \{\bar {\boldsymbol {\epsilon }}_{\mathrm {d}}, \bar {p}\}\) and \(p^{\mathrm {s}}\{\bar {\boldsymbol {\epsilon }}_{\mathrm {d}}, \bar {p}\}\). Finally, the canonical form results in the identical responses \(\bar {\boldsymbol {\sigma }}_{\mathrm {d}}\{{\bar {\boldsymbol {\epsilon }}}_{\mathrm {d}}, \bar {p}\}\) and \(\bar {e}\{{\bar {\boldsymbol {\epsilon }}}_{\mathrm {d}}, \bar {p}\}\) as expressed in (61). Since the macroscopic response variables are identical, the VCMC is still fulfilled.
An illustrative comparison between the two RVE formulations are shown Figure 3. Despite the different displacement fields, the homogenized responses, \(\bar {\boldsymbol {\sigma }}_{\mathrm {d}}\) and \(\bar {e}\), are the same.
Remark.
Finescale incompressibility within the RVE is defined by , which gives . As a result, . In practice, there is (of course) no need to establish s a separate system, since \(\bar {e}=0\) is obtained directly as part of the solution of (71) in this case.
Variational properties of the RVEproblem – Energy bounds from RVEfunctional
Variational properties for the canonical formulation of the RVEproblem – The RVEpotential
In order to establish bounds on the homogenized properties based on the results from the Dirichlet and Neumann problems, we shall introduce an appropriate “RVEpotential” as follows:
where is the “intrinsic” energy potential that was defined in (22).
We now define the volumespecific “macroscale energy density”^{d} as the value of at the following generalized saddlepoint:
A stationary point of is defined by the relations
It is readily seen that the system in (85) is precisely that of (71), and we may parameterize its solution as
At the stationary point, we may use the result in (85c) to conclude that
whereby the energy at the stationary point is deduced to become
We now deduce that serves as the “macroscale energy density” for \(\bar {\boldsymbol {\sigma }}_{\mathrm {d}}\) and \(\bar {e}\) in the sense that we have the macroscale constitutive relations
Details of the proof are given in Appendix ‘Homogenized macroscale energy’.
From the solution of the RVEproblem, \(\bar {e}\) is obtained directly as a primary variable. The deviatoric stress \(\bar {\boldsymbol {\sigma }}_{\mathrm {d}}\) is obtained by postprocessing:
Variational properties for Dirichlet boundary conditions
In order to establish a suitable variational setting we replace the solution space with defined as
where
while are the same spaces as for the generic problem. The generalized saddlepoint problem (84) can now be rephrased as
where
The only possibility to obtain a finite value of while evaluating the sup over is to set \(\check {\boldsymbol {\epsilon }}_{\mathrm {d}} = \bar {\boldsymbol {\epsilon }}_{\mathrm {d}}\) and \(\check {e} = \hat {\bar {e}}\). Hence, we replace the saddlepoint problem by
where
A stationary point of is defined by the relations
and this system of equations takes the explicit form: For given values \(\bar {\boldsymbol {\epsilon }}_{\mathrm {d}}, \bar {p}\), find the subscale fields that solve the system
Finally, the macroscale energy density at the stationary point becomes
From the RVEproblem in (98), \(\bar {e}\) is obtained directly as a primary variable. The deviatoric stress \(\bar {\boldsymbol {\sigma }}_{\mathrm {d}}\) is obtained by postprocessing:
However in practice \(\bar {\boldsymbol {\sigma }}_{\mathrm {d}}\) is conveniently obtained as the reaction forces associated with the prescribed \(\bar {\boldsymbol {\epsilon }}_{\mathrm {d}}\).
Remark.
The VMCM in (64) is still valid for the restricted space . Since , (64a) is satisfied for all \(\delta \boldsymbol {u}^{\mathrm {s}}\in \mathbb {U}^{{\mathrm {D}},\mathrm {s}}\).
Variational properties for Neumann boundary conditions
The Neumann condition represents the weakest possible way of enforcing the microperiodicity condition. Here it is considered as a model assumption; however, it is also possible to view this choice as a (crude) FEapproximation of the deviatoric traction field. The pertinent RVEproblem is obtained from the general format upon restricting the space , i.e. introducing defined as
while and are left unrestricted. Clearly, the choice of restricts the tractions to become piecewise constant on each of the three positive boundary faces of the RVEcube. The generalized saddlepoint problem (84) can then be rephrased as
where
The only possibility to obtain a finite value of while evaluating the inf over \(\hat {\bar {e}}\in \mathbb {R}\) is to set \(\check {p} = \bar {p}\). As a direct consequence the variable \(\hat {\bar {e}}\) disappears in the resulting expression, cf. (105) below. Moreover, from (90) we see that for tractions we obtain .
Hence, we replace the saddlepoint problem by
where
A stationary point of is defined by the relations
and this system of equations takes the explicit form: For given values \(\bar {\boldsymbol {\epsilon }}_{\mathrm {d}}, \bar {p}\), find the subscale fields that solve the system
Finally, we obtain the macroscale energy density as
From the RVEproblem in (107), \(\bar {\boldsymbol {\sigma }}_{\mathrm {d}}\) is obtained directly as a primary variable. The volumetric strain \(\bar {e}\) is obtained by postprocessing:
Remark.
The VMCM in (64) is still valid for the restricted space . Since , (64a) is valid for all .
Macroscale tangent relations
In order to solve the (generally nonlinear) macroscale problem (37) in the FE^{2}setting using Newton iterations, we need the pertinent tangent operators. Linearizing the implicit relations \(\bar {\boldsymbol {\sigma }}_{\mathrm {d}}\{{\bar {\boldsymbol {\epsilon }}}_{\mathrm {d}}, \bar {p}\}\) and \(\bar {e}\{{\bar {\boldsymbol {\epsilon }}}_{\mathrm {d}}, \bar {p}\}\) we obtain
thereby defining the tangents \(\bar {\textbf {\textsf {E}}}\) (4th order), \(\bar {\boldsymbol {E}}\) (2nd order), \(\bar {\boldsymbol {C}}\) (2nd order), and \(\bar {C}\) (scalar). Since a potential exists, it follows that \(\bar {\textbf {\textsf {E}}}\) possesses major symmetry and that \(\bar {\boldsymbol {E}} = \bar {\boldsymbol {C}}\), cf. Öhman et al. [14].
In order to compute the tangent operators, it is necessary to first compute the relevant sensitivity fields w.r.t. changes of the macroscale variables \(\bar {\boldsymbol {\epsilon }}_{\mathrm {d}}\) and \(\bar {p}\), and they are solved from the pertinent tangent problem. On should then note that the particular sensitivity fields that are actually exploited (and needed) and the corresponding explicit formulation of the tangent problem depend on the chosen formulation of the RVEproblem (in terms of the boundary conditions: Weakly Periodic, Dirichlet, Neumann). Here, we shall give details only on the “generic” choice of the weakly periodic conditions.
Remark.
In the case of macroscale incompressibility, \(\bar {\boldsymbol {C}}\) and \(\bar {C}\) will vanish.
As a point of departure for computing the tangent tensors, we note that \(\bar {e}\{{\bar {\boldsymbol {\epsilon }}}_{\mathrm {d}}, \bar {p}\}\) is a primary variable in the RVEproblem, whereas \(\bar {\boldsymbol {\sigma }}_{\mathrm {d}}\{{\bar {\boldsymbol {\epsilon }}}_{\mathrm {d}}, \bar {p}\}\) is postprocessed via (90). We thus need to compute sensitivities of \(\bar {e}\) and t from the tangent problem. To begin with, we establish the linearized form of (71) at the solution state to obtain
which must be valid for any given perturbations \(\mathrm {d}\bar {\boldsymbol {\epsilon }}_{\mathrm {d}}\) and \(\mathrm {d}\bar {p}\) giving rize to the corresponding perturbations du, dp, dt, and \(\mathrm {d}\bar {e}\) in the solution fields.
Next, we introduce orthonormal base dyads G _{ i }, i=1,2,…,8 to ensure that the deviatoric tensors ε _{d} and σ _{d} are, indeed, deviatoric in character. In other words we do not use the conventional base dyads e _{ i }⊗e _{ j }, i,j=1,2,3, but we rather introduce the new set of base dyads such that the requirement G _{ i }:I=0 for i=1,2,…,8 is fulfilled. Examples of such (generally unsymmetric) dyads with Cartesian components, are
with respect to this basis, we have the representation
Next, we express the pertinent sensitivities via the representations
We thus obtain the following sets of tangent problems:

\(\mathrm {d}\bar {\boldsymbol {\epsilon }}_{\mathrm {d}} = \boldsymbol {G}_{k}\) while \(\mathrm {d}\bar {p} = 0\): For k=1,…,8, solve for the sensitivities \(\boldsymbol {u}_{\mathrm {d}}^{(k)}\), \(p_{\mathrm {d}}^{(k)}\), \(\boldsymbol {t}_{\mathrm {d}}^{(k)}\), \(\bar {e}_{\mathrm {d}}^{(k)}\) from the system
((116a))((116b))((116c))((116d)) 
\(\mathrm {d}\bar {p} = 1\) while \(\mathrm {d}\bar {\boldsymbol {\epsilon }}_{\mathrm {d}} = \boldsymbol {0}\): Solve for the sensitivities u _{p}, p _{p}, t _{p}, \(\bar {e}_{\mathrm {p}}\) from the system
((117a))((117b))((117c))((117d))
Now, upon linearizing (90) and using the representation for dt in (115c), we obtain
Directly from (115d), we obtain
Macroscale tangent relations for the Dirichlet and Neumann boundary conditions are detailed in Appendix ‘Sensitivity problem’.
Bounds on the macroscale energy density
Since we have derived the Dirichlet and Neumann problems by respectively restricting the solution spaces as
it follows from (93), (102), and (84) that
In other words, the Dirichlet and Neumann boundary conditions represent upper and lower bounds on the macroscale energy density that is obtained with periodic boundary conditions for any given realization of the RVE and for a given macroscale state \((\bar {\boldsymbol {\epsilon }}_{\mathrm {d}}, \bar {p})\). We note that choosing any discretization of in (84) leads to a lower bound of \(\bar {\psi }\{{\bar {\boldsymbol {\epsilon }}}_{\mathrm {d}}, \bar {p}\}\).
In the case of linear elasticity, the tangent operators represent the upscaled (=homogenized) constant operators in the representations
whereby we obtain
Results and discussion
Preliminaries
Computations were carried out for a random composite with spherical particles embedded in a matrix, whereby both constituents are assumed to be homogeneous and isotropic linear elastic. The intrinsic properties are thus defined by the free energy expressions
where G and C (=K ^{−1}) are the shear stiffness and bulk compliance respectively. This choice corresponds to the standard relations \(\hat {\boldsymbol {\sigma }}_{\mathrm {d}}(\boldsymbol {\epsilon }_{\mathrm {d}}) = 2 G\,\boldsymbol {\epsilon }_{\mathrm {d}}\) and \(\hat {e}(p) = C\, p\). The parameter set associated with the matrix and particles are (G _{mat},C _{mat}) and (G _{part},C _{part}), respectively.
All computational results were obtained for cubic Statistical Volume Elements (SVEs). The target volume fraction of (nonoverlapping, spherical particles) if 10% and their sizedistribution, in terms of diameter, is uniform in the range \(\left [\frac {2}{3},\frac {4}{3}\right ]\), thus with a mean value of 1. Two examples of realizations with SVEsize and are shown in Figure 4 and Figure 5, respectively.
Now, for sufficiently large size of the SVE (for any given realization of a random composite), we may assume that the macroscale response is isotropic corresponding to the effective moduli \(\bar {G}\) and \(\bar {C}\) (\(=\bar {K}^{1}\)). However, even for a smaller SVEsize, it is possible to compute an “apparent” value of \(\bar {G}\) by the minimization
where I _{d} is the deviatoric fourth order identity tensor such that the isotropic part of \(\bar {\textbf {\textsf {E}}}\) can be expressed as \(\bar {\textbf {\textsf {E}}}_{\text {iso}} = 2 \bar {G} \textbf {\textsf {I}}_{\mathrm {d}}\) in standard fashion.
For the finite element analysis of the SVEproblems, tetrahedral elements with socalled Minielement approximations are utilized. This type of element is defined by linear approximation plus a bubble function for u and linear approximation for p, cf. [19].
The RVE‐problems with the different boundary conditions are implemented in the open source C++ code OOFEM (www.oofem.org) [20] and are available under the name MixedGradientPressure. However, due to the difficulty of constructing , the present implementation of the weakly periodic boundary conditions is based on global polynomials for discretizing . Thus, only a lower bound for the true periodicity is obtained. Due to the high computational cost, only the Neumann and Dirichlet boundary conditions are used in the numerical examples in this paper.
Homogenization of macroscopically incompressible response — A convergence study
The first series of SVEcomputations were carried out for macroscopically incompressible response, which is obtained when both matrix and particles are completely incompressible. Taking the shear modulus for the matrix, G _{mat}, as the reference modulus, we choose G _{part}=5 G _{mat}, whereas C _{mat}=C _{part}=0. Convergence results for the two extremes of Dirichlet and Neumann boundary conditions are shown in Figure 6. The mean value and variance of the homogenized shear modulus \(\bar {G}\) are shown for increasing SVEsize, represented by . Here, a certain number of realizations are used for each value of , starting with 200 realizations for the smallest SVE size and ending up with only 4 realizations for the largest SVEsize that has approximately 500 thousand degrees of freedom. The Salome platform was used for generating the SVEs and the decrease of the number of realizations was due to the meshing code failing when inclusions badly intersect the boundary, which becomes increasingly likely with increasing . However, as seen in Figure 6, the variance is small in the homogenized results for and the mean values are still accurate. An RVE is obtained when the variance goes to zero, and the mean value converges, which is the case when the SVEsize is sufficiently large. In the present case, for engineering purposes, we consider sufficiently large to qualify as an RVE.
Seamless transition from macroscopically compressible to incompressible response
In the second series of computations, we show how the homogenization scheme can seamlessly handle the transition from the macroscopically compressible to the incompressible response. Here, it is assumed that incompressible particles, defined by G _{part}=5 G _{mat} and C _{part}=0 (as before), are embedded in a matrix material with C _{mat}≥0. Only the situation with Dirichlet boundary conditions is considered (without affecting the generality of the main conclusion from the results). In Figure 7 it is shown how the homogenized properties are affected by the matrix compliance, C _{mat}, for a given (fixed) realization and given SVEsize , cf. Figure 5), thus defining an RVE. Clearly, rather than choosing C _{mat} as the variable material property, one may choose Poisson’s ratio ν _{mat} via the elementary relation
which is also used in Figure 7.
The computational results verify that the homogenization framework is able to handle the transition from macroscale compressibility to incompressibility when C _{mat}→0 in a seamless fashion without any algorithmic changes. That macroscopical incompressibility is verified by the computed value \(\bar {C} = 0\). It is also noted in Figure 7 that the affect on \(\bar {G}\) is more pronounced, however quite limited, when incompressibility is approached.
Conclusions
In this paper we have introduced a variationally consistent homogenization scheme for heterogeneous solids, whose constituents may be incompressible in the extreme case. The mixed control of the homogenized quantities, \((\bar {\boldsymbol {\epsilon }}_{\mathrm {d}}, \bar {p}) \to (\bar {\boldsymbol {\sigma }}_{\mathrm {d}}, \bar {e})\), is what enables the macroscopically incompressible response, whereas traditional straincontrolled homogenization, \(\bar {\boldsymbol {\epsilon }} \to \bar {\boldsymbol {\sigma }}\), leads to a singular problem when applied to fully incompressible microstructures. The transition from macroscopically compressible to incompressible states (which is the extreme result when all constituents are incompressible) is handled in a completely seamless fashion. Based on a mixed formulation on the fine scale, it has been demonstrated how the concept of variationally consistent homogenization can be adopted to derive the pertinent twoscale problem. The proposed canonical formulation of the problem on an underlying Representative Volume Element (RVE) is an extension of the formulation presented in Öhman et al. (2013), allowing for a general choice of boundary conditions. Moreover, the canonical formulation has been shown to satisfy the generalized macrohomogeneity condition. The FEsetting on both scales (macroscale and RVE) is based on the standard (u,p)formulation, and the Minielement approximation with a bubble function is adopted for the RVEproblem. All implementation and numerical results pertain to 3Dcubes.
It was shown that theoretical bounds on the “homogenized energy density” for weakly periodic boundary conditions on the RVE are obtained by imposing Dirichlet and Neumann boundary conditions. The computational results from a statistical analysis verify the convergence of these bounds with increasing SVEsize. As to the computational cost of solving the RVEproblem with the different types of boundary conditions for the standard compressible problem and the present (in)compressible one, it is concluded that the Neumann boundary condition is quite costneutral. However, applying the Dirichlet boundary condition to the canonical RVEformat means a noticeable additional cost, since the additional (global) variable \(\bar {e}\) is required in order to account for the (possibly vanishing) macroscale volumetric deformation. In this context, we remark that imposing weak periodicity would infer a further increased computational cost associated with the boundary discretization of tractions in ; hence, it is conjectured that exploiting the standard Dirichlet and Neumann conditions computations and drawing conclusions about their bounding character would be a most competitive approach in FE^{2}computations.
As to future developments, it is desirable to further exploit the concept of weakly periodic boundary conditions; however, constructing basis functions for in 3D is not a trivial task. Of course, it is also of interest to compare with the classical condition of strongly periodic fluctuations, which is defined by replacing (92) with
and performing the corresponding steps up to (98). However, this method has the definite drawback that the meshing software must produce strictly periodic meshes.
In a forthcoming paper, the proposed theoretical framework will be applied to the problem of sintering, a characteristic of which is that the microstructure evolves from being highly porous to becoming completely dense. The “compaction” process is driven by surface tension along the particlepores boundaries until the pore have completely disappeared, cf. Öhman et al. (2013).
Endnotes
^{a} * indicates “complementary energy”
^{b} Curly brackets {(∙)} indicate implicit and/or nonlocal functional dependence on (∙).
^{c} Double arguments, i.e. \(\boldsymbol {u}(\bar {\boldsymbol {x}},\boldsymbol {x})\), are used to explicitly point out the underlying scale separation.
^{d} , the “effective energy density”, for sufficiently large RVE.
Appendix
Variationally Consistent Homogenization (VCH)
VMSapproach – Scale separation and homogenization
Classical modelbased homogenization can be formulated in the spirit of the Variational MultiScale method (VMS), cf. [21,22]. We introduce the abbreviated notation \(z = (\boldsymbol {u}, p) \in \mathbb {Z} = \mathbb {U} \times \mathbb {P}\), where represents the space of finescale (nonhomogenized) solutions to the system (3), which may conveniently be abbreviated in abstract form as the residual equation^{e}
where \(\mathbb {Z}^{0} = \mathbb {U}^{0} \times \mathbb {P}\) is the test space and where each \(\boldsymbol {u}\in \mathbb {U}^{0}\) vanishes on the Dirichlet part of Γ.
A multiscale formulation of (129) is defined by the hierarchical split \(\mathbb {Z} = \mathbb {Z}^{\mathrm {M}} \oplus \mathbb {Z}^{\mathrm {s}}\), where \(\mathbb {Z}^{\mathrm {M}}\) contains smooth macroscale functions and \(\mathbb {Z}^{\mathrm {s}}\) is the hierarchical complement of \(\mathbb {Z}^{\mathrm {M}}\) that, typically, represents the finescale features. It is assumed that each \(z\in \mathbb {Z}\) can be split uniquely as z=z ^{M}+z ^{s} such that \(z^{\mathrm {M}} \in \mathbb {Z}^{\mathrm {M}}\) and \(z^{\mathrm {s}} \in \mathbb {Z}^{\mathrm {s}}\). Therefore, solve \(z^{\mathrm {M}} \in \mathbb {Z}^{\mathrm {M}}\), \(z^{\mathrm {s}} \in \mathbb {Z}^{\mathrm {s}}\) such that (129) can be represented by the set of equations
Without introducing further assumptions (approximations), the dimension of the original problem has not changed, i.e. (130) represent two global problems whose solution requires the same computational effort as does (129). In order to reduce the problem dimension based on homogenization (via the assumption of scale separation), we introduce the following key assumptions:

The integrands in the pertinent volume integrals are replaced by a running volume average on RVE’s of the type introduced in (11) such that, typically, the residual in (130a) can be rewritten in terms of the contributions defined on each RVE as
((131))where is the RVEresidual that is localized to the Representative Volume Element (RVE). In practice (in FEanalysis), quadrature is used such that the evaluation of is carried out only in the Gauss points.
Furthermore, for the sake of simplicity we assume smoothness of boundary terms, such that R _{ Γ }(z;δ z ^{M})≈R _{ Γ }(z ^{M};δ z ^{M}), i.e. no boundary homogenization is necessary.

Local approximations for the fluctuation field z ^{s} are introduced in the spirit of VMS. This means that is the approximate solution^{f} of the finescale equation (130b) for given z ^{M}, i.e. (130b) is replaced by “closed” RVEproblems (in the macroscale quadrature points) associated with a particular choice of boundary conditions on . In this paper we obtain such “closed” RVEproblems by choosing weakly periodic boundary conditions.
Returning to (130a), we now replace this problem by the approximate, homogenized, problem
which has the same dimension as (130a). We note that (132) represents a valid homogenization problem for any given choice of ; however, to preserve typical Galerkin properties, such as symmetry of the macroscale tangent operator when such symmetry is inherent in the underlying finescale problem, it is crucial to satisfy the VCMC.
Variationally Consistent Macrohomogeneity Condition (VCMC)
We shall assume that there exists a potential Π(z) such that (132) represents the stationary point of Π(z), i.e. it is assumed that
Next, we introduce the crucial approximation (restriction) \(z \approx z^{\mathrm {M}} + \tilde {z}^{\mathrm {s}}\{z^{\mathrm {M}}\}\) before evaluation of the stationarity conditions, whereby we obtain
where \((\tilde {z}^{\mathrm {s}})'\{z^{\mathrm {M}};\delta z^{\mathrm {M}}\}\) denotes the sensitivity (or directional derivative) of \(\tilde z^{\mathrm {s}}\) for a variation δ z ^{M} of the macroscale solution z ^{M}. Hence, the choice of test function in (134) is restricted as compared to (133), and this restriction represents a “generalized Galerkin property” in terms of the underlying macroscale functions in \(\mathbb {Z}^{\mathrm {M}}\). Moreover, and most importantly, (134) is completely equivalent to the homogenized problem (132) if it is possible, for any given \(z^{\mathrm {M}}\in \mathbb {Z}^{\mathrm {M}}\), to satisfy the constraint
or, equivalently,
where the test space \(\tilde {\mathbb {Z}}'^{\mathrm {s}}(z^{\mathrm {M}})\), which is the tangent space to the approximation space, is defined as
Hence, any \(\mathrm {d} \tilde {z}^{\mathrm {s}}\in \tilde {\mathbb {Z}}'^{\mathrm {s}}(z^{\mathrm {M}})\) is a sensitivity of the fluctuation field \(\tilde {z}^{\mathrm {s}}\) for a differential change of the macroscale field z ^{M} within the considered RVE. It is computed from the tangent problem that is associated with the RVEproblem. We refer to (135) or (136) as a “Variationally Consistent (generalized) Macrohomogeneity Condition” (VCMC). Obviously, a sufficient condition for these identities to hold true is to require the RVEresidual to vanish on each RVE (in each quadrature point), i.e. to ensure that
An even stronger condition is to require that for any δ z ^{s} in a given set of functions that is defined locally for the considered RVE without requiring any implicit (or explicit) coupling to the sensitivity field \((\tilde {z}^{\mathrm {s}})'\{z^{\mathrm {M}};\delta z^{\mathrm {M}}\}\), which obviously defines a restricted choice of test functions. In such a case, the VCMC can be identified as precisely the classical HillMandel macrohomogeneity condition.
Finally, we remark that the VCMC ensures that the macroscale tangent operator becomes symmetrical, since it holds that
With the introduced approximation \(z^{\mathrm {s}} \approx \tilde {z}^{\mathrm {s}}\{z^{\mathrm {M}}\}\), the test functions in (139) are chosen as
Homogenized macroscale energy
In order to show that serves as the “macroscale energy density” for \(\bar {\boldsymbol {\sigma }}_{\mathrm {d}}\) and \(\bar {e}\), expressed by the relations (89), we first recall the identity
where the RVEpotential was defined in (83). Working out the total differential w.r.t. \(\bar {\boldsymbol {\epsilon }}_{\mathrm {d}}\) and \(\bar {p}\) at equilibrium, we obtain
where we introduced the sensitivities w.r.t. changes in \(\bar {\boldsymbol {\epsilon }}_{\mathrm {d}}\) and \(\bar {p}\) as follows:
Combining these with the (85), we can see that the last four terms in (142) vanish, i.e.. What remains of the total differential is thus
Now, we may use (85a) and choose \(\delta \boldsymbol {u} = \mathrm {d}\bar {\boldsymbol {\epsilon }}\cdot [\!\boldsymbol {x}  \bar {\boldsymbol {x}}]\) to obtain
Finally, combining this result with (147), we obtain
which proves (89).
Sensitivity problem
Dirichlet boundary condition
Using the perturbations \(\bar {\boldsymbol {\epsilon }}_{\mathrm {d}} + \mathrm {d}\bar {\boldsymbol {\epsilon }}_{\mathrm {d}}\) and \(\bar {p} + \mathrm {d}\bar {p}\) in the linearized form of (98) at equilibrium we obtain
which must hold for any given \(\mathrm {d}\bar {\boldsymbol {\epsilon }}_{\mathrm {d}}\) and \(\mathrm {d}\bar {p}\). We thus consider the cases:

\(\mathrm {d}\bar {\boldsymbol {\epsilon }}_{\mathrm {d}} = \boldsymbol {G}_{k}\) while \(\mathrm {d}\bar {p} = 0\) (with \(\boldsymbol {u}^{{\mathrm {M}}(k)}_{\mathrm {d}} \overset {\text {def}}{=} \boldsymbol {G}_{k} \cdot [\!\boldsymbol {x}  \bar {\boldsymbol {x}}]\)): For k=1,…,8, solve for the sensitivities \(\boldsymbol {u}_{\mathrm {d}}^{\mathrm {s}(k)}\), \(p_{\mathrm {d}}^{(k)}\), \(\bar {e}_{\mathrm {d}}^{(k)}\) from the system
((151a))((151b))((151c)) 
\(\mathrm {d}\bar {p} = 1\) while \(\mathrm {d}\bar {\boldsymbol {\epsilon }}_{\mathrm {d}} = \boldsymbol {0}\): Solve for the sensitivities \(\boldsymbol {u}_{\mathrm {p}}^{\mathrm {s}}\), p _{p}, \(\bar {e}_{\mathrm {p}}\) from the system
((152a))((152b))((152c))
Upon using the identity , we deduce the tangent operators from
Tangent operators associated with \(\bar {e}\) are obtained directly from the sensitivity analysis
Neumann boundary condition
Using the perturbations \(\bar {\boldsymbol {\epsilon }}_{\mathrm {d}} + \mathrm {d}\bar {\boldsymbol {\epsilon }}_{\mathrm {d}}\) and \(\bar {p} + \mathrm {d}\bar {p}\) in the linearized form of (107) at equilibrium we obtain
which must hold for any given \(\bar {\boldsymbol {\epsilon }}_{\mathrm {d}}\) and \(\mathrm {d}\bar {p}\). We thus consider the cases:

\(\mathrm {d}\bar {\boldsymbol {\epsilon }}_{\mathrm {d}} = \boldsymbol {G}_{k}\) while \(\mathrm {d}\bar {p} = 0\): For k=1,…,8, solve for the sensitivities \(\boldsymbol {u}_{\mathrm {d}}^{(k)}\), \(p_{\mathrm {d}}^{(k)}\), \(\bar {\boldsymbol {\sigma }}_{\mathrm {d},\mathrm {d}}^{(k)}\) from the system
((156a))((156b))((156c)) 
\(\mathrm {d}\bar {p} = 1\) while \(\mathrm {d}\bar {\boldsymbol {\epsilon }}_{\mathrm {d}} = \boldsymbol {0}\): Solve for the sensitivities u _{p}, p _{p}, \(\bar {\boldsymbol {\sigma }}_{\mathrm {d},\mathrm {p}}\) from the system
((157a))((157b))((157c))
Directly from the solution we obtain tangent operators associated with \(\bar {\boldsymbol {\sigma }}_{\mathrm {d}}\):
For tangent operators associated with \(\bar {e}\), we obtain
Abbreviations
 RVE:

Representative volume element
 SVE:

Statistical volume element
 VCH:

Variationally consistent homogenization
 VMS:

Variational multiScale
 VCMC:

Variationally consistent macrohomogeneity condition
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The work was funded by the Swedish Research Council.
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The theory represents joint work by all authors. MÖ developed the computer code for the numerical simulations. All authors read and approved the final manuscript.
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Öhman, M., Runesson, K. & Larsson, F. On the variationally consistent computational homogenization of elasticity in the incompressible limit. Adv. Model. and Simul. in Eng. Sci. 2, 1 (2015). https://doi.org/10.1186/s4032301400171
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Keywords
 Multiscale
 Computational homogenization
 Incompressibility
 Mixed variational formulations