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
$$ \boldsymbol{\sigma}_{\mathrm{d}} = \hat{\boldsymbol{\sigma}}_{\mathrm{d}}(\boldsymbol{\epsilon}_{\mathrm{d}}[\!\boldsymbol{u}]\!), \quad \boldsymbol{u}\cdot\boldsymbol{\nabla} = e = \hat{e}(p) $$
((1))
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:
$$\begin{array}{*{20}l} \left[\hat{\boldsymbol{\sigma}}_{\mathrm{d}}(\boldsymbol{\epsilon}_{\mathrm{d}}[\!\boldsymbol{u}]\!)p\boldsymbol{I}\right]\cdot\boldsymbol{\nabla} & = \boldsymbol{f} \quad\qquad\,\text{in}\,\, \Omega \end{array} $$
((2a))
$$\begin{array}{*{20}l} \boldsymbol{u}\cdot\boldsymbol{\nabla} + \hat{e}(p) & = 0 \qquad\quad\,\text{in}\,\, \Omega \end{array} $$
((2b))
$$\begin{array}{*{20}l} \boldsymbol{u} & = \boldsymbol{u}_{\text{pre}} \quad\quad\text{on}\,\, \Gamma^{\mathrm{D}} \end{array} $$
((2c))
$$\begin{array}{*{20}l} \boldsymbol{t}\overset{\text{def}}{=}\left[\hat{\boldsymbol{\sigma}}_{\mathrm{d}}(\boldsymbol{\epsilon}_{\mathrm{d}} [\!\boldsymbol{u}]\!)p\boldsymbol{I}\right]\cdot\boldsymbol{n} & = \boldsymbol{t}_{\text{pre}} \quad \quad\,\text{on}\,\, \Gamma^{\mathrm{N}} \end{array} $$
((2d))
The corresponding weak format is: Find \(\boldsymbol {u}\in \mathbb {U}, p\in \mathbb {P}\) s.t.
$$\begin{array}{*{20}l} a(\boldsymbol{u};\delta\boldsymbol{u}) + b(p,\delta\boldsymbol{u}) &= l(\delta\boldsymbol{u}) &\quad& \forall \delta\boldsymbol{u} \in \mathbb{U}^{0} \end{array} $$
((3a))
$$\begin{array}{*{20}l} b(\delta p,\boldsymbol{u}) + c^{*}(p;\delta p) &= 0 &\quad& \forall \delta p \in \mathbb{P} \end{array} $$
((3b))
where
$$\begin{array}{*{20}l} a(\boldsymbol{v};\boldsymbol{w}) &\overset{\text{def}}{=} \int_{\Omega} \boldsymbol{\epsilon}_{\mathrm{d}}[\!\boldsymbol{w}]: \hat{\boldsymbol{\sigma}}_{\mathrm{d}}(\boldsymbol{\epsilon}_{\mathrm{d}}[\!\boldsymbol{v}]\!) \mathrm{d} V \end{array} $$
((4))
$$\begin{array}{*{20}l} b(q,\boldsymbol{v}) &\overset{\text{def}}{=}  \int_{\Omega} q\,\boldsymbol{v}\cdot\boldsymbol{\nabla} \mathrm{d} V \end{array} $$
((5))
$$\begin{array}{*{20}l} c^{*}(q;r) &\overset{\text{def}}{=} \int_{\Omega} r\,\hat{e}(q) \mathrm{d} V \end{array} $$
((6))
$$\begin{array}{*{20}l} l(\boldsymbol{v}) &\overset{\text{def}}{=} \int_{\Omega} \boldsymbol{v}\cdot\boldsymbol{f} \mathrm{d} V + \int_{\Gamma^{\mathrm{N}}} \boldsymbol{v}\cdot \boldsymbol{t}_{\text{pre}} \mathrm{d} S \end{array} $$
((7))
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)
$$ \Pi(\boldsymbol{u},p) \overset{\text{def}}{=} \Lambda(\boldsymbol{u},p)  l(\boldsymbol{u}) \quad\text{with}\quad \Lambda(\boldsymbol{u},p) \overset{\text{def}}{=} \int_{\Omega} \left[\psi_{\mathrm{u}}(\boldsymbol{\epsilon}_{\mathrm{d}}[\!\boldsymbol{u}]\!)  p\,\boldsymbol{u}\cdot\boldsymbol{\nabla} + \psi_{\mathrm{p}}^{*}(p)\right] \mathrm{d} V $$
((8))
where ψ
_{
u
}(ε
_{d}) and \(\psi _{p}^{*}(p)\) are constitutive energy densities^{a} such that
$$ \hat{\boldsymbol{\sigma}}_{\mathrm{d}}(\boldsymbol{\epsilon}_{\mathrm{d}})= \frac{\partial\psi_{\mathrm{u}}(\boldsymbol{\epsilon}_{\mathrm{d}})}{\partial\boldsymbol{\epsilon}_{\mathrm{d}}}, \quad \hat{e}(p)=\frac{\partial\psi_{\mathrm{p}}^{*}(p)}{\partial p} $$
((9))
The stationarity conditions of Π(u,p) are
$$\begin{array}{*{20}l} \Pi'_{u}(\boldsymbol{u},p;\delta\boldsymbol{u}) &= a(\boldsymbol{u};\delta\boldsymbol{u}) + b(p,\delta\boldsymbol{u})  l(\delta\boldsymbol{u}) =0 &\quad& \forall \delta\boldsymbol{u} \in \mathbb{U}^{0} \end{array} $$
((10a))
$$\begin{array}{*{20}l} \Pi'_{p}(\boldsymbol{u},p;\delta p) &= b(\delta p,\boldsymbol{u}) + c^{*}(p;\delta p) = 0 &\quad& \forall \delta p \in \mathbb{P} \end{array} $$
((10b))
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
_{
#
}
$$ y \to \langle y \rangle_{\#} \overset{\text{def}}{=} \frac{1}{\Gamma_\#} \int_{\Gamma_\#} y\, \mathrm{d}S $$
((12))
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
$$\begin{array}{*{20}l} a\!\left(\tilde{\boldsymbol{u}}\!\left\{\boldsymbol{u}^{\mathrm{M}},p^{\mathrm{M}}\right\};\delta\boldsymbol{u}^{\mathrm{M}}\right) + b\!\left(\tilde{p}\!\left\{\boldsymbol{u}^{\mathrm{M}},p^{\mathrm{M}}\right\},\delta\boldsymbol{u}^{\mathrm{M}}\right) &= l\!\left(\delta\boldsymbol{u}^{\mathrm{M}}\right) &\;\;& \forall \delta\boldsymbol{u}^{\mathrm{M}} \in \mathbb{U}^{{\mathrm{M}},0} \end{array} $$
((24a))
$$\begin{array}{*{20}l} b\!\left(\delta p^{\mathrm{M}},\tilde{\boldsymbol{u}}\!\left\{\boldsymbol{u}^{\mathrm{M}},p^{\mathrm{M}}\right\}\right) + c^{*}\!\left(\tilde{p}\!\left\{\boldsymbol{u}^{\mathrm{M}},p^{\mathrm{M}}\right\};\delta p^{\mathrm{M}}\right) &= 0 &\;\;& \forall \delta p^{\mathrm{M}} \in \mathbb{P}^{\mathrm{M}} \end{array} $$
((24b))
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
$$\begin{array}{*{20}l} \bar{a}(\bar{\boldsymbol{u}},\bar{p};\delta\bar{\boldsymbol{u}}) + \bar{b}(\bar{p},\delta\bar{\boldsymbol{u}}) &= \bar{l}(\delta\bar{\boldsymbol{u}}) \quad\quad \forall \delta\bar{\boldsymbol{u}} \in \bar{\mathbb{U}}^{0} \end{array} $$
((37a))
$$\begin{array}{*{20}l} \bar{b}(\delta\bar{p},\bar{\boldsymbol{u}}) + \bar{c}^{*}(\bar{\boldsymbol{u}},\bar{p};\delta\bar{p}) &= 0 \quad\quad\qquad\!\! \forall \delta\bar{p} \in \bar{\mathbb{P}} \end{array} $$
((37b))
where
$$\begin{array}{*{20}l} \bar{a}(\bar{\boldsymbol{u}},\bar{p};\bar{\boldsymbol{w}}) &\overset{\text{def}}{=} \int_{\Omega} \boldsymbol{\epsilon}_{\mathrm{d}}[\!\bar{\boldsymbol{w}}]: \bar{\boldsymbol{\sigma}}_{\mathrm{d}}\{\bar{\boldsymbol{u}}, \bar{p}\}\, \mathrm{d} V \end{array} $$
((38))
$$\begin{array}{*{20}l} \bar{b}(\bar{q},\bar{\boldsymbol{u}}) &\overset{\text{def}}{=}  \int_{\Omega} \bar{q}\,\bar{\boldsymbol{u}}\cdot\boldsymbol{\nabla}\, \mathrm{d} V \end{array} $$
((39))
$$\begin{array}{*{20}l} \bar{c}^{*}(\bar{\boldsymbol{u}},\bar{p};\bar{r}) &\overset{\text{def}}{=} \int_{\Omega} \bar{r}\,\bar{e}\{\bar{\boldsymbol{u}}, \bar{p}\}\, \mathrm{d} V \end{array} $$
((40))
$$\begin{array}{*{20}l} \bar{l}(\bar{\boldsymbol{u}}) &\overset{\text{def}}{=} \int_{\Omega} \bar{\boldsymbol{u}}\cdot\bar{\boldsymbol{f}}\, \mathrm{d} V + \int_{\Gamma^{\mathrm{N}}} \bar{\boldsymbol{u}}\cdot\bar{\boldsymbol{t}}_{\text{pre}}\, \mathrm{d} S \end{array} $$
((41))
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
$$ \boldsymbol{t} = \bar{\boldsymbol{\tau}}[\!\boldsymbol{t}]\cdot\boldsymbol{n}\, + \,[\!\boldsymbol{t}  \bar{\boldsymbol{\tau}}[\!\boldsymbol{t}]\cdot\boldsymbol{n}] $$
((56))
and the fact that
$$ \bar{\boldsymbol{\tau}}[\!\boldsymbol{t}  \bar{\boldsymbol{\tau}}[\boldsymbol{t}]\cdot\boldsymbol{n}] = \bar{\boldsymbol{\tau}}[\!\boldsymbol{t}]  \bar{\boldsymbol{\tau}}[\!\bar{\boldsymbol{\tau}}[\boldsymbol{t}]\cdot\boldsymbol{n}] = \bar{\boldsymbol{\tau}}[\!\boldsymbol{t}]  \bar{\boldsymbol{\tau}}[\!\boldsymbol{t}] = \boldsymbol{0} $$
((57))
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
$$  \boldsymbol{u}\cdot\boldsymbol{\nabla} + \hat{e}(p) + \bar{\lambda} = 0, \quad \bar{\lambda}\in\mathbb{R} $$
((62))
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
$$\begin{array}{*{20}l} \boldsymbol{u}^{\mathrm{s}}\{\bar{\boldsymbol{\epsilon}}_{\mathrm{d}}, \bar{p}\} &= \boldsymbol{u}\{\bar{\boldsymbol{\epsilon}}_{\mathrm{d}}, \bar{p}\}  \bar{\boldsymbol{\epsilon}}_{\mathrm{d}} \cdot[\!\boldsymbol{x}  \bar{\boldsymbol{x}}]  \bar{e}\{\bar{\boldsymbol{\epsilon}}_{\mathrm{d}}, \bar{p}\}\,\boldsymbol{x}_{\mathrm{m}} \end{array} $$
((74a))
$$\begin{array}{*{20}l} p^{\mathrm{s}}\{\bar{\boldsymbol{\epsilon}}_{\mathrm{d}}, \bar{p}\} &= p\{\bar{\boldsymbol{\epsilon}}_{\mathrm{d}}, \bar{p}\}  \bar{p} \end{array} $$
((74b))
$$\begin{array}{*{20}l} \boldsymbol{t}^{\mathrm{s}}\{\bar{\boldsymbol{\epsilon}}_{\mathrm{d}}, \bar{p}\} &= \boldsymbol{t}\{\bar{\boldsymbol{\epsilon}}_{\mathrm{d}}, \bar{p}\}  \left[\bar{\boldsymbol{\sigma}}_{\mathrm{d}}\{\bar{\boldsymbol{\epsilon}}_{\mathrm{d}}, \bar{p}\}  \bar{p}\boldsymbol{I}\right]\cdot \boldsymbol{n} \end{array} $$
((74c))
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
$$ \delta\check{\boldsymbol{\tau}}:\left[\check{\boldsymbol{h}}  \bar{\boldsymbol{\epsilon}}_{\mathrm{d}}  \frac13\bar{e}\boldsymbol{I}\right] = 0 \quad \forall \delta\check{\boldsymbol{\tau}} \in \mathbb{R}^{3\times 3} \quad\implies\quad \check{\boldsymbol{h}} = \bar{\boldsymbol{\epsilon}}_{\mathrm{d}} + \frac13 \bar{e}\,\boldsymbol{I}. $$
((79))
Next, testing (71a) with \(\delta \check {\boldsymbol {h}}\) we obtain
which gives
$$ \delta\check{\boldsymbol{h}} :[\!\bar{\boldsymbol{\sigma}}_{\mathrm{d}}  \check{p} \boldsymbol{I}  \check{\boldsymbol{\tau}}] = 0 \quad \forall \delta\check{\boldsymbol{h}} \in {\mathbb{R}}^{3\times 3} \quad\implies\quad \check{\boldsymbol{\tau}} = \bar{\boldsymbol{\sigma}}_{\mathrm{d}}  \check{p} \boldsymbol{I}. $$
((81))
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
$$ \boldsymbol{u}=\boldsymbol{u}\{{\bar{\boldsymbol{\epsilon}}}_{\mathrm{d}}, \bar{p}\}, \,\, p=p\{{\bar{\boldsymbol{\epsilon}}}_{\mathrm{d}}, \bar{p}\}, \,\, \boldsymbol{t}=\boldsymbol{t}\{{\bar{\boldsymbol{\epsilon}}}_{\mathrm{d}}, \bar{p}\}, \,\, \bar{e}=\bar{e}\{{\bar{\boldsymbol{\epsilon}}}_{\mathrm{d}}, \bar{p}\} $$
((86))
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
$$ \mathrm{d}\bar{\boldsymbol{\sigma}}_{\mathrm{d}} = \bar{\textbf{\textsf{E}}} : \mathrm{d}\bar{\boldsymbol{\epsilon}}_{\mathrm{d}} + \bar{\boldsymbol{E}}\, \mathrm{d}\bar{p}, \quad \mathrm{d}\bar{e} = \bar{\boldsymbol{C}} : \mathrm{d}\bar{\boldsymbol{\epsilon}}_{\mathrm{d}}  \bar{C}\, \mathrm{d}\bar{p} $$
((110))
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
$$ \boldsymbol{G}_{1} = \frac{1}{\sqrt{6}}\left[\begin{array}{ccc} 2 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{array}\right],\; \boldsymbol{G}_{2} = \left[\begin{array}{ccc} 0 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right],\; \boldsymbol{G}_{3} = \left[\begin{array}{ccc} 0 & 1 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right],\; \ldots,\; \boldsymbol{G}_{8} = \left[\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 1 & 0\end{array}\right]. $$
((112))
with respect to this basis, we have the representation
$$ \boldsymbol{\epsilon}_{\mathrm{d}} = \sum\limits_{k} (\boldsymbol{\epsilon}_{\mathrm{d}})_{k} \, \boldsymbol{G}_{k},\quad (\boldsymbol{\epsilon}_{\mathrm{d}})_{k} = \boldsymbol{G}_{k} : \boldsymbol{\epsilon}_{\mathrm{d}} $$
((113))
$$ \boldsymbol{\sigma}_{\mathrm{d}} = \sum\limits_{k} (\boldsymbol{\sigma}_{\mathrm{d}})_{k} \, \boldsymbol{G}_{k},\quad (\boldsymbol{\sigma}_{\mathrm{d}})_{k} = \boldsymbol{G}_{k} : \boldsymbol{\sigma}_{\mathrm{d}} $$
((114))
Next, we express the pertinent sensitivities via the representations
$$\begin{array}{*{20}l} \mathrm{d}\boldsymbol{u} &= \sum\limits_{k} \boldsymbol{u}_{\mathrm{d}}^{(k)} (\mathrm{d}\bar{\boldsymbol{\epsilon}}_{\mathrm{d}})_{k} + \boldsymbol{u}_{\mathrm{p}}\,\mathrm{d}\bar{p} \end{array} $$
((115a))
$$\begin{array}{*{20}l} \mathrm{d} p &= \sum\limits_{k} p_{\mathrm{d}}^{(k)} (\mathrm{d}\bar{\boldsymbol{\epsilon}}_{\mathrm{d}})_{k} + p_{\mathrm{p}}\,\mathrm{d}\bar{p} \end{array} $$
((115b))
$$\begin{array}{*{20}l} \mathrm{d}\boldsymbol{t} &= \sum\limits_{k} \boldsymbol{t}_{\mathrm{d}}^{(k)} (\mathrm{d}\bar{\boldsymbol{\epsilon}}_{\mathrm{d}})_{k} + \boldsymbol{t}_{\mathrm{p}}\,\mathrm{d}\bar{p} \end{array} $$
((115c))
$$\begin{array}{*{20}l} \mathrm{d}\bar{e} &= \sum_{k} \bar{e}_{\mathrm{d}}^{(k)} (\mathrm{d}\bar{\boldsymbol{\epsilon}}_{\mathrm{d}})_{k} + \bar{e}_{\mathrm{p}}\,\mathrm{d}\bar{p} \end{array} $$
((115d))
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

\(\mathrm {d}\bar {p} = 1\) 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
Now, upon linearizing (90) and using the representation for dt in (115c), we obtain
Directly from (115d), we obtain
$$ \mathrm{d}\bar{e} = \underbrace{\sum\limits_{k} \bar{e}_{\mathrm{d}}^{(k)}\, \boldsymbol{G}_{k}}_{\bar{\boldsymbol{C}}} : \mathrm{d} \bar{\boldsymbol{\epsilon}}_{\mathrm{d}} + \underbrace{\bar{e}_{\mathrm{p}}}_{\bar{C}}\,\mathrm{d}\bar{p} $$
((120))
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
$$ \bar{\boldsymbol{\sigma}}_{\mathrm{d}} = \bar{\textbf{\textsf{E}}} : \bar{\boldsymbol{\epsilon}}_{\mathrm{d}} + \bar{\boldsymbol{E}}\, \bar{p}, \quad \bar{e} = \bar{\boldsymbol{C}}:\bar{\boldsymbol{\epsilon}}_{\mathrm{d}}  \bar{C}\,\bar{p} $$
((123))
whereby we obtain