Novel multiscale models in a multicontinuum approach to divide and conquer strategies

Abstract

This contribution presents a comprehensive, in-depth analysis of the solution of the mechanical equilibrium problem for a generic solid with microstructure. The exact solution to this problem, referred to here as the reference solution, corresponds to the full-scale model of the problem that takes into account the kinematics and constitutive behavior of its entire microstructure. The analysis is carried out based on the Principle of Multiscale Virtual Power (PMVP) previously proposed by the authors. The PMVP provides a robust theoretical setting whereby the strong links between the reference solution and solutions of the mechanical equilibrium obtained using coarser scale models are brought to light. In this context, some fundamental properties of coarser scale solutions are identified by means of variational arguments. These findings unveil a new homogenization landscape for Representative Volume Element (RVE) multiscale theories, leading to the construction of new Minimal Kinematical Restriction (MKR)-based models where either displacements or tractions may be prescribed on the RVE boundary. A careful observation of the aforementioned landscape leads naturally to the proposal of a new, multicontinuum strategy (a generalized continuum counterpart of multigrid strategies) to approximate the reference solution at low computational cost. In the proposed strategy, the mechanical interactions among neighboring microcells are accounted for in an iterative fashion by means of suitably chosen boundary conditions enforced alternately on the new MKR-based models. The proposed developments are presented assuming a classical continuum at all scales, but the results are equally valid when different kinematical and constitutive assumptions are made at different scales.

Appendices

Appendix A: The MKR multiscale model

According to Blanco et al. (2014, 2016b), Taroco et al. (2020), a kinematically admissible displacement $\textbf{u}_\mu \in \mathcal {V}_\mu$ is characterized by the Minimal Kinematical Restriction multiscale model (MKR model), which satisfies the Principle of Multiscale Kinematical Admissibility. This implies that the following relations are satisfied: $\textbf{u}_M\vert _{\textbf{x}_i}=\mathcal {H}_\mu ^{\mathcal {V}}(\textbf{u}_\mu )$ and $\textbf{G}_M\vert _{\textbf{x}_i}= \mathcal {H}_\mu ^{\mathcal {W}}(\nabla \textbf{u}_\mu )$, leading to the kinematical restrictions

$$\begin{aligned} \mathcal {H}_\mu ^{\mathcal {V}}(\widetilde{\textbf{u}}_\mu )= & {} \frac{1}{\vert \mathcal {P}_\mu ^{i,s}\vert }\int \limits _{\mathcal {P}_\mu ^{i,s}} \widetilde{\textbf{u}}_\mu \, d\Omega _\mu = \textbf{0}, \end{aligned}$$

(A1)

$$\begin{aligned} \mathcal {H}_\mu ^{\mathcal {W}}(\nabla \widetilde{\textbf{u}}_\mu )= & {} \frac{1}{\vert \mathcal {P}_\mu ^{i,s}\vert } \left[ \int \limits _{\mathcal {P}_\mu ^{i,s}}\nabla \widetilde{\textbf{u}}_\mu \, d\Omega _\mu - \int \limits _{\partial \mathcal {P}_\mu ^{i, s, v_i}} \widetilde{\textbf{u}}_\mu \otimes \textbf{n}_\mu ^{v_i} \, d\partial \Omega _\mu \right. \nonumber \\{} & {} \left. - \int \limits _{\partial \mathcal {P}_\mu ^{i, s, v_b}} \widetilde{\textbf{u}}_\mu \otimes \textbf{n}_\mu ^{v_b} \, d\partial \Omega _\mu - \int \limits _{\partial \mathcal {P}_\mu ^{i,s, b}} \widetilde{\textbf{u}}_\mu \otimes \overline{\textbf{n}}_\mu \, d\partial \Omega _\mu \right] \nonumber \\= & {} \frac{1}{\vert \mathcal {P}_\mu ^{i,s}\vert } \int \limits _{\partial \mathcal {P}_\mu ^{i,s, b}} \widetilde{\textbf{u}}_\mu \otimes (\textbf{n}_\mu - \overline{\textbf{n}}_\mu ) \, d\partial \Omega _\mu \nonumber \\= & {} \mathcal {H}_{\mu ,\partial \mathcal {P}_\mu ^{i,s, b}}^{\mathcal {W}}(\widetilde{\textbf{u}}_\mu \vert _{\partial \mathcal {P}_\mu ^{i,s, b}}) = \textbf{O}. \end{aligned}$$

(A2)

Then, the kinematically admissible displacements for the MKR model live in the linear manifold $Kin_{\textbf{u}_\mu }^{MKR}$ characterized by

$$\begin{aligned} Kin_{\textbf{u}_\mu }^{MKR}= & {} \{ \textbf{u}_\mu \in \mathcal {V}_\mu ; \mathcal {H}_\mu ^{\mathcal {V}}(\textbf{u}_\mu )=\textbf{u}_M\vert _{\textbf{x}_i}, \mathcal {H}_\mu ^{\mathcal {W}}(\nabla \textbf{u}_\mu )= \textbf{G}_M\vert _{\textbf{x}_i} \} \nonumber \\= & {} \{ \textbf{u}_\mu \in \mathcal {V}_\mu ; \textbf{u}_\mu = \textbf{u}_M\vert _{\textbf{x}_i} + \textbf{G}_M\vert _{\textbf{x}_i} (\textbf{y} - \textbf{y}_G) + \widetilde{\textbf{u}}_\mu , \nonumber \\{} & {} \mathcal {H}_\mu ^{\mathcal {V}}(\widetilde{\textbf{u}}_\mu )= \textbf{0}, \mathcal {H}_{\mu ,\partial \mathcal {P}_\mu ^{i,s, b}}^{\mathcal {W}}(\widetilde{\textbf{u}}_\mu \vert _{\partial \mathcal {P}_\mu ^{i,s, b}}) = \textbf{O} \} \nonumber \\= & {} \textbf{u}_\mu ^0 + Var_{\widetilde{\textbf{u}}_\mu }^{MKR}, \end{aligned}$$

(A3)

where $\textbf{u}_\mu ^0$ is an arbitrary element of $Kin_{\textbf{u}_\mu }^{MKR}$, and $Var_{\widetilde{\textbf{u}}_\mu }^{MKR}$ is given by

$$\begin{aligned} Var_{\widetilde{\textbf{u}}_\mu }^{MKR}= & {} \{ \textbf{v}_\mu \in \mathcal {V}_\mu ; \mathcal {H}_\mu ^{\mathcal {V}}(\textbf{v})= \textbf{0}, \mathcal {H}_{\mu ,\partial \mathcal {P}_\mu ^{i,s, b}}^{\mathcal {W}}(\textbf{v}\vert _{\partial \mathcal {P}_\mu ^{i,s, b}}) = \textbf{O} \}. \end{aligned}$$

(A4)

Appendix B: Principle of multiscale virtual power

The energetic consistency between scales is satisfied through the formulation of the Principle of Multiscale Virtual Power (PMVP). This balance of power between macro- and microscales was originally proposed in Mandel (1971), Hill (1972) through the so-called Hill–Mandel principle of Macrohomogeneity, which was claimed to hold for the true powers exerted at both scales. In Blanco et al. (2014, 2016b), Taroco et al. (2020), the PMVP was postulated by re-casting the Hill–Mandel principle in a variational setting.

The PMVP applied to the isolated microcell $\mathcal {P}_\mu ^i$ within the context of the MKR Model, is given by

$$\begin{aligned} \textbf{P}_M\vert _{\textbf{x}_i} \cdot \widehat{\textbf{G}}_M\vert _{\textbf{x}_i} - \textbf{b}_M\vert _{\textbf{x}_i} \cdot \widehat{\textbf{u}}_M\vert _{\textbf{x}_i}{} & {} = \frac{1}{\vert \mathcal {P}_\mu ^i\vert } \bigg [ \int \limits _{\mathcal {P}_\mu ^{i,s}} (\textbf{P}_\mu (\textbf{u}_\mu ) \cdot (\widehat{\textbf{G}}_M\vert _{\textbf{x}_i} + \nabla \textbf{v}) \nonumber \\{} & {} \qquad - \textbf{b}_\mu \cdot (\widehat{\textbf{u}}_M\vert _{\textbf{x}_i} + \widehat{\textbf{G}}_M\vert _{\textbf{x}_i}(\textbf{y} - \textbf{y}_G) + \textbf{v})) \bigg ] d\Omega _\mu , \nonumber \\{} & {} \forall (\widehat{\textbf{u}}_M\vert _{\textbf{x}_i}, \widehat{\textbf{G}}_M\vert _{\textbf{x}_i}, \textbf{v}) \in \mathbb {R}_{\mathcal {V}_M}^{\textbf{x}_i} \times \mathbb {R}_{\mathcal {W}_M}^{\textbf{x}_i} \times Var_{\widetilde{\textbf{u}}_\mu }^{MKR}.\qquad \end{aligned}$$

(B5)

Using standard variational arguments, (B5) yields

• $\textbf{P}_M\vert _{\textbf{x}_i}$-Homogenization ($(\textbf{0}, \forall \widehat{\textbf{G}}_M\vert _{\textbf{x}_i}, \textbf{0})\in \mathbb {R}_{\mathcal {V}_M}^{\textbf{x}_i} \times \mathbb {R}_{\mathcal {W}_M}^{\textbf{x}_i} \times Var_{\widetilde{\textbf{u}}_\mu }^{MKR}$)

  $$\begin{aligned}{} & {} \textbf{P}_M\vert _{\textbf{x}_i} = \frac{1}{\vert \mathcal {P}_\mu ^i\vert } \left[ \int \limits _{\mathcal {P}_\mu ^{i,s}} (\textbf{P}_\mu (\textbf{u}_\mu ) - \textbf{b}_\mu \otimes (\textbf{y} - \textbf{y}_G)) \, d\Omega _\mu \right] . \end{aligned}$$

  (B6)

• $\textbf{b}_M\vert _{\textbf{x}_i}$-Homogenization ($(\forall \widehat{\textbf{u}}_M\vert _{\textbf{x}_i}, \textbf{O},\textbf{0}) \in \mathbb {R}_{\mathcal {V}_M}^{\textbf{x}_i} \times \mathbb {R}_{\mathcal {W}_M}^{\textbf{x}_i} \times Var_{\widetilde{\textbf{u}}_\mu }^{MKR}$)

  $$\begin{aligned} \textbf{b}_M\vert _{\textbf{x}_i} = \frac{1}{\vert \mathcal {P}_\mu ^i\vert } \int \limits _{\mathcal {P}_\mu ^{i,s}} \textbf{b}_\mu \, d\Omega _\mu . \end{aligned}$$

  (B7)

• Equilibrium of the isolated microcell $\mathcal {P}_\mu ^i$ given by the following variational problem: Find $\textbf{u}_\mu ^i \in Kin_{\textbf{u}_\mu }^{MKR}$ (or equivalent find $\widetilde{\textbf{u}}_\mu ^i \in Var_{\widetilde{\textbf{u}}_\mu }^{MKR}$) such that satisfies the following variational equation:

  $$\begin{aligned} \int \limits _{\mathcal {P}_\mu ^{i,s}} [\textbf{P}_\mu (\textbf{u}_\mu ^i)\cdot \nabla \textbf{v} - \textbf{b}_\mu \cdot \textbf{v}] d\Omega _\mu = 0 \quad \forall \textbf{v} \in Var_{\widetilde{\textbf{u}}_\mu }^{MKR}. \end{aligned}$$

  (B8)

  Let ${\varvec{\Theta }}_\mu ^{MKR}$ and ${\varvec{\Lambda }}_\mu ^{MKR}$ be the Lagrange multipliers corresponding to the kinematical restrictions $\mathcal {H}_\mu ^{\mathcal {V}}(\textbf{v})= \textbf{0}$ and $\mathcal {H}_{\mu ,\partial \mathcal {P}_\mu ^{i,s, b}}^{\mathcal {W}}(\textbf{v}\vert _{\partial \mathcal {P}_\mu ^{i,s, b}}) = \textbf{O}$ in $Var_{\widetilde{\textbf{u}}_\mu }^{MKR}$, then the above variational problem can be rewritten as follows: Given $\textbf{u}_M\vert _{\textbf{x}_i}$ and $\textbf{G}_M\vert _{\textbf{x}_i}$ find $\widetilde{\textbf{u}}_\mu ^i \in \mathcal {V}_\mu , {\varvec{\Theta }}_\mu ^{MKR} \in \mathbb {R}_{\mathcal {V}_M}^{\textbf{x}_i}$ and ${\varvec{\Lambda }}_\mu ^{MKR} \in \mathbb {R}_{\mathcal {W}_M}^{\textbf{x}_i}$ such that satisfy the following variational equation

  $$\begin{aligned}{} & {} \int \limits _{\mathcal {P}_\mu ^{i,s}} [\textbf{P}_\mu (\textbf{u}_\mu ^i)\cdot \nabla \textbf{v} - (\textbf{b}_\mu + {\varvec{\Theta }}_\mu ^{MKR})\cdot \textbf{v} ] d\Omega _\mu \nonumber \\{} & {} \quad -{\varvec{\Lambda }}_\mu ^{MKR} \cdot \int \limits _{\partial \mathcal {P}_\mu ^{i,s, b}} \textbf{v} \otimes (\textbf{n}_\mu - \overline{\textbf{n}}_\mu ) \, d\partial \Omega _\mu \nonumber \\{} & {} \quad - \widehat{{\varvec{\Theta }}}_\mu ^{MKR} \cdot \int \limits _{\mathcal {P}_\mu ^{i,s}} \widetilde{\textbf{u}}_\mu ^i \, d\Omega _\mu - \widehat{{\varvec{\Lambda }}}_\mu ^{MKR} \cdot \int \limits _{\partial \mathcal {P}_\mu ^{i,s, b}} \widetilde{\textbf{u}}_\mu ^i \otimes (\textbf{n}_\mu - \overline{\textbf{n}}_\mu ) \, d\partial \Omega _\mu = 0, \nonumber \\{} & {} \quad \quad \forall (\widehat{{\varvec{\Theta }}}_\mu ^{MKR}, \widehat{ {\varvec{\Lambda }}}_\mu ^{MKR},\textbf{v})\in \mathbb {R}_{\mathcal {V}_M}^{\textbf{x}_i} \times \mathbb {R}_{\mathcal {W}_M}^{\textbf{x}_i}\times \mathcal {V}_\mu . \end{aligned}$$

  (B9)

  The Euler–Lagrange equations associated with the above variational equilibrium problem are given by

  $$\begin{aligned}{} & {} \mathcal {H}_\mu ^{\mathcal {V}}(\widetilde{\textbf{u}}_\mu ^i)= \textbf{0}, \end{aligned}$$

  (B10)
  $$\begin{aligned}{} & {} \mathcal {H}_{\mu ,\partial \mathcal {P}_\mu ^{i,s, b}}^{\mathcal {W}}(\widetilde{\textbf{u}}_\mu ^i\vert _{\partial \mathcal {P}_\mu ^{i,s, b}}) = \textbf{O}, \end{aligned}$$

  (B11)
  $$\begin{aligned}{} & {} {\text {div}} \textbf{P}_\mu (\textbf{u}_\mu ^i) + \textbf{b}_\mu + {\varvec{{\Theta }}}_\mu ^{MKR}= \textbf{0} \hbox { in } \mathcal {H}_\mu ^{i,k} \; k = 1, \dots , n_i, \end{aligned}$$

  (B12)
  $$\begin{aligned}{} & {} \llbracket \textbf{P}_\mu (\textbf{u}_\mu ^i) \textbf{n}_\mu \rrbracket = \textbf{0} \hbox { on } \partial \mathcal {H}_\mu ^{i,k}\cap \partial \mathcal {H}_\mu ^{i,m}, k\ne m, k, m = 1, \dots , n_i, \end{aligned}$$

  (B13)
  $$\begin{aligned}{} & {} \textbf{P}_\mu (\textbf{u}_\mu ^i) \textbf{n}_\mu = {\varvec{\Lambda }}_\mu ^{MKR} (\textbf{n}_\mu - \overline{\textbf{n}}_\mu )\hbox { on } \partial \mathcal {P}_\mu ^{i,s,b}, \end{aligned}$$

  (B14)
  $$\begin{aligned}{} & {} \textbf{P}_\mu (\textbf{u}_\mu ^i) \textbf{n}_\mu = \textbf{0} \hbox { on } \partial \mathcal {P}_\mu ^{i,v}. \end{aligned}$$

  (B15)

  Since (B9) must be satisfied for all $\textbf{v} \in \mathcal {V}_\mu$ and in particular for an arbitrary constant vector $\textbf{v} = \textbf{c}$, we obtain an additional Euler–Lagrange equation characterizing the Lagrange multiplier ${\varvec{\Theta }}_\mu ^{MKR} \in \mathbb {R}_{\mathcal {V}_M}^{\textbf{x}_i}$

  $$\begin{aligned} {\varvec{\Theta }}_\mu ^{MKR} = - \frac{1}{\vert \mathcal {P}_\mu ^{i,s}\vert }\int \limits _{\mathcal {P}_\mu ^{i,s}} \textbf{b}_\mu d\Omega _\mu . \end{aligned}$$

  (B16)

  Furthermore, considering variations of the form $\textbf{v} = \textbf{A} (\textbf{y} - \textbf{y}_G)$ characterized by any constant second-order tensor $\textbf{A} \in Lin$, we also obtain

  $$\begin{aligned}{} & {} \textbf{A} \cdot \bigg [\int \limits _{\mathcal {P}_\mu ^{i,s}} [\textbf{P}_\mu (\textbf{u}_\mu ^i) - \textbf{b}_\mu \otimes (\textbf{y} - \textbf{y}_G)] d\Omega _\mu \nonumber \\{} & {} \quad - {\varvec{\Lambda }}_\mu ^{MKR} \int \limits _{\partial \mathcal {P}_\mu ^{i,s,b}} (\textbf{n}_\mu -\overline{\textbf{n}}_\mu ) \otimes (\textbf{y} - \textbf{y}_G) d\partial \Omega _\mu \bigg ] = 0 \quad \forall \textbf{A} \in Lin. \end{aligned}$$

  (B17)

  From the above expression, we obtain the Euler–Lagrange equation that characterizes the Lagrange multiplier ${\varvec{\Lambda }}_\mu ^{MKR}$

  $$\begin{aligned} {\varvec{\Lambda }}_\mu ^{MKR} \textbf{B}_\mu = \int \limits _{\mathcal {P}_\mu ^{i,s}} [\textbf{P}_\mu (\textbf{u}_\mu ^i) - \textbf{b}_\mu \otimes (\textbf{y} - \textbf{y}_G)] d\Omega _\mu , \end{aligned}$$

  (B18)

  where $\textbf{B}_\mu$ is given by

  $$\begin{aligned} \textbf{B}_\mu= & {} \int \limits _{\partial \mathcal {P}_\mu ^{i,s,b}} (\textbf{n}_\mu -\overline{\textbf{n}}_\mu ) \otimes (\textbf{y} - \textbf{y}_G) d\partial \Omega _\mu . \end{aligned}$$

  (B19)

For the previous developments, spaces $\mathbb {R}_{\mathcal {V}_M}^{\textbf{x}_i}$ and $\mathbb {R}_{\mathcal {W}_M}^{\textbf{x}_i}$ depend on the spatial dimension of the problem. For three-dimensional problems, we have $\mathbb {R}_{\mathcal {V}_M}^{\textbf{x}_i}\rightarrow {\mathbb {R}}^3$ and $\mathbb {R}_{\mathcal {W}_M}^{\textbf{x}_i}\rightarrow {\mathbb {R}}^{3\times 3}$.

Appendix C: Isolated microcell equilibrium

The equilibrium of the isolated microcell $\mathcal {P}_\mu ^i$ submitted to the prescribed displacement $\textbf{u}_\mu ^{*,D}$ at the boundary $\partial \mathcal {P}_\mu ^{i,s,b}$ and to a force system given by $\{\textbf{b}_\mu \}$ is characterized by the following variational problem: Find $\textbf{u}_\mu ^*\vert _{\mathcal {P}_\mu ^{i,s}} \in Kin_{\textbf{u}_\mu }^{*,D}$ such that satisfies the following variational equation:

$$\begin{aligned} \int \limits _{\mathcal {P}_\mu ^{i,s}} [\textbf{P}_\mu (\textbf{u}_\mu ^*)\cdot \nabla \textbf{v} - \textbf{b}_\mu \cdot \textbf{v}] d\Omega _\mu = 0 \quad \forall \textbf{v} \in Var_{\textbf{u}_\mu }^{*,D}, \end{aligned}$$

(C20)

where

$$\begin{aligned} Kin_{\textbf{u}_\mu }^{*,D} = \{\textbf{u}_\mu \in \mathcal {V}_\mu ; \textbf{u}_\mu \vert _{\partial \mathcal {P}_\mu ^{i,s,b}}= \textbf{u}_\mu ^{*,D} \}=\textbf{u}_\mu ^*\vert _{\mathcal {P}_\mu ^{i,s}} + Var_{\textbf{u}_\mu }^{*,D}, \end{aligned}$$

(C21)

and

$$\begin{aligned} Var_{\textbf{u}_\mu }^{*,D} = \{ \textbf{v} \in \mathcal {V}_\mu ; \textbf{v}\vert _{\partial \mathcal {P}_\mu ^{i,s,b}} = \textbf{0}\}. \end{aligned}$$

(C22)

The variational problem (C20) can be redefined by relaxing the kinematical restriction $\textbf{u}_\mu \vert _{\partial \mathcal {P}_\mu ^{i,s,b}}= \textbf{u}_\mu ^{*,D}$. This procedure leads to the following equivalent variational problem: Find $\textbf{u}_\mu ^*\vert _{\mathcal {P}_\mu ^{i,s}} \in \mathcal {V}_\mu$ such that satisfies the following variational equation

$$\begin{aligned}{} & {} \int \limits _{\mathcal {P}_\mu ^{i,s}} [\textbf{P}_\mu (\textbf{u}_\mu ^*)\cdot \nabla \textbf{v} - \textbf{b}_\mu \cdot \textbf{v} ] d\Omega _\mu - \int \limits _{\partial \mathcal {P}_\mu ^{i,s,b}}\textbf{t}_\mu ^{*,i} \cdot \textbf{v} d\partial \Omega _\mu \nonumber \\{} & {} \quad - \int \limits _{\partial \mathcal {P}_\mu ^{i,s,b}} \hat{\textbf{t}}_\mu \cdot (\textbf{u}_\mu ^*\vert _{\partial \mathcal {P}_\mu ^{i,s,b}} - \textbf{u}_\mu ^{*,D}) d\partial \Omega _\mu = 0 \nonumber \\{} & {} \quad \forall (\textbf{v}, \hat{\textbf{t}}_\mu ) \in \mathcal {V}_\mu \times \mathcal {V}_\mu '(\partial \mathcal {P}_\mu ^{i,s,b}), \end{aligned}$$

(C23)

where $\textbf{t}_\mu ^{*,i}$ is the (Lagrange multiplier) vector traction field over $\partial \mathcal {P}_\mu ^{i,s,b}$ associated by duality with the kinematical restriction $\textbf{u}_\mu \vert _{\partial \mathcal {P}_\mu ^{i,s,b}}= \textbf{u}_\mu ^{*,D}$, and $\hat{\textbf{t}}_\mu$ is its virtual variation.

The Euler–Lagrange equations associated to the variational problem (C23) are given by

• Taking $\textbf{v}=\textbf{0}$ and for all $\hat{\textbf{t}}_\mu \in \mathcal {V}_\mu '$, we have

  $$\begin{aligned} \textbf{u}_\mu ^*\vert _{\partial \mathcal {P}_\mu ^{i,s,b}}= \textbf{u}_\mu ^{*,D} \hbox { on } \partial \mathcal {P}_\mu ^{i,s,b}. \end{aligned}$$

  (C24)

• Taking $\hat{\textbf{t}}_\mu = \textbf{0}$ and for all $\textbf{v} \in \mathcal {V}_\mu$, we have

  $$\begin{aligned}{} & {} {\text {div}} \textbf{P}_\mu (\textbf{u}_\mu ^*) + \textbf{b}_\mu = \textbf{0} \hbox { in } \mathcal {H}_\mu ^{i,k} \; k = 1, \dots , n_i, \end{aligned}$$

  (C25)
  $$\begin{aligned}{} & {} \llbracket \textbf{P}_\mu (\textbf{u}_\mu ^*) \textbf{n}_\mu \rrbracket = \textbf{0} \hbox { on } \partial \mathcal {H}_\mu ^{i,k}\cap \partial \mathcal {H}_\mu ^{i,m}, k\ne m, k, m = 1, \dots , n_i, \end{aligned}$$

  (C26)
  $$\begin{aligned}{} & {} \textbf{P}_\mu (\textbf{u}_\mu ^*) \textbf{n}_\mu = \textbf{t}_\mu ^{*,i} \hbox { on } \partial \mathcal {P}_\mu ^{i,s,b}, \end{aligned}$$

  (C27)
  $$\begin{aligned}{} & {} \textbf{P}_\mu (\textbf{u}_\mu ^*) \textbf{n}_\mu = \textbf{0} \hbox { on } \partial \mathcal {P}_\mu ^{i,v}, \end{aligned}$$

  (C28)

Now, since (C23) must be satisfied for all $\textbf{v} \in \mathcal {V}_\mu$, it particularly holds for any arbitrary constant vector field $\textbf{v} = \textbf{c}$. Then, it results

$$\begin{aligned} \int \limits _{\partial \mathcal {P}_\mu ^{i,s,b}} \textbf{t}_\mu ^{*,i} d\partial \Omega _\mu = -\int \limits _{\mathcal {P}_\mu ^{i,s}} \textbf{b}_\mu d\Omega _\mu . \end{aligned}$$

(C29)

Also, (C23) must be satisfied for all fields of the form $\textbf{v} = \textbf{A} (\textbf{y} - \textbf{y}_G)$ characterized by any constant second-order tensor $\textbf{A} \in Lin$. Then,

$$\begin{aligned}{} & {} \textbf{A} \cdot \bigg [\int \limits _{\mathcal {P}_\mu ^{i,s}} [\textbf{P}_\mu (\textbf{u}_\mu ^*) - \textbf{b}_\mu \otimes (\textbf{y} - \textbf{y}_G)] d\Omega _\mu \nonumber \\{} & {} \quad - \int \limits _{\partial \mathcal {P}_\mu ^{i,s,b}} \textbf{t}_\mu ^{*,i} \otimes (\textbf{y} - \textbf{y}_G) d\partial \Omega _\mu \bigg ] = 0 \quad \forall \textbf{A} \in Lin. \end{aligned}$$

(C30)

From the above expression, we obtain

$$\begin{aligned} \int \limits _{\partial \mathcal {P}_\mu ^{i,s,b}} \textbf{t}_\mu ^{*,i} \otimes (\textbf{y} - \textbf{y}_G) d\partial \Omega _\mu = \int \limits _{\mathcal {P}_\mu ^{i,s}} [\textbf{P}_\mu (\textbf{u}_\mu ^*) - \textbf{b}_\mu \otimes (\textbf{y} - \textbf{y}_G)] d\Omega _\mu . \end{aligned}$$

(C31)