Unlocking the Potential of Second-order Computational Homogenisation: An Overview of Distinct Formulations and a Guide for their Implementation

Abstract

Multi-scale models enriched with the second gradient of the displacement field and based on the concept of representative volume element (RVE) are examined and discussed in detail in this paper. The development of new models relies on the Method of Multi-Scale Virtual Power from which equilibrium equations and homogenisation relations are derived employing variational arguments. Two classes of enhanced multi-scale constitutive models are analysed: standard second-order homogenisation and fully second-order homogenisation. On the computational side, the numerical treatment of second gradient equilibrium problems with a mixed approach is addressed systematically within an implicit finite element discretisation in conjunction with the full Newton–Raphson scheme for the iterative solution of the corresponding non-linear systems of equations. Different formulations available in the literature for standard second-order homogenisation are presented, along with a critical appraisal of their assumptions and limitations. A macroscopic second-gradient constitutive law with an orthotropic length scale is developed based on numerical findings. The impact of finite strains, the micro constituents’ behaviour, and the RVE morphology on the resulting macroscopic length scale parameter’s evolution is assessed. A fully second-order homogenisation-based model at finite strains is also presented. The suitability of this class of models to capture size effects due to the micro-constituent’s size is illustrated by an FE² simulation.

Appendices

Appendix 1: Determination of the Strong Form of a Second-gradient Equilibrium Problem

The strong form of the equilibrium problem may be determined from its weak version defined in Eq. (2) by developing each of the terms with integration by parts. This procedure is also described by Blanco et al. [23], Kouznetsova [88], Lesičar [97], Luscher [105].

The first term is developed as in a standard Cauchy continuum:

$$\begin{aligned} \int _{\varOmega } {\varvec{ P}}: \delta {\varvec{ F}}\ \mathrm {d}V = \int _{\varOmega } {\varvec{ P}}: \nabla _X\delta {\varvec{u}}\ \mathrm {d}V = \int _{\partial \varOmega }\left( {\varvec{ P}}\cdot {\varvec{N}}\right) \cdot \delta {\varvec{u}}\ \mathrm {d}A - \int _{\varOmega }{{\,\mathrm{div}\,}}{\varvec{ P}}\cdot \delta {\varvec{u}}\ \mathrm {d}V. \end{aligned}$$

(301)

The development of the term related to the second gradient is performed in index notation:

$$\begin{aligned} \int _{\varOmega } {\mathsf{\bf Q}}\vdots \nabla _X\nabla _X\delta {\varvec{u}}\ \mathrm {d}V&\Rightarrow \int _{\varOmega } Q_{ijk} \dfrac{\partial ^2 \delta u_i}{\partial X_j \partial X_k} \nonumber \\&= \int _{\varOmega } \left[ \dfrac{\partial }{\partial X_k} \left( Q_{ijk} \dfrac{\partial \delta u_i}{\partial X_j}\right) - \dfrac{\partial Q_{ijk}}{\partial X_k}\dfrac{\partial \delta u_i}{\partial X_j} \right] \ \mathrm {d}V \nonumber \\&= \int _{\partial \varOmega } Q_{ijk} \dfrac{\partial \delta u_i}{\partial X_j} N_k \ \mathrm {d}A - \int _{\varOmega }\dfrac{\partial Q_{ijk}}{\partial X_k}\dfrac{\partial \delta u_i}{\partial X_j} \ \mathrm {d}V \end{aligned}$$

(302)

$$\begin{aligned}&= \int _{\partial \varOmega } Q_{ijk} \dfrac{\partial \delta u_i}{\partial X_j} N_k \ \mathrm {d}A - \int _{\varOmega }\left[ \dfrac{\partial }{\partial X_j}\left( \dfrac{\partial Q_{ijk}}{\partial X_k} \delta u_i \right) - \dfrac{\partial }{\partial X_j}\left( \dfrac{\partial Q_{ijk}}{\partial X_k}\right) \delta u_i\right] \ \mathrm {d}V \nonumber \\&= \int _{\partial \varOmega } Q_{ijk} \dfrac{\partial \delta u_i}{\partial X_j} N_k \ \mathrm {d}A - \int _{\partial \varOmega } \dfrac{\partial Q_{ijk}}{\partial X_k} \delta u_i N_j \ \mathrm {d}A + \int _{\varOmega } \dfrac{\partial }{\partial X_j}\left( \dfrac{\partial Q_{ijk}}{\partial X_k}\right) \delta u_i \ \mathrm {d}V \nonumber \\&\Rightarrow \int _{\partial \varOmega } \left( {\mathsf{\bf Q}}\cdot {\varvec{N}}\right) : \nabla _X\delta {\varvec{u}}\ \mathrm {d}A - \int _{\partial \varOmega } {{\,\mathrm{div}\,}}{\mathsf{\bf Q}}\cdot {\varvec{N}} \cdot \delta {\varvec{u}}\ \mathrm {d}A + \int _{\varOmega } {{\,\mathrm{div}\,}}\left( {{\,\mathrm{div}\,}}{\mathsf{\bf Q}}\right) \cdot \delta {\varvec{u}}\ \mathrm {d}V, \nonumber \\ \end{aligned}$$

(303)

hence

$$\begin{aligned} \int _{\varOmega } {\mathsf{\bf Q}}\vdots \delta {\mathsf{\bf G}}\ \mathrm {d}V&= \int _{\varOmega } {\mathsf{\bf Q}}\vdots \nabla _X\nabla _X\delta {\varvec{u}}\ \mathrm {d}V \nonumber \\&=\int _{\partial \varOmega } \left( {\mathsf{\bf Q}}\cdot {\varvec{N}}\right) : \nabla _X\delta {\varvec{u}}\ \mathrm {d}A - \int _{\partial \varOmega } {{\,\mathrm{div}\,}}{\mathsf{\bf Q}}\cdot {\varvec{N}} \cdot \delta {\varvec{u}}\ \mathrm {d}A + \int _{\varOmega } {{\,\mathrm{div}\,}}\left( {{\,\mathrm{div}\,}}{\mathsf{\bf Q}}\right) \cdot \delta {\varvec{u}}\ \mathrm {d}V. \end{aligned}$$

(304)

The gradient $\nabla _X\delta {\varvec{u}}$ is not independent from $\delta {\varvec{u}}$ thus it must be decomposed into its normal and surface components before proceeding:

$$\begin{aligned} \nabla _X\delta {\varvec{u}}&= \nabla _X^t\delta {\varvec{u}}+ D_0\delta {\varvec{u}}\otimes {\varvec{N}} \nonumber \\&= \nabla _X\delta {\varvec{u}}\cdot \left( {\varvec{I}}-{\varvec{N}}\otimes {\varvec{N}}\right) + \nabla _X\delta {\varvec{u}}\cdot {\varvec{N}} \otimes {\varvec{N}}. \end{aligned}$$

(305)

As a consequence, the same decomposition can be applied to the divergence operator:

$$\begin{aligned} {{\,\mathrm{div}\,}}\delta {\varvec{u}}= {{\,\mathrm{div}\,}}^t\delta {\varvec{u}}+ {{\,\mathrm{div}\,}}^n\delta {\varvec{u}}, \end{aligned}$$

(306)

where ${{\,\mathrm{div}\,}}\delta {\varvec{u}}= \nabla _X\delta {\varvec{u}}: {\varvec{I}}$, ${{\,\mathrm{div}\,}}^t\delta {\varvec{u}}= \nabla _X^t\delta {\varvec{u}}: {\varvec{I}}$ and ${{\,\mathrm{div}\,}}^n\delta {\varvec{u}}= \left( D_0\delta {\varvec{u}}\otimes {\varvec{N}}\right) : {\varvec{I}}$.

Therefore, the term $\left( {\mathsf{\bf Q}}\cdot {\varvec{N}}\right) : \nabla _X\delta {\varvec{u}}$ is developed according to this decomposition. Assuming that ${\varvec{A}} = {\mathsf{\bf Q}}\cdot {\varvec{N}}$, for the sake of simplification, then

$$\begin{aligned} {\varvec{A}} : \nabla _X\delta {\varvec{u}}= {\varvec{A}} : \nabla _X^t\delta {\varvec{u}}+ {\varvec{A}} : \left( D_0\delta {\varvec{u}}\otimes {\varvec{N}}\right) . \end{aligned}$$

(307)

The second term of the right-hand-side may be simply rewritten by

$$\begin{aligned} {\varvec{A}} : \left( D_0\delta {\varvec{u}}\otimes {\varvec{N}}\right)&= D_0\delta {\varvec{u}}\cdot {\varvec{A}} \cdot {\varvec{N}}. \end{aligned}$$

(308)

Regarding the first term, it is developed as follows:

$$\begin{aligned} {\varvec{A}} : \nabla _X^t\delta {\varvec{u}}&= {\varvec{A}} : \left[ \nabla _X\delta {\varvec{u}}\cdot \left( {\varvec{I}}-{\varvec{N}}\otimes {\varvec{N}}\right) \right] = A_{ij}\dfrac{\partial u_i}{\partial X_k} \left( \delta _{kj} - N_k N_j\right) {\varvec{E}}_i\otimes {\varvec{E}}_j \nonumber \\&= A_{ij}\dfrac{\partial u_i}{\partial X_k} \left( \delta _{kj} - N_k N_j\right) {\varvec{E}}_i\otimes {\varvec{E}}_j \nonumber \\&= \dfrac{\partial \left( u_i A_{ij}\right) }{\partial X_k} \left( \delta _{kj} - N_k N_j\right) - u_i \dfrac{\partial A_{ij}}{\partial X_k} \left( \delta _{kj} - N_k N_j\right) {\varvec{E}}_i\otimes {\varvec{E}}_j \nonumber \\ \Leftrightarrow {\varvec{A}} : \nabla _X^t\delta {\varvec{u}}&= {{\,\mathrm{div}\,}}^t\left( \delta {\varvec{u}}\cdot {\varvec{A}}\right) - \delta {\varvec{u}}\cdot {{\,\mathrm{div}\,}}^t{\varvec{A}}. \end{aligned}$$

(309)

According to Brand [30]

$$\begin{aligned} \int _{\partial \varOmega } {{\,\mathrm{div}\,}}^t\left( \delta {\varvec{u}}\cdot {\varvec{A}}\right) \ \mathrm {d}A = \int _{\partial \varOmega } {{\,\mathrm{div}\,}}^t\left( {\varvec{N}}\right) {\varvec{N}}\cdot \left( \delta {\varvec{u}}\cdot {\varvec{A}}\right) \ \mathrm {d}A + \oint _{\partial \varGamma } {\varvec{m}}\cdot \left( \delta {\varvec{u}}\cdot {\varvec{A}}\right) dL, \end{aligned}$$

(310)

where ${\varvec{m}}$ denotes the external unit vector normal to the closed boundary curve $\varGamma$ and tangent to the surface, obtained as ${\varvec{m}} = {{\boldsymbol{\tau }}} \times {\varvec{N}}$, with ${{\boldsymbol{\tau }}}$ denoting the unit vector tangent to the curve. The last term vanishes in the presence of a surface $\partial \varOmega$ without the presence of edges or corners. If the surface has $n_e$ edges, then the line integral can be expressed as [88]

$$\begin{aligned} \oint _{\varGamma } {\varvec{m}}\cdot \left( \delta {\varvec{u}}\cdot {\varvec{A}}\right) dL = \sum _{i=1}^{n_e} \oint _{\varGamma ^i} \left\Vert {\varvec{m}}\cdot \left( \delta {\varvec{u}}\cdot {\varvec{A}}\right) \right\Vert dL. \end{aligned}$$

(311)

Therefore, the integral

$$\begin{aligned} \int _{\partial \varOmega } \left( {\mathsf{\bf Q}}\cdot {\varvec{N}}\right) : \nabla _X\delta {\varvec{u}}\ \mathrm {d}A =&\int _{\partial \varOmega } {{\,\mathrm{div}\,}}^t\left( {\varvec{N}}\right) \left( \left( {\mathsf{\bf Q}}\cdot {\varvec{N}}\right) \cdot {\varvec{N}} \right) \cdot \delta {\varvec{u}}\ \mathrm {d}A \nonumber \\ & - \int _{\partial \varOmega } {{\,\mathrm{div}\,}}^t\left( {\mathsf{\bf Q}}\cdot {\varvec{N}}\right) \cdot \delta {\varvec{u}}\ \mathrm {d}A + \sum _{i=1}^{n_e} \oint _{\varGamma ^i} \left\Vert \left( {\mathsf{\bf Q}}\cdot {\varvec{N}}\right) \cdot {\varvec{m}}\right\Vert \cdot \delta {\varvec{u}}dL \nonumber \\ & +\int _{\partial \varOmega } \left( {\mathsf{\bf Q}}\cdot {\varvec{N}}\right) \cdot {\varvec{N}} \cdot D_0\delta {\varvec{u}}\ \mathrm {d}A. \end{aligned}$$

(312)

Finally, including the above developments, Eq. (2) is rewritten as

$$\begin{aligned}&\int _{\varOmega } \left[ {{\,\mathrm{div}\,}}\left( {{\,\mathrm{div}\,}}{\mathsf{\bf Q}}\right) - {{\,\mathrm{div}\,}}{\varvec{ P}}\right] \cdot \delta {\varvec{u}}\ \mathrm {d}V \nonumber \\ & + \int _{\partial \varOmega ^N} \left[ {\varvec{ P}}\cdot {\varvec{N}} - {{\,\mathrm{div}\,}}{\mathsf{\bf Q}}\cdot {\varvec{N}} + {{\,\mathrm{div}\,}}^t{\varvec{N}}\left( {\mathsf{\bf Q}}\cdot {\varvec{N}}\right) \cdot {\varvec{N}} - {{\,\mathrm{div}\,}}^t\left( {\mathsf{\bf Q}}\cdot {\varvec{N}}\right) \right] \cdot \delta {\varvec{u}}\ \mathrm {d}A \nonumber \\ & + \int _{\partial \varOmega } \left( {\mathsf{\bf Q}}\cdot {\varvec{N}}\right) \cdot {\varvec{N}} \cdot D_0\delta {\varvec{u}}\ \mathrm {d}A + \sum _{i=1}^{n_e} \oint _{\varGamma ^i} \left\Vert \left( {\mathsf{\bf Q}}\cdot {\varvec{N}}\right) \cdot {\varvec{m}}\right\Vert \cdot \delta {\varvec{u}}dL \nonumber \\ =&\int _{\varOmega } {\varvec{b}}_{0}\cdot \delta {\varvec{u}}\ \mathrm {d}V + \int _{\partial \varOmega ^N}{\varvec{t}}_0\cdot \delta {\varvec{u}}\ \mathrm {d}A + \int _{\partial \varOmega ^N}{\varvec{r}}_{0}\cdot D_0\delta {\varvec{u}}\ \mathrm {d}A + \int _{\varGamma ^N} {\varvec{s}}_0 \cdot \delta {\varvec{u}}\ \mathrm {d}A. \end{aligned}$$

(313)

By standard variational arguments, since Eq. (313) must be satisfied for any admissible displacement variation $\delta {\varvec{u}}$, the strong form of the equilibrium problem is expressed by the following set of equations:

$$\begin{aligned} {\varvec{b}}_0&= {{\,\mathrm{div}\,}}\left( {{\,\mathrm{div}\,}}{\mathsf{\bf Q}}\right) - {{\,\mathrm{div}\,}}{\varvec{ P}}, \quad&\text {in } \varOmega \end{aligned}$$

(3)

$$\begin{aligned} {\varvec{t}}_0&= {\varvec{ P}}\cdot {\varvec{N}} - {{\,\mathrm{div}\,}}{\mathsf{\bf Q}}\cdot {\varvec{N}} + {{\,\mathrm{div}\,}}^t{\varvec{N}}\left( {\mathsf{\bf Q}}\cdot {\varvec{N}}\right) \cdot {\varvec{N}} - {{\,\mathrm{div}\,}}^t\left( {\mathsf{\bf Q}}\cdot {\varvec{N}}\right) , \quad&\text {on } \partial \varOmega \end{aligned}$$

(4)

$$\begin{aligned} {\varvec{r}}_0&= \left( {\mathsf{\bf Q}}\cdot {\varvec{N}}\right) \cdot {\varvec{N}}, \quad&\text {on } \partial \varOmega \end{aligned}$$

(5)

$$\begin{aligned} {\varvec{s}}_0&= \left\Vert \left( {\mathsf{\bf Q}}\cdot {\varvec{N}}\right) \cdot {\varvec{m}}\right\Vert , \quad&\text {on } \varGamma \end{aligned}$$

(6)

Appendix 2: Determination of the Strong Form of the Micro-scale Equilibrium Problem

1.1 General Formulation

The strong form of the equilibrium problem is determined by setting $\delta {\varvec{L}}= {\bf 0}$ and $\delta {\mathsf{\bf M}}= {\bf 0}$ in Eq. (228), which leads to

$$\begin{aligned} \int _{\varOmega _{\mu }} \left( {\varvec{ P}}_\mu - {\varvec{L}}\right) : \nabla _Y \delta \tilde{{\varvec{u}}}\ \mathrm {d}V + \int _{\varOmega _{\mu }} \left( {\mathsf{\bf Q}}_\mu - {\mathsf{\bf M}}\right) \vdots \nabla _Y\nabla _Y\delta \tilde{{\varvec{u}}}\ \mathrm {d}V = 0. \end{aligned}$$

(314)

Integrating by parts, the first term is developed to:

$$\begin{aligned} \int _{\varOmega _{\mu }} \left( {\varvec{ P}}_\mu - {\varvec{L}}\right) : \nabla _Y \delta \tilde{{\varvec{u}}}\ \mathrm {d}V = \int _{\partial \varOmega _{\mu }} \left( {\varvec{ P}}_\mu \cdot {\varvec{N}} - {\varvec{L}}\cdot {\varvec{N}}\right) \cdot \delta \tilde{{\varvec{u}}}\ \mathrm {d}A - \int _{\varOmega _{\mu }} {{\,\mathrm{div}\,}}{{\varvec{ P}}_\mu } \cdot \delta \tilde{{\varvec{u}}}\ \mathrm {d}V. \end{aligned}$$

(315)

Similarly, the second term may be expressed as:

$$\begin{aligned} \int _{\varOmega _{\mu }} \left( {\mathsf{\bf Q}}_\mu - {\mathsf{\bf M}}\right) \vdots \nabla _Y\nabla _Y\delta \tilde{{\varvec{u}}}\ \mathrm {d}V =&\int _{\partial \varOmega _{\mu }} \left[ \left( {\mathsf{\bf Q}}_\mu - {\mathsf{\bf M}}\right) \cdot {\varvec{N}}\right] :\nabla _Y\delta \tilde{{\varvec{u}}}\ \mathrm {d}A \nonumber \\ &- \int _{\partial \varOmega _{\mu }} {{\,\mathrm{div}\,}}\left( {\mathsf{\bf Q}}_\mu - {\mathsf{\bf M}}\right) \cdot {\varvec{N}} \cdot \delta \tilde{{\varvec{u}}}\ \mathrm {d}A + \int _{\varOmega _{\mu }} {{\,\mathrm{div}\,}}\left[ {{\,\mathrm{div}\,}}\left( {\mathsf{\bf Q}}_\mu - {\mathsf{\bf M}}\right) \right] \cdot \delta \tilde{{\varvec{u}}}\ \mathrm {d}V. \end{aligned}$$

(316)

The first term of the above right-hand-side is further developed through integration by parts and decomposition of the gradient into its normal and surface components [155], leading to:

$$\begin{aligned} \int _{\partial \varOmega _{\mu }} \left[ \left( {\mathsf{\bf Q}}_\mu - {\mathsf{\bf M}}\right) \cdot {\varvec{N}}\right] :\nabla _Y\delta \tilde{{\varvec{u}}}\ \mathrm {d}A =&\int _{\partial \varOmega _{\mu }} {{\,\mathrm{div}\,}}^t{\varvec{N}}\left\{ \left[ \left( {\mathsf{\bf Q}}_{\mu }-{\mathsf{\bf M}}\right) \cdot {\varvec{N}}\right] \cdot {\varvec{N}}\right\} \cdot \delta \tilde{{\varvec{u}}}\ \mathrm {d}A \nonumber \\ &- \int _{\partial \varOmega _{\mu }} {{\,\mathrm{div}\,}}^t\left[ \left( {\mathsf{\bf Q}}_\mu -{\mathsf{\bf M}}\right) \cdot {\varvec{N}}\right] \cdot \delta \tilde{{\varvec{u}}}\ \mathrm {d}A \nonumber \\ & + \int _{\partial \varOmega _{\mu }} \left\{ \left[ \left( {\mathsf{\bf Q}}_\mu -{\mathsf{\bf M}}\right) \cdot {\varvec{N}}\right] \cdot {\varvec{N}}\right\} \cdot D_0\delta \tilde{{\varvec{u}}}\ \mathrm {d}A \nonumber \\ & +\sum _{i}^{n_e} \oint _{\varGamma ^i} \left\Vert \left[ \left( {\mathsf{\bf Q}}_\mu -{\mathsf{\bf M}}\right) \cdot {\varvec{N}}\right] \cdot {\varvec{m}}\right\Vert \cdot \delta \tilde{{\varvec{u}}}dL. \end{aligned}$$

(317)

Including Eqs. (315)–(317) into (314), taking into account that ${\mathsf{\bf M}}$ is constant over the RVE, thus ${{\,\mathrm{div}\,}}{\mathsf{\bf M}}= 0$, results in

$$\begin{aligned}&\int _{\varOmega _{\mu }} \left( {{\,\mathrm{div}\,}}{{\,\mathrm{div}\,}}{\mathsf{\bf Q}}_\mu -{{\,\mathrm{div}\,}}{{\varvec{ P}}_\mu }\right) \cdot \delta \tilde{{\varvec{u}}}\ \mathrm {d}V \nonumber \\ &+ \int _{\partial \varOmega _{\mu }} \left\{ {\varvec{ P}}_\mu \cdot {\varvec{N}} - {\varvec{L}}\cdot {\varvec{N}} - {{\,\mathrm{div}\,}}{\mathsf{\bf Q}}_\mu \cdot {\varvec{N}} + {{\,\mathrm{div}\,}}^t{\varvec{N}}\left\{ \left[ \left( {\mathsf{\bf Q}}_{\mu }-{\mathsf{\bf M}}\right) \cdot {\varvec{N}}\right] \cdot {\varvec{N}}\right\} - {{\,\mathrm{div}\,}}^t\left[ \left( {\mathsf{\bf Q}}_\mu - {\mathsf{\bf M}}\right) \cdot {\varvec{N}}\right] \right\} \cdot \delta \tilde{{\varvec{u}}}\ \mathrm {d}A \nonumber \\ + &\int _{\partial \varOmega _{\mu }} \left\{ \left[ \left( {\mathsf{\bf Q}}_\mu -{\mathsf{\bf M}}\right) \cdot {\varvec{N}}\right] \cdot {\varvec{N}}\right\} \cdot D_0\delta \tilde{{\varvec{u}}}\ \mathrm {d}A \nonumber \\ & +\sum _{i}^{n_e} \oint _{\varGamma ^i} \left\Vert \left[ \left( {\mathsf{\bf Q}}_\mu -{\mathsf{\bf M}}\right) \cdot {\varvec{N}}\right] \cdot {\varvec{m}}\right\Vert \cdot \delta \tilde{{\varvec{u}}}dL = 0. \end{aligned}$$

(318)

Therefore, by variational arguments, the strong form of this equilibrium equation is defined as:

$$\begin{aligned} {{\,\mathrm{div}\,}}{{\,\mathrm{div}\,}}{\mathsf{\bf Q}}_\mu - {{\,\mathrm{div}\,}}{\varvec{ P}}_\mu = {\bf 0}, \quad&\text {in } \varOmega _{\mu } \end{aligned}$$

(229)

$$\begin{aligned} {\varvec{ P}}_\mu \cdot {\varvec{N}} - {\varvec{L}}\cdot {\varvec{N}} - {{\,\mathrm{div}\,}}{\mathsf{\bf Q}}_\mu \cdot {\varvec{N}} + {{\,\mathrm{div}\,}}^t{\varvec{N}}\left\{ \left[ \left( {\mathsf{\bf Q}}_{\mu }-{\mathsf{\bf M}}\right) \cdot {\varvec{N}}\right] \cdot {\varvec{N}}\right\} - {{\,\mathrm{div}\,}}^t\left[ \left( {\mathsf{\bf Q}}_\mu - {\mathsf{\bf M}}\right) \cdot {\varvec{N}}\right] = {\bf 0}, \quad&\text {on } \partial \varOmega _\mu \end{aligned}$$

(230)

$$\begin{aligned} \left[ \left( {\mathsf{\bf Q}}_\mu -{\mathsf{\bf M}}\right) \cdot {\varvec{N}}\right] \cdot {\varvec{N}} = {\bf 0}, \quad&\text {on } \partial \varOmega _\mu \end{aligned}$$

(231)

$$\begin{aligned} \left\Vert \left[ \left( {\mathsf{\bf Q}}_\mu -{\mathsf{\bf M}}\right) \cdot {\varvec{N}}\right] \cdot {\varvec{m}}\right\Vert = {\bf 0}, \quad&\text {on } \varGamma _\mu . \end{aligned}$$

(232)

1.2 Mixed Formulation

The strong form of the equilibrium problem is obtained by defining $\delta {\boldsymbol{{\lambda }}} = {\bf 0}$, $\delta {\varvec{L}}= {\bf 0}$ and $\delta {\mathsf{\bf M}}= {\bf 0}$ in Eq. (268), and developing the remaining terms as shown next. The standard decomposition of the first term is repeated here:

$$\begin{aligned} \int _{\varOmega _{\mu }} {\varvec{ P}}_\mu : \nabla _Y \delta \tilde{{\varvec{u}}}\ \mathrm {d}V = \int _{\partial \varOmega _\mu }\left( {\varvec{ P}}\cdot {\varvec{N}}\right) \cdot \delta \tilde{{\varvec{u}}}\ \mathrm {d}A - \int _{\varOmega _\mu }{{\,\mathrm{div}\,}}{\varvec{ P}}\cdot \delta \tilde{{\varvec{u}}}\ \mathrm {d}V. \end{aligned}$$

(319)

Similarly, the second term is rewritten as:

$$\begin{aligned} \int _{\varOmega _{\mu }} {\mathsf{\bf Q}}_\mu \vdots \nabla _Y^s\delta \hat{\tilde{{\varvec{ F}}}} \ \mathrm {d}V = \int _{\partial \varOmega _\mu }\left( {\mathsf{\bf Q}}\cdot {\varvec{N}}\right) : \delta \hat{\tilde{{\varvec{ F}}}} \ \mathrm {d}A - \int _{\varOmega _\mu }{{\,\mathrm{div}\,}}{\mathsf{\bf Q}}: \delta \hat{\tilde{{\varvec{ F}}}} \ \mathrm {d}V. \end{aligned}$$

(320)

Thereafter:

$$\begin{aligned} -\int _{\varOmega _{\mu }} {\boldsymbol{{\lambda }}} : \nabla _Y \delta \tilde{{\varvec{u}}}\ \mathrm {d}V = - \int _{\partial \varOmega _\mu }\left( {\boldsymbol{{\lambda }}}\cdot {\varvec{N}}\right) \cdot \delta \tilde{{\varvec{u}}}\ \mathrm {d}A + \int _{\varOmega _\mu }{{\,\mathrm{div}\,}}{\boldsymbol{{\lambda }}}\cdot \delta \tilde{{\varvec{u}}}\ \mathrm {d}V. \end{aligned}$$

(321)

The terms related to the micro-scale constraints are expressed as

$$\begin{aligned} - {\varvec{L}}: \int _{\partial \varOmega _{\mu }}\delta \tilde{{\varvec{u}}}\otimes {\varvec{N}} \ \mathrm {d}A = - \int _{\partial \varOmega _{\mu }}\left( {\varvec{L}}\cdot {\varvec{N}}\right) \cdot \delta \tilde{{\varvec{u}}}\ \mathrm {d}A, \end{aligned}$$

(322)

and

$$\begin{aligned} - {\mathsf{\bf M}}\vdots \int _{\partial \varOmega _{\mu }}\delta \hat{\tilde{{\varvec{ F}}}}\otimes {\varvec{N}} \ \mathrm {d}A = - \int _{\partial \varOmega _{\mu }}\left( {\mathsf{\bf M}}\cdot {\varvec{N}}\right) : \delta \hat{\tilde{{\varvec{ F}}}}\ \mathrm {d}A. \end{aligned}$$

(323)

Finally, including these decompositions, the following expression arises:

$$\begin{aligned}& \int _{\partial \varOmega _\mu }\left( {\varvec{ P}}\cdot {\varvec{N}} - {\boldsymbol{{\lambda }}}\cdot {\varvec{N}} - {\varvec{L}}\cdot {\varvec{N}}\right) \cdot \delta \tilde{{\varvec{u}}}\ \mathrm {d}A + \int _{\varOmega _\mu }\left( -{{\,\mathrm{div}\,}}{\varvec{ P}}+ {{\,\mathrm{div}\,}}{\boldsymbol{{\lambda }}}\right) \cdot \delta \tilde{{\varvec{u}}}\ \mathrm {d}V \nonumber \\ &\quad +\int _{\partial \varOmega _\mu }\left( {\mathsf{\bf Q}}\cdot {\varvec{N}} - {\mathsf{\bf M}}\cdot {\varvec{N}}\right) : \delta \hat{\tilde{{\varvec{ F}}}} \ \mathrm {d}A + \int _{\varOmega _\mu }\left( - {{\,\mathrm{div}\,}}{\mathsf{\bf Q}}+ {\boldsymbol{{\lambda }}}\right) : \delta \hat{\tilde{{\varvec{ F}}}} \ \mathrm {d}V&= 0. \end{aligned}$$

(324)

The corresponding differential form is given by:

$$\begin{aligned} - {{\,\mathrm{div}\,}}{\varvec{ P}}_\mu + {{\,\mathrm{div}\,}}{\boldsymbol{{\lambda }}} = {\bf 0}, \quad&\text {in } \varOmega _{\mu } \end{aligned}$$

(269)

$$\begin{aligned} {\varvec{ P}}_\mu \cdot {\varvec{N}} - {\varvec{L}}\cdot {\varvec{N}} - {\boldsymbol{{\lambda }}}\cdot {\varvec{N}} = {\bf 0}, \quad&\text {on } \partial \varOmega _\mu \end{aligned}$$

(270)

$$\begin{aligned} - {{\,\mathrm{div}\,}}{\mathsf{\bf Q}}_\mu + {\boldsymbol{{\lambda }}} = {\bf 0}, \quad&\text {in } \varOmega _{\mu } \end{aligned}$$

(271)

$$\begin{aligned} {\mathsf{\bf Q}}_\mu \cdot {\varvec{N}}-{\mathsf{\bf M}}\cdot {\varvec{N}} = {\bf 0}, \quad&\text {on } \partial \varOmega _\mu . \end{aligned}$$

(272)