2. Benchmark solution for MASSIF

2.1. Calculation method used for the reference solution

\(\{\begin{array}{c}{\mathrm{\chi }}^{\mathit{kh}}\in {V}_{\mathit{pér}}^{\mathit{CA}}\hfill \\ {\int }_{{Z}_{e}}\epsilon (\vec{{\mathrm{\chi }}^{\mathit{kh}}})\text{.}A\text{.}\epsilon (\vec{\mathrm{\nu }})\phantom{\rule{0.5em}{0ex}}\mathit{dz}=-{\int }_{{Z}_{e}}{\epsilon }_{\mathit{kh}}^{\text{0}}\text{.}A\text{.}\epsilon (\vec{\mathrm{\nu }})\phantom{\rule{0.5em}{0ex}}\mathit{dz},\forall \vec{\mathrm{\nu }}\in {V}_{\mathit{pér}}^{\mathit{CA}}\end{array}\) eq 2.1-1 For a 3D linear elasticity problem, we determine the 6 correctors that verify the equation:

From the calculation of the correctors, the homogenized coefficients of the matrix are determined according to the following formula:

math:

`{epsilon} _ {mathit {ij}}} ^ {text {0}}text {.} {A} ^ {mathit {name}}text {.} {epsilon} _ {mathit {kh}}} ^ {text {0}}} =frac {1} {| {Z} _ {e} |} {\ int} _ {Z} _ {e} _ {e}} {\ epsilon}} {\ epsilon}} {\ epsilon}} _ {\ epsilon}} _ {\ epsilon}} _ {\ epsilon} _ {\ epsilon}} _ {\ epsilon} _ {\ epsilon}} _ {\ epsilon} _ {\ epsilon} _ {\ epsilon}} _ {\ epsilon} _ {\ epsilon} _ {\ epsilon}} _ {\ epsilon} _ {\ epsilon} _ {\ mathit {kh}}} ^ {\ text {0}}\ phantom {\ rule {0.5em} {0ex}}\ mathit {dz} -\ frac {1} {| {| {Z} {| {Z}} _ {e} |} {e} |} {int} |} {int} _ {e}}epsilon (vec {{mathrm {mathrm {chi}}} ^ {chi}}} ^ {mathit {ij}}})text {.} Atext {.} epsilon (vec {{mathrm {chi}}} ^ {mathit {kh}}})phantom {rule {0.5em} {0ex}}mathit {chi}}mathit {dz}} `eq 2.1-2

where \({\epsilon }^{0}\) is the given macroscopic deformation and \({Z}_{e}\) is the total volume of the base cell.

The first integral of the equation is the law of stiffness mixtures while the second integral is the contribution of correctors. For the anisotropic case, there will be at most 21 independent elasticity coefficients. On the other hand, in our case study, this number is reduced due to the symmetries and the invariance according to Oz of the base cell, which makes it possible to arrive at 12 relationships on homogenized elastic stiffness coefficients:

\(\begin{array}{c}{A}_{\text{1111}}^{\text{hom}}={A}_{\text{2222}}^{\text{hom}}\phantom{\rule{2em}{0ex}};\phantom{\rule{2em}{0ex}}{A}_{\text{1133}}^{\text{hom}}={A}_{\text{2233}}^{\text{hom}}\phantom{\rule{2em}{0ex}};\phantom{\rule{2em}{0ex}}{A}_{\text{3131}}^{\text{hom}}={A}_{\text{2323}}^{\text{hom}}\\ {A}_{\text{1112}}^{\text{hom}}={A}_{\text{2212}}^{\text{hom}}={A}_{\text{1113}}^{\text{hom}}={A}_{\text{2213}}^{\text{hom}}={A}_{\text{2223}}^{\text{hom}}={A}_{\text{1123}}^{\text{hom}}={A}_{\text{3313}}^{\text{hom}}={A}_{\text{3323}}^{\text{hom}}={A}_{\text{3312}}^{\text{hom}}=0\end{array}\) eq 2.1-3

Therefore, only 9 coefficients of homogenized elasticity need to be determined: the homogenized material is orthotropic elastic. On the other hand, in, we show certain relationships between homogenized coefficients that reduce the number of independent coefficients.

math:

`{A} _ {text {3333}}} ^ {text {3333}}} ^ {e}} ^ {e} |} {\ int} _ {Z} _ {Z} _ {e}}} E\ phantom {\ rule {3333}}} {e}} ^ {e}}}\ phantom {\ rule {0.5em} {0ex}}} =\ frac {1} {| {Z} _ {2} _ {e}} _ {e}}} Ephantom {rule {0.5em} {0ex}}}mathit {dz} +2 {nu} ^ {2}}{Z} _ {2} _ {e}} _ {e}}} Ephantom {rule {0.5em} {0ex} left ({A} _ {text {1111}}} ^ {text {hom}}} + {text {1122}}} ^ {text {home}}}right) `eq 2.1-4

\({A}_{\text{1133}}^{\text{hom}}={A}_{\text{2233}}^{\text{hom}}=\nu \text{.}\left({A}_{\text{1111}}^{\text{hom}}+{A}_{\text{1122}}^{\text{hom}}\right)\) eq 2.1-5

Likewise, for thermal, three homogenized coefficients are calculated, two of which are independent \({K}_{11}^{\text{hom}}\text{et}{K}_{33}^{\text{hom}}\). In, a relationship between the homogenized transverse shear stiffness coefficients and the homogenized thermal coefficients is also determined.

\({A}_{\text{3}\mathrm{\alpha }\text{3}\mathrm{\alpha }}^{\text{hom}}=\frac{2\mathrm{\mu }}{k}{K}_{\text{11}}^{\text{hom}}\) eq 2.1-6

All these relationships will allow us to verify that the finite element calculation produces correct results by verifying and comparing the values obtained numerically. For the calculations of homogenized coefficients, we compare our results to those obtained in VOLDOIRE93, which brings together the numerical results of the calculation of the homogenized elastic, thermal and thermoelastic behavior of perforated plates.