5. Examples

You can consult the test cases:

name	title	documentation
HPLV105	Homogenized parameters of GV tube plates	[V7.03.105]
HPLV106	Homogenized parameters of a fibreglass/resin composite	[V7.03.106]
HPLV107	Homogenized parameters of N4 tube plates on a whole cell	[V7.03.107]
HPLV108	Homogenized parameters of a pierced plate with unitary materials and comparison	[V7.03.108]

We can also perform a trivial check with test case HPLV101a entitled « Homogenization of a homogeneous material » [V7.03.101], in thermal: isotropic stationary and isotropic in elasticity.

5.1. laminated elastic plate

We consider a material laminated in 3 layers of different thicknesses of isotropic materials. On note in particular the thicknesses of the successive layers: \({e}_{K}={z}_{K}-{z}_{K-1}\), see r5.03.37-fig-stratifie. The layers are assumed to be symmetric with respect to the sheet. medium, so as to cancel the flexion-membrane coupling.

In this case, the base cell has any periodicity on the directions of the tangential plane \((X,Y)\). Therefore, the fields of membrane correctors \({\chi }_{11}^{m}\), \({\chi }_{22}^{m}\) are finely piecewise dependent on the position \(Z\) in the thickness of the plate, while the concealer in Membrane \({\chi }_{12}^{m}\) associated with distortion sucks everywhere. And the fields of flexure correctors \({\chi }_{11}^{f}\), \({\chi }_{22}^{f}\) are parabolic by pieces in \(Z\) in the thickness of the plate, while the concealer in Flexion \({\chi }_{12}^{f}\) sucks everywhere.

The analytical resolution of elementary problems in membrane and then in flexure provides the analytical expressions of the following homogenized linear elastic characteristics:

\[\]

begin {array} {c}

{A} _ {text {name} 1111}} ^ {text {memb}} = {A} _ {text {name} 2222}} ^ {text {memb}} = sum _ {K} {e} _ {K}\,frac {{E} _ {K}} {1- {nu} _ {K} ^ {2}}\ {A} _ {text {name} 1122}} ^ {text {memb}} = sum _ {K} {e} _ {K}\,frac {{nu} _ {K} {E} _ {K}} {1- {nu} _ {K} _ {K}} ^ {2}}\ {A} _ {text {name} 1212}} ^ {text {memb}} = sum _ {K} {e} _ {K}\,frac {{E} _ {K}} {2 (1+ {nu}} _ {K})}

end {array}

\[\]

begin {array} {c}

{A} _ {text {name} 1111}} ^ {text {flex}} = {A} _ {text {home} 2222}} ^ {text {flex}} = frac {1} {3}sum_ {K} ({z} {K} _ {K} _ {K} ^ {3} - {z} _ {K-1} ^ {3}),frac {{E} _ {K} _ {K} _ {K} ^ {2}}\ {A} _ {text {home} 1122}} ^ {text {flex}} = frac {1} {3}sum _ {K} ({z}} ({z} _ {K} _ {1}} ^ {3}),frac {{nu} _ {K} {K} {E} {E} _ {E} _ {E} _ {K}} {K} {2}} _ {K} {2}}\ {A} _ {text {home} 1212}} ^ {text {flex}} = frac {1} {3}sum _ {K} ({z}} ({z} _ {K} _ {K} {3}),frac {{E} _ {K}} {K}} {2 (1+ {nu} _ {3})})}

end {array}

This analysis can also be carried out using transverse shear.