2. Reference solution#

2.1. Calculation method used for the reference solution#

  • In thermal: we solve the stationary thermal problem:

_images/Object_4.svg

Note:

The boundary conditions chosen here are not those necessary for the homogenization method: we would in fact find

_images/Object_5.svg

everywhere.

The solution is then (verifying the conditions defined in [§1.3]):

_images/Object_6.svg

The potential energy is then in equilibrium:

_images/Object_7.svg
  • In mechanics: we solve the elastostatic problem:

_images/Object_8.svg

for cases:

3D loading

membrane

3D loading

of bending

2D loading

plane constraints

_images/Object_9.svg _images/Object_10.svg _images/Object_11.svg

The solutions are:

in 3D,**membrane* loading and isotropic elasticity:

_images/Object_12.svg

the potential energy at equilibrium is:

_images/Object_13.svg

in 3D,**membrane* loading and orthotropic elasticity:

_images/Object_14.svg

with

either

because the local coordinate system is not confused with the global coordinate system (the nautical angles are all equal to 90°).

_images/100016AA000069D500001389E75473AD8F381DD8.svg

where

_images/Object_15.svg

and

_images/10000476000069D500001020293ECDA7851C214D.svg

in 3D, loading**flexion*:

_images/Object_16.svg

;

_images/Object_17.svg

in 3D,**flexure load* and orthotropic elasticity:

_images/Object_18.svg

with

_images/Object_19.svg

either

_images/Object_20.svg

because the local coordinate system is not confused with the global coordinate system (the nautical angles are all equal to 90°).

_images/Object_21.svg

where

_images/Object_22.svg

and

_images/Object_23.svg
  • in 2D, plane loading:

    _images/Object_24.svg

;

_images/Object_25.svg