Shells made up of homogeneous layers
=======================================




Diaper description
-----------------------

We consider the shell consisting of a stack of :math:`{N}_{\text{couch}}` layers (parallel to the tangential plane) in the thickness :math:`]-h,h[` each consisting of one of the :math:`{M}_{\text{mater}}` orthotropic homogeneous materials (laminated shell [:ref:`Figure 4.1-a <Figure 4.1-a>`]).


.. image:: images/10000200000001040000009591580DAD68BCE5C7.png
    :width: 3.611in
    :height: 2.0693in


.. _RefImage_10000200000001040000009591580DAD68BCE5C7.png:

**Figure 4.1-a**

A :math:`n` layer is defined by:


* its :math:`{e}_{n}` thickness with the ordinates of the lower and upper interfaces:


.. math::
 :label: eq-35

    {x} _ {3} ^ {n-1} =-h+\ sum _ {j=1} ^ {n-1} {e} _ {j}


* the constituent material :math:`m`, and its physical characteristics,


* the angle :math:`{\phi }_{n}` of the first orthotropy direction (noted :math:`L`) in the tangential plane (:math:`T`) with respect to the reference direction :math:`{e}_{\text{réf}}` (see figure).

**Note:**

*In the case of a layer made up of fibers in a resin matrix, the first direction of orthotropy corresponds to the direction of the fibers.*


.. image:: images/1000020000000743000008001E0FBD1542900386.png
    :width: 2.5in
    :height: 1.2201in


.. _RefImage_1000020000000743000008001E0FBD1542900386.png:

Figure 4.1-1: Defining the coordinate system on an orthotropic layer




Thermal
---------

The expression of the vectors :math:`{A}^{\text{ij}}((i,j)\in {\left\{m,s,i\right\}}^{2},i\le j)` defined in [§ :ref:`3.2.1 <RefNumPara__4292_753263007>`] is obtained from the conductivities :math:`{k}_{m}` of the material :math:`m` constituting the layers :math:`n`. In the cases of orthotropy :math:`(L\text{,}T)` of the material :math:`m`, the conductivity coefficients are:

.. math::
 :label: eq-36

    {k} _ {(L, T)}} = (\ begin {array} {c} {k} _ {L}\\ {k} _ {T}\\\ 0\ end {array})

In the case of a transverse isotropic material, the coefficient :math:`{k}_{\text{33}}` is equal to :math:`{k}_{T}`. To have the expression of :math:`{A}^{\text{ij}}` in the coordinate system of the :math:`({V}_{1},{V}_{2})` element, we must apply the following rotation, from the orthotropy coordinate system to the element coordinate system, as explained in [§ :ref:`3 <RefNumPara__4306_753263007>`]:

.. math::
 :label: eq-37

    {k} ^ {(m)} = (\ begin {array} {c} {c} {k} _ {\ text {11}}\\ {k} _ {\ text {22}}\\ {k} _ {\ text {12}}}\\ text {12}}}}\ end {12}}}\\ {12}}\\ text {12}}}\\ {12}}\\ text {12}}}\\ {12}}\\ text {12}}}\ end {array}}) = (\ begin {array}}) = (\ begin {array}} {array}) = (\ begin {array}} {cc} {C} ^ {2} & {S} ^ {2}} _ {\ text {12}}} _ {\ text {12}}}} & {C} ^ {2}\\\ text {CS} & -\ text {CS} & -\ text {CS}\ end {array}) {(\ begin {array} {c} {k} _ {L}\\ {k}\\ {k} _ {k} _ {k} _ {k} _ {k} _ {k} _ {k} _ {k} _ {k} _ {k} _ {k} _ {k} _ {k} _ {k} _ {k} _ {k} _ {k} _ {k} _ {k} _ {k} _ {

The vectors :math:`{A}^{\text{ij}}` can then be expressed by integration into the thickness of the layer contributions:

.. math::
 :label: eq-38

    {A} ^ {\ text {ij}} =\ sum _ {n=1} ^ {n=1} ^ {{N} _ {\ text {couch}}} {\ int} _ {3} ^ {n-1}} ^ {n-1}}}} ^ {x}}} {n} _ {3}}} ^ {x} _ {3}}} ^ {x} _ {3}}} ^ {x} _ {3}}} ^ {n}}} ^ {x} _ {3}}} ^ {x} - {n-1}}}} ^ {x} - {n-1}}}} ^ {x} - {n-1}}}} ^ {x} - {n}}} {3}} ^ {x} _ {1}}} ^ {x} - {n}}}} _ {3})\ cdot {k} _ {(m)}\ cdot {\ text {xx}} _ {3}

The :math:`{B}^{\text{ij}}((i,j)\in {\left\{\mathrm{2,}3\right\}}^{2},i\le j)` terms are:

.. math::
 :label: eq-39

    {B} ^ {\ text {ij}} =\ sum _ {n=1} ^ {n=1} ^ {{N} _ {\ text {couch}}} {\ int} _ {3} ^ {n-1}}} ^ {n-1}}} ^ {n-1}}} ^ {n-1}}} ^ {n}} ^ {3}}} ^ {x}}} {\ partial {x}} _ {3}}\ cdot\ frac {\ partial {P} _ {p} _ {P} _ {P} _ {P} _ {3})} {\ partial {x} _ {3}}\ cdot {k} _ {\ text {k} _ {k} _ {3} _ {3} _ {3} _ {3}

Likewise for :math:`{C}^{\text{ij}}`:

.. math::
 :label: eq-40

    {C} ^ {\ text {ij}} =\ sum _ {n=1} ^ {n=1} ^ {{N} _ {\ text {couch}}} {\ int} _ {3} ^ {n-1}} ^ {n-1}}}} ^ {x}}} {n} _ {3}}} ^ {x} _ {3}}} ^ {x} _ {3}}} ^ {x} _ {3}}} ^ {n-1}}} ^ {x}}} {n}} {3}} ^ {x} _ {3}}} ^ {n}}} ^ {x} _ {3}}} ^ {x} -1}}} ^ {n-1}}} ^ {x} _ {1}}} ^ {x} _ {1}}} ^ {x}} _ {3})\ cdot {\ mathrm {\ mathrm {\ rho C}}} _ {(m)}\ cdot {\ text {xx}}} _ {3}


.. _RefNumPara__4140_753263007:




Thermomechanics
---------------




Behavioral relationship
~~~~~~~~~~~~~~~~~~~~~~~~~~

In the case of laminated shells, it is shown that the relationship between the deformations :math:`\varepsilon` and the stress :math:`\sigma` in the layer :math:`n` depends on the constants of the orthotropic material :math:`m`. For the elastic coefficients, we have :math:`{E}_{\text{LL}}^{(m)}`, :math:`{E}_{\text{TT}}^{(m)}`, :math:`{\nu }_{\text{LT}}^{(m)}`,, :math:`{G}_{\text{LT}}^{(m)}`,, :math:`{G}_{\text{LZ}}^{(m)}` and :math:`{G}_{\text{TZ}}^{(m)}` and the expansion coefficients :math:`{d}_{}^{(m)}` and :math:`{d}_{\text{TT}}^{(m)}`. In the orthotropy axes :math:`(L,T)` of the material :math:`m`, the flexibility matrix :math:`S` is expressed by:

.. math::
 :label: eq-41

    {S} _ {(m)} {\ mid} _ {(L, T)} _ {(L, T)}} = {\ left [\ begin {array} {ccc}}\ frac {{E} _ {\ text {LL}}}} {\ text {LL}}} {\ text {\ array}} {{E}} _ {\ mathrm {TT}} _ {\ text {LL}}}} {\ text {LL}}}} & -\ frac {LL}}}} & -\ frac {{LL}}}} & -\ frac {{\nu}} _ {\ text {\nu}} _ {\ text {\\nu}} _ {\ text {\\nu}} _ {\ text {\\nu}} _ {\ text {\ ac {{\nu} _ {\ text {TL}}}} {{E}}} {{E}} _ {\ mathrm {TT}}} &\ frac {1} {{E} _ {\ mathrm {TT}}}}}} & 0\\ 0&\ 0& 0& 0&\ 0& 0&\ 0& 0&\ frac {1} {{G}} _ {{G}} _ {{G}}}\ end {array}\ right]} _ {(m)}}

Stiffness :math:`{\Lambda }_{(m)}={S}_{(m)}^{-1}` being:

.. math::
 :label: eq-42

    {\ Lambda} _ {(m)} {\ mid} _ {\ mid} _ {(L, T)} _ {\ left [\ begin {array} {ccc}}\ frac {{E} _ {\ text {LL}}}}} {\ text {LL}}}} {\ text {LL}}}}} {\ text {LL}}}}} {1- {\nu}}} {1- {\nu} _ {\ text {LL}}}}} {1- {\nu}}} {1- {\nu} _ {\ mathrm {TL}} _ {\ text {LL}}}}} {1- {\nu}}} {1- {\nu} _ {\ text {LL}}}}} {1- {\nu}}} _ {\ text {TL}}\ cdot {E}} _ {\ text {E}} _ {\ text {LL}}}} {1- {\nu} _ {\ text {LT}}} _ {\ text {LT}}}}} & 0\\\ frac {{LL}}}} {1- {\nu} _ {\ mathrm {TL}}}\ cdot {\nu}} _ {\ text {LT}}} &\ frac {{E} _ {\ mathrm {TT}}}} {1- {\nu}} _ {\ mathrm {\nu}} _ {\ text {TT}}}} {\ mathrm {TL}}}} {1- {\nu} _ {\ mathrm {TL}} _ {\ mathrm {TL}} _ {\ mathrm {TL}}} _ {\ mathrm {TL}} _ {\ mathrm {TL}} _ {\ mathrm {TL}} _ {\ mathrm {TL}} _ {\ mathrm {TL}} _ {\ mathrm {TL}} {\ text {LT}}\ end {array}\ right]} _ {(m)}

For its part, transverse shear stiffness is expressed as follows:

.. math::
 :label: eq-43

    {\ Lambda} _ {\ tau (m)} {\ mid} _ {\ mid} _ {(L, T)}}\ text {} =\ text {} {\ left [\ begin {array} {cc} {G}} _ {\ mathrm {LZ}} _ {\ mathrm {LZ}} _ {\ mathrm {LZ}}}\ end {array} {G}} _ {\ mathrm {LZ}}} _ {\ mathrm {LZ}}} _ {(m)}

By placing ourselves in the coordinate system of element :math:`({V}_{1},{V}_{2})`, we use the transition matrix :math:`{P}^{(m)}` from the deformation tensor defined at [:ref:`§3 <§3>`] of :math:`({V}_{1},{V}_{2})` to the orthotropy coordinate system:

.. math::
 :label: eq-44

    {P} ^ {(m)} =\ left [\ begin {array} {\ begin {array} {ccc} {C} {C} ^ {2} & 2\ text {CS}\\ {S} ^ {2} & {C} ^ {2} & {C} ^ {2} & {C} ^ {2} & {C} ^ {2} & {C} ^ {2} & {C} ^ {2} & {C} ^ {2} & {C} ^ {2} & {C} ^ {2} & {C} ^ {2} & {C} ^ {2} & {C} ^ {2} & {C} ^ {2} & {C} ^ {2} & {C} ^ {2} & {C} ^ {2} & {C} ^ {2} & {2}\ end {array}\ right]

Likewise, the expansion vector is expressed in coordinate system :math:`({V}_{1},{V}_{2})`:

.. math::
 :label: eq-45

    {d} ^ {(m)} = (\ begin {array} {c} {d} {d} _ {\ text {11}}\\ {d} _ {\ text {22}}\\ {d} _ {\ text {12}}}\\ text {12}}}\ end {12}}}\ end {array} {12}}} {\ text {12}}}\\ text {12}}}\\ {d} _ {\ text {12}}}\ end {array}}}\ end {12}}}\ end {array}} {\ text {12}}}\ end {array}}}\ end {12}}}\ end {array}} {\ text {12}}}\ end {array}}}\ end {12}}}\ end {array}}}}\\ {d} _ {\ text {TT}}}\\ 0\ end {array}})} _ {(L, T)} = (\ begin {array} {cc} {C} ^ {2} & {S} ^ {2} & {S} ^ {2}}\\ {S} ^ {2}}\\ 2\ text {CS} {2}\ text {CS} {2} & -2\ text {CS} & -2\ text {CS} & {S} ^ {2} & {S} ^ {2}\ text {CS} & -2\ text {CS} & {S} ^ {2}\ text {CS} & -2\ text {CS} & {S} ^ {2}\ text {CS} & -2\ text {CS}} & {S} ^ {2}}\) {(\ begin {array} {c} {d} _ {d} _ {LL}\\ {d} _ {\ mathrm {TT}}\ end {array})}} _ {(L, T)}

     So in layer :math:`n` (material: :math:`m`), in :math:`{x}_{3}`, we have:

.. math::
 :label: eq-46

    {s} _ {(n)} = {P} _ {s} ^ {{} ^ {{} ^ {(m)} ^ {-1}}}}\ text {.} \ Lambda {\ mid} _ {(L, T)}\ text {.} {P} ^ {(m)}\ text {.} (e (u) - {e} ^ {\ text {th}}) = {} ^ {T}\ text {} {P} ^ {(m)}}\ text {.} \ Lambda {\ mid} _ {(L, T)}\ text {.} {P} ^ {(m)}\ text {.} (e (u) - {e} ^ {\ text {th}}) = {\ text {th}}) = {\ Lambda} _ {(m)}} (e (u) - {e}} ^ {\ text {th}}) = {\ text {th}}) = {\ text {th}})

With:

.. math::
 :label: eq-47

    \ epsilon\ left (u\ right) =\ left (\ begin {array} {c} {E} {E} _ {11}\ {E} _ {22}\\ {E} _ {12}\ end {array}\\\\\\\ right) + {x}}\\ {right) + {x} _ {right) + {x} _ {right) + {x} _ {right) + {x} _ {3}\ _ {3}\ {3}\\ {K}\ {K}} _ {12}\ end {array}\ right)

**Note:**

*In the code, we chose to perform the transition from the orthotropy coordinate system to the element coordinate system in two steps. A first step concerns the transition from the orthotropy coordinate system to the coordinate system defined by* *ANGL_REP. The data of* *DEFI_MATERIAU is thus transformed during this first pass. The equivalent material is then treated as would be done with conventional plate elements.*

*The treatment of thermal expansion is done in the form of a contribution to the second member of the matrix equation to be solved based on the principle of virtual work. This post reads:* :math:`{\sigma }^{{\text{th}}_{\left(n\right)}}=-{}^{T}\text{}{P}^{\left(m\right)}\text{.}\Lambda {\mid }_{\left(L,T\right)}\text{.}\left\{\begin{array}{c}{d}_{\text{LL}}\Delta T\\ {d}_{\text{TT}}\Delta T\\ 0\end{array}\right\}`.


.. _RefNumPara__7167_753263007:




Transverse shear
~~~~~~~~~~~~~~~~~~~~~~~~~

The transverse shear stiffness of each layer is written in coordinate system :math:`({V}_{1},{V}_{2})` in the same way as the expansion:

.. math::
 :label: eq-48

    {\ Lambda} _ {t (m)} {\ mid} _ {({V} _ {1}, {V} _ {2})} = {} ^ {t}\ text {} {P} _ {P} _ {2} _ {2} _ {2}} ^ {(m)}\ text {.} {\ Lambda} _ {t (m)}\ text {.} {P} _ {2} ^ {(m)}

With :math:`{P}_{2}^{(m)}=\left[\begin{array}{cc}C& S\\ -S& C\end{array}\right]` the vector transition matrix from :math:`({V}_{1},{V}_{2})` to the orthotropy coordinate system. The overall transverse shear stiffness of the shell :math:`\left[{R}_{c}\right]` calculated so as to be equal to that given by the law of three-dimensional elasticity [:ref:`bib2 <bib2>`], the matrix :math:`\left[{R}_{c}\right]` is defined so that the surface density of transverse shear energy :math:`{U}_{2}` obtained for a three-dimensional distribution of the stresses :math:`{\sigma }_{\text{13}}` and :math:`{\sigma }_{23}` is identical to that associated with the Reissner-Mindlin plate model noted :math:`{U}_{2}`. In the end:

.. math::
 :label: eq-49

    {U} _ {1} =\ frac {1} {2} {2} {\ int} _ {\ int} _ {-h}\ langle\ tau\ rangle {\ left [{\ Lambda} _ {\ tau\ left (m\ left (m\ right)} {m\ right)}\ right]}\ right]} ^ {-1}\ left\ {\ tau\ right\} {d} _ {3}

And:

.. math::
: label: eq-50

    {U} _ {2} =\ frac {1} {2} {2} V {\ left [{R} _ {c}\ right]} ^ {-1} V=\ frac {1} {2}\ left ({\ int} _ {int} _ {int} _ {int} _ {2} _ {c} _ {c}\ right]} ^ {-1}\ frac {1} {2}\ left ({\ int} _ {-h} ^ {h}\ left\ {\ tau\ right\}} {d} _ {3}\ right)

With :math:`\langle \tau \rangle =\langle {\sigma }_{13}{\sigma }_{23}\rangle`. With the equilibrium equations:

.. math::
 :label: eq-51

    \ {\ begin {array} {c} {\ sigma} _ {\ sigma} _ {\ text {13}} _ {-h} ^ {{x} _ {3}} ({\ sigma} _ {\ text {11}\ text {11}}\ mathrm {11}}\ mathrm {11}}\\ {11}} _ {11}}\\ {sigma} _ {\ text {23}} =- {\ int} _ {\ int} _ {-h} _ {3}} ({\ sigma} _ {\ text {12}\ mathrm {,1}}} + {s}} + {s}} _ {s}} _ {s}} _ {1}} + {s}} _ {1}}} + {s}} _ {1}} + {s}} _ {1}} + {s}} _ {s}} _ {1}} + {s}} _ {s}} _ {1}} + {s}} _ {s}} _ {1}} + {s}} _ {s}} + {s}

And the conditions:

.. math::
: label: eq-52

    0= {\ sigma} _ {\ text {13}}} = {\ sigma} _ {\ text {23}}}

The plane stresses :math:`{\sigma }_{11}`, :math:`{\sigma }_{22}` and :math:`{\sigma }_{12}` are expressed as a function of the resulting forces by assuming pure flexure and the absence of membrane/flexure coupling. The result is that:

.. math::
 :label: eq-53

    \ sigma ({x} _ {3}) = {x} _ {3}\ text {.} {\ Lambda} _ {(m)} ({x} _ {3}) {P} ^ {-1}\ text {.} M

Where :math:`P` is the flexural stiffness matrix of the entire multilayer defined by the first equation of (). These calculations, as well as the following ones, are to be performed in a single coordinate system. In *Code_Aster*, we choose the coordinate system intrinsic to the element. It is therefore necessary to transform the matrix A into this coordinate system. We then have:

.. math::
 :label: eq-54

    \ left\ {\ tau ({x} _ {3})\ right\})\ right\} = {D} _ {1} ({x} _ {3}) V+ {D} _ {2} ({x} _ {3})\ left\ {\ lambda\ right\}

With:

.. math::
 :label: eq-55

    V=\ langle {M} _ {\ mathrm {11.1}}} + {M} _ {\ mathrm {12.2}}; {M} _ {\ mathrm {12.1}}} + {M}} _ {\ mathrm {22.2}}\ rangle

     And:

.. math::
 :label: eq-56

    \ langle\ lambda\ rangle =\ langle {M} _ {\ mathrm {11.1}} - {M} _ {\ mathrm {12.2}}; {M} _ {\ mathrm {12.1}} - {M}} - {M}} _ {\ mathrm {22.2}}; {M} _ {\ mathrm {22.1}}}; {M} _ {\ mathrm {12.1}}} - {M} _ {\ mathrm {12.1}}} - {M} _ {\ mathrm {12.1}}} - {M}} thrm {11,2}}\ rangle

And:

.. math::
 :label: eq-57

    \ begin {array} {c} {D} _ {1} = {\ int} _ {\ int} _ {- {x} _ {3}} ^ {h} -\ frac {z} {2}\ left [\ begin {array} {cc} {cc} {cc} {cc} {cc}} {cc} {CC} {CC} {CC} {A} _ {\ text {13}} {cc} {cc} {CC} {A} _ {A} _ {\ text {13}} {CC} {CC} {CC} {A} _ {CC} {CC} {A} _ {\ array} {cc} {CC} {A}} = {\ text {13}} {CC} {CC} {A} = {\ array} {1} = {\ array} {1} _ {\ text {32}}\\ {A} _ {\ text {33}} _ {\ text {31}} _ {\ text {23}}} & {A} _ {\ text {22}} + {A} _ {\ text {33}}} _ {33}}} _ {\ text {33}}} _ {\ text {33}}} _ {3}}\ end {array}}} _ {33}} _ {3}}\ end {array}}} _ {3}} ^ {h} -\ frac {z} {2}\ left [\ begin {array} {\ begin {array} {cccc} {A} _ {\ text {33}} & {A}} & {A} _ {\ text {13}}}\ left [\ text {13}}}\ left [\ text {13}}}\ left [\ text {13}}}\ left [\ text {13}}}\ left [\ text {13}}}\ left [\ text {13}}}\ left [\ text {13}}}\ left [\ text {13}}}\ left [\ text {13}}}\ left [\ text {13}}}\ left [\ text {13}}}\ left [\ text {13}}\ left [\ text {13}}\ left thrm {2A}} _ {\ text {31}}}\\ {A}}\\ {A} _ {\ text {31}} _ {\ text {23}} & {A} _ {\ text {33}} - {A} - {A} _ {\ text {22}}} _ {\ text {22}}} _ {\ text {22}}} & {\ mathrm {2A}}} _ {\ text {32}} & {\ mathrm {2A}}} - {\ text {33}}} - {A}} - {A}} - {A}} - {A}} - {A}} _ {\ text {33}} - {A}} - {A}} - {A}} - {A}} - {A}} - {A}} {21}}\ end {array}\ right]\ text {dz}\ end {array}

So :math:`{U}_{1}` is written as:

.. math::
 :label: eq-58

    {U} _ {1} =\ frac {1} {2}\ langle V\ mid\ mid\ lambda\ rangle\ left [\ begin {array} {cc} {C} _ {\ text {11}}} & {C}} & {C}} & {C}} & {\ text {11}}} & {\ text {11}}} & {\ text {11}}} & {\ text {11}}} & {C}} & {\ text {11}}} & {C}} & {\ text {11}}} & {C}} & {\ text {11}}} & {\ text {11}}} & {C}}} & {\ text {11}}} & {C}}} & {\ text {11}}} & {C}} end {array}\ right]\ left\ {\ frac {V} {\ lambda}\ right\}

With:

.. math::
 :label: eq-59

    \ begin {array} {}\ underset {2\ times 2} {{C}} {{C}} _ {\ text {11}}}} = {\ int} _ {{h} {{D} _ {{1} ^ {T}} ^ {T}}} {\ T}}}} {\ Lambda}} _ {\ Lambda} _ {3}\\ underset} _ {3}\\ underset 2\ times 4} {{C} _ {\ text {12}}}} = {\ int}}} = {\ int} _ {\ int} _ {{1} ^ {T}} {\ Lambda} _ {\ Lambda} _ {\ tau (m)} _ {\ tau (m)}}}}}} = {\ int}}}} = {\ int} _ {1} ^ {T}}} {\ Lambda} _ {\ Lambda} _ {\ Lambda} _ {\ Lambda} _ {\ Lambda} _ {\ Lambda} _ {\ Lambda} _ {\ Lambda} _ {\ Lambda} _ {\ text {22}}} = {\ int} _ {-h} _ {-h} ^ {h} ^ {D} _ {{2} ^ {T}} {\ Lambda} _ {\ tau (m)}}}} ^ {-1}} {-1} {-1} {D} {D} _ {3}\ end {array}}}}} ^ {-1} {-1} {1} {D} {D} _ {3}\ end {array}}

Hence the final trips:

.. math::
 :label: eq-60

    {U} _ {1} = {U} _ {2}\ iff\ langle V\ mid\ lambda\ rangle\ left [\ begin {array} {cc} {C} _ {\ text {11}}} - {{H}}} - {{H} _ {c}}} - {{H} _ {H} _ {{H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {H} _ {}} & {C} _ {\ text {22}}}\ end {array}\ right]\ left\ {\ frac {V} {\ lambda}\ right\} =0\ forall V,\ left\ {\ array}\ right\}

So we propose solution :math:`{H}_{c}={C}_{{\text{11}}^{-1}}`. The transverse shear correction coefficients correspond to the ratio of the terms of :math:`{H}_{c}` to the integral over the thickness of the laminate of the terms of :math:`{\Lambda }_{\tau (m)}`.




Generalized efforts
~~~~~~~~~~~~~~~~~~~~~

The generalized efforts put into a vector form are obtained by integration into the thickness of the shell by summing the contributions of the layers (of thickness :math:`{e}_{n}={x}_{3}^{n}-{x}_{3}^{n-1}`):

.. math::
 :label: eq-61

    \ begin {array} {c} M= (\ begin {array} {c} {c} {M} _ {\ text {11}}\\ {M} _ {\ text {22}}\\ {M} _ {\ text {12}}}\ end {12}}}\ end {12}}}\ end {array}}}\ end {array}}) = {\ int} {array}) = {\ int} _ {l}\ sigma\ cdot {x} _ {3}\ cdot {\ text {12}}} _ {3} =\ sum _ {n=1} ^ {{1} ^ {N} _ {\ text {n} _ {\ text {couch}}}} {\ int} _ {3}} ^ {x} _ {3} ^ {3} ^ {n}} {n}}} {\ n}}} {\ sigma}}} {\ sigma}} _ {\ sigma}} _ {3}\\ begin {array} {}\\ N= (\ begin {array} {c} {c} {N}} _ {\ text {11}}\\ {N} _ {\ text {22}}\\ {N} _ {\ text {12}}}\ end {12}}}\ end {array}}}\ end {array}}}\ end {array}}) = {\ end {array}}) = {\ int} {array}) = {\ int} _ {array}) = {\ int} _ {array}) = {\ int} _ {array}) = {\ int} _ {array}) = {\ int} _ {array}) = {\ int} _ {array}) = {\ int} _ {array}) = {\ int} _ {array}) = {\ int} {{N} _ {\ text {couch}}}} {\ int}} {\ int} _ {{x} _ {3} ^ {n-1}} ^ {x} _ {3} ^ {n}} {\ sigma}} {\ sigma}} _ {(n)}}\ cdot {\ text {xx}} _ {3}\ end {array} _ {3}\ end {array}\ end {n}}} {\ sigma}} _ {\ sigma}} _ {(n)} _ {3}\ end {array}} {\ sigma}} _ {\ sigma}} _ {\ sigma}} _ {\ sigma}} _ {\ sigma}} _ {

If we express as before (with :math:`m` material from layer :math:`n`):

.. math::
: label: eq-62

    {\ sigma} _ {(n)} = {\ Lambda} _ {(m)}}\ cdot (E+ {x} _ {3}\ cdot K- {d} _ {(m)}} (T ({x} _ {3})} (T ({x} _ {3})) - {T} ^ {{} _ {\ text {ref}}}))

We can note the widespread efforts in the form of:

.. math::
 :label: eq-63

    \ begin {array} {} M- {M} ^ {\ text {th}} ^ {\ text {th}} =P\ cdot K+Q\ cdot E\\ N- {N}} ^ {\ text {th}}} =Q\ cdot K+R\ cdot K+R\ cdot E\ end {array}

With :math:`P,Q,R` 3 x 3 matrices expressed by:

.. math::
 :label: eq-64

    \ begin {array} {cc} P=\ sum _ {n=1}} ^ {{n}} _ {\ text {couch}}} {\ Lambda} _ {\ int} _ {{x} _ {3} _ {3} _ {3}} ^ {n-1}}} ^ {n-1}}} ^ {3}} {x}} _ {3} ^ {2} _ {3} ^ {2} _ {3} ^ {2} _ {3} ^ {2} _ {3} ^ {2} _ {3} ^ {2} _ {3}} _ {\ x} _ {2}\ cdot {\ text {xx}}} _ {3} & =\ sum _ {n=1} ^ {n=1} ^ {1} ^ {1}} ^ {1}} ^ {3}\ cdot\ frac {1} {3}\ cdot ({({x} _ {3} _ {3} _ {3} _ {3} _ {3} {3} {3} {3} {3} {3} {3} {3}} {3} {3}}\ cdot ({x}} {3} {3} {3}} {3} {3}}\ cdot ({x}} {3} {3} {3}} {3} {3}} {3} {3}} {3} {3}}\ cdot ({x} 1}} {3} {3} {3}})\\ Q=\ sum _ {n=1} ^ {n} ^ {n} _ {\ text {couch}}} {\ Lambda} _ {(m)} {\ int} _ {{x} _ {3} ^ {n-1}}} ^ {n-1}}}} ^ {n}} _ {3}} _ {3}\ cdot {\ text {xx}} _ {3} ^ {n-1}}}} ^ {n-1}}} ^ {x}}} ^ {x}}} ^ {x} _ {3} & =\ sum _ {n=1} ^ {{N} _ {\ text {couch}}} {\ text {couch}}}} {\ Lambda} _ {(m)}\ cdot {1} {2}\ cdot ({({x} _ {3} _ {3} _ {x} _ {1}} {2})\ ({x}} _ {n-1}) {({x} _ {2})\\ R= sum\ _ {n=1} ^ {{N} _ {\ text {couch}}}} {\ Lambda} _ {(m)}\ cdot ({x} _ {3} ^ {n} _ {3} ^ {n} - {x} _ { 3} ^ {n-1}) & =\ sum _ {n=1} ^ {{n} _ {\ text {couch}}} {\ Lambda} _ {(m)}\ cdot {e}}\ cdot {e} _ {n} _ {n}\ end {array}

The shear force :math:`V` is obtained by deriving the moment [§ :ref:`4.3.2 <RefNumPara__7167_753263007>`]. The generalized forces of thermal origin are calculated directly:

.. math::
 :label: eq-65

    \ begin {array} {} {M} ^ {\ text {th}} ^ {\ text {th}}} =\ sum _ {{n} _ {\ text {couch}}}} {\ Lambda}} {\ Lambda} _ {\ text {couch}}} {\ Lambda} _ {\ Lambda} _ {(m)} _ {(m)}\ text {.} {\ int} _ {{x} _ {3}} ^ {n-1}} ^ {n-1}}} ^ {3} ^ {n}} {x} _ {3} _ {3}\ text {.} (T ({x} _ {3}) - {T} ^ {\ text {ref}})\ text {.})\ text {.} {d} _ {(m)}\ text {.} {\ text {dx}} _ {3}\\ {N}} ^ {\ text {th}} ^ {\ text {th}} =\ sum _ {{n} _ {\ text {couch}}}}} {\ Lambda}} {\ Lambda} _ {(m)}\ text {.} {\ int} _ {{x} _ {3} _ {3} ^ {n-1}}} ^ {n-1}}} ^ {n-1}} ^ {3}} (T ({x} _ {3}) - {T} ^ {3}) - {T} ^ {\ text {ref}})\ text {.} {d} _ {(m)}\ text {.} {\ text {xx}} _ {3}\ end {array}




Location of constraints (post-processing)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Conversely, following a finite element calculation and obtaining deformations :math:`E` and variations in curvature :math:`K`, it is then possible to calculate the stress field :math:`{\sigma }_{(n)}(n=\mathrm{1,}{N}_{\text{couch}})` in each layer of the element.

In each layer :math:`(n)`, it is necessary to calculate the matrix :math:`{\Lambda }_{(m)}` and the terms :math:`(T({x}_{3})-{T}^{\text{réf}})\text{.}{d}_{(m)}` (cf. [§ :ref:`3.2 <RefNumPara__7181_753263007>`]) (:math:`m={\text{mat}}_{n}` represents the material characteristics of the layer :math:`n`).

The constraints :math:`{\sigma }_{\alpha \beta }` to an ordinate :math:`{x}_{3}\in ]{x}_{3}^{n-1},{x}_{3}^{n}[` in the layer (:math:`n`) are then:

.. math::
 :label: eq-66

    {\ sigma} _ {(n)} ({x} _ {3}) = {\ Lambda} _ {(m)}\ text {.} \ left [E+ {x} _ {3}\ text {.} K- {d} _ {(m)} (T ({x} _ {3}) - {T} _ {3}) - {T} _ {\ text {ref}}})\ right]

And the transverse shear:

.. math::
 :label: eq-67

    {\ tau} _ {(n)} ({x} _ {3}) = {D} _ {3}) = {D} _ {3})\ text {.} V+ {D} _ {2} _ {2} ({x} _ {3})\ text {.} \ Lambda

**Note:**


+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|*In the code, post-treatments of plate elements are generally defined in the coordinate system associated with* *ANGL_REP* * *. The constraints in the intrinsic coordinate system of the element are thus brought back into the manifold coordinate system. We have:*                                                                                                                                                                                                                                                                                                                                                                   |
+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`{(\begin{array}{c}{\sigma }_{\text{11}}\\ {\sigma }_{\text{22}}\\ {\sigma }_{\text{12}}\end{array})}_{\text{eref}}=(\begin{array}{ccc}{C}^{2}& {S}^{2}& +2\text{CS}\\ {S}^{2}& {C}^{2}& -2\text{CS}\\ -\text{CS}& +\text{CS}& {C}^{2}-{S}^{2}\end{array}){(\begin{array}{c}{\sigma }_{\text{11}}\\ {\sigma }_{\text{22}}\\ {\sigma }_{\text{12}}\end{array})}_{n}`|*where* :math:`\begin{array}{c}C=\text{cos}({\varphi }_{0})\\ S=\text{sin}({\varphi }_{0})\text{}(\text{cf}\text{.}\left[\S 4\text{.}1\right])\end{array}` *where* :math:`{\varphi }_{0}` *is the angle between* :math:`{V}_{1}` *and* :math:`{e}_{\text{réf}}`|
+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+




Calculation of breakage criteria in diapers (post-treatment)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The breaking stress limit values depend on the material of the layer, the direction and the direction of the stress (for a group of elements corresponding to the same material field):

:math:`\begin{array}{}\begin{array}{ccc}\begin{array}{c}{\text{mat}}_{n}\end{array}& \begin{array}{c}X\text{: limite en traction dans le sens L}\\ {X}^{\text{'}}\text{: limite en compression dans le sens L}\\ Y\text{: limite en traction dans le sens T}\\ {Y}^{\text{'}}\text{: limite en compression dans le sens T}\\ S\text{: limite en cisaillement dans le sens LT}\end{array}& \begin{array}{c}(\text{1ère direction orthotropie : sens des fibres})\\ (\text{1ère direction orthotropie : sens des fibres})\\ (\text{2ème direction orthogonale à la 1 ère})\\ (\text{2ème direction orthogonale à la 1 ère})\\ \end{array}\end{array}\\ \end{array}`

It is necessary to calculate the stresses in the layer coordinate system (defined by the orthotropy axes) from the constraints in the element coordinate system. The angle between :math:`{V}_{1}` and :math:`{e}_{\text{réf}}` is :math:`{\varphi }_{0}`, and the angle between :math:`{e}_{\text{réf}}` and the orthotropy coordinate system is :math:`{\varphi }_{n}`:

.. math::
 :label: eq-68

    {\ left (\ begin {array} {c} {\ sigma} _ {\ sigma} _ {L}\\ {\ sigma} _ {T}\\ {array}\ right)} _ {n} =\ left (\ begin {array} {\ sigma} =\ left (\ begin {array} {sigma} _ {sigma} _ {2} _ {array}\ {n} =\ {n} =\ left (\ begin {array}} {sigma} _ {n} =\ left (\ begin {array}} {sigma} _ {n} =\ left (\ begin {array}} {sigma} _ {n} =\ left (\ begin {array}} {sigma} _ {n} =\ left (\ begin {array}} {sigma} _ {n} =\ left (\ begin {array} {sigma} 2} & {C} ^ {2} & -2\ text {CS}\\ -\ -\ text {CS} & +\ text {CS} & {C} ^ {2} - {S} ^ {2}\ end {array}\\ right) {\ right) {\ left) {\ left) {\ left (\ left (\ begin {array}}\ {array} {\ array}\ right) {\ left (\ begin {array} {\ array}\ right) {\ left (\ begin {array} {c} {c}} {\ sigma}} _ {\ text {22}}\ right) {\ text {22}}\ right) {\ left (\ begin {array}}\\ {\ sigma} _ {\ text {12}}}\ end {array}\ right)} _ {n}

For the maximum stress criterion, the following five criteria are calculated per layer:

.. math::
 :label: eq-69

    \ textrm {For}
    n=\ mathrm {1,} N-\ text {couch}

The Tsai-Hill criterion is written in each layer as follows:

.. math::
 :label: eq-70

    {C} _ {\ text {TH}}} =\ frac {{\ sigma} _ {L (n)} ^ {2}} {{X} _ {({\ text {mat}} _ {n})} _ {n})}}} ^ {n})}} ^ {2}}} -\ frac {{\ sigma} _ {L (n)\ text {.}} {\ sigma} _ {T (n)}} {{X} _ {{X} _ {({\ text {mat}}} _ {n})} ^ {2}}\ frac {{\ sigma} _ {T (n)} _ {T (n)}} ^ {2}} _ {2}} _ {n}}\ frac {{\ sigma} _ {\ text {LT} (n)}} ^ {2}} {{S} _ {({\ text {mat}}} _ {n})}} ^ {2}}

The material is broken when :math:`{C}_{\text{TH}}\ge 1`. The values :math:`X` and :math:`Y` are replaced by :math:`{X}^{\text{'}}` and :math:`{Y}^{\text{'}}` when the corresponding :math:`({\sigma }_{L(n)}\text{,}{\sigma }_{T(n)})` constraints are negative.