Specific models
===================




Global models
----------------




Overview of global models
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In this context, a so-called global plate behavior model, or structural element model in general, means that the law of behavior is written directly in terms of the relationship between generalized stresses and generalized deformations. The global approach to modeling the behavior of structures applies in particular to composite structures, for example reinforced concrete (see Figure 1), and represents an alternative to so-called local or semi-global approaches, which are finer but more expensive models (see [:ref:`bib6 <bib6>`] and [:ref:`bib7 <bib7>`]). In the local approach, fine modeling is used for each of the phases (steel, concrete) and their interactions (adhesion) and in the semi-global approach, the slenderness of the structure is exploited to simplify the description of the kinematics, which results in models PMF (Multi-Fiber Beam) or multi-layer shells.

The advantage of the global model lies in the fact that the corresponding finite element only requires one integration point in the thickness and especially in obtaining homogenized behavior. This advantage is even more important in reinforced concrete analysis, since the location problems encountered when modeling unreinforced concrete are bypassed. Obviously, a global model represents local phenomena in a crude way.


.. image:: images/Object_143.svg
    :width: 493
    :height: 235


.. _RefImage_Object_143.svg:

Figure 3.1-a. Reinforced concrete slab.




Implementation of the GLRC_DM/VMIS_ coupling
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The most important hypothesis that is made in this particular coupling is that elasto-plasticity can only be activated by stressing the membrane plate. In other words, in pure bending the structure will never plasticize. On the one hand, this simplification is justified by the target applications, where the stresses are greater in membranes than in flexure and where dominant influences from flexure are therefore not expected. On the other hand, the hypothesis is necessary for technical reasons, because currently there is no suitable elasto-plastic "global" model for coupling with a "global" damage model.

The global models are formulated in terms of generalized constraints and deformations, which requires us, in the coupler (see §2), to replace the constraints, :math:`\sigma`, and deformations, :math:`\epsilon`, by:


.. math::
 :label: eq-3

    \ sigma\ to\ Sigma =\ left (\ begin {array} {c} {c} {n}} _ {\ mathit {xx}}\\ {N}}\\ {N}}\\ {N} _ {\ mathit {xy}}} {\ mathit {xy}}}\\ mathit {xy}} _ {\ mathit {xy}} _ {\ mathit {xy}}}\\ {M}} _ {mathit {xy}}}\\ {M}} _ {mathit {xy}}}\\ {M}} _ {mathit {xy}}}\\ {M} _ {mathit {xy}}}\\ {M} _ {mathit {xy}}}\\ {M} _ {mathit {xy}}}\\ {M} _ it {xy}}\ end {array}\ right)

where generalized deformations, :math:`{\rm E}`, break down into membrane extensions :math:`ϵ` and curvatures :math:`\kappa` while generalized stresses are composed of membrane forces :math:`N` and bending moments :math:`M`.

It is important to note that in this framework, the hypothesis is made of thin plates where the transverse distortion and the shear force are negligible. Below we recall the Hencky-Mindlin kinematics (see [:ref:`bib9 <bib9>`] for details) for shells and plates as well as the definition of generalized stresses:


.. math::
 :label: eq-3

    (\ begin {array} {} {U} _ {1} _ {1} ({x} _ {\ mathrm {1,}} {x} _ {\ mathrm {2,}} z)\\ {U} _ {2} _ {2}} ({x} _ {x} _ {\ mathrm {1,}}} z)\\ {U} _ {z} ({x} _ {\ mathrm {1,}} {x}} _ {\ mathrm {2,}} z)\ end {array}) =\ underset {\ begin {array} {}\ mathrm {kinematic}\ mathrm {kinematic}\ mathrm {of}\ mathrm {of}} _ {s}\ mathrm {of}}\ mathrm {of}}\ mathrm {of}\ mathrm {of}}\ mathrm {of}\ mathrm {of}}\ mathrm {of}\ mathrm {of}\ mathrm {of}}\ mathrm {of}\ mathrm {of}}\ mathrm {of}\ mathrm {of}}\ mathrm {of}} {\ underset {\ underbrace {}} {(\ begin {array} {} {array} {} {u} _ {\ mathrm {1,}} {x} _ {2})\\ {u} _ {2})\\ {u} _ {2})\\ {u} _ {2})\\ {u} _ {2})\\ {u} _ {2})\\ {u} _ {2})\\ {u} _ {2})\\ {u} _ {2})\\ {u} _ {2})\\ {u} _ {2})\\ {u} _ {2})\\ {u} _ {2})\\ {u} _ {2})\\ {u} _ {2})\\ {} _ {\ mathrm {1,}} {x} _ {2})\ end {array}) +z (\ begin {array} {} {\ theta} _ {2} ({x} _ {\ mathrm {1,}} {1,} _ {2})\\ - {\ theta} _ {2})\\ - {\ theta} _ {1}} ({x} _ {\ mathrm {1,}} {x} _ {2})\\ 0\ end {array})}}} +\ underset {\ begin { array} {}\ mathrm {displacement}\ mathrm {complementary}\\ {u}\\ {u} ^ {u} ^ {c}\ in {V} _ {c}\ end {array}} {\ underset {\ underbrace {}}}} {} {}} {\ underbrace {}}} {}} {\ underbrace {}}} {}} {\ underbrace {}}} {}} {\ underbrace {}}} {}} {\ underbrace {}}} {}} {\ underbrace {}}} {}} {\ underbrace {}}} {}} {\ underbrace {}}} {}} {\ underbrace {}}} {}} {\ underbrace {}}} {}} {mathrm {2,}} z)\\ {u} _ {2} ^ {c} _ {x} _ {\ mathrm {1,}} {x} _ {\ mathrm {2,}} z)\\ {u}} _ {u} _ {z} _ {z} _ {z} _ {\ mathrm {2,}} _ {\ mathrm {2,}} _ {\ mathrm {2,}} _ {\ mathrm {2,}}} z)\ end {array})}}

where :math:`U={({U}_{1}{U}_{2}{U}_{z})}^{T}` is the 3D displacement field, :math:`u={({u}_{1}{u}_{2}{u}_{z})}^{T}` the displacement of the middle sheet and :math:`{\theta }_{x}`, :math:`{\theta }_{y}` its rotations. Thus, the strain tensor, defined as:


.. math::
 :label: eq-3

    {\ varepsilon} _ {\ mathrm {ij}} =\ frac {1} {2}} (\ frac {\ partial {U} _ {i}}} {\ partial {x} _ {j}}} +\ frac {\ partial {j}}} +\ frac {\ partial {i}}} +\ frac {\ partial {j}}} +\ frac {\ partial {j}}} +\ frac {\ partial {j}}} +\ frac {\ partial {i}}} +\ frac {\ partial {j}}} +\ frac {\ partial {i}}} +\ frac {\ partial {j}}} +\ frac {\ partial {i}}} +\ frac {\ partial {j}}} +\

is also written like:


.. math::
 :label: eq-3

    \ begin {array} {} {\ varepsilon} _ {11} =\ underset {{\ varepsilon} _ {11} ^ {s}} {\ underset {\ underbrace {}}} {{\ epsilon}} {{\ epsilon}} _ {\ epsilon} _ {\ epsilon} _ {\ epsilon}} _ {\ epsilon} _ {\ epsilon} _ {11}} + {\ kappa} _ {11}}} + {\ mathrm {1,1}}} ^ {c}\\ {\ varepsilon} _ {22} =\ underset {{\ varepsilon} _ {22} ^ {s}} {\ underset {\ underbrace {}} {{\ epsilon} _ {\ epsilon} _ {22}} + {\ kappa} _ {22}}} + {\ kappa} _ {22}}} + {u} _ {\ mathrm {2,2}}} ^ {\ epsilon} _ {\ epsilon} _ {\ epsilon} _ {22}} + {\ kappa} _ {22}}} + {\ kappa} _ {22}}} + {\ kappa} _ {22}}} + {u} _ {\ mathrm {2,2}}} ^ {\ epsilon}} arepsilon} _ {12} =\ underset {{\ varepsilon} _ {12} ^ {s}} {\ underset {\ underbrace {}} {{\ epsilon} _ {12}} +z {\ kappa}} +z {\ kappa}} _ {12}}} +\ frac {1} {2}} ({u} _ {\ mathrm {2,1}}} {12}} +z {\ kappa}} _ {12}} +z {\ kappa} _ {12}} +\ frac {12}}} {12}}}} +\ frac {1} {2}} ({u} _ {\ mathrm {2,1}}} {12}} +z {\ kappa}} _ {12}} + + {u} _ {\ mathrm {1.2}}} ^ {c})\\ {\ varepsilon} _ {\ mathrm {1z}} = {\ varepsilon} _ {\ mathrm {1z}}} ^ {z}}} ^ {z}}} ^ {1}}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c}} ^ {c} \ varepsilon} _ {\ mathrm {2z}} = {\ varepsilon} _ {\ mathrm {2z}} ^ {c} =\ frac {1} {2} {u} {u} _ {\ mathrm {2z}} _ {\ mathrm {2z}} {\ mathrm {2z}} = {\ mathrm {zz}} {u} _ {\ mathrm {2} {u}} _ {\ mathrm {2} {u}} _ {\ mathrm {2} {u} _ {\ mathrm {2} {u}} _ {\ mathrm {2} {u}} _ {\ mathrm {2} {u}} _ {\ mathrm {2} {u silon} _ {\ mathrm {zz}}} ^ {c}} = {u} _ {\ mathrm {3,} z} ^ {c}\ end {array}

where :math:`\epsilon` is the membrane extension tensor, defined in the plane:


.. math::
 :label: eq-3

    {\ epsilon} _ {\ mathrm {ij}} =\ frac {1} {2}} (\ frac {\ partial {u} _ {i}} {\ partial {x} _ {j}}} +\ frac {\ partial {j}}} +\ frac {\ partial {u}}} +\ frac {\ partial {u}}} +\ frac {\ partial {u}} _ {j}}} +\ frac {\ partial {u}}} +\ frac {\ partial {u}} _ {j}} +\ frac {\ partial {u}}} +\ frac {\ partial {u}} _ {j}} +\ frac {\ partial {u}}} +\ frac {\ partial {u}}}

and :math:`\kappa` the curvature variation tensor, defined in the plane:


.. math::
 :label: eq-3

    {\ kappa} _ {11}\ mathrm {=}\ frac {\ mathrm {\ partial} {\ theta} _ {2}} {\ mathrm {\ partial} {\ partial} {x} _ {1}}

relationships to which is added the hypothesis of plane constraints :math:`{\sigma }_{\mathit{zz}}\mathrm{=}0`, :math:`{\sigma }_{\mathrm{1z}}\mathrm{=}0`, :math:`{\sigma }_{\mathrm{2z}}\mathrm{=}0` which will determine the complementary displacement field :math:`{\mathrm{u}}^{c}\mathrm{\in }{V}_{c}`. In the theory used here, only two rotation components :math:`{\theta }_{x}` and :math:`{\theta }_{y}` are introduced, which implies that the curvature variation tensor is 2D and has only 3 independent components.

Among the two coupled models, GLRC_DM is a global model and therefore formulated directly in terms of :math:`\Sigma \mathrm{=}\Sigma (E)`, but the VMIS_ISOT_LINE and VMIS_CINE_LINE models are the classical 3D laws of behavior. Since it is mainly desired that the elasto-plastic part be represented as a membrane, the law of behavior VMIS_ is applied only to the relationship :math:`N\mathrm{=}N(\epsilon )`, then it is assumed that the flexural behavior remains linear elastic. So we have:

**Damage**:


.. math::
 :label: eq-3

    {\ Sigma} ^ {d} (\ epsilon,\ kappa)\ mathrm {=} {\ Sigma} _ {\ text {GLRC\ _DM}} (\ epsilon,\ kappa)

**Elasto-plasticity**:


.. math::
 :label: eq-3

    {\ Sigma} ^ {p} (\ epsilon,\ kappa)\ mathrm {=} (\ begin {array} {c} {N} _ {\ mathit {VMIS}} (\ epsilon)\\ {H}} _\\ {H}} _ {\ mathit {ELAS}}\ mathrm {:}\ mathit {}}} (\ mathit {}}} (\ epsilon)\\ {H} _ {\ H} _ {\ H} _ {\ mathit {}}}\ mathrm {:}\ kappa\ end {array})

where :math:`{\Sigma }_{\text{GLRC\_DM}}(\epsilon ,\kappa )` represents the global law of behavior GLRC_DM, :math:`{N}_{\mathit{VMIS}}(\epsilon )` the law of Von Mises under plane stresses and :math:`{H}_{\mathit{ELAS}}` the elastic tensor acting on the curvature (see [:ref:`bib9 <bib9>`] for its exact shape). These two modules, damage and elasto-plasticity, are then used in the coupler (see §2) by replacing :math:`{\sigma }^{p}\to {\Sigma }^{p}` and :math:`{\sigma }^{d}\to {\Sigma }^{d}`.




Internal variables
~~~~~~~~~~~~~~~~~~~

The internal variables of the coupled model are stored in series: the first correspond to the damage model, which are followed by that of the elasto-plastic model. The coupler needs six (6) internal variables to store the deformation-type tensor, :math:`{\stackrel{ˉ}{\varepsilon }}^{\mathit{ed}}`, which drives the internal loop (see §2.3). In the case of the GLRC_DM/VMIS_ coupling, four (4) are added because of the algorithm making it possible to satisfy the condition of the plane constraints. These four internal variables have the same meaning as those used for method DEBORST (see [:ref:`bib12 <bib12>`]).

For GLRC_DM/VMIS_ we therefore have 24 internal variables: V1-V7: GLRC_DM; V8-V7:; V8-V14: VMIS_ (we only use V5 and V6 for VMIS_ISOT_LINE); V15-V20: deformation tensor :math:`{\stackrel{ˉ}{\epsilon }}^{\text{ed}}`; V21-V24: method DEBORST.




Validation
~~~~~~~~~~

This model is validated by tests SSNS106F, G (see [:ref:`bib8 <bib8>`]).


.. _Ref71535038: