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====== Methods to integrate stresses ======

Following the classical approach (Cauchy stresses and out of conservative schemes), stresses over an element can be computed using 4 different methods.

===== Standard formulation=====

When using the standard formulation (''CAUCHYMECHVOLINTMETH = VES_CMVIM_STD''), deviatoric stresses and volume stresses (pressure) are computed at each integration point. Using this method can lead to pressure locking issues when undergoing plasticity (the element is not able to take into account the incompressibility constraint associated with plasticity). A typical solution used to prevent locking is to use a Full Reduced Integration, but then Hourglass modes could appear (spurious strains which does not lead to an increase in stiffness).

===== Selective Reduced Integration =====

The classical solution to the locking issue, experienced when using the [[#Standard formulation]], is to use elements with selective reduced integration (''CAUCHYMECHVOLINTMETH = VES_CMVIM_SRI'') : the pressure is seen as constant over the element, and computed in its center (which means that the pressure is computed on one integration point only).

<note> **Selective Reduced Integration** is different from **Full Reduced Integration**. Indeed, when using **Full Reduced Integration**, both pressure and deviatoric stresses are integrated on one integration point, located at the center of the element. This integration method induces //hourglass// modes, which are purely numerical modes leading to a degradation of the mesh. 
</note>

When using **Selective Reduced Integration**, pressure is computed at the element center but deviatoric stresses are computed using 4 integration points in 2D and 8 in 3D. This way, //Hourglass// modes are prevented.

The integration of internal forces is done with the formula:

$$ F^{int} = \underbrace{\int_{V(t)}{ [B]^{T} {s} \ } dV}_{4 \ integration \ points \ in \ 2D - 8 \ in \ 3D} + 
             \underbrace{\int_{V(t)}{ p [B]^{T} {I} \ } dV}_{1 \ integration \ points} $$ 

where 

  * $s$ is the stress deviator
  * p is the pressure
  * $[B]^T$ is the "strain-displacement" matrix

The evaluation of the pressure integral using only one integration point may lead to imprecision (for example, on elements situated near the axis in a axisymmetric case). A solution consists in estimating the pressure at the element center, then, reporting it on each deviatoric integration point and calculating the pressure integral with 4 points (8 in 3D) instead of only one (see [[#Selective Reduced Integration with Pressure Report]]).

===== Selective Reduced Integration with Pressure Report =====

Integrating the pressure using only one integration point leads to inaccuracies due to an erroneous estimation of the element volume (this can become quite significant in axisymmetric near the revolution axis or when the mesh is highly distorted). The solution consists in calculating the pressure at the element center. Then, since its value is constant over the element, it can be integrated at each integration point used to compute deviatoric stresses. Therefore, the pressure is reported at these integration point to compute the pressure integral using these four points. This method (''CAUCHYMECHVOLINTMETH = VES_CMVIM_SRIPR'') is the default one.

===== EAS Formulation=====

Another method which can avoid locking issues is the EAS integration (''CAUCHYMECHVOLINTMETH = VES_CMVIM_EAS''). Computationaly more expensive, this method handles locking by adding deformation modes (which can take pressure and shear locking into account) to the strain field. This method requires the introduction of more specific parameters in the ''[[doc:user:elements:general:def_element_properties|ElementProperties]]'' of the [[doc:user:elements:volumes:volumeelement|volume element]] (see [[doc:user:elements:volumes:volumeelement#Parameters]]):
 - integer parameters : ''EASS'', ''EASV'', ''KEAS'', ''UEAS'', ''IEAS'', ''TEAS'', ''EEAS''
 - double parameters (''PEAS''). 

<note warning>EAS is not implemented for 2D axisymmetric problems!</note>