Methods to integrate stresses

Following the classical approach (Cauchy stresses and out of conservative schemes), four methods to integrate stresses on an element exist.

Standard formulation

When using the standard formulation (CAUCHYMECHVOLINTMETH = VES_CMVIM_STD), deviatoric stresses and volume stresses (pressure) at each integration point. This method can experience pressure locking issues when undergoing plasticity (the element is not capable to take into account the incompressibility constraint associated to plasticity). Consequently, it is possible that the integration is not done properly and Hourglass modes could appear (deformations which do not stiffen the element).

Selective Reduced Integration

The classical solution to the locking issue, see in 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 calculated in is center (which means that the pressure is calculated on one integration point only).l

Selective Reduced Integration is different from Full Reduced Integration (method which is not implemented in Metafor). 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 non physical deformation modes inducing stresses but null strains, leading to a degradation of the mesh.

When using Selective Reduced Integration, pressure is calculated at the element center but deviatoric stresses are calculated using f integration point in 2D and 8 in 3D. This way, Hourglass modes are prevented.

The integration of internal forces is done with the formulae:

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

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 calculate deviatoric stresses. Therefore, the pressure is reported at these integration point to calculate the pressure integral using these four points. This method (CAUCHYMECHVOLINTMETH = VES_CMVIM_SRIPR) is the default one.

EAS Formulation

Another method which can avoir locking issues is the EAS integration (CAUCHYMECHVOLINTMETH = VES_CMVIM_EAS). Computationaly more expense, 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 ElementProperties of the volume element (see parametres_des_elements_de_volume): - integer parameters : EASS, EASV, KEAS, UEAS, IEAS, TEAS, EEAS - double parameters (PEAS).