ADVANCED ELEMENT FORMULATIONS
In nearly Incompressible materials (0.4 < v < 0.49) an over stiff response has been observed. The ideal solution for elements which suffer from incompressibility constraints is to break up the strain displacement matrix into two parts, one dilatational and one deviatoric.
In this method , volumetric strain at the Gauss integration point is replaced with the average volumetric strain of the elements. The average volumetric strain of the element is evaluated at the element center. This method is referred as the B-Bar method. See Hughes T.J.R.
In SCIFESOL, B-Bar method is implemented for plain strain analysis in Q4 and H8 elements.
Isoparametric lower and higher order elements based on pure displacement formulations are only efficient for general analysis. Shear locking in bending dominated problems and volumetric locking in nearly incompressible materials require specialized element formulations to overcome these problems and provide accurate solutions.
Enhanced Assumed Strain (EAS) method (Simo and Rifai 1990) is based upon Hu-Washizu variational principle.
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In this method the strain field is split into two parts namely a compatible part and an incompatible part which is called the enhanced strain field.
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In SCIFESOL , four parameter (EAS-4) enhanced assumed strain interpolation is implemented for plain stress, plain strain and axisymmetric analysis in four node quadrilateral element.
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For 8 node hexahedral element EAS-models namely 3D.EAS-3, 3D.EAS-9, 3D.EAS-15, 3D.EAS-21, 3D.EAS-30 have been implemented.
In mixed u/p formulation, pressure is interpolated independently as primary variable along with displacement variable. Pressure variable is discontinuous across element boundaries and condensed out during assembly of global stiffness matrix.
The basic equations for formulating mixed u/p elements are formulated in standard manner.
where
BD = Deviatoric Strain-Displacement matrix
BV = Volumetric Strain-Displacement matrix
C’ = Deviatoric Material matrix
HP = Pressure interpolation matrix
The pressure is obtained directly from element matrices instead of being calculated from volumetric strain. Hence, it is more robust for nearly incompressible materials.
The interpolation function of pressure is determined according to the order of elements. For lower order elements constant pressure interpolation is used and for quadratic elements linear interpolation function is used.
In SCIFESOL, mixed u/p formulation is implemented for plain strain & axisymmetric analysis in Q4,Q8 and for 3D H8 elements