hex8sr

The hex8sr element defines a 8-noded hexahedral element with a selective reduced integration.

Stiffness Matrix

The local stiffness matrix, defined by the integral \(\mathbf{K}_{e} = \int_{V_{e}} \mathbf{B}^{T} \mathbf{D} \mathbf{B} dV\), is computed numerically using 2 Gauss points for each isoparametric coordinate for the deviatoric part and 1 Gauss point for the volumetric one [Hughes, 2000]:

\(\mathbf{K}_{e}=\sum_{i=1}^{2}\sum_{j=1}^{2}\sum_{k=1}^{2}\mathbf{B}_{ijk}^{T} \mathbf{D_{q}}_{ijk} \mathbf{B}_{ijk} w_{i}w_{j}w_{k}|\mathbf{J}_{ijk}| + \mathbf{\bar{B}}^{T} \mathbf{D_{p}} \mathbf{\bar{B}}\)

where \(\mathbf{\bar{B}}\) is the average of \(\mathbf{B}\) over the element volume and \(\mathbf{D_{q}}\) and \(\mathbf{D_{p}}\) are the deviatoric and volumetric parts of the elasticiy matrix \(\mathbf{D}\), respectively.

Input Notes

• In the input file topol_list must be set the element type ELE_TYPE = 3.

• In the input file topol_file must be set for each element : the element index ELE_IND, the material index ELE_MAT and the nodal connections ELE_CON[NC_TYPE] with NC_TYPE = 8.

• The nodal connections must have a specific sorting (n1,n2,n3,n4,n5,n6,n7,n8) :

             n1 ----------- n2
            /|              /|
           / |             / |
          /  |            /  |
         n4 ----------- n3   |
         |   |           |   |
         |   n5 ---------|- n6
         |  /            |  /
         | /             | /
         |/              |/
         n8 ----------- n7

where (n1,n2,n3,n4) are the nodes of any face of the hexahedron and (n5,n6,n7,n8) are the nodes of the mirror face. The following connections must be present: n1-n5, n2-n6, n3-n7 and n4-n8. (n1,n2,n3,n4) must be sorted following the edges and not diagonals, the sorting can be either clockwise or counterclockwise.

• The parameters needed to assembly \(\mathbf{D_{q}}\) and \(\mathbf{D_{p}}\) are defined by the material related to MAT_TYPE[ELE_MAT] index (see topol_file and materials input files).

Output Notes

• The output stress tensor in the out.DistStress file is the average over the element volume.