## 4. Membrane patch test

The standard displacement-controlled membrane patch test checks whether the elements can exactly reproduce a constant in-plane strain state for any configuration. If this is the case, then the element will converge to the mathematically exact solution (assuming that the materials are elastic and deformations are small), as the mesh is refined. However, depending on the element type and problem, convergence may be too slow for practical purposes.

The displacement field is described by the bilinear equations (From Belytschko, pp 461-463):

ux = ax0 + ax1 x + ax2 y

uy = ay0 + ay1 x + ay2 y

where axi and ayi are set by the user. All constants should be non-zero. The above displacements are prescribed on all boundary nodes of the patch. The displacement of the internal nodes and the strains and stresses inside the elements are to be obtained with a linear static analysis. The displacements of the internal nodes should follow the same displacement field. Figure 117. Patch geometry and displacement field example

Table 22. Nodal coordinates

 Node X Y Z 1 0.0 0.0 0.0 2 9.9 0.0 0.0 3 8.0 8.0 0.0 4 0.0 6.0 0.0 5 2.5 2.0 0.0 6 6.5 1.5 0.0 7 5.5 5.0 0.0 8 2.5 5.0 0.0

Table 23. Displacement function coefficients

 ax0 0.0001 ax1 0.0061 ax2 0.0049 ay0 -0.0005 ay1 0.0042 ay2 0.0038

The analytical solution for the strains are obtained by deriving the expressions of the displacement field with respect to x and y:

εxx = ax1

εyy = ay2

εxy = ax2 + ay1

The test case supports quadrilateral and triangular shell and 2D elements of first and second order.