# Linear Elastic Cook's Membrane: `cooksMembrane`

Prepared by Ivan Batistić

## Tutorial Aims

- Demonstrate how to perform a solid-only analysis in solids4foam.
- To demonstrate solver performance under both shear and bending conditions in combination with a skew numerical mesh.

A finite-strain elastoplastic version of this tutorial is also available in solids4foam at tutorials/elastoplasticity/cooksMembrane

## Case Overview

Cook's membrane is a well-known bending-dominated benchmark case. The tapered panel (trapezoid) is fixed on one side and subjected to uniform shear traction on the opposite side. The prescribed shear traction is 6250 Pa. The vertices of the trapezoid (in mm) are (0, 0), (48, 44), (48, 60), and (0, 44), as shown in Figure 1. The Young's modulus is 70 Pa, and the Poisson's ratio is 1/3. Gravitation effects are neglected, and there are no body forces. The problem is solved as static, using one loading increment.

**Figure 1: Problem geometry**

## Expected Results

- Around the top-left corner, the elastic body is squeezed the most (positive hydrostatic compression).
- The body is most stretched near the bottom side (negative hydrostatic compression).
- There is no known analytical solution for this problem, but results from various codes and procedures are available in the literature [1, 2, 3, 4].

Table 1 summarises predictions from various finite element codes for the top-right corner vertical displacement [1]. For the mesh consisting of \(20 \times 20\) cells, the `solids4foam`

prediction is 32.12 mm (using `foam-extend-4.1`

). As the mesh is refined, the `solids4foam`

prediction converges to the values reported in Table 1.

**Table 1: Finite element predictions for the top-right corner vertical displacement, reported at CoFEA [1]**

Solver | Very Fine Mesh quadratic (Hexahedral mesh) [in mm] | Very Fine Mesh quadratic (Tetrahedral mesh) [in mm] |
---|---|---|

Calculix | 32.27 | 32.27 |

Code_Aster | 32.20 | 32.20 |

Elmer | 32.28 | 32.27 |

In the `solids4foam`

case, the vertical displacement at the top right corner is extracted using the `solidPointDisplacement`

function object placed in the `controlDict`

:

```
functions
{
pointDisp
{
type solidPointDisplacement;
point (0.048 0.060 0);
}
}
```

The `solidPointDisplacement`

function finds the mesh vertex nearest to the specified `point`

and writes the displacement of this vertex to `postProcessing/0/solidPointDisplacement_pointDisp.dat`

.

## Running the Case

The tutorial case is located at `solids4foam/tutorials/solids/linearElasticity/cooksMembrane`

. The case can be run using the included `Allrun`

script, i.e. `> ./Allrun`

. In this case, the Allrun consists of creating the mesh using `blockMesh`

(`> ./blockMesh`

) followed by running the `solids4foam`

solver (`> ./solids4Foam`

).

## References

[1] https://cofea.readthedocs.io/en/latest/benchmarks/002-cook-membrane/results.html