Cook's tapered cantilever with a hyperelastic material, benchmark 1 (Sect. 2.3) of the selection of solid mechanics benchmarks of the DFG priority programme SPP 1748, Schröder et al. (2021) [89]. The membrane combines bending and shear. Its compressibility is reduced in five steps from Poisson's ratio 0.35 to 0.4999, which shows the volumetric locking of displacement-only elements of low order. The tests solve it with the mixed displacement–pressure formulation with the volumetric–isochoric split (Hyperelasticity with volumetric-isochoric split), in 2D (this folder) and in 3D (test/solidmechanics/hyperelastic_voliso/cooks_membrane_3d).

Problem

Geometry and boundary conditions of Cook's membrane, dimensions in mm

The membrane has the corners (0, 0), (48, 44), (48, 60) and (0, 44) mm and the thickness 1 mm. The left edge \( x = 0 \) is clamped and the right edge \( x = 48 \) mm carries the shear traction \( p_0 = 20 \) MPa in \( +y \) direction (dead load); the other edges are free. The material is

\begin{align}\end{align}

with \( \mu = 185.185 \) MPa and \( \lambda \) = 432.099, 750, 9260, 92,600 and 926,000 MPa (Poisson's ratio 0.35, 0.401, 0.4902, 0.499 and 0.4999); every test runs all five values. The quantity of interest is the vertical displacement \( u_y \) of the upper right corner \( A = (48, 60) \) mm.

2D: plane strain.
3D: the front and back faces are free. Half of the thickness, \( 0 \leq z \leq 0.5 \) mm, is modelled with \( u_z = 0 \) on the mid-plane \( z = 0 \); \( A = (48, 60, 0) \) mm.

The plane-strain and the 3D values differ by about 7 % because the 3D membrane is not constrained in thickness direction.

Discretization

Test	Displacement	Pressure
test_hyperelastic_cooks_membrane_mini, test_hyperelastic_cooks_membrane_mini_3d	P1 with bubble	P1
test_hyperelastic_cooks_membrane_taylorhood, test_hyperelastic_cooks_membrane_taylorhood_3d	P2	P1

The 2D mesh cooksmembrane.msh has 233 triangles, the 3D mesh cooksmembrane3d.msh 714 tetrahedra. Grid.Refinement refines them by conforming bisection, which roughly doubles the number of elements per level (535 and 1166 triangles on levels 1 and 2). The full load is applied in one Newton solve with line search for each \( \lambda \).

Results

2D, plane strain

Deformed membrane on Taylor–Hood refinement 4 for Poisson's ratios 0.35 and 0.4999, coloured by the displacement magnitude; dashed: undeformed membrane

Tip displacement \( u_y(A) \) in mm for Poisson's ratios 0.35 ( \( \lambda = 432.099 \) MPa) and 0.4999 ( \( \lambda = 926,000 \) MPa), and the number of degrees of freedom (displacement and pressure, including the constrained ones). The last row is the Richardson extrapolation of the last three even and the last three odd refinement levels:

Refinement	MINI, dofs	MINI, \( \nu = 0.35 \)	MINI, \( \nu = 0.4999 \)	Taylor–Hood, dofs	Taylor–Hood, \( \nu = 0.35 \)	Taylor–Hood, \( \nu = 0.4999 \)
1	1979	10.4676	8.8703	2583	10.5672	9.0009
2	4228	10.5291	8.9490	5490	10.5837	9.0239
3	8894	10.5535	8.9784	11,516	10.5847	9.0315
4	18,559	10.5749	9.0106	23,975	10.5912	9.0415
5	38,457	10.5817	9.0222	49,609	10.5910	9.0455
6	79,085	10.5888	9.0363	101,913	10.5937	9.0500
7	161,493	10.5906	9.0409	207,965	10.5933	9.0525
8	327,992	10.5930	9.0475
extrapolated		10.5947–10.5948	9.0549–9.0563		10.5947–10.5949	9.0577–9.0592

Relative error of the tip displacement at Poisson's ratio 0.35 with respect to the p-FEM value of Schröder et al. (2021); dotted: deviation of the extrapolated value

Schröder et al. report 10.5953 mm (p-FEM with plane-strain constraints, 18,213 dofs) and 10.5919 mm (quadratic triangles T2, 115,680 dofs) (Table 3 of [89]). Grid.Refinement bisects every triangle once per level, so the mesh size halves every second level and the tip displacement changes in steps of two levels. At Poisson's ratio 0.35, the change over two levels decreases by a factor of 2.7 to 3.3 from one pair of levels to the next for both elements, roughly \( N^{-3/4} \) in the number of degrees of freedom \( N \). Both elements extrapolate to 10.5947 to 10.5949 mm, 0.0004 to 0.0006 mm below the p-FEM value and 0.0028 to 0.0030 mm above the T2 value. Taylor–Hood on refinement 6 (101,913 dofs) is 0.0016 mm below the p-FEM value, T2 with 115,680 dofs 0.0034 mm.

At Poisson's ratio 0.4999, the factor is 2.0 to 2.3 and the values extrapolate to 9.0549 to 9.0592 mm. The finest meshes are 0.06 % to 0.07 % (Taylor–Hood, refinement 7) and 0.08 % to 0.10 % (MINI, refinement 8) below the extrapolated values of the same element.

All five values of \( \lambda \) on the finest meshes and extrapolated (mm):

\( \lambda \) [MPa]	MINI, refinement 8	Taylor–Hood, refinement 7	extrapolated, both elements
432.099	10.5930	10.5933	10.5947–10.5949
750	10.1082	10.1094	10.1109–10.1122
9260	9.1599	9.1642	9.1665–9.1705
92,600	9.0580	9.0629	9.0653–9.0696
926,000	9.0475	9.0525	9.0549–9.0592

Schröder et al. give no plane-strain reference for \( \lambda > 432.099 \) MPa.

3D

Tip displacement \( u_y(A) \) in mm on the finest meshes, with the p-FEM reference values of Schröder et al. (Table 9 of [89]):

\( \lambda \) [MPa]	reference	MINI, refinement 3	Taylor–Hood, refinement 2
432.099	11.381	11.183	11.349
750	11.156	10.933	11.119
9260	10.783	10.420	10.735
92,600	10.746	10.332	10.698
926,000	10.741	10.321	10.694

At Poisson's ratio 0.4999, MINI reaches 95 %, 95 % and 96 % of the reference on refinement levels 1, 2 and 3, and Taylor–Hood 99.4 % and 99.6 % on levels 1 and 2.

Running the tests

./test_hyperelastic_cooks_membrane_taylorhood params_mixed.input -Grid.Refinement 3

./test_hyperelastic_cooks_membrane_mini_3d params_mixed3d.input -Grid.Refinement 3

The 2D results and figures are reproduced in the build directory of the 2D tests with (requires numpy, matplotlib and pyvista)

python3 plot_results.py ./test_hyperelastic_cooks_membrane_mini ./test_hyperelastic_cooks_membrane_taylorhood

🛠 Edit above doc on GitLab