A nearly incompressible hyperelastic block under a partial surface load, benchmark 2 (Sect. 3) of the selection of solid mechanics benchmarks of the DFG priority programme SPP 1748, Schröder et al. (2021) [89]. Two line singularities of the load intersect in a point, and the material is close to incompressible, so displacement-only elements of low order lock. The tests solve it with the mixed displacement–pressure formulation with the volumetric–isochoric split (Hyperelasticity with volumetric-isochoric split).

Problem

Modelled quarter of the block with its boundary conditions, dimensions in mm

The block has the width and length 100 mm and the height 50 mm. Because of the symmetry of the loading, one quarter, \( [0, 50]^3 \) mm, is modelled:

bottom \( z = 0 \): \( u_z = 0 \),
top \( z = 50 \) mm: \( u_x = u_y = 0 \),
symmetry planes \( x = 50 \) mm and \( y = 50 \) mm: \( u_x = 0 \) and \( u_y = 0 \), respectively,
the part \( 25 \leq x, y \leq 50 \) mm of the top face (the quarter of the 50 mm × 50 mm loaded area at the centre of the block) carries the dead load \( q = 3 \) MPa in \( -z \) direction.

The strain energy density is \( W = \frac{\mu}{2}(\operatorname{tr}\mathbf{C} - 3 - 2 \ln J) + \frac{\lambda}{2}(J - 1)^2 \) with \( \lambda = 499.92568 \) MPa and \( \mu = 1.61148 \) MPa (Poisson's ratio 0.4983). The quantity of interest is the vertical displacement \( |u_z(P)| \) of the centre \( P = (50, 50, 50) \) mm of the loaded area. Schröder et al. report a converged value of 20.0 mm, and for the trilinear displacement element (H1) 7.78 mm (260 dofs), 13.17 mm (1800 dofs), 17.54 mm (13,328 dofs) and 19.52 mm (102,432 dofs) (Table 17 of [89]).

Discretization

Test	Displacement	Pressure	Mesh
test_hyperelastic_incompressible_block_p1b_p1	P1 with bubble (MINI)	P1, box	tetrahedra, block_delaunay.msh
test_hyperelastic_incompressible_block_p2_p1	P2, hybrid control-volume / finite-element scheme	P1, box	tetrahedra, block_delaunay.msh
test_hyperelastic_incompressible_block_p2fe_p1fe	P2, finite elements	P1, finite elements	tetrahedra, block_delaunay.msh
test_hyperelastic_incompressible_block_q2_q1	Q2, hybrid	Q1, box	4 × 4 × 4 hexahedra

The load is applied in five steps (LoadStepping.NSteps).

Results

Vertical displacement \( |u_z(P)| \) in mm:

Test	dof positions (displacement + pressure)	\( \lvert u_z(P) \rvert \)
P1-bubble–P1 (MINI)	818	19.26
P2–P1, hybrid	1112	20.02
P2–P1, finite elements	1112	19.97
Q2–Q1, hybrid	854	20.10

With about a thousand dof positions, the mixed formulations are within 3.7 % of the converged value 20.0 mm. The trilinear displacement element of Schröder et al. gives 13.17 mm with 1800 dofs.

Deformed block after the last load step, Q2–Q1 on 8 × 8 × 8 hexahedra per quarter, coloured by the displacement magnitude; grey lines: undeformed block

On the 8 × 8 × 8 hexahedral mesh (5642 dof positions), Q2–Q1 gives \( |u_z(P)| \) = 20.14 mm.

The CTests compare the displacement field with a reference. The figure is produced in the build directory with (requires numpy, matplotlib and pyvista)

python3 plot_results.py ./test_hyperelastic_incompressible_block_q2_q1

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