1.5.2 Patch test for three-dimensional solid elements

Products: ABAQUS/Standard ABAQUS/Explicit

Elements tested

C3D4 C3D4H C3D4T C3D6 C3D6H C3D6T

C3D8 C3D8H C3D8I C3D8IH C3D8R C3D8RH C3D8RT

C3D10 C3D10H C3D10M C3D10MH C3D10MHT C3D10MT

C3D15 C3D15H C3D15V C3D15VH

C3D20 C3D20H C3D20R C3D20RH C3D27 C3D27H C3D27R C3D27RH

Problem description

Material:

Linear elastic, Young's modulus = 1.0 × 10⁶, Poisson's ratio = 0.25.

For coupled temperature-displacement elements dummy thermal properties are prescribed to complete the material definition.

Loading for Step 1:

Displacement boundary conditions at all exterior nodes: 10^–3(2)/2, 10^–3( 2)/2, 10^–3( 2z)/2.

In the ABAQUS/Explicit simulations this step is followed by an intermediate step in which the model is returned to its unloaded state.

Loading for Step 2:

Uniform pressure load: 10000. (Rigid body motion is constrained.)

Loading for Step 3:

Displacement boundary conditions at all exterior nodes: 10^–3(2)/2, 10^–3( 2)/2, 10^–3( 2z)/2, where x, y, and z are the coordinates of the undeformed geometry.

In the ABAQUS/Standard simulations this step is defined as a perturbation step; in the ABAQUS/Explicit simulations a velocity boundary condition that gives rise to the perturbation is specified instead.

Reference solution

The analytical results for each step are presented below.

Step 1: PERTURBATION

2000.
400.
10^–3.
10^–3.

Step 2: NLGEOM

10000.
0.
5.0 × 10^–3.
0.

In the ABAQUS/Explicit simulations this is the third step. (The second step in the ABAQUS/Explicit simulations returns the model to its unloaded state.)

Step 3: PERTURBATION

1990.
398.0.
9.95 × 10^–4.
9.95 × 10^–4.
In the ABAQUS/Explicit simulations this is the fourth step. The results from the third step in the ABAQUS/Explicit simulations must be subtracted from the results of the fourth step to obtain the perturbation about the loaded state.

Results and discussion

All elements except C3D27R and C3D27RH yield exact solutions. These elements use a special 14-point reduced-integration scheme since Gaussian 2 × 2 × 2 integration leaves too many kinematic nodes. The stiffness matrix is not integrated exactly with the employed integration rule, leading to small discrepancies in the results. The wedge elements and the quadratic reduced-integration brick elements pass only a restricted patch test; i.e., such elements with midside nodes on any edges will pass the patch test only if those edges are straight.

The *SECTION FILE and *SECTION PRINT output requests are used in the input files with C3D8H, C3D10MH, and C3D27RH elements to output accumulated quantities in different sections through the model.