1.5.5 Patch test for axisymmetric elements with twist

Product: ABAQUS/Standard

Elements tested

CGAX3 CGAX3H CGAX3HT CGAX3T CGAX4 CGAX4H CGAX4HT CGAX4R CGAX4RH CGAX4T CGAX6 CGAX6H CGAX6M CGAX6MH CGAX8 CGAX8H CGAX8HT CGAX8R CGAX8RH CGAX8RHT CGAX8RT CGAX8T

Problem description

Material:

Linear elastic, Young's modulus = 1.0 × 10⁶, Poisson's ratio = 0.25, conductivity = 4.85 × 10^–4.

Loading for Step 1:

A twist of 0.01 per unit length applied to face CD.

1.0 × 10^–2.

Loading for Step 2:

Displacement boundary conditions applied to all exterior nodes: 10^–3r, 10^–3, 0.

Nonuniform body force: To maintain a constant shear stress 400 and preserve equilibrium, an equilibrating body force, BZNU, is defined in user subroutine DLOAD as BZNU = –400, where r is the radius of the integration point.

Loading for Step 3:

Displacement boundary conditions applied to all exterior nodes: 10^–2r, 10^–2z, 0.

Loading for Step 4:

Displacement boundary conditions applied to the deformed geometry of Step 2 at all exterior nodes: 10^–3r, 10^–3, 0.

Nonuniform body force (as described for Step 2): BZNU = –400.

Loading for Step 5:

The displacement boundary conditions are the same as those applied in Step 3.

Temperatures are prescribed at every node along the boundary of the mesh. , where T is the temperature, and are arbitrary constants, and r, z denote spatial location.

Nonuniform distributed flux: To maintain a uniform heat flux, q, a distributed heat flux, BFNU, is defined in user subroutine DFLUX as BFNU = , where r is the radius of the integration point and k is the conductivity.

Reference solution

The analytical results for each step are presented below.

Step 1: PERTURBATION

Shear stress, , where r is the radial distance from the axis of symmetry and G is the shear modulus.

Resultant moment, 2 = 6283.2.

Step 2: PERTURBATION

2000.
400.
10^–3.
10^–3.

Step 3: NLGEOM

19900.
0.
9.95 × 10^–3.
0.

Step 4: PERTURBATION

2000.
400.
1 × 10^–3.
1 × 10^–3.

Step 5: *COUPLED TEMPERATURE-DISPLACEMENT

This step is applied only in tests of coupled temperature-displacement elements (CGAXxT).

Stresses and strains are the same as in Step 3. ; .

Results and discussion

The results agree well with the analytical solution for all elements.

The *SECTION FILE and *SECTION PRINT output requests are used in the input files with CGAX8RH elements to output accumulated quantities in different sections through the model.