Patch test for axisymmetric elements

This problem contains basic test cases for one or more Abaqus elements and features.

The following topics are discussed:

Elements tested
Problem description
Reference solution
Results and discussion
Input files

ProductsAbaqus/StandardAbaqus/Explicit

Elements tested

CAX3
CAX3H
CAX3T
CAX4
CAX4H
CAX4I
CAX4IH
CAX4R
CAX4RH
CAX4RHT
CAX4RT

CAX6
CAX6H
CAX6M
CAX6MH
CAX8
CAX8H
CAX8R
CAX8RH

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 applied to all exterior nodes: $u_{r} =$ 10⁻³r, $u_{z} =$ 10⁻³ $(r + z)$ .

Nonuniform body force: To maintain a constant shear stress $σ_{r z} =$ 400 and preserve equilibrium, an equilibrating body force, BZNU, is defined in user subroutine DLOAD as BZNU $= - σ_{r z} / r =$ −400 $/ r$ , where r is the radius of the integration point.

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

Displacement boundary conditions applied to all exterior nodes: $u_{r} =$ 10⁻²r, $u_{z} =$ 10⁻²z.

Loading for Step 3

Displacement boundary conditions applied to the deformed geometry of Step 2 at all exterior nodes: $u_{r} =$ 10⁻³r, $u_{z} =$ 10⁻³ $(r + z)$ .

Nonuniform body force (as described for Step 1): BZNU $= - σ_{r z} / r =$ −400 $/ r$ .

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

$σ_{r} = σ_{z} = σ_{θ} =$ 2000.
$σ_{r z} =$ 400.
$ϵ_{r} = ϵ_{z} = ϵ_{θ} =$ 10⁻³.
$γ_{r z} =$ 10⁻³.

Step 2: NLGEOM

$σ_{r} = σ_{z} = σ_{θ} =$ 19900.
$σ_{r z} =$ 0
$ϵ_{r} = ϵ_{z} = ϵ_{θ} =$ 9.95 × 10⁻³.
$γ_{r z} =$ 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

$σ_{r} = σ_{z} = σ_{θ} =$ 2000.
$σ_{r z} =$ 400.
$ϵ_{r} = ϵ_{z} = ϵ_{θ} =$ 1 × 10⁻³.
$γ_{r z} =$ 1 × 10⁻³.

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 yield exact solutions.

Input files

Abaqus/Standard input files

eca3sfp5.inp: CAX3 elements.
eca3sfp5.f: User subroutine DLOAD used in eca3sfp5.inp.
eca3shp5.inp: CAX3H elements.
eca3shp5.f: User subroutine DLOAD used in eca3shp5.inp.
eca4sfp5.inp: CAX4 elements.
eca4sfp5.f: User subroutine DLOAD used in eca4sfp5.inp.
eca4shp5.inp: CAX4H elements.
eca4shp5.f: User subroutine DLOAD used in eca4shp5.inp.
eca4sip5.inp: CAX4I elements.
eca4sip5.f: User subroutine DLOAD used in eca4sip5.inp.
eca4sjp5.inp: CAX4IH elements.
eca4sjp5.f: User subroutine DLOAD used in eca4sjp5.inp.
eca4srp5.inp: CAX4R elements.
eca4srp5.f: User subroutine DLOAD used in eca4srp5.inp.
eca4syp5.inp: CAX4RH elements.
eca4syp5.f: User subroutine DLOAD used in eca4syp5.inp.
eca4typ5.inp: CAX4RHT elements.
eca4typ5.f: User subroutine DLOAD used in eca4typ5.inp.
eca4trp5.inp: CAX4RT elements.
eca4trp5.f: User subroutine DLOAD used in eca4trp5.inp.
eca6sfp5.inp: CAX6 elements.
eca6sfp5.f: User subroutine DLOAD used in eca6sfp5.inp.
eca6shp5.inp: CAX6H elements.
eca6shp5.f: User subroutine DLOAD used in eca6shp5.inp.
eca6skp5.inp: CAX6M elements.
eca6skp5.f: User subroutine DLOAD used in eca6skp5.inp.
eca6slp5.inp: CAX6MH elements.
eca6slp5.f: User subroutine DLOAD used in eca6slp5.inp.
eca8sfp5.inp: CAX8 elements.
eca8sfp5.f: User subroutine DLOAD used in eca8sfp5.inp.
eca8shp5.inp: CAX8H elements.
eca8shp5.f: User subroutine DLOAD used in eca8shp5.inp.
eca8srp5.inp: CAX8R elements.
eca8srp5.f: User subroutine DLOAD used in eca8srp5.inp.
eca8syp5.inp: CAX8RH elements.
eca8syp5.f: User subroutine DLOAD used in eca8syp5.inp.

Abaqus/Explicit input files

stresspatch_xpl_cax3t.inp: CAX3T elements.
stresspatch_xpl_cax4rt.inp: CAX4RT elements.
stresspatch_xpl_cax.f: User subroutine VDLOAD for the Abaqus/Explicit simulations.