Patch test for three-dimensional solid 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

C3D4
C3D4H
C3D4T
C3D5
C3D6
C3D6H
C3D6T

C3D8
C3D8H
C3D8I
C3D8IH
C3D8R
C3D8RH
C3D8RT
C3D8T

C3D10
C3D10H
C3D10HS
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: $u_{x} = $ 10⁻³(2 $x + y + z$ )/2, $u_{y} = $ 10⁻³( $x +$ 2 $y + z$ )/2, $u_{z} = $ 10⁻³( $x + y +$ 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: $u_{x} = $ 10⁻³(2 $x + y + z$ )/2, $u_{y} = $ 10⁻³( $x +$ 2 $y + z$ )/2, $u_{z} = $ 10⁻³( $x + y +$ 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

$σ_{x} = σ_{y} = σ_{z} = $ 2000.
$σ_{x y} = σ_{y z} = σ_{x z} = $ 400.
$ϵ_{x} = ϵ_{y} = ϵ_{z} = $ 10⁻³.
$γ_{x y} = γ_{y z} = γ_{x z} = $ 10⁻³.

Step 2: NLGEOM

$σ_{x} = σ_{y} = σ_{z} = $ 10000.
$σ_{x y} = σ_{y z} = σ_{x z} = $ 0.
$ϵ_{x} = ϵ_{y} = ϵ_{z} = $ 5.0 × 10⁻³.
$γ_{x y} = γ_{y z} = γ_{x 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_{x} = σ_{y} = σ_{z} = $ 1990.
$σ_{x y} = σ_{y z} = σ_{x z} = $ 398.0.
$ϵ_{x} = ϵ_{y} = ϵ_{z} = $ 9.95 × 10⁻⁴.
$γ_{x y} = γ_{y z} = γ_{x z} = $ 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.

Section output requests to the results (.fil) file and to the data (.dat) file are used in the input files with C3D8H, C3D10MH, and C3D27RH elements to output accumulated quantities in different sections through the model.

Input files

Abaqus/Standard input files

ec34sfp2.inp: C3D4 elements.
ec34shp2.inp: C3D4H elements.
ec35sfp2.inp: C3D5 elements.
ec36sfp2.inp: C3D6 elements.
ec36shp2.inp: C3D6H elements.
ec38sfp2.inp: C3D8 elements.
ec38shp2.inp: C3D8H elements.
ec38sip2.inp: C3D8I elements.
ec38sjp2.inp: C3D8IH elements.
ec38srp2.inp: C3D8R elements.
ec38syp2.inp: C3D8RH elements.
ec3asfp2.inp: C3D10 elements.
ec3ashp2.inp: C3D10H elements.
ec3asip2.inp: C3D10HS elements.
ec3askp2.inp: C3D10M elements.
ec3aslp2.inp: C3D10MH elements.
ec3atlp2.inp: C3D10MHT elements.
ec3atkp2.inp: C3D10MT elements.
ec3fsfp2.inp: C3D15 elements.
ec3fshp2.inp: C3D15H elements.
ec3isfp2.inp: C3D15V elements.
ec3ishp2.inp: C3D15VH elements.
ec3ksfp2.inp: C3D20 elements.
ec3kshp2.inp: C3D20H elements.
ec3ksrp2.inp: C3D20R elements.
ec3ksyp2.inp: C3D20RH elements.
ec3rsfp2.inp: C3D27 elements.
ec3rshp2.inp: C3D27H elements.
ec3rsrp2.inp: C3D27R elements.
ec3rsyp2.inp: C3D27RH elements.

Abaqus/Explicit input files

stresspatch_xpl_c3d4t.inp: C3D4T elements.
stresspatch_xpl_c3d6t.inp: C3D6T elements.
stresspatch_xpl_c3d8rt.inp: C3D8RT elements.
stresspatch_xpl_c3d8t.inp: C3D8T elements.