Generalized plane strain elements with relative motion of bounding planes

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

This page discusses:

ProductsAbaqus/Standard

Elements tested

CPEG3

CPEG3H

CPEG3HT

CPEG3T

CPEG4

CPEG4H

CPEG4HT

CPEG4I

CPEG4IH

CPEG4R

CPEG4RH

CPEG4RHT

CPEG4RT

CPEG4T

CPEG6

CPEG6H

CPEG6M

CPEG6MH

CPEG6MHT

CPEG6MT

CPEG8

CPEG8H

CPEG8HT

CPEG8R

CPEG8RH

CPEG8RHT

CPEG8T

Problem description



Material:

Linear elastic, Young's modulus = 30 × 106, Poisson's ratio = 0.3.

Boundary conditions:

uxC=uyC=0,uyD=0.

Step 1 (Perturbation)

An out-of-plane displacement of 0.01 units (motion of one bounding plane relative to the other) is applied to degree of freedom 3 of the reference node, which is the change in fiber length degree of freedom.

uzA​=0.01ϕxA​=0.0ϕyA​=0.0
Analytical solution:
Stresses

At every node σz⁢z=​ 3.0 × 105.

Strains

At every node εx⁢x=εy⁢y=​ −3.0 × 10−3, εz⁢z=​ 1.0 × 10−2.

Step 2 (Perturbation)

A relative rotation of 0.01 radians about the y-axis is applied to degree of freedom 5 of the reference node (the rotation degree of freedom of one bounding plane relative to the other).

uzA​=0.0ϕxA​=0.0ϕyA​=0.01
Analytical solution:
Stresses

Maximum tensile stress σz⁢zm⁢a⁢x=​ 1.5 × 105.

Strains

Maximum tensile strain εz⁢zm⁢a⁢x=​ 5 × 10−3.

Results and discussion

For Step 1, all element types yield the exact solution. The results for Step 2 are given in the following table:

Element typeσz⁢zm⁢a⁢xεz⁢zm⁢a⁢x
CPEG3 1.264 × 105 4.167 × 10−3
CPEG3H 1.264 × 105 4.167 × 10−3
CPEG3HT 1.264 × 105 4.167 × 10−3
CPEG3T 1.264 × 105 4.167 × 10−3
CPEG4 1.131 × 105 3.750 × 10−3
CPEG4H 1.131 × 105 3.750 × 10−3
CPEG4HT 1.131 × 105 3.750 × 10−3
CPEG4I 1.500 × 105 5.000× 10−3
CPEG4IH 1.500 × 105 5.000 × 10−3
CPEG4R 1.125 × 105 3.750 × 10−3
CPEG4RH 1.125 × 105 3.750 × 10−2
CPEG4RHT 1.125 × 105 3.750 × 10−2
CPEG4RT 1.125 × 105 3.750 × 10−3
CPEG4T 1.131 × 105 3.750 × 10−3
CPEG6 1.500× 105 5.000× 10−3
CPEG6H 1.500× 105 5.000 × 10−3
CPEG6M 1.504× 105 5.000 × 10−3
CPEG6MH 1.504 × 105 5.000× 10−3
CPEG6MHT 1.504 × 105 5.000× 10−3
CPEG6MT 1.504 × 105 5.000× 10−3
CPEG8 1.500 × 105 5.000 × 10−3
CPEG8H 1.500 × 105 5.000 × 10−3
CPEG8HT 1.500 × 105 5.000 × 10−3
CPEG8R 1.500 × 105 5.000 × 10−3
CPEG8RH 1.500 × 105 5.000 × 10−3
CPEG8RHT 1.500 × 105 5.000 × 10−3
CPEG8T 1.500 × 105 5.000 × 10−3

Second-order quadrilateral elements, first-order incompatible mode elements, and quadratic triangles yield the exact solutions. Modified triangles yield nearly exact solutions. Other element types exhibit stiff response.