NL3: Hardening with two variables under load control

This problem provides evidence that Abaqus can reproduce the result from the benchmark defined by NAFEMS and cited as the reference solution.

This page discusses:

Elements tested
Problem description
Reference solution
Results and discussion
Input files

ProductsAbaqus/Standard

Elements tested

T2D2

Problem description

Model:

AE = 5.0 × 10⁷, L = 2500, $α$ L = 25, K₁= 1.5, K₂ = AE/L(1 + $α ^{2}$ )^1/2 = 19999.0, K₃ = 2.0.

Boundary conditions:

$u_{y} = $ 0 at node 1, $u_{x} = $ 0 at node 9.

Loading:

A load is applied to node 1 in the x-direction.

Reference solution

This is a test recommended by the National Agency for Finite Element Methods and Standards (U.K.): Test NL3 from NAFEMS Publication NNB, Rev. 1, “NAFEMS Non-Linear Benchmarks,” October 1989.


Step	P	$u_{1}$	v
1	1100	0.1803	10.37
2	2200	0.7160	35.45
3	3300	8.515	180.7
4	4400	1339	2189
5	5500	3468	2280
6	6600	4986	236.3

Results and discussion

The results are shown in the following table. The values enclosed in parentheses are percentage differences with respect to the reference solution.


Step	P	$u_{1}$	v
1	1100	0.1802 (−0.06%)	10.37 (0.0%)
2	2200	0.7157 (−0.0%)	35.44 (−0.03%)
3	3300	8.564 (+0.58%)	181.3 (+0.33%)
4	4400	1339 (0.0%)	2189 (0.0%)
5	5500	3469 (+0.03%)	2280 (0.0%)
6	6600	4985 (−0.02%)	244.7 (+3.55%)

Input files

nnl3xf2x.inp: T2D2 elements.