Test No. VNL05 Find residual deformations in a bar
axially loaded beyond plasticity.
Definition
Bar dimensions are 10 x 10 x 200 mm. Strainstress curve of the bar material is
defined by the power law:
(1)
$$\sigma =K{\epsilon}^{n}$$
Where,

$K$
 Strength coefficient.

$n$
 Must be in the range [0,1].

$n$
=0
 Material is perfectly plastic.

$n$
=1
 Material is elastic.
The left end of the bar is clamped and the right end is loaded with force F.
The material properties are:
 Properties
 Value

$K$
 530 MPa

$n$
 0.26
 Poisson's Ratio
 0
Note: Elasticity modulus defined by the first two points of the strainstress
curve is E=2.67324e+10 Pa.
The study was performed for the following load F values: 20000 N, 25000 N, and 30000
N
Reference Solution
Onedimensional analytical reference solution is described here.
At strain
$\epsilon $
and stress
$\sigma $
, the residual strain is:
(2)
$$\epsilon r=\epsilon \epsilon e=\left(\raisebox{1ex}{$\sigma $}\!\left/ \!\raisebox{1ex}{$K$}\right.\right)\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$n$}\right.\raisebox{1ex}{$\sigma $}\!\left/ \!\raisebox{1ex}{$E$}\right.=\left(\raisebox{1ex}{$F$}\!\left/ \!\raisebox{1ex}{$\left(K\ast A\right)$}\right.\right)\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$n$}\right.\left(\raisebox{1ex}{$F$}\!\left/ \!\raisebox{1ex}{$\left(E\ast A\right)$}\right.\right)$$
Where,

$\epsilon $
 Total strain in the bar.

$\epsilon e$
 Elastic component of the total strain.

$\epsilon r$
 Residual strain.

$A$
 Bar crosssection area.
Then residual displacement at the right end of the bar.
(3)
$$Ur=\underset{0}{\overset{L}{{\displaystyle \int}}}(\sqrt[n]{\frac{F}{K*A}}F/\left(E*A\right))dx$$
Results
The bar was modeled as a 3D solid. The left end of the solid was fixed and the right
end loaded with axial force (
Figure 3).
The following table summarizes the residual deformations results.
Force F [N] 
SOL Reference, Residual
Displacement [mm] 
SimSolid, Residual Displacement
[mm] 
% Difference 
20000 
3.22 
3.43 
6.78% 
25000 
9.16 
9.172 
0.13% 
30000 
19.077 
19.14 
0.33% 