# SS-V: 5040 Residual Deformations in an Axially Loaded Plastic Bar

Test No. VNL05Find residual deformations in a bar axially loaded beyond plasticity.

## Definition

Bar dimensions are 10 x 10 x 200 mm. Strain-stress 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 strain-stress 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

One-dimensional analytical reference solution is described here.

At strain $\epsilon$ and stress $\sigma$ , the residual strain is:(2)
$\epsilon r=\epsilon -\epsilon e=\left(\sigma }{K}\right)1}{n}-\sigma }{E}=\left(F}{\left(K\ast A\right)}\right)1}{n}-\left(F}{\left(E\ast A\right)}\right)$
Where,
$\epsilon$
Total strain in the bar.
$\epsilon e$
Elastic component of the total strain.
$\epsilon r$
Residual strain.
$A$
Bar cross-section area.
Then residual displacement at the right end of the bar.(3)

## 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%