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

Test No. VNL05 Find 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:

$$\sigma =K{\epsilon}^{n}$$

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:

$$\epsilon r=\epsilon -\epsilon e=\left(\raisebox{1ex}{$\sigma $}\!\left/ \!\raisebox{-1ex}{$K$}\right.\right)\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.-\raisebox{1ex}{$\sigma $}\!\left/ \!\raisebox{-1ex}{$E$}\right.=\left(\raisebox{1ex}{$F$}\!\left/ \!\raisebox{-1ex}{$\left(K\ast A\right)$}\right.\right)\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.-\left(\raisebox{1ex}{$F$}\!\left/ \!\raisebox{-1ex}{$\left(E\ast A\right)$}\right.\right)$$

Where,

- $\epsilon $
- Total strain in the bar.
- $\epsilon e$
- Elastic component of the total strain.
- $\epsilon r$
- Residual strain.
- $A$
- Bar cross-section area.

Then residual displacement at the right end of the bar.

$$Ur=\underset{0}{\overset{L}{{\displaystyle \int}}}(\sqrt[n]{\frac{F}{K*A}}-F/\left(E*A\right))dx$$

## 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 | 3.3 | 2.48% |

25000 | 9.16 | 9.23 | 0.76% |

30000 | 19.077 | 20.2 | 5.89% |