Automated Calculation of Strain Hardening Parameters from Tensile Stress vs. Strain Data for Low Carbon Steel Exhibiting Yield Point Elongation

Abstract

Existing guidance from ASTM standards on extracting mechanical properties from uniaxial tension test data is not suitable for high-volume applications because it lacks automation. The Pacific Gas and Electric Company (PG&E) has performed over 450 uniaxial tension tests on samples extracted from dozens of distinct natural gas pipeline features in support of materials verification efforts; 144 of these tests recorded the entire deformation response up to fracture and were subsequently analyzed herein. Algorithms were developed to enable automatic, batch post-processing of the tensile data. Among the mechanical properties that this software extracts from the tensile data are the power-law hardening exponent and strength coefficient. A novel algorithm was developed to calculate these power-law parameters while accommodating the range of yielding and hardening behaviors present among the low carbon pipeline steels tested in this effort. The algorithm, presented in this brief technical note, first identifies the lower limit of the hardening region by calculating the tangent modulus of the stress–strain curve, which reaches its maximum value at the onset of strain hardening. The end of uniaxial hardening coincides with the ultimate tensile stress and the end of uniform deformation. From there the algorithm computes the power-law hardening parameters using conventional linear regression. Analysis of the data shows that this approach is more accurate than two other approaches: (1) regressing from 0.2% plastic strain to the limit load, and (2) iteratively regressing to identify the region that minimizes the regression error. The advantage of this approach, over existing methods, is observed for materials that exhibit a yield plateau. To further demonstrate the utility of the strain hardening information collected during this effort, the relationship between strain hardening exponent and ratio of yield strength to ultimate tensile strength from this data was compared to those reported in previous literature pertaining to remaining life assessments of steel line pipe. Overall, the strain hardening data obtained in this algorithm indicates greater hardening in pipeline steels than some previously published results. This algorithm has been made available as part of this paper’s supplementary materials.

Appendices

Appendix 1: Analytical Derivation of the Relationship Between R and n

The ratio of YS to UTS was defined, $R\equiv YS/UTS$. Relating YS to true stress at yielding, ${\sigma }_{YS}$:

$$YS=\frac{{\sigma }_{YS}}{\left(1+{e}_{YS}\right)}$$

Substituting the power law model (${\sigma }_{YS}=K({{\varepsilon }_{YS})}^{n}$):

$$YS=\frac{K({{\varepsilon }_{YS})}^{n}}{\left(1+{e}_{YS}\right)}=\frac{K{\left(\mathrm{ln}\left(1+{e}_{YS}\right)\right)}^{n}}{\left(1+{e}_{YS}\right)}$$

Given that engineering strain at yielding, ${e}_{YS}\equiv 0.005$:

$$YS=\frac{K{\left(\mathrm{ln}\left(1+0.005\right)\right)}^{n}}{\left(1+0.005\right)}\cong K{\left(0.005\right)}^{n}$$

Relating UTS to true stress at UTS, ${\sigma }_{UTS}$, at the strain at UTS, ${e}_{UTS}$:

$$UTS=\frac{{\sigma }_{UTS}}{\left(1+{e}_{UTS}\right)}$$

Relating engineering strain to true strain at UTS:

$${e}_{UTS}=\mathrm{exp}\left({\varepsilon }_{UTS}\right)-1$$

$$\therefore UTS=\frac{{\sigma }_{UTS}}{\mathrm{exp}\left({\varepsilon }_{UTS}\right)}$$

According to Considère’s Criterion, $\sigma ={\sigma }_{UTS}$ when the following equation is satisfied:

$$\frac{d\sigma }{d\varepsilon }=\sigma$$

Thus, differentiating the power-law model ($\sigma =K{\varepsilon }^{n}$):

$$\frac{d\sigma }{d\varepsilon }=nK{\varepsilon }^{n-1}$$

Equating the results per Considère’s Criterion:

$$nK{({\varepsilon }_{UTS})}^{n-1}=K({{\varepsilon }_{UTS})}^{n}$$

$$\therefore n={\varepsilon }_{UTS}$$

$$\therefore UTS=\frac{{\sigma }_{UTS}}{\mathrm{exp}\left(n\right)}$$

Substituting the power law model (${\sigma }_{UTS}=K({{\varepsilon }_{UTS})}^{n}$):

$$UTS=\frac{K({{\varepsilon }_{UTS})}^{n}}{\mathrm{exp}\left(n\right)}=\frac{K{n}^{n}}{\mathrm{exp}\left(n\right)}=K{\left(\frac{n}{\mathrm{exp}\left(1\right)}\right)}^{n}$$

Therefore:

$$R=\frac{YS}{UTS}\cong K{\left(0.005\right)}^{n}\times \frac{1}{K}{\left(\frac{\mathrm{exp}\left(1\right)}{n}\right)}^{n}={\left(\frac{0.005\mathrm{exp}\left(1\right)}{n}\right)}^{n}$$

Performing a linear regression over the range $0.03<n<0.25$:

$$\therefore n\cong 0.453-0.432R$$

Appendix 2: Algorithm for Automatic Calculation of Strain Hardening Parameters (R Language)