# Critical Distance Default Values for Structural Steels and a Simple Formulation to Estimate the Apparent Fracture Toughness in U-Notched Conditions

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## Abstract

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## 1. Introduction

_{mat}):

## 2. The Theory of Critical Distances

_{mat}is the material fracture toughness (derived in cracked conditions) and σ

_{0}is a characteristic strength parameter, known as the inherent strength, which is generally higher than the ultimate tensile strength (σ

_{u}) and requires calibration. When the material behavior is fully linear-elastic, σ

_{0}is equal to σ

_{u}, and obtaining L is straightforward once the material fracture toughness and ultimate tensile strength are known.

_{mat}, derived in cracked conditions), the notch radius (ρ), and the material critical distance (L). Analogously, when considering the LM (Equation (4)), together with Creager-Paris stress distribution (Equation (5)), the result is an even simpler equation [1]:

_{mat}. Accordingly, fracture takes place when:

_{mat}with ${K}_{\mathrm{mat}}^{N}$ in the definition of the K

_{r}coordinate of the assessment point. This coordinate is defined as the ratio between the applied stress intensity factor (K

_{I}) and the material fracture resistance (K

_{mat}for cracks and ${K}_{\mathrm{mat}}^{N}$ for notches) [20,21,22]. This work is not focused on the FAD approach, but the application of the ${K}_{\mathrm{mat}}^{N}$ values generated here to FAD analyses would be straightforward.

## 3. Materials and Methods

_{0}values were directly obtained from Equation (2) once K

_{mat}and L were known. The total number of tests is 394, with L values varying from 0.0028 mm up to 0.0198 mm. Thus, the experimental results collected here represent a wide range of situations.

## 4. Derivation of Default Values of L and Apparent Fracture Toughness Estimations

_{0}have been derived (through Equation (2)), and provided that σ

_{u}values are known for each material condition, it is proposed to establish the (non-dimensional) relation between the inherent strength (σ

_{0}) and the ultimate tensile strength (σ

_{u}):

_{u}), may substitute the inherent strength (σ

_{0}) in Equation (2), resulting in a default (conservative) critical distance (L

_{d}) that does not need to be calibrated and allows the TCD to be applied safely:

_{u}in MPa):

_{u}in MPa):

_{mat}) against (ρ/L

_{d})

^{1/2}plot.

_{mat}considered in the analysis to be introduced in Equation (13): the average value of the experimental results (K

_{mat}) for each particular steel and testing temperature, and K

_{mat,0.95}, which is associated to a 95% confidence level. The latter has been derived assuming a normal distribution of the experimental results obtained in cracked conditions (K

_{mat}), and would be the prediction required for structural integrity assessment purposes, whereas the former is the prediction that should better capture the physics of the phenomenon being analyzed. The above referred confidence level is, therefore, limited for the fracture results obtained in cracked conditions (ρ = 0 mm). For notched conditions (i.e., for the rest of the curve), using K

_{mat,0.95}provides a more conservative estimation of the apparent fracture toughness than that obtained with the average value of the fracture toughness (K

_{mat}), but the corresponding predictions are not necessarily associated to a 95% confidence level.

- The LM predictions derived from the proposed default values of the material critical distance (L
_{d}) capture a significant part of the physics of the notch effect, given that the LM prediction adequately follows the tendency of the experimental results, which have been obtained for a wide variety of structural steels and conditions. The results are particularly accurate considering fitting Equations (11) and (12) and the average value of the material fracture toughness (K_{mat}) for each particular steel and working temperature. These fitting equations significantly reduce the conservatism obtained from the lower bound values proposed in [31]. - The most conservative results have been obtained in Lower Shelf conditions when using the lower bound value of m (m = 1.6). Even under such circumstances, the resulting apparent fracture toughness estimations may be significantly higher than the corresponding fracture toughness obtained in cracked conditions, so the potential reduction of conservatism is still important.
- If the LM evaluations are to be used in structural integrity assessments, although the use of K
_{mat}(average value of the material fracture toughness obtained in cracked conditions) captures most of the notch effect adequately, it may be unsafe on many occasions due to the high scatter of the fracture processes. This means that it sometimes provides apparent fracture toughness values higher than those measured experimentally (see Figure 4 and Figure 6). In order to provide a fracture analysis tool to be used in structural integrity assessments, it is necessary to propose a methodology that is capable of providing safe predictions of the apparent fracture toughness. With this purpose, it is proposed to use a 95% confidence level value of the fracture toughness (K_{mat,0.95}). - Thus, the most accurate, yet conservative, methodology for the apparent fracture toughness estimation arises from the combination of K
_{mat,0.95}(as the material fracture toughness) and the m values derived from Equations (11) and (12), for LS and DBTZ conditions respectively.

## 5. Summary

_{d}) to be obtained. The m factors have been provided as fitting curves depending exclusively on the material ultimate tensile strength. Results are also shown when using previously proposed m lower bound values, which provide more conservative estimations of the material apparent fracture toughness. The methodology has been applied to four structural steels (S275JR, S355J2, S460M, and S690Q) tested on notched conditions (U-shaped notches) and operating at both the Lower Shelf and the Ductile-to-Brittle Transition Zone. The experimental values of the apparent fracture toughness (${K}_{\mathrm{mat}}^{N}$) and the notch radii values have been normalized by the corresponding fracture toughness (K

_{mat}or K

_{mat,0.95}) obtained in cracked conditions and by the default values derived for the critical distance (L

_{d}) respectively, representing the 394 tests in (${K}_{\mathrm{mat}}^{N}$/K

_{mat}) − (ρ/L

_{d})

^{1/2}plots.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Experimental notch fracture toughness results, Line Method (LM) fitting and derivation of the corresponding critical distance. Steel S275JR at −120 °C.

**Figure 3.**m values versus tensile strength for structural steels operating at the Ductile-to-Brittle Transition Zone. Derivation of a safe estimation (m = 13.0).

**Figure 4.**Apparent fracture toughness predictions (m derived from Equation (11)) and comparison with the experimental results. Structural steels operating at Lower Shelf temperatures.

**Figure 5.**Apparent fracture toughness predictions (m = 1.6) and comparison with the experimental results. Structural steels operating at Lower Shelf temperatures.

**Figure 6.**Apparent fracture toughness predictions (m derived from Equation (12)) and comparison with the experimental results. Structural steels operating at the DBTZ.

**Figure 7.**Apparent fracture toughness predictions (m = 13.0) and comparison with the experimental results. Structural steels operating at the DBTZ.

Steel | Number of Tests | K_{mat} (MPa·m^{1/2}) | L (mm) | σ_{u} (MPa) | σ_{0} (MPa) |
---|---|---|---|---|---|

S275JR (−120 °C, LS) | 23 | 48.80 | 0.0137 | 614 | 7438 |

S275JR (−90 °C, LS) | 24 | 62.72 | 0.0062 | 597 | 14,211 |

S275JR (−50 °C, DBTZ) | 24 | 80.60 | 0.0049 | 565 | 20,543 |

S275JR (−30 °C, DBTZ) | 24 | 100.70 | 0.0061 | 549 | 23,003 |

S275JR (−10 °C, DBTZ) | 34 | 122.80 | 0.0083 | 536 | 24,048 |

S355J2 (−196 °C, LS) | 24 | 31.27 | 0.0198 | 923 | 3965 |

S355J2 (−150 °C, DBTZ) | 21 | 60.56 | 0.0084 | 758 | 11,789 |

S355J2 (−120 °C, DBTZ) | 22 | 146.60 | 0.0168 | 672 | 20,179 |

S355J2 (−100 °C, DBTZ) | 35 | 157.40 | 0.0140 | 647 | 23,734 |

S460M (−140 °C, DBTZ) | 24 | 45.60 | 0.0028 | 795 | 15,375 |

S460M (−120 °C, DBTZ) | 24 | 88.29 | 0.0075 | 759 | 18,189 |

S460M (−100 °C, DBTZ) | 33 | 88.58 | 0.0053 | 727 | 21,708 |

S690Q (−140 °C, DBTZ) | 24 | 69.11 | 0.0069 | 1112 | 14,844 |

S690Q (−120 °C, DBTZ) | 24 | 103.80 | 0.0131 | 1061 | 16,180 |

S690Q (−100 °C, DBTZ) | 34 | 125.40 | 0.0170 | 1016 | 17,159 |

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**MDPI and ACS Style**

Cicero, S.; Fuentes, J.D.; Procopio, I.; Madrazo, V.; González, P.
Critical Distance Default Values for Structural Steels and a Simple Formulation to Estimate the Apparent Fracture Toughness in U-Notched Conditions. *Metals* **2018**, *8*, 871.
https://doi.org/10.3390/met8110871

**AMA Style**

Cicero S, Fuentes JD, Procopio I, Madrazo V, González P.
Critical Distance Default Values for Structural Steels and a Simple Formulation to Estimate the Apparent Fracture Toughness in U-Notched Conditions. *Metals*. 2018; 8(11):871.
https://doi.org/10.3390/met8110871

**Chicago/Turabian Style**

Cicero, Sergio, Juan Diego Fuentes, Isabela Procopio, Virginia Madrazo, and Pablo González.
2018. "Critical Distance Default Values for Structural Steels and a Simple Formulation to Estimate the Apparent Fracture Toughness in U-Notched Conditions" *Metals* 8, no. 11: 871.
https://doi.org/10.3390/met8110871