# Finite Fracture Mechanics Assessment in Moderate and Large Scale Yielding Regimes

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## Abstract

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## 1. Introduction

_{c}(G

_{c}$\propto $ K

_{Ic}

^{2}, K

_{Ic}being the fracture toughness) or the tensile strength σ

_{u}, respectively. The crack advance l results to be a function of K

_{Ic}and σ

_{u}, and more in general it becomes a structural parameter in the case of coupled criteria, i.e., when the energy and stress conditions are coupled together [5,6]. It is worthwhile to mention that under the small scale yielding regime, the generalized stress intensity factor (GSIF, governing the asymptotic expansions of both the stress field and the SIF related to a virtual crack stemming from the notch tip providing the crack driving force) results to be the dominating failure parameter: Fracture can thus be supposed to take place when it reaches its critical value, also known as generalized (or notch) fracture toughness.

_{u}obtained by tensile testing hourglass-shape samples can be lower than the real one σ

_{0}. The ratio σ

_{0}/σ

_{u}can vary sensibly from material to material. Taylor [1] observed by fitting the value of σ

_{0}through the simple point stress criterion that the ratio σ

_{0}/σ

_{u}approaches the unit value for ceramics, it is comprised between one and two for polymers, and can be sensibly larger (even up to eight) for metals. Of course, in this latter case, it is difficult to refer to σ

_{0}as a tensile strength, and some attempts to provide a clear physical/mechanics were recently put forward [7]. It should be also mentioned that in order to avoid the problems raised when dealing with plain specimens, Seweryn [3,4] decided to test blunt notched samples where the root radius was large enough to provide a nearly constant stress field: The maximum tensile stress at the notch tip was estimated equal to σ

_{0.}

_{0}was estimated through a fitting procedure on the stress criterion, namely through the intersection point of the failure stress fields for a blunt and a sharp notched structure. Finally, it was concluded that prediction accuracy slightly increased by using a numerical elasto-plastic analysis, but not so much to justify the huge computational effort.

_{0}was evaluated through the Equivalent Material Concept (EMC) theory proposed by Torabi [15]. Once the fracture toughness K

_{Ic}is fixed, the EMC idea is to compare the behaviour of a ductile material with that of a virtual brittle material possessing a different tensile strength σ

_{0}. Its values are determined by considering identical values for the strain energy density (i.e., the area below the stress-strain curve) required by the ductile material under investigation and by the virtual brittle one. In formulae:

_{Y}is the yield strength, E is the Young’s modulus, K is the strain-hardening coefficient, n is the strain-hardening exponent, and ε

_{u}is the true plastic strain at maximum load. Thus, differently from the previous approaches where σ

_{0}was no more than a fitting parameter

_{,}according to EMC it recovers a precise physical meaning, and its value can be determined starting from the real experimental properties. More recently also the strain energy density (SED) criterion [2] has been exploited and applied to the experimental results as before [16,17].

## 2. FFM Criteria

_{I}(c) is the SIF related to a crack of length c stemming from the notch root (Figure 1). An analytical relationship for the function K

_{I}(c) was proposed by Sapora et al. [27]

_{y}(x) together with the material tensile strength σ

_{u}. It can be either a punctual requirement, ${\sigma}_{y}\left(x\right)\hspace{0.17em}\ge \hspace{0.17em}{\sigma}_{u}\hspace{0.17em}$ for $0\le x\le l\hspace{0.17em}$, which can be expressed as [5]

_{c}, and the apparent generalized fracture toughness ${K}_{Ic}^{V,\rho}$ (i.e., the failure load).

_{Ic}, σ

_{u}) and a given geometry (ω, ρ), the critical crack advancement ${l}_{\hspace{0.17em}c}$ can be evaluated from the former equation in (14). This value must then be inserted into the latter equation to obtain the apparent generalized fracture toughness ${K}_{Ic}^{V,\rho}$.

^{®}code is less than 5 s for each geometry, showing the potentiality of the present semi-analytical approach.

## 3. Comparison with Experimental Results

_{Ic}was derived experimentally, the tensile strength σ

_{u}was obtained through the EMC by means of Equation (1) (and thus the parameter is re-termed as σ

_{0}, Section 1): Table 2 reports the values implemented in the present analysis taken from [12,13]. The ratio σ

_{0}/σ

_{u}is equal to 3.16 for Al 7075-T6 and to 3.65 for Al 6061-T6, reflecting the fact failure of the samples made of Al 6061-T6 involved a larger amount of plasticity.

_{ch}≈ 0.73 mm for Al7075-T6, and l

_{ch}≈ 1.3 mm for Al6061-T6. It can be seen that the apparent generalized fracture toughness increases as the radius increases and/or the angle decreases. Thus, the cracked configuration is the most affected one by the presence of a radius ρ ≠ 0. As concerns the crack advance l

_{c}, it results a function of the material (through l

_{ch}) and of the radius, tending to 2/(π·1.12

^{2}) as ρ tends to infinite (smooth elements): From Figure 7 it can be seen that the values related to the crack case (ω = 0°) are the lowest ones for non-negligible radii. The analytical expressions for ${K}_{Ic}^{V}$ and l

_{c}when ρ = 0 (sharp case) can be found in [6].

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Blunt V-notch geometry. The presence of a “virtual” crack of length c stemming from the notch tip is also depicted.

**Figure 2.**U-notched structures (ω = 0°), apparent fracture toughness: Predictions by punctual FFM (dashed line) and average FFM (continuous line) related to experimental data on Al 7075-T6 plates (circles) and on Al 6061-T6 plates (triangles).

**Figure 3.**Blunt V-notched structures (ω = 30°), apparent generalized fracture toughness: Predictions by punctual FFM (dashed line) and average FFM (continuous line) related to experimental data on Al 7075-T6 plates (circles) and on Al 6061-T6 plates (triangles).

**Figure 4.**Blunt V-notched structures (ω = 60°), apparent generalized fracture toughness: Predictions by punctual FFM (dashed line) and average FFM (continuous line) related to experimental data on Al 7075-T6 plates (circles) and on Al 6061-T6 plates (triangles).

**Figure 5.**Blunt V-notched structures (ω = 90°), apparent generalized fracture toughness: Predictions by punctual FFM (dashed line) and average FFM (continuous line) related to experimental data on Al 7075-T6 plates (circles) and on Al 6061-T6 plates (triangles).

**Figure 6.**Average FFM predictions: Percentage discrepancy with respect to experimental data referring to blunt V-notched structures made of Al 7075-T6 (circles), and made of Al 6061-T6 (triangles).

**Figure 7.**Average FFM: Dimensionless apparent generalized fracture toughness (

**a**) and dimensionless critical crack extension (

**b**) compared to dimensionless notch root radius.

ω | λ | β | η | μ | m |
---|---|---|---|---|---|

0° | 0.5000 | 1.000 | 1.000 | −0.5000 | 1.820 |

30° | 0.5015 | 1.005 | 1.034 | −0.4561 | 1.473 |

60° | 0.5122 | 1.017 | 0.9699 | −0.4057 | 1.338 |

90° | 0.5445 | 1.059 | 0.8101 | −0.3449 | 1.314 |

Material | Al 7075-T6 | Al 6061-T6 |
---|---|---|

K_{Ic} (MPa √m) | 50 | 38 |

σ_{0} (MPa) | 1845 | 1066 |

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**MDPI and ACS Style**

Torabi, A.R.; Berto, F.; Sapora, A.
Finite Fracture Mechanics Assessment in Moderate and Large Scale Yielding Regimes. *Metals* **2019**, *9*, 602.
https://doi.org/10.3390/met9050602

**AMA Style**

Torabi AR, Berto F, Sapora A.
Finite Fracture Mechanics Assessment in Moderate and Large Scale Yielding Regimes. *Metals*. 2019; 9(5):602.
https://doi.org/10.3390/met9050602

**Chicago/Turabian Style**

Torabi, Ali Reza, Filippo Berto, and Alberto Sapora.
2019. "Finite Fracture Mechanics Assessment in Moderate and Large Scale Yielding Regimes" *Metals* 9, no. 5: 602.
https://doi.org/10.3390/met9050602