# Investigation by Digital Image Correlation of Mixed Mode I and II Fracture Behavior of Metallic IASCB Specimens with Additive Manufactured Crack-Like Notch

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{I}and K

_{II}) and T-Stress without requiring a complicated loading fixture.

_{IC}of specimens. In general, two main approaches emerges in literature concerning the fitting of theoretical models on the full displacement fields: the first one is based on guessing a general form of an analytical function and fitting this to the displacement experimental data: an example is the complex function analysis of Muskhelishvili [16], which was further developed to calculate mixed mode (I + II) [17] and generalized to cases without restrictions in boundary conditions or symmetry [18]. The second approach is based on the Williams’ model [19,20,21,22], where the fitting on the experimental data is done by considering the first n term of the Williams expansion for the displacement. In this work this latter approach was used. The SIFs obtained by the DIC fitting were compared to the ones obtained with the conventional critical load method. Finally, the resulting SIFs were described by the mixed mode local stress criterium.

## 2. Materials and Methods

#### 2.1. Specimen Fabrication by Additive Manufacturing

#### 2.2. IASB Specimens

- (1)
- S
_{1}/R and S_{2}/R (the ratio between the support spans and the specimen radius); - (2)
- a/R (the ratio between the crack length and the specimen radius);
- (3)
- α (the angle between the crack line and the load direction).

_{2}/R and α were varied. S

_{1}was maintained fixed at 42 mm, whereas the values of t, R and a were the same for all the specimens (t = 6mm, R= 60 mm, a = 24 mm) The various combinations of these two, indeed, can cover Mode I, Mode II and mixed modes I-II. When the bottom supports are located symmetrically to the crack line (i.e., when S

_{1}= S

_{2}) and the crack line is in the same direction as the load (i.e., when α = 0°), the specimen is subjected to Mode I (opening mode). To obtain mixed mode I-II or Mode II (sliding mode), an appropriate combination of S

_{2}/R and α should be chosen.

_{Imin}/K

_{Imax}= 0.3) until the crack propagated for 2 mm. In Figure 2 micrographs of the notch induced with the two different methods are shown. The crack-like notch induced by the AM technique has a radius of about 90 μm, whereas the notch induced by SM has a radius of about 150 μm. The AM specimens were manufactured with an angle α of 0° and 10° and tested with different support spans S

_{1}and S

_{2}(see Figure 1), in order to obtain different mode combinations. The AM specimens with crack-like notch angle α of 0° were only tested with symmetric support spans to pre-crack and then monotonic loaded in Mode I. Table 2 reports the specimens tested with the method employed to induce cracks, the values of α, the supports spans and the mixed mode ratio expressed as ${M}^{e}=2/\pi \mathrm{atan}({K}_{I}/{K}_{II})$.

#### 2.3. Experimental Setup

## 3. Results

#### 3.1. Evaluation of Stress Intensity Factors by the Critical Fracture Load Method

_{I}and Y

_{II}are geometry factors corresponding to Mode I and Mode II, respectively (reported in Table 2 of reference [10]). Concerning the crack length a, for the SM specimens it was measured from the bottom of the sharp notch to the pre-crack tip, whereas for the AM specimens it was considered as the length of the crack-like notch. For each test, the critical fracture load (P

_{cr}) from the load-displacement curves was evaluated at the instant in which the crack propagation was visually observed, by the camera pointing at the crack tip. Then, by Equations (1) and (2) the critical stress intensity factors (K

_{I}and K

_{II}) of the tested IASCB specimens were calculated. The obtained fracture parameters from the critical fracture load method (henceforth named PCR) are reported in Table 3.

#### 3.2. Evaluation of Stress Intensity Factors by the DIC Full Displacement Field

^{®}R2019b (MathWorks Inc., Natick, MA, USA) along with the frame of reference used, with the origin located into the crack tip and the axis oriented as shown in Figure 4c.

_{n}and b

_{n}are the model parameters. In this analysis, the specimen was considered subjected to plane stress condition, which is an assumption justified by the specimen geometry and loading conditions [20].

_{1}, …, a

_{n}and b = b

_{1}, …, b

_{n}were considered, but also four additional parameters were accounted to compensate rigids motions, which is an approach used also in [19] and [20]. In particular, the three translations and the rotation around the z-axis (perpendicular to the crack plane) were considered. As explained in [19], this approach allows one to tackle situations in which the material exhibits small-scale plastic deformation, but makes the fitting problem non-linear in the fitting parameters, because of the x and y translations x

_{0}and y

_{0}, are embedded in the calculus of the polar coordinates (r, θ) as shown in Equations (11) and (12). This approach was particularly useful in the case of the SM specimens, because the fact that they were pre-cracked would have made difficult a precise manual identification of the crack tip.

^{®}by the authors, which exploits the function “lsqcurvefit”. The code is made publicly available at the link provided in the “Supplementary Materials” section. To perform the fitting, it was decided to utilize the y-displacement v, since is the one generally used to determinate the SIFs for Mode I, and for mixed mode problems the dominant displacement component for the crack cannot be known in advance [19].

_{IC}measured for the specimens with the sintered crack-like notch (AM), compared to the pre-cracked one (SM), is overestimated of about 40% and 30% for the PCR and the DIC method, respectively. The reason is attributed to the larger radius of the crack-like notch compared to the one, significantly smaller, of the case of the pre-cracked specimens. However, with this specimen geometry, the pre-crack could be induced without having a deviation from its principal axis only for Mode I, and for this reason mixed mode SIFs were investigated only with crack-like notch directly induced by the AM process. For Mode I, an overestimation of about 70% of the DIC results with respect to conventional PCR method can be observed. Going from Mode I to Mode II, this discrepancy becomes negligible.

#### 3.3. Generalized Mixed-Mode Local Stress Criterium

_{I}- K

_{II}, normalized to K

_{IC}, which is the critical SIF obtained for Mode I. As a multiaxial fracture model, the local stress criterium (LS), proposed by Yongming Liu in [27] was exploited. Classical criteria based on maximum tangential stress, as the generalized maximum tangential stress one (GMTS) [28], can only predict fracture toughness of brittle materials. On the other hand, a major advantage of the LS criterium is that it can be applied to different materials (brittle and ductile), which experience either shear or tensile dominated crack propagation. Therefore, this model can be suitable for the strong, but at the same time ductile, maraging steel. The LS mixed mode criterium is described by Equation (13), with s equal to K

_{IIC}/K

_{IC}. The material parameter s is related to the material ductility and affects the critical plane orientation. If the parameter s is higher than 1, as in this case, A = 9 (s

^{2}−1), B = s, Υ = 0. The equation 13 represents the implicit curve of the LS criterium, and the coefficient s can be determined numerically by fitting the experimental SIFs normalized to the K

_{IC}.

## 4. Discussion

^{0.5}and 79.1 MPa m

^{0.5}were obtained with PCR methodology for Mode I and Mode II, respectively.

## 5. Conclusions

## Supplementary Materials

^{®}code used for the calculation of the DIC method is online at https://www.mdpi.com/2075-4701/10/3/400/s1.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Kruth, J.; Mercelis, P.; van Vaerenbergh, J.; Froyen, L.; Rombouts, M. Binding mechanisms in selective laser sintering and selective laser melting. Rapid Prototyp. J.
**2005**, 11, 26–36. [Google Scholar] [CrossRef] [Green Version] - Zhang, W.; Melcher, R.; Travitzky, N.; Bordia, R.K.; Greil, P. Three-dimensional printing of complex-shaped alumina/glass composites. Adv. Eng. Mater.
**2009**, 11, 1039–1043. [Google Scholar] [CrossRef] - Brugo, T.; Palazzetti, R.; Ciric-Kostic, S.; Yan, X.T.; Minak, G.; Zucchelli, A. Fracture Mechanics of laser sintered cracked polyamide for a new method to induce cracks by additive manufacturing. Polym. Test.
**2016**, 50, 301–308. [Google Scholar] [CrossRef] [Green Version] - Williams, J.G.; Ewing, P.D. Fracture under complex stress—The angled crack problem. Int. J. Fract. Mech.
**1972**, 8, 441–446. [Google Scholar] [CrossRef] - Papadopoulos, G.A.; Poniridis, P.I. Crack initiation under biaxial loading with higher-order approximation. Eng. Fract. Mech.
**1989**, 32, 351–360. [Google Scholar] [CrossRef] - Silva, A.L.L.; de Jesus, A.M.P.; Xavier, J.; Correia, J.A.F.O.; Fernandes, A.A. Combined analytical-numerical methodologies for the evaluation of mixed-mode (I + II) fatigue crack growth rates in structural steels. Eng. Fract. Mech.
**2017**, 185, 124–138. [Google Scholar] [CrossRef] - Aliha, M.R.M.; Ayatollahi, M.R. Brittle fracture evaluation of a fine grain cement mortar in combined tensile-shear deformation. Fatigue Fract. Eng. Mater. Struct.
**2009**, 32, 987–994. [Google Scholar] [CrossRef] - Lim, I.L.; Johnston, I.W.; Choi, S.K.; Boland, J.N. Fracture testing of a soft rock with semi-circular specimens under three-point bending. Part 2—Mixed-mode. Int. J. Rock Mech. Min. Sci. Geomech. Abstr.
**1994**, 31, 199–212. [Google Scholar] [CrossRef] - Ayatollahi, M.R.; Aliha, M.R.M.; Saghafi, H. An improved semi-circular bend specimen for investigating mixed mode brittle fracture. Eng. Fract. Mech.
**2011**, 78, 110–123. [Google Scholar] [CrossRef] - Darban, H.; Haghpanahi, M.; Assadi, A. Determination of crack tip parameters for ascb specimen under mixed mode loading using finite element method. Comput. Mater. Sci.
**2011**, 50, 1667–1674. [Google Scholar] [CrossRef] - Saghafi, H.; Monemian, S.A. New fracture toughness test covering mixed-mode conditions and positive and negative t-stresses. Int. J. Fract.
**2010**, 165, 135–138. [Google Scholar] [CrossRef] - Saghafi, H.; Zucchelli, A.; Minak, G. Evaluating fracture behavior of brittle polymeric materials using an IASCB specimen. Polym. Test.
**2013**, 32, 133–140. [Google Scholar] [CrossRef] - Moore, A.J.; Tyrer, J.R. The evaluation of fracture mechanics parameters from electronic speckle pattern interferometric fringe patterns. Opt. Lasers Eng.
**1993**, 19, 325–336. [Google Scholar] [CrossRef] - Peters, H.W.; Ranson, F.W. Digital imaging techniques in experimental stress analysis. Opt. Eng.
**1982**, 21, 427–431. [Google Scholar] - McNeill, S.R.; Peters, W.H.; Sutton, M.A. Estimation of stress intensity factor by digital image correlation. Eng. Fract. Mech.
**1987**, 28, 101–112. [Google Scholar] [CrossRef] - Muskhelishvili, N.I. Some Basic Problems of the Mathematical Theory of Elasticity, 4th ed.; Springer: Berlin, Germany, 1977. [Google Scholar]
- Nurse, A.D.; Patterson, E.A. Determination of predominantly mode II stress intensity factors from isochromatic data. Fatigue Fract. Eng. Mater. Struct.
**1993**, 16, 1339–1354. [Google Scholar] [CrossRef] - Lopez-Crespo, P.; Shterenlikht, A.; Patterson, E.A.; Yates, J.R.; Withers, P.J. The stress intensity of mixed mode cracks determined by digital image correlation. J. Strain Anal. Eng. Des.
**2008**, 43, 769–780. [Google Scholar] [CrossRef] - Yoneyama, S.; Ogawa, T.; Kobayashi, Y. Evaluating mixed-mode stress intensity factors from full-field displacement fields obtained by optical methods. Eng. Fract. Mech.
**2007**, 74, 1399–1412. [Google Scholar] [CrossRef] - Yates, J.R.; Zanganeh, M.; Asquith, D.; Tai, Y.H. Quantifying Crack Tip Displacement Fields: T-Stress and CTOA. In Proceedings of the Crack Paths, Vicenza, Italy, 23–25 September 2009. [Google Scholar]
- Beretta, S.; Patriarca, L.; Rabbolini, S. Stress intensity factor calculation from displacement fields. Frat. ED Integrità Strutt.
**2017**, 11, 269–276. [Google Scholar] [CrossRef] [Green Version] - Bonniot, T.; Doquet, V.; Mai, S.H. Determination of effective stress intensity factors under mixed-mode from digital image correlation fields in presence of contact stresses and plasticity. Strain
**2020**, 56, e12332. [Google Scholar] [CrossRef] - Croccolo, D.; de Agostinis, M.; Fini, S.; Olmi, G.; Vranic, A.; Ciric-Kostic, S. Influence of the build orientation on the fatigue strength of eos maraging steel produced by additive metal machine: How the build direction affects the fatigue strength of additive manufacturing processed parts. Fatigue Fract. Eng. Mater. Struct.
**2016**, 39, 637–647. [Google Scholar] [CrossRef] - Palanca, M.; Brugo, T.M.; Cristofolini, L. Use of digital image correlation to investigate the biomechanics of the vertebra. J. Mech. Med. Biol.
**2015**, 15, 1540004. [Google Scholar] [CrossRef] - Williams, M.L. On the stress distribution at the base of a stationary crack. J. Appl. Mech.
**1957**, 24, 109–114. [Google Scholar] [CrossRef] - Tan, C.; Zhou, K.; Ma, W.; Zhang, P.; Liu, M.; Kuang, T. Microstructural evolution, nanoprecipitation behavior and mechanical properties of selective laser melted high-performance grade 300 maraging steel. Mater. Des.
**2017**, 134, 23–34. [Google Scholar] [CrossRef] - Liu, Y.; Mahadevan, S. Threshold stress intensity factor and crack growth rate prediction under mixed-mode loading. Eng. Fract. Mech.
**2007**, 74, 332–345. [Google Scholar] [CrossRef] - Smith, D.J.; Ayatollahi, M.R.; Pavier, M.J. The role of t-stress in brittle fracture for linear elastic materials under mixed-mode loading. Fatigue Fract. Eng. Mater. Struct.
**2001**, 24, 137–150. [Google Scholar] [CrossRef] - Anderson, T.L. Fracture Mechanics: Fundamentals and Applications, 3rd ed.; CRC Press, Taylor & Francis Group: Boca Raton, FL, USA, 2005. [Google Scholar]
- Abd-Elhady, A.A. Mixed mode I/II stress intensity factors through the thickness of disc type specimens. Eng. Solid Mech.
**2013**, 1, 119–128. [Google Scholar] [CrossRef]

**Figure 1.**Geometrical parameters and loading conditions of the IASCB (Inclined edge cracked Semi-Circular Bend) specimen.

**Figure 2.**Micrographs of the crack tip;

**(a)**crack-like notch manufactured during the additive manufacturing process (AM) and (

**b**) crack induced by subtractive manufacturing (SM) and subsequent pre-cracking (indicated by red arrows).

**Figure 3.**(

**a**) Experimental setup; (

**b**) crack axis normal displacement and (

**c**) strain field correlated by the GOM software around the crack tip, for Mode I.

**Figure 4.**(

**a**) x-displacement (u) and (

**b**) y-displacement (v) of the AM-0-42-42-1 specimen obtained by the DIC (digital image correlation).

**(c)**Position and orientation of the frame of reference used.

**Figure 5.**Stress intensity factors for values of n ranging from 2 to 15 in the case of AM-0-42-42-1 specimen.

**Figure 6.**Comparison between the v-displacements field measured by the DIC and the one fitted by the Williams’ model for various specimens and loading condition: (

**a**) SM-0-42-42 (Mode I, pre-cracked); (

**b**) AM-0-42-42 (Mode I); (

**c**) AM-10-42-42 (mixed mode I-II); (

**d**) AM-10-42-18 (mixed mode I-II); (

**e**) AM-10-42-10.2 (Mode II).

**Figure 7.**u-displacements field measured by the DIC and the one fitted by the Williams’ model in the case of the specimen AM-10-42-10.2 (Mode II).

Property | Value |
---|---|

Sintered Density | 8.0–8.1 g/cm³ |

Elastic Modulus | 180 ± 20 GPa |

Yield Strength | min. 1862 MPa |

Tensile Strength | min. 1930 MPa |

Hardness | typ. 50–56 HRC |

Ductility (Notched Charpy Impact Test) | 11 ± 4 J |

Type | α (deg) | S_{2} (mm) | S_{1} (mm) | n | Mode | M^{e} |
---|---|---|---|---|---|---|

SM | 0 | 42 | 42 | 3 | Mode I | 1 |

AM | 0 | 42 | 42 | 3 | Mode I | 1 |

AM | 10 | 42 | 42 | 3 | Mixed I-II | 0.77 |

AM | 10 | 42 | 18 | 3 | Mixed I-II | 0.48 |

AM | 10 | 42 | 10.2 | 3 | Mode II | 0.12 |

**Table 3.**Values of the SIFs obtained by critical fracture load and by fitting the displacement field.

Specimen Configuration | By Critical Fracture Load (PCR) | By Crack Tip Displacement Field (DIC) | DIC vs. PCR | |||
---|---|---|---|---|---|---|

${\mathit{K}}_{\mathit{I}}\text{}(\mathbf{MPa}\sqrt{\mathbf{m}})$ | ${\mathit{K}}_{\mathbf{II}}\text{}(\mathbf{MPa}\sqrt{\mathbf{m}})$ | K_{I} (MPa$\sqrt{\mathbf{m}})$ | K_{II} (MPa$\sqrt{\mathbf{m}})$ | Δ K_{I} (%) | Δ K_{II} (%) | |

SM-0-42-42-1 | 32.1 | 0.0 | 58.3 | 1.2 | 81 | - |

SM-0-42-42-2 | 30.4 | 0.0 | 56.1 | 2.9 | 85 | - |

SM-0-42-42-3 | 33.7 | 0.0 | 60.5 | 2.3 | 80 | - |

AM-0-42-42-1 | 42.5 | 0.0 | 73.5 | 0.2 | 73 | - |

AM-0-42-42-2 | 44.9 | 0.0 | 76.4 | 0.1 | 70 | - |

AM-0-42-42-3 | 46.6 | 0.0 | 75.2 | 0.2 | 61 | - |

AM-10-42-42-1 | 44.4 | 16.5 | 70.5 | 20.5 | 59 | 25 |

AM-10-42-42-2 | 44.6 | 16.5 | 66.2 | 18.3 | 49 | 11 |

AM-10-42-42-3 | 45.1 | 16.7 | 70.2 | 18.3 | 56 | 10 |

AM-10-42-18-1 | 33.7 | 36.1 | 43.7 | 42.2 | 30 | 17 |

AM-10-42-18-2 | 36.3 | 38.8 | 39.2 | 43.4 | 8 | 12 |

AM-10-42-18-3 | 34.4 | 36.9 | 44.2 | 45.7 | 28 | 24 |

AM-10-42-10.2-1 | 15.3 | 83.5 | 14.9 | 77.1 | -2 | -8 |

AM-10-42-10.2-2 | 13.6 | 74.2 | 14.4 | 75.5 | 5 | 2 |

AM-10-42-10.2-3 | 14.2 | 79.4 | 14.9 | 80.8 | 5 | 2 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Campione, I.; Brugo, T.M.; Minak, G.; Tomić, J.J.; Bogojević, N.; Kostić, S.Ć.
Investigation by Digital Image Correlation of Mixed Mode I and II Fracture Behavior of Metallic IASCB Specimens with Additive Manufactured Crack-Like Notch. *Metals* **2020**, *10*, 400.
https://doi.org/10.3390/met10030400

**AMA Style**

Campione I, Brugo TM, Minak G, Tomić JJ, Bogojević N, Kostić SĆ.
Investigation by Digital Image Correlation of Mixed Mode I and II Fracture Behavior of Metallic IASCB Specimens with Additive Manufactured Crack-Like Notch. *Metals*. 2020; 10(3):400.
https://doi.org/10.3390/met10030400

**Chicago/Turabian Style**

Campione, Ivo, Tommaso Maria Brugo, Giangiacomo Minak, Jelena Janković Tomić, Nebojša Bogojević, and Snežana Ćirić Kostić.
2020. "Investigation by Digital Image Correlation of Mixed Mode I and II Fracture Behavior of Metallic IASCB Specimens with Additive Manufactured Crack-Like Notch" *Metals* 10, no. 3: 400.
https://doi.org/10.3390/met10030400