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

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Specimen Fabrication

#### 2.2. IASB Specimens

_{1}and S

_{2}and the crack tilt angle α, it is possible to obtain a large variety of mixed I/II modes. The values of t, R, and a were identical for all the specimens (t = 6 mm, R = 60 mm, a = 24 mm), and S

_{1}was kept fixed at 42 mm. The different mode mixity was obtained by choosing the suitable combination of S

_{2}/R and α. In particular, when the bottom supports are located symmetrically to the crack line (i.e., when S

_{1}= S

_{2}) and the crack line is parallel to the load (i.e., when α = 0°), the specimen is subjected to pure opening mode I. With different combinations of S

_{2}/R and α, mixed-mode I/II can be obtained. In Figure 1b, a different test configuration used to determine the pure indentation behavior is shown.

_{2}, while S

_{1}was kept at 42 mm for all the tests to obtain different mode I/II mixity. The AM specimens with crack-like notch angle α of 0° were only tested with symmetric configuration (pure mode I). Table 2 presents the specimens tested along with their typology, the values of α; the supports spans S

_{1}and S

_{2}, the number n of tested specimens and the mixed-mode ratio calculated as ${M}^{e}=2/\pi \mathrm{atan}\left({K}_{I}/{K}_{II}\right)$.

#### 2.3. Experimental Setup

#### 2.4. J-Integral Computation via the LDC Method

#### 2.4.1. FEM Calculation of the J-Integral Geometry Factors

^{2}, and ${U}_{FEM}$ is the work done by the external load, expressed in mJ, both evaluated by the FEM software.

#### 2.4.2. J-Integral Computation from the Load–Displacement Curve Results

_{T}the indentation energy U

_{i}. The total energy is obtained by integrating the load–displacement curve up to the load at which the crack was visually observed to propagate of 1 mm, on the images acquired by the camera used for the DIC. The indentation energy U

_{i}was obtained by integrating the load–displacement curve measured in the indentation test up to the displacement that corresponds to the maximum load at which the total energy U

_{T}was evaluated. The indentation test was carried out on a specimen with the same geometry of the reference one but without the notch and an adjacent central support span, as described in Figure 1b.

#### 2.5. J-Integral Computation via the DIC Method

_{1}and Path Γ

_{5}: $\widehat{n}=\left(-1,0\right),\Delta s=-\left|\Delta gy\right|,\Delta y=-\left|\Delta gy\right|$ where $\left|\Delta gy\right|$ is the distance (constant and ≥0) between 2 points of the DIC grid along the y-direction:

_{2}: $\widehat{n}=\left(0,-1\right),\Delta s=\left|\Delta gx\right|,\Delta y=0$ where $\left|\Delta gx\right|$ is the distance (constant and ≥0) between two points of the DIC grid along the x-direction:

_{3}: $\widehat{n}=\left(+1,0\right),\Delta s=\left|\Delta gy\right|,\Delta y=\left|\Delta gy\right|$

_{4}: $\widehat{n}=\left(0,+1\right),\Delta s=-\left|\Delta gx\right|,\Delta y=0$

_{xz}and ε

_{yz}are equally null:

## 3. Results

#### 3.1. J-Integral Results from the LDC Method

_{1}/R = 0.7, by varying the support span S

_{2}and the crack angle α. In this way, it was possible to cover the mode I-II mixity, ranging from pure mode I to pure mode II. The results are reported in Table 3, and the geometry factors are plotted in Figure 5 as a function of the S

_{2}/R for different crack angles. As is clear from Figure 5, η increases as the S

_{2}support span increases, while for a constant value of S

_{2}/R, η decreases for higher values of crack angle.

_{1}= 42 mm and S

_{2}= 18 mm (AM-10-42-18) is reported in Figure 6. The red curve represents the notched specimen, and the blue curve the unnotched one; the latter used to evaluate the indentation energy U

_{i}. The black dotted curve represents the 5% augmented tangent line with respect to the initial slope of the notched specimen, which is normally used for the stress intensity factor calculation. In the case study, however, the maximum load for the notched specimen curve is higher than 10% of its intersection with the dotted tangent line, which makes this method invalid according to the ASTM D5045-14 standard [29]. For this reason, the crack propagation start was visually evaluated by examining the frames recorded by the DIC camera. In compliance with the standard ASTM D6068-96 [20], the crack was considered to start to propagate when it reached an extension of 1 mm. The load corresponding to that time instant was then read from the load–displacement curve (roughly 6000 N in the case of Figure 6). The notched and the indentation curves are then integrated up to this load to compute the energy U

_{T}and U

_{i}, respectively, used for the calculation of the J-integral according to Equation (2).

#### 3.2. J-Integral Computation via the DIC-Based Method

## 4. Discussion

^{e}as the x-axis.

^{e}, it can be observed that the curve has a minimum at a mixity ratio equal to 0.77 and then it increases up to pure mode II. Generally, in the literature [4], higher fracture toughness values are experimentally observed for mode II with respect to mode I. This is due to a wider stress state distribution for mode II than mode I, which leads to higher fracture toughness values.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Geometrical features and loading conditions of the inclined asymmetrical semicircular specimen subjected to three points loading (IASCB) specimen; (

**a**) fracture test; (

**b**) indentation test.

**Figure 2.**(

**a**) Experimental setup; (

**b**) y-strain field around the crack tip obtained by the digital image correlation (DIC) for a 10–42–42 test configuration.

**Figure 6.**Fracture test curve (red), indentation curve (blue) and compliance line (black dotted line) in the case of the AM-10-42-18-2 test configuration.

**Figure 8.**The x and y displacement (u) strain (ε) maps for different test configuration: (

**a**) SM-0–42–42 (mode I); (

**b**) AM-0–42–42 (mode I); (

**c**) AM-10–42–42; (

**d**) AM-10–42–18; (

**e**) AM-10–42–10.2 (mode II).

**Figure 9.**J-integral on different paths for the various specimen configurations. The lower the index, the shorter the path (referring to Figure 4).

**Figure 10.**Comparison between the results obtained with the DIC method (blue) and the ones obtained with the LDC method (red). (

**a**) Bar plot for the different test configurations; (

**b**) J-integral plotted as a function of the mode mixity ratio M

^{e}.

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

Flexural Young’s modulus | 2.1 ± 0.1 GPa |

Flexural Yield strength | 55 ± 3 MPa |

Flexural strength | 68 ± 2 MPa |

Type | α (°) | S_{1} (mm) | S_{2} (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.**Geometry factor η for different values of S

_{2}/R and crack angle α obtained with values of a/R and S

_{1}/R fixed, respectively, to 0.4 and 0.7.

S_{2}/R | η | S_{2}/R | η | S_{2}/R | η | S_{2}/R | η | S_{2}/R | η | S_{2}/R | η |
---|---|---|---|---|---|---|---|---|---|---|---|

α = 0° | α = 10° | α = 20° | α = 30° | α = 40° | α = 50° | ||||||

0.7 | 1.564 | 0.7 | 1.474 | 0.7 | 1.235 | 0.7 | 0.916 | 0.7 | 0.598 | 0.7 | 0.342 |

0.6 | 1.449 | 0.6 | 1.323 | 0.6 | 1.048 | 0.6 | 0.709 | 0.65 | 0.501 | 0.67 | 0.293 |

0.5 | 1.276 | 0.5 | 1.111 | 0.5 | 0.809 | 0.5 | 0.480 | 0.6 | 0.400 | 0.64 | 0.244 |

0.4 | 1.032 | 0.4 | 0.837 | 0.4 | 0.557 | 0.45 | 0.389 | 0.55 | 0.305 | 0.6 | 0.183 |

0.3 | 0.719 | 0.3 | 0.558 | 0.3 | 0.465 | 0.4 | 0.356 | 0.5 | 0.240 | 0.55 | 0.129 |

0.1 | 0.358 | 0.17 | 0.522 | 0.26 | 0.554 | 0.36 | 0.407 | 0.45 | 0.245 | 0.52 | 0.125 |

**Table 4.**Results of the J-integral computed with the load–displacement curves (LDC) method and with the DIC method.

Specimen Configuration | M^{e} | LDC Method J (mJ/mm^{2}) | J_{mean} | σ(J) | DIC Method J (mJ/mm^{2}) | J_{mean} | σ(J) | LDC vs. DIC Δ% |
---|---|---|---|---|---|---|---|---|

SM-0-42-42-1 | 1 | 5.40 | 5.14 | 0.29 | 4.95 | 4.66 | 0.30 | −8.33 |

SM-0-42-42-2 | 4.82 | 4.68 | −2.90 | |||||

SM-0-42-42-3 | 5.20 | 4.36 | −16.15 | |||||

AM-0-42-42-1 | 1 | 5.09 | 5.77 | 0.59 | 6.42 | 5.99 | 0.45 | 26.13 |

AM-0-42-42-2 | 6.12 | 5.52 | −9.80 | |||||

AM-0-42-42-3 | 6.10 | 6.03 | −1.15 | |||||

AM-10-42-42-1 | 0.77 | 4.30 | 4.50 | 0.25 | 4.98 | 4.64 | 0.30 | 15.81 |

AM-10-42-42-2 | 4.78 | 4.55 | −4.81 | |||||

AM-10-42-42-3 | 4.42 | 4.40 | −0.45 | |||||

AM-10-42-18-1 | 0.48 | 5.56 | 5.79 | 0.21 | 6.16 | 6.12 | 0.38 | 10.79 |

AM-10-42-18-2 | 5.95 | 6.48 | 8.91 | |||||

AM-10-42-18-3 | 5.87 | 5.72 | −2.56 | |||||

AM-10-42-10.2-1 | 0.12 | 9.03 | 9.13 | 0.92 | 9.59 | 10.44 | 0.82 | 6.20 |

AM-10-42-10.2-2 | 8.26 | 10.50 | 27.12 | |||||

AM-10-42-10.2-3 | 10.10 | 11.23 | 11.19 |

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**MDPI and ACS Style**

Brugo, T.M.; Campione, I.; Minak, G.
Investigation by Digital Image Correlation of Mixed-Mode I and II Fracture Behavior of Polymeric IASCB Specimens with Additive Manufactured Crack-Like Notch. *Materials* **2021**, *14*, 1084.
https://doi.org/10.3390/ma14051084

**AMA Style**

Brugo TM, Campione I, Minak G.
Investigation by Digital Image Correlation of Mixed-Mode I and II Fracture Behavior of Polymeric IASCB Specimens with Additive Manufactured Crack-Like Notch. *Materials*. 2021; 14(5):1084.
https://doi.org/10.3390/ma14051084

**Chicago/Turabian Style**

Brugo, Tommaso Maria, Ivo Campione, and Giangiacomo Minak.
2021. "Investigation by Digital Image Correlation of Mixed-Mode I and II Fracture Behavior of Polymeric IASCB Specimens with Additive Manufactured Crack-Like Notch" *Materials* 14, no. 5: 1084.
https://doi.org/10.3390/ma14051084