# Measuring the Cohesive Law in Mode I Loading of Eucalyptus globulus

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{I}) that were obtained from experimental research reported in the literature [8]. Although proposals of other different analytical models considering G

_{I}as input material parameter have been compiled in the literature, [9] there is, at present, no general agreement between the results. This empirical research technique is time consuming and not-always cost-efficient. It may be more efficient to achieve these objectives through numerical simulations, which show advantages in terms of effectiveness with regard to time, cost, the exactitude of results, and the possibility of conducting parametric studies, e.g., [10,11].

_{I}) was explicitly derived from the load-displacement curves that were obtained in each test by applying the Compliance-Based Beam Method (CBBM). This method has the advantage of not requiring measurements of crack propagation during the test, which would be too difficult in practice given the material heterogeneity. An equivalent crack length (a

_{eq}) was considered instead. The cohesive law, defined as the relationship between cohesive traction tension (σ

_{I}) and crack tip opening displacement (w

_{I}), was determined by differentiating the G

_{I}-w

_{I}relationship and applying least-squares regression analysis. The w

_{I}-parameter was measured by digital image correlation (DIC) technique. This fracture behavior was studied for two crack propagation systems, RL and TL, where the first letter indicates the direction normal to the crack plane and the second letter refers to the crack propagation direction (Longitudinal, Radial, and Tangential).

## 2. Materials and Methods

#### 2.1. Raw Material

_{L}) resulting from edgewise bending tests under four-point loading according to EN 408:2011 [23].

_{R}= 1820 MPa, the tangential modulus of elasticity E

_{T}= 821 MPa, the shear modulus of elasticity in the LR plane G

_{LR}= 1926 MPa, and the shear modulus of elasticity in the LT plane G

_{LT}= 969 MPa, are taken from [24] using Galician Eucalyptus globulus with a similar density to the boards used in this study. These parameters were obtained by compression tests coupled with a stereovision system (DIC 3D). DCB specimens were prepared from these boards according to the specifications shown in Section 2.3.

#### 2.2. Compliance-Based Beam Method (CBBM)

_{I}), the crack tip opening displacement (w

_{I}), and the traction tension (σ

_{I}), according to Equation (1).

_{I}evolution as a function of w

_{I}in the course of an experimental fracture test (in this case a DCB test, see details in Section 2.3). The classical data reduction schemes used for this purpose are based on beam theory or compliance calibration and require crack length (a) measuring during testing [25]. However, the fracture process zone (FPZ) ahead of the crack tip in wood involves toughening mechanisms, such as microcracking, crack-branching, or fiber-bridging, hindering the identification of the crack tip and therefore also the a-measurement. To overcome this problem, the Compliance Based Beam method (CBBM) [20,26] is shown to be a suitable alternative. It is based on Timoshenko beam theory and it introduces the concept of an equivalent crack length (a

_{eq}), accounting for the FPZ effect given by a

_{eq}= a + Δ + Δa

_{FPZ}. Accordingly, compliance for a DCB specimen during crack propagation can be written as

_{LR}is the shear modulus in the LR plane; B and h the specimen dimensions; and, E

_{f}the corrected flexural modulus (instead of E

_{L}) to take into account the cross-section rotation effects at the crack tip during testing and local stress concentrations. E

_{f}can be estimated from Equation (4) when considering the initial compliance (C

_{0}) and a corrected initial crack length (a

_{0}+ Δ)

_{f}is reached. It must be noted that E

_{T}and G

_{LT}values should be used instead of E

_{R}and G

_{LR}in Equations (3)–(7) when the TL crack propagation system is considered.

_{eq}, that meets the specimen compliance recorded during propagation is evaluated from a polynomial function solved with Matlab

^{®}(Mathworks, Madrid, Spain), according to [20].

_{I}) is obtained by combining Equations (3) and (7). It represents the resistance curve (R-curve) of the material to the crack growth.

_{I}without crack length monitoring, making it less sensitive to experimental errors. The G

_{I}is then correlated with crack tip opening displacement in mode I (w

_{I}), measured by the digital image correlation (DIC) technique during the test (see details in Section 2.3) and its derivative yields in the cohesive law expressed in Equation (2). It is therefore important to accurately evaluate the ${G}_{\mathrm{I}}=\mathrm{f}({w}_{\mathrm{I}})$ relationship. This was performed in two ways for further comparison and discussion: (a) a smoothing spline using Matlab

^{®}was adjusted to the experimental curve in order to soften the noise before differentiation; (b) the G

_{I}-w

_{I}data were fitted, in the least-square sense, by a continuous approximation function (logistic function), as follows,

_{1}, A

_{2}, p, and w

_{I,0}are constants determined by regression analysis. Although this function has no particular physical meaning, it is simply a tool for the analytical differentiation that is required to obtain the cohesive law. The A

_{2}parameter must provide an estimation of the critical strain release, as

#### 2.3. Double Cantilever Beam (DCB) Test Coupled with Digital Image Correlation

_{1}× 2h × B mm (250 mm × 20 mm × 20 mm) nominal dimension, as schematically shown in Figure 1. A mid-height pre-cracked surface of 100 mm in length and 1 mm thickness was initially performed. This initial notch was then lengthened a few millimeters with a band saw in order to guarantee a sharp initial crack. The actual a

_{0}value for each specimen was measured after testing. A symmetrical pair of 3 mm diameter holes were drilled at 10 mm from the specimen end, where the load (P) perpendicular to the pre-cracked surface was applied. The applied load was transferred to the specimen by means of two 3 mm diameter steel pins that were inserted into the holes.

_{I}) was obtained by post-processing the displacements monitored by DIC. The initial crack length was firstly identified in the undeformed image. The relative displacement between a pair of subsets selected close to the crack tip is evaluated afterwards. The value of w

_{I}is calculated as the Eucledian norm, as shown in Equation (9) [33,34].

## 3. Results and Discussion

#### 3.1. Resistance Curve from CBBM

_{0}is calculated using Matlab

^{®}, as this is the result that provides the maximum R

^{2}in every P-δ curve. A representative example of a DCB specimen is shown in Figure 5. The C

_{0}values resulting from all the tests are also included in Table 3 and Table 4 for both crack propagation systems. There are minor quantitative differences between both orientations, while scatter is within the expected range for wood.

_{Ic}) and it represents the material´s toughness to crack-growth.

_{I,Pmax}) could be assumed as critical strain energy release rate. Both values are reported in Table 3 and Table 4, together with the maximum load reached in the tests, the corrected flexural modulus of elasticity from every specimen and the initial compliance. The last three values are input parameters in the CBBM formulation detailed in Section 2.2. The wide dispersion of R-curves may be due to local variability of wood microstructure at the crack tip among the specimens, e.g., earlywood and latewood [26].

_{Ic}values that were obtained from both crack propagation systems were found to be the same: 0.77 N/mm. This value is considerably higher than that for other species. In particular, DCB specimens of Pinus pinaster gave a mean G

_{Ic}value of 0.31 N/mm when applying the same data reduction method in [25]. Eucalyptus globulus also displays higher fracture energy than other hardwood species. For instance, average G

_{f}values of 0.48 and 0.40 N/mm were obtained for beech and ash, respectively, in previous work by the author [40].

#### 3.2. Cohesive Law

_{I}evolution was correlated with the crack tip opening displacement (CTOD) values that were provided by DIC during testing in order to obtain the cohesive law in mode I. Normal and transverse CTOD with respect to the crack plane were determined. Representative normal and transverse CTOD-δ curves are shown in Figure 7. As can be seen, CTOD in mode II (w

_{II}) was found to be negligible in DCB mode I tests.

_{1}, A

_{2}, p, and w

_{I0}), the area circumscribed by the cohesive laws (G

_{law,I}), the maximum stress (σ

_{I,u}), and the relative displacements (w

_{Iu}and w

_{Ic}) are all included in Table 5 and Table 6 for RL and TL crack propagation systems, respectively.

_{2}depicts an estimation of the critical strain energy release rate, G

_{Ic}, and it acquires the values of 0.78 and 0.76 N/mm in the RL and TL crack propagation systems, respectively. These values are quite close to the mean ones obtained from the horizontal asymptote of the R-curves, that is 0.77 N/mm in both propagation systems (see Table 3 and Table 4).

## 4. Conclusions

_{I}) from the load-displacement curves when considering an equivalent crack length (a

_{eq}) instead of the actual one, which would be difficult to measure. The G

_{I}was correlated with the crack tip opening displacements measured by digital image correlation technique to obtain the cohesive laws.

_{Ic}value of 0.77 N/mm was obtained for RL and TL crack propagation systems from the horizontal asymptote of the R-curves. The estimation of G

_{Ic}from the logistic function parameters that was used to attain the cohesive law was also within the same range (0.78 and 0.76 N/mm in RL and TL, respectively).

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 7.**Normal and transverse crack tip opening displacements (CTOD) measured by DIC from a representative DCB test in RL (

**left**) and TL (

**right**) crack propagation systems.

**Figure 8.**Characteristic G

_{I}-w

_{I}curves in RL (

**left**); experimental G

_{I}-w

_{I}curve of “189-2-RL” specimen and least-square regression with the logistic function (

**right**).

**Figure 9.**Characteristic G

_{I}-w

_{I}curves in TL (

**left**); experimental G

_{I}-w

_{I}curve of “140-1-TL” specimen and least-square regression with the logistic function (

**right**).

Board Ref | ρ (kg/m^{3}) | E_{L} (MPa) |
---|---|---|

140 | 781 | 19,863 |

144 | 765 | 19,234 |

161 | 867 | 19,658 |

176 | 779 | 19,359 |

189 | 748 | 19,114 |

192 | 815 | 20,612 |

mean | 793 | 19,640 |

SD | 43 | 551 |

CoV (%) | 5.4 | 2.8 |

CCD Camera | Settings |
---|---|

Model | Baumer Optronic FWX20 (8 bits, 1624 × 1236 pixels, 4.4 μm/pixel) |

Shutter time | 0.7 ms |

Acquisition frequency | 1 Hz |

Lens | |

Model | Opto Engineering Telecentric lens TC 23 36 |

Magnification | 0.243 ± 3% |

Field of view (1/1.8″) | 29.3 × 22.1 mm^{2} |

Working distance | 103.5 ± 3 mm |

Working F-number | f/8 |

Field depth | 11 mm |

Conversion factor | 0.018 mm/pixel |

Lighting | Raylux 25 white-light LED |

DIC measurements | |

Subset size | 15 × 15 pixel^{2} (0.270 × 0.270 mm^{2}) |

Subset step | 13 × 13 pixel^{2} (0.234 × 0.234 mm^{2}) |

Resolution | 1–2 × 10^{−2} pixel (0.18 × 0.36 μm) |

**Table 3.**Fracture energy obtained from DCB specimens oriented in RL by means of Compliance-Based Beam Method (CBBM).

Specimen Ref | E_{f} (MPa) | P_{max} (N) | C_{0} (mm/N) | G_{I,Pmax} (N/mm) | G_{Ic} (N/mm) |
---|---|---|---|---|---|

140-1-RL | 14,250 | 191.70 | 0.042 | 1.07 | 1.01 |

144-1-RL | 15,203 | 173.00 | 0.040 | 0.81 | 0.84 |

144-2-RL | 13,254 | 123.08 | 0.048 | 0.52 | 0.48 |

161-1-RL | 12,266 | 172.35 | 0.047 | 0.97 | 0.95 |

161-3-RL | 14,593 | 176.29 | 0.039 | 0.85 | 0.85 |

176-1-RL | 14,557 | 151.70 | 0.042 | 0.65 | 0.61 |

176-2-RL | 12,335 | 156.59 | 0.049 | 0.82 | 0.76 |

176-3-RL | 11,577 | 183.95 | 0.048 | 1.10 | 1.02 |

189-1-RL | 15,087 | 168.16 | 0.039 | 0.75 | 0.70 |

189-2-RL | 12,293 | 177.06 | 0.045 | 0.95 | 0.92 |

192-1-RL | 16,707 | 162.01 | 0.038 | 0.65 | 0.63 |

192-2-RL | 14,399 | 147.76 | 0.044 | 0.70 | 0.65 |

192-3-RL | 16,103 | 153.99 | 0.038 | 0.56 | 0.57 |

Mean | 14,048 | 164.43 | 0.043 | 0.80 | 0.77 |

SD | 1590 | 18.03 | 0.004 | 0.19 | 0.18 |

CoV (%) | 11 | 11 | 10 | 23 | 23 |

Specimen Ref | E_{f} (MPa) | P_{max} (N) | C_{0} (mm/N) | G_{I,Pmax} (N/mm) | G_{Ic} (N/mm) |
---|---|---|---|---|---|

140-1-TL | 14,491 | 182.31 | 0.046 | 1.03 | 0.96 |

140-2-TL | 11,295 | 157.74 | 0.055 | 0.94 | 0.89 |

144-1-TL | 14,080 | 154.80 | 0.049 | 0.75 | 0.72 |

161-1-TL | 12,742 | 188.28 | 0.049 | 1.28 | 1.09 |

176-1-TL | 12,466 | 146.01 | 0.054 | 0.84 | 0.82 |

176-2-TL | 13,516 | 160.37 | 0.049 | 0.93 | 0.84 |

176-3-TL | 12,103 | 147.33 | 0.053 | 0.81 | 0.78 |

189-1-TL | 12,482 | 162.12 | 0.050 | 0.85 | 0.84 |

189-2-TL | 14,087 | 160.69 | 0.046 | 0.77 | 0.76 |

192-1-TL | 15,538 | 126.49 | 0.044 | 0.45 | 0.47 |

192-2-TL | 14,421 | 139.78 | 0.048 | 0.61 | 0.62 |

192-3-TL | 13,220 | 114.81 | 0.050 | 0.47 | 0.44 |

Mean | 13,370 | 153.39 | 0.049 | 0.81 | 0.77 |

SD | 1207 | 20.76 | 0.003 | 0.23 | 0.19 |

CoV (%) | 9 | 14 | 7 | 29 | 24 |

**Table 5.**Logistic function parameters (A

_{1}, A

_{2}, p, and w

_{I0}), maximum stress (σ

_{Iu}) and relative displacement (w

_{Iu}), as determined by CBBM equations, from specimens with the RL crack system.

Ref | A_{1} (N/mm) | A_{2} (N/mm) | p (-) | w_{I0} (mm) | G_{law,I} (N/mm) | σ_{Iu} (MPa) | w_{Iu} (mm) |
---|---|---|---|---|---|---|---|

140-1-RL | 0.044 | 1.04 | 2.93 | 0.030 | 1.00 | 26.94 | 0.024 |

144-1-RL | 0.039 | 0.82 | 2.80 | 0.019 | 0.78 | 32.21 | 0.015 |

144-2-RL | 0.024 | 0.49 | 2.30 | 0.023 | 0.46 | 14.21 | 0.015 |

161-1-RL | 0.029 | 0.93 | 2.20 | 0.023 | 0.91 | 27.19 | 0.014 |

161-3-RL | 0.016 | 0.78 | 2.02 | 0.020 | 0.75 | 25.39 | 0.011 |

176-1-RL | 0.026 | 0.67 | 2.46 | 0.018 | 0.64 | 26.35 | 0.013 |

176-2-RL | 0.010 | 0.78 | 1.42 | 0.026 | 0.76 | 18.00 | 0.008 |

176-3-RL | 0.025 | 1.10 | 1.58 | 0.074 | 1.05 | 8.86 | 0.029 |

189-1-RL | 0.019 | 0.73 | 2.29 | 0.051 | 0.70 | 9.79 | 0.034 |

189-2-RL | 0.028 | 0.97 | 1.72 | 0.044 | 0.92 | 13.11 | 0.020 |

192-1-RL | 0.024 | 0.63 | 2.37 | 0.016 | 0.61 | 27.41 | 0.011 |

192-2-RL | 0.043 | 0.66 | 3.60 | 0.019 | 0.62 | 32.11 | 0.016 |

192-3-RL | 0.035 | 0.58 | 2.69 | 0.012 | 0.55 | 34.78 | 0.009 |

Mean | 0.028 | 0.78 | 2.34 | 0.029 | 0.75 | 22.80 | 0.017 |

SD | 0.010 | 0.18 | 0.59 | 0.018 | 0.18 | 8.90 | 0.008 |

CoV (%) | 37 | 23 | 25 | 61 | 24 | 39 | 47 |

**Table 6.**Logistic function parameters (A

_{1}, A

_{2}, p, and w

_{I0}), maximum stress (σ

_{Iu}) and relative displacement (w

_{Iu}), as determined by CBBM equations, from specimens with a TL crack system.

Ref | A_{1} (N/mm) | A_{2} (N/mm) | p (-) | w_{I0} (mm) | G_{law,I} (N/mm) | σ_{I,u} (MPa) | w_{Iu} (mm) |
---|---|---|---|---|---|---|---|

140-1-TL | 0.027 | 0.96 | 1.93 | 0.038 | 0.93 | 15.65 | 0.021 |

140-2-TL | 0.012 | 0.94 | 1.83 | 0.056 | 0.92 | 10.31 | 0.029 |

144-1-TL | 0.016 | 0.80 | 2.25 | 0.034 | 0.78 | 15.86 | 0.022 |

161-1-TL | 0.023 | 1.16 | 1.92 | 0.056 | 1.13 | 12.92 | 0.031 |

176-1-TL | 0.013 | 0.64 | 1.74 | 0.031 | 0.61 | 12.33 | 0.015 |

176-2-TL | 0.007 | 0.73 | 1.73 | 0.043 | 0.70 | 10.31 | 0.020 |

176-3-TL | 0.025 | 0.80 | 1.97 | 0.074 | 0.76 | 6.77 | 0.042 |

189-1-TL | 0.015 | 0.87 | 1.70 | 0.063 | 0.85 | 8.35 | 0.029 |

189-2-TL | 0.028 | 0.79 | 2.07 | 0.061 | 0.76 | 8.28 | 0.037 |

192-1-TL | 0.023 | 0.45 | 2.26 | 0.036 | 0.43 | 8.35 | 0.024 |

192-2-TL | 0.011 | 0.61 | 1.98 | 0.080 | 0.58 | 4.83 | 0.045 |

192-3-TL | 0.007 | 0.42 | 2.26 | 0.042 | 0.41 | 6.78 | 0.027 |

Mean | 0.017 | 0.76 | 1.97 | 0.051 | 0.74 | 10.06 | 0.028 |

SD | 0.008 | 0.21 | 0.21 | 0.016 | 0.21 | 3.52 | 0.009 |

CoV (%) | 44 | 28 | 10 | 32 | 28 | 35 | 32 |

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**MDPI and ACS Style**

Majano-Majano, A.; Lara-Bocanegra, A.J.; Xavier, J.; Morais, J.
Measuring the Cohesive Law in Mode I Loading of *Eucalyptus globulus*. *Materials* **2019**, *12*, 23.
https://doi.org/10.3390/ma12010023

**AMA Style**

Majano-Majano A, Lara-Bocanegra AJ, Xavier J, Morais J.
Measuring the Cohesive Law in Mode I Loading of *Eucalyptus globulus*. *Materials*. 2019; 12(1):23.
https://doi.org/10.3390/ma12010023

**Chicago/Turabian Style**

Majano-Majano, Almudena, Antonio José Lara-Bocanegra, José Xavier, and José Morais.
2019. "Measuring the Cohesive Law in Mode I Loading of *Eucalyptus globulus*" *Materials* 12, no. 1: 23.
https://doi.org/10.3390/ma12010023