Experimental Evaluation of Mode II fracture Properties of Eucalyptus globulus L.

Eucalyptus globulus Labill is a hardwood species of broad growth in temperate climates, which is receiving increasing interest for structural applications due to its high mechanical properties. Knowing the fracture behaviour is crucial to predict, through finite element models, the load carrying capacity of engineering designs with possibility of brittle failures such as elements with holes, notches, or certain types of joints. This behaviour can be adequately modelled on a macroscopic scale by the constitutive cohesive law. A direct identification of the cohesive law of Eucalyptus globulus L. in Mode II was performed by combining end-notched flexure (ENF) tests with digital image correlation (DIC) for radial-longitudinal crack propagation system. The critical strain energy release for this fracture mode, which represents the material toughness to crack-growth, was determined by applying the Compliance Based Beam Method (CBBM) as data reduction scheme and resulted in a mean value of 1.54 N/mm.


Introduction
For a policy of sustainability, engineering applications employing wood and wood-based products are receiving major interest in recent years due mainly to their environmental advantages. Consequently, softwood species such as spruce and pine have been deeply studied because of their wide availability. On the contrary, research concerning hardwood species is scarce. However, the increase in forest area dedicated to temperate hardwoods and the need to increase timber supply in the market are encouraging different European countries to explore the potential of these species for use in construction and, especially, for application as structural elements.
The use of timber and its derivative products for structural purposes requires knowing the response of the material to all possible loading situations, especially those that may produce brittle failures. There are many engineering solutions in timber, such as connections loaded at an angle to the grain, beams with holes, or beams with notches at the supports, which generate perpendicular to the grain and shear stresses that could cause a sudden failure of the element. Regarding the analysis of these situations, an energetic approach, in the framework of fracture mechanics, results very suitable. In fact, various current design codes of timber structures include practical expressions, based on fracture mechanics, for the verification of the load carrying capacity of this type of solutions [1,2]. The engineering elastic constants of the material are required for the data reduction method followed in the present work, as detailed in Section 2.2. Since it is focused on RL (radial-longitudinal) fracture system (common fracture system in the context of structural application of timber), the required properties in addition to E L , are the radial modulus of elasticity (E R ) and the shear modulus of elasticity in LR plane (G LR ). The corresponding values were taken from Crespo et al. [14], who used Galician E. globulus with similar density to the boards in the present study. Mean values of E R = 1820 MPa and G LR = 1926 MPa were determined from compression tests coupled with a digital image correlation system (DIC) to measure the strain fields.
End-notched flexure (ENF) specimens were prepared from these boards according to the specifications detailed in Section 2.3. The specimens were named using the board reference from which they were obtained.

Data Reduction: Compliance-Based Beam Method (CBBM)
The cohesive law in Mode II is defined as the relationship between shear stresses, σ II , and crack tip opening displacements in Mode II (displacements in the direction of the crack propagation associated to the upper and the lower cracked surface), w II , that is the function: σ II = f (w II ).
The data reduction method applied to determine this cohesive law in Mode II corresponds to a direct method. It requires differentiating the relationship between the strain energy release rates, G II , and crack tip opening displacements, w II , as being G II accordingly expressed as [4]: To directly solve Equation (1), the evolution of G II with regard to w II in the course of a fracture test must be determined using suitable methodologies. End-notched flexure (ENF) tests (see details in Section 2.3) were performed. The classical data reduction schemes used to evaluate G II are based on beam theory or compliance calibration. Considering the Irwin-Kies equation [19], G II is expressed as with B denoting the specimen width, C the specimen compliance and a the crack length. Therefore, it requires crack-length monitoring during testing [20][21][22][23], which is quite a difficult task in wood considering the toughening mechanisms such as microcracking, crack-branching or fiber-bridging produced at the fracture process zone (FPZ) ahead of the crack tip. Therefore, the classical data reduction methods based on crack length measurements can induce important errors on the G II evaluation.
To overcome this limitation, a powerful method based on an equivalent crack length (a eq ) concept is followed instead, which is called Compliance-based beam method (CBBM). It makes possible the determination of G II by explicitly processing the global load-displacement curve obtained from the test without requirements of crack length monitoring. This method is therefore less sensitive to experimental inaccuracies.
Following CBBM and assuming beam theory with consideration of shear effects, the specimen compliance during crack propagation is written as [24] where G LR is the shear modulus of the material; A = 2hB is the cross-section area; I = 8Bh 3 /12 is the second moment of area; a eq = a + ∆a FPZ denotes the equivalent crack length, with ∆a FPZ the crack length correction accounting for the FPZ effect; and E f is a corrected flexural modulus which replaces E L to take into account the variability in elastic properties of wood. E f can be estimated from Equation (4) following the next expression: being a 0 the initial crack length and C 0 the initial compliance of the specimen.
Accordingly, the equivalent crack length during propagation can be determined as follows where Considering the parameters shown in Equations (4)-(7) by applying the CBBM, the general expression to determine the energy release rate in Mode II (Equation (3)) is particularised as G II represents de resistance curve (R-curve) of the material to the crack growth. As shown above, CBBM does not require crack length monitoring during testing, which evinces the great potential and advantages of the method for this purpose. The only experimental measures needed are the loads and corresponding displacements which are directly recorded in every test. The w II parameter, needed to be correlated with G II (see Equation (1)), was measured by means of image processing based on the digital image correlation (DIC) technique. Speckled images were recorded during the fracture tests, with suitable spatial and temporal resolutions. The numerical image correlation provided the calculation of the displacement fields around the region of interest. Post-processing the displacements at the crack tip allowed an estimation of the crack tip opening displacement during the fracture test. The initial crack tip location (x c i , y c i ) was firstly identified in a reference image before crack propagation. At that coordinate location, the uy(:,y c i ) displacement profile was evaluated in a direction perpendicular to the crack growth. The crack tip opening displacement was then evaluated as the major different along the profile, corresponding to the real jump in terms of displacements at the two cracked surfaces. This approach allowed systematically considering any path deviation of the crack propagation during the test. The G II -w II curve is finally differentiated to obtain the cohesive law in Mode II. In the process, a continuous approximation function (logistic function) was used to fit the G II -w II experimental data according to Equation (9).
where A 1 , A 2 , p and w II,0 constants were obtained by least-square regression analysis [25]. Even if this function does not have a particular physical meaning, it is used instead to provide filtering and analytical differentiation in the reconstruction of the cohesive law [26]. From such relationship, the A 2 parameter should provide an estimation of the critical strain release in Mode II, as

End-notched Flexure Tests
End-notched flexure (ENF) tests were proposed in this work. They consist of pre-cracked beam specimens which are loaded in a three-point bending configuration. Ten ENF specimens were cut from dried boards of E. globulus. The nominal dimensions were L 1 mm × 2h mm × B mm (500 mm × 20 mm × 20 mm), as schematically shown in Figure 1. All specimens were oriented along the RL crack propagation system (Radial loading direction and Longitudinal crack propagation direction). A mid-height sharp pre-cracked surface of around 162 mm in length was initially performed by a band saw of 1 mm thickness and lengthened around 1-2 mm afterwards using an impacted blade. The actual value of initial crack length (a 0 ) was more clearly measured after testing once the specimen was broken and thus divided in two parts.
Materials 2020, 13, x FOR PEER REVIEW 5 of 13 displacement was then evaluated as the major different along the profile, corresponding to the real jump in terms of displacements at the two cracked surfaces. This approach allowed systematically considering any path deviation of the crack propagation during the test.
The GII-wII curve is finally differentiated to obtain the cohesive law in Mode II. In the process, a continuous approximation function (logistic function) was used to fit the GII-wII experimental data according to Equation (9).
where A1, A2, p and wII,0 constants were obtained by least-square regression analysis [25]. Even if this function does not have a particular physical meaning, it is used instead to provide filtering and analytical differentiation in the reconstruction of the cohesive law [26]. From such relationship, the A2 parameter should provide an estimation of the critical strain release in Mode II, as

End-notched Flexure Tests
End-notched flexure (ENF) tests were proposed in this work. They consist of pre-cracked beam specimens which are loaded in a three-point bending configuration. Ten ENF specimens were cut from dried boards of E. globulus. The nominal dimensions were L1 mm × 2h mm × B mm (500 mm × 20 mm × 20 mm), as schematically shown in Figure 1. All specimens were oriented along the RL crack propagation system (Radial loading direction and Longitudinal crack propagation direction). A midheight sharp pre-cracked surface of around 162 mm in length was initially performed by a band saw of 1 mm thickness and lengthened around 1-2 mm afterwards using an impacted blade. The actual value of initial crack length (a0) was more clearly measured after testing once the specimen was broken and thus divided in two parts. Prior to testing, the specimens were stabilised at laboratory conditions of 20 °C and 65% relative humidity.
Fracture tests were performed using an INSTRON 1125 universal testing machine (Instron, Barcelona, Spain) with a load cell of 5 kN maximum capacity and 200 N/V gain. The specimens were placed on two cylindrical rollers with 2L = 460 mm span and the load was applied at mid-span using a cylindrical actuator to minimise indentation. To reduce frictional forces during testing, a Teflon film was placed along the pre-cracked surfaces of the specimen. The tests were carried out under displacement control with a cross-head velocity of 3 mm/min.
A non-contact optical system, ARAMIS 2D (GOM mbH, Braunschweig, Germany) [27], was coupled with the test device in order to measure crack tip opening displacements at the specimens during fracture tests. This technique applies the principles of digital image correlation (DIC) and makes possible full-field measurements, which lead to more robust results in comparison with conventional methods. ARAMIS 2D system is composed by an eight-bit charge-coupled device Prior to testing, the specimens were stabilised at laboratory conditions of 20 • C and 65% relative humidity.
Fracture tests were performed using an INSTRON 1125 universal testing machine (Instron, Barcelona, Spain) with a load cell of 5 kN maximum capacity and 200 N/V gain. The specimens were placed on two cylindrical rollers with 2L = 460 mm span and the load was applied at mid-span using a cylindrical actuator to minimise indentation. To reduce frictional forces during testing, a Teflon Materials 2020, 13, 745 6 of 13 film was placed along the pre-cracked surfaces of the specimen. The tests were carried out under displacement control with a cross-head velocity of 3 mm/min.
A non-contact optical system, ARAMIS 2D (GOM mbH, Braunschweig, Germany) [27], was coupled with the test device in order to measure crack tip opening displacements at the specimens during fracture tests. This technique applies the principles of digital image correlation (DIC) and makes possible full-field measurements, which lead to more robust results in comparison with conventional methods. ARAMIS 2D system is composed by an eight-bit charge-coupled device (CCD) camera with a telecentric lens, and two cold light sources to illuminate the specimens (Figure 2, left). Details of the DIC setting parameters are shown in Table 2. A subtle speckle pattern of black ink points on a white matte surface was applied at the crack tip area of every specimen using an airbrush IWATA, model CM-B (Anesta Iwata Iberica SL, Barcelona, Spain) in order to have proper granulometry contrast and isotropy at the magnification scale (Figure 2, right). Both the speckle pattern quality and the DIC setting parameters must be property selected to achieve a suitable compromise in terms of spatial resolution and accuracy [28][29][30]. In particular, a spatial resolution of 0.270 mm was decided in this work for a displacement accuracy around 1-2 × 10 −2 pixel (0.18-0.36 µm 2 ). The accuracy in displacements was estimated experimentally by statistically analysing noisy fields obtained by processing images recorded for rigid-body translation tests. P-δ data were recorded in all tests with an acquisition rate of 5 Hz. The DIC images acquisition was chosen as 1 Hz frequency. Crack tip opening displacements in Mode II (w II ) were determined by post-processing the displacements monitored by DIC, as specified in Section 2.2.
Materials 2020, 13, x FOR PEER REVIEW 6 of 13 2, left). Details of the DIC setting parameters are shown in Table 2. A subtle speckle pattern of black ink points on a white matte surface was applied at the crack tip area of every specimen using an airbrush IWATA, model CM-B (Anesta Iwata Iberica SL, Barcelona, Spain) in order to have proper granulometry contrast and isotropy at the magnification scale (Figure 2, right). Both the speckle pattern quality and the DIC setting parameters must be property selected to achieve a suitable compromise in terms of spatial resolution and accuracy [28][29][30]. In particular, a spatial resolution of 0.270 mm was decided in this work for a displacement accuracy around 1-2 × 10 −2 pixel (0.18-0.36 μm 2 ). The accuracy in displacements was estimated experimentally by statistically analysing noisy fields obtained by processing images recorded for rigid-body translation tests. P-δ data were recorded in all tests with an acquisition rate of 5 Hz. The DIC images acquisition was chosen as 1 Hz frequency. Crack tip opening displacements in Mode II (wII) were determined by post-processing the displacements monitored by DIC, as specified in Section 2.2.

Resistance Curve
The load-displacement curves resulted from the ENF tests are shown in Figure 3. The curves show quite consistent and similar behaviour considering inherent variability of this heterogeneous material. The effect of such heterogeneity may also explain the uneven and gradual decrease of load after the peak [31]. The non-linear behaviour observed before peak load is related to the development

Resistance Curve
The load-displacement curves resulted from the ENF tests are shown in Figure 3. The curves show quite consistent and similar behaviour considering inherent variability of this heterogeneous material. The effect of such heterogeneity may also explain the uneven and gradual decrease of load after the peak [31]. The non-linear behaviour observed before peak load is related to the development of the FPZ ahead of the crack tip. The maximum loads reached in the fracture tests are summarised in Table 3. Even if they show some scatter typical of the material, all values remain approximately in a similar range with an average value of 706 N.
From every P-δ curve, the initial compliance C0 was calculated using MATLAB ® software as the value that gives the maximum R 2 . An example of the compliance-R 2 relationship and the corresponding P-δ curve for a particular ENF specimen is illustrated in Figure 4. The C0 results from all tests are included in Table 3. The R-curves were then evaluated from the experimental P-δ curves by applying the CBBM data reduction method detailed in Section 2.2. They are shown in Figure 5 as relationship between equivalent crack length and energy release rate. These curves are characterised by an initial rising domain meaning that the FPZ is developing, and a horizontal asymptote afterwards which represents the material toughness to crack-growth, the so-called strain energy release rate (GIIc). This is one of the most important parameters that define the material fracture behaviour in Mode II.
The R-curves scatter is within the expected range for wood and may be attributed to the The maximum loads reached in the fracture tests are summarised in Table 3. Even if they show some scatter typical of the material, all values remain approximately in a similar range with an average value of 706 N. Table 3. Flexural modulus (E f ), maximum load (P max ), initial compliance (C 0 ), strain energy release rate at maximum load (G II,Pmax ) and critical strain energy release rate (G IIc ) in Mode II by CBBM.  From every P-δ curve, the initial compliance C 0 was calculated using MATLAB ® software as the value that gives the maximum R 2 . An example of the compliance-R 2 relationship and the corresponding P-δ curve for a particular ENF specimen is illustrated in Figure 4. The C 0 results from all tests are included in Table 3. some scatter typical of the material, all values remain approximately in a similar range with an average value of 706 N.
From every P-δ curve, the initial compliance C0 was calculated using MATLAB ® software as the value that gives the maximum R 2 . An example of the compliance-R 2 relationship and the corresponding P-δ curve for a particular ENF specimen is illustrated in Figure 4. The C0 results from all tests are included in Table 3. The R-curves were then evaluated from the experimental P-δ curves by applying the CBBM data reduction method detailed in Section 2.2. They are shown in Figure 5 as relationship between equivalent crack length and energy release rate. These curves are characterised by an initial rising domain meaning that the FPZ is developing, and a horizontal asymptote afterwards which represents the material toughness to crack-growth, the so-called strain energy release rate (GIIc). This is one of the most important parameters that define the material fracture behaviour in Mode II.
The R-curves scatter is within the expected range for wood and may be attributed to the variability of the material microstructure at the initial crack tip [26]. However, most of the specimens showed a plateau for a given crack extent. Therefore, GIIc could be derived as a mean value over data covering the plateau domain. These values are included in Table 3 for each specimen as well as the strain energy release rate corresponding to the maximum load, GII,Pmax. The R-curves were then evaluated from the experimental P-δ curves by applying the CBBM data reduction method detailed in Section 2.2. They are shown in Figure 5 as relationship between equivalent crack length and energy release rate. These curves are characterised by an initial rising domain meaning that the FPZ is developing, and a horizontal asymptote afterwards which represents the material toughness to crack-growth, the so-called strain energy release rate (G IIc ). This is one of the most important parameters that define the material fracture behaviour in Mode II.   The R-curves scatter is within the expected range for wood and may be attributed to the variability of the material microstructure at the initial crack tip [26]. However, most of the specimens showed a plateau for a given crack extent. Therefore, G IIc could be derived as a mean value over data covering the plateau domain. These values are included in Table 3 for each specimen as well as the strain energy release rate corresponding to the maximum load, G II,Pmax .
There is no good correlation (R 2 = 0.10) between flexural modulus obtained by the ENF tests (also included in Table 3) and the G IIc values. The mean G IIc value resulted in 1.54 N/mm. This value is twice the G Ic value (0.77 N/mm) obtained from DCB tests under Mode I using also Eucalyptus globulus of the same quality in previous work by the authors [16]. E. globulus also shows higher G IIc values than other species. For instance, Pinus pinaster subjected to the same ENF tests and applying CBBM as data reduction method led to an average G IIc of 1.15 N/mm [26]. G IIc values ranging from 0.39 to 0.55 N/mm depending of the analysis method were obtained for Sitka spruce from ENF tests [32]. Studies on Western hemlock derived in G IIc values in the range of approximately 0.20-0.55 N/mm depending on the initial crack length in ENF specimens [33].
As can be seen in the table, the G II,Pmax values resulted similar to those of G IIc . Therefore, the first one could be assumed as critical strain energy release rate value in Mode II in a practical way or in cases where R-curve does not show a clear horizontal asymptote. It would mean that a critical energy would be already consumed at maximum load on the onset of a fully developed FPZ just before steady-stage crack propagation [22,24].

Cohesive Law
The cohesive laws in Mode II of the E. globulus specimens were obtained from the relationship between G II and transverse crack tip displacements in Mode II (w II ) measured by DIC during testing. Figure 6 shows the evolution of transverse crack tip displacements in Mode II (sliding) and also normal crack tip displacements corresponding to a Mode I during testing of an ENF specimen. As derived from the results, crack tip displacements were found to be predominant in transverse direction following Mode II for a representative extension of the crack propagation. However, some Mode I crack opening can eventually be observed in most of the specimens mainly at the end of the ENF test because there may not be a perfect match between the specimen neutral plane and the initial crack surface.
Materials 2020, 13, x FOR PEER REVIEW 9 of 13 direction following Mode II for a representative extension of the crack propagation. However, some Mode I crack opening can eventually be observed in most of the specimens mainly at the end of the ENF test because there may not be a perfect match between the specimen neutral plane and the initial crack surface. Typical macroscopic fracture behaviour of the ENF specimens before and after crack propagation can be observed in Figure 7 (left and right, respectively), which clearly evidence the predominant sliding Mode II of fracture mentioned before. GII values were finally correlated with wII, as shown in Figure 8 (left). The experimental results were filtering by least-square regression using a logistic function. Figure 8 (right) illustrates both experimental and logistic approximation data for a ENF specimen. As can be seen, a reasonably accurate fitting between the curves is displayed. Typical macroscopic fracture behaviour of the ENF specimens before and after crack propagation can be observed in Figure 7 (left and right, respectively), which clearly evidence the predominant sliding Mode II of fracture mentioned before.
Materials 2020, 13, x FOR PEER REVIEW 9 of 13 direction following Mode II for a representative extension of the crack propagation. However, some Mode I crack opening can eventually be observed in most of the specimens mainly at the end of the ENF test because there may not be a perfect match between the specimen neutral plane and the initial crack surface. Typical macroscopic fracture behaviour of the ENF specimens before and after crack propagation can be observed in Figure 7 (left and right, respectively), which clearly evidence the predominant sliding Mode II of fracture mentioned before. GII values were finally correlated with wII, as shown in Figure 8 (left). The experimental results were filtering by least-square regression using a logistic function. Figure 8 (right) illustrates both experimental and logistic approximation data for a ENF specimen. As can be seen, a reasonably accurate fitting between the curves is displayed. G II values were finally correlated with w II , as shown in Figure 8 (left). The experimental results were filtering by least-square regression using a logistic function. Figure 8 (right) illustrates both experimental and logistic approximation data for a ENF specimen. As can be seen, a reasonably accurate fitting between the curves is displayed. The cohesive laws in Mode II expressed by σII-wII relationship were obtained by analytical differentiation of the GII-wII curves. The results for each specimen are shown in Figure 9. The parameters describing the logistic function of every cohesive law (A1, A2, p and wII0 in Equation (9)), the area circumscribed by the cohesive laws (Glaw,II), the maximum stress (σIIu) and the relative displacements in Mode II corresponding to maximum stress (wIIu) are summarised in Table 4.   The cohesive laws in Mode II expressed by σ II -w II relationship were obtained by analytical differentiation of the G II -w II curves. The results for each specimen are shown in Figure 9. The parameters describing the logistic function of every cohesive law (A 1 , A 2 , p and w II0 in Equation (9)), the area circumscribed by the cohesive laws (G law,II ), the maximum stress (σ IIu ) and the relative displacements in Mode II corresponding to maximum stress (w IIu ) are summarised in Table 4.
The mean value of A 2 parameter provides an estimation of the critical strain energy release rate, G IIc . In the present case, a value of G Iic = 1.63 N/mm was obtained, which is just 5.6% higher than the G IIc value determined from the R-curves (1.54 N/mm, see Table 3) even considering the scatter exhibited by the results. A similar pattern in terms of dispersion was observed for other species in a related work on Pinus pinaster [26].
The mean experimental cohesive law in Mode II for Eucalyptus globulus L. can be finally built from the mean parameters presented in Table 4. It is highlighted by a bold curve in Figure 9. This cohesive law could be implemented in finite element cohesive zone models to simulate the development of the FPZ and crack growth, and thus study the fracture behaviour of timber structures with possibility of brittle failures with Mode II component. The cohesive laws in Mode II expressed by σII-wII relationship were obtained by analytical differentiation of the GII-wII curves. The results for each specimen are shown in Figure 9. The parameters describing the logistic function of every cohesive law (A1, A2, p and wII0 in Equation (9)), the area circumscribed by the cohesive laws (Glaw,II), the maximum stress (σIIu) and the relative displacements in Mode II corresponding to maximum stress (wIIu) are summarised in Table 4.

Conclusions
The fracture properties in Mode II of Eucalyptus globulus L. in RL crack propagation system were experimentally determined by coupling end-notched flexure (ENF) tests with digital image correlation.
The resistance curves (R-curves) of the material are presented, which were derived by applying the CBRM considering an equivalent crack length and thus overcoming inherent difficulties of measuring the actual crack length during propagation of a highly heterogeneous and orthotropic material.
The mean value of critical strain energy release rate in Mode II (G IIc ) resulted in 1.54 N/mm, which is twice the mean value reported for Mode I in previous work by the authors on the same wood species. The G IIc value obtained for eucalyptus was also greater than that of other species reported in the literature.
The fracture cohesive laws of eucalyptus, expressed as the relationship between stresses and relative displacements in Mode II, measured by means of digital image correlation, are shown. The laws definition could be accurately implemented in finite element models to predict the crack growth along a fracture process zone.
The fracture properties in Mode II presented make it possible to quantify the fracture behaviour of this potential species in timber engineering situations involving this type of failure.