Introduction

Timber has played an important role in the construction field since the dawn of civilization and, as climate issues emerge, wooden buildings have gained interest for their low carbon emission feature and sustainability [1,2,3]. Research on the structural performance of timber such as strength, stiffness, withdrawal resistance, and fire resistance has been carried out, and these studies play a key role in the design and stability assessment of timber structures [4,5,6,7].

Among the key material properties of wood, critical strain energy release rate (Gc) and fracture energy (Gf) are widely utilized in fracture mechanics. The critical strain energy release rate (Gc) is the amount of energy dissipated from external loads and deformed body to the crack tip during crack propagation, and the fracture energy (Gf) is the consumed energy per area of newly formed crack surface [8]. Although these two characteristics seem similar, they should be distinguished from one another [9]. Since the emergence of fracture mechanics, numerous researchers have investigated these properties to enhance the understanding of wood failure behavior, especially for mode II fracture due to its hazardous cracking characteristics [10]. The fracture parameters Gc and Gf serve as fundamental parameters for assessing interfacial failure [11], including the strength of Glued-in-Rod (GiR) [12, 13], splitting failure of timber [14, 15] and predicting failure loads in the presence of flaws, such as notches, holes, or cracks [16]. Moreover, fracture mechanics can provide a rational framework for explaining the counterintuitive phenomenon in which the overall system weakens when adhesives with higher strength are used [17, 18]. Given their significance, investigating the fracture-related properties of timber could enhance the reliability of timber structures.

In the case of perfectly brittle materials, Gc and Gf are equal [8]. Wood is usually considered a brittle material so it might be reasonable to adopt linear elastic fracture mechanics (LEFM) for timber. However, wood is not a perfectly brittle material and there is evidence that crack-resisting processes, such as fiber bridging retard the fracture of timber [19]. This fracture process zone (FPZ) hinders the applicability of traditional LEFM to wood [20, 21]. To address this problem, the equivalent LEFM approach is commonly used for measuring wood’s fracture properties, because it can reflect the effect of FPZ [21,22,23]. On the other hand, for quasi-brittle materials like timber, the non-linear fracture mechanics (NLFM) approach is also typically utilized for characterizing material properties [9, 24, 25]. However, there are limited studies comparing these two approaches.

The main objective of this study is to experimentally compare the two representative approaches for timber mode II fracture (i.e., Gc in equivalent LEFM and Gf in NLFM). Through investigating the potential equivalence between these two approaches, this study aims to provide researchers and engineers with guidance for selecting the optimal testing method based on their specific experimental capabilities and requirements. While LEFM-based methods offer simpler experimental procedures but require complex result interpretation, NLFM-based methods involve more complex experimental setups but provide straightforward specimen preparation and result analysis. This study is expected to lay the foundation for applying fracture mechanics to wood, particularly in assessing the strength performance of components, where loads are primarily transferred through shear forces, such as Glued-in-Rod (GiR) joints, timber shear connections, and laminated timber interfaces.

Previous studies

Research on Gc and Gf in wood presents inherent challenge due to its quasi-brittle behavior. The quasi-brittle behavior, which is the representative characteristics of wood in shear, tension, and bending, leads to unstable load–displacement curves, making it difficult to evaluate accurate fracture properties of wood [26, 27]. This instability complicates precise assessments and highlights the complexities associated with characterizing the fracture properties of timber. Despite the difficulty of measuring fracture properties of wood, many studies have been conducted to measure Gc and Gf of timber.

Investigations for mode II G f

The Gf (fracture energy) of wood was evaluated using the compact shear specimen with two notches [28, 29]. However, the specimens showed an unstable curve after the peak load which might lead to inaccurate results. Beam-shaped configurations also have been utilized for evaluating Gf [30]. The total consumed energy was calculated from the loading–unloading curve of specimens and divided by the final crack area. However, this method requires measuring the actual crack length, which demands considerable effort. Another test setup was applied using an elaborate fixture with small wood specimens [31]. They used sophisticated steel hardware to avoid moment and unintended movement of the specimen. Wood specimens were attached to the steel fixtures by epoxy adhesive to prevent rotation, springs or rubber bands prevented the specimens from moving horizontally, and a steel ball was inserted between steel components. The specimens were loaded until complete separation, which means that it does not require measuring the actual crack length.

Investigations for mode II G c

The commonly used testing configurations for measuring Gc of wood include end-notched flexure (ENF) and end-loaded split (ELS), both of which induce shear stresses through bending of the beam. A key advantage of these beam-based tests is that they theoretically allow for stable crack propagation when the crack length exceeds a certain threshold [32, 33]. Moreover, a study stated that stable crack propagation can be seen even with short initial crack lengths, because “the sample behaves as if it has a longer crack” [23]. However, one of the primary challenges of this method is the difficulty in monitoring the crack length, which is needed for calculating Gc. Mode II cracks have unclear crack tip and lack a “clear opening,” making direct monitoring problematic [34]. To address this issue, the compliance-based beam method (CBBM) was introduced, incorporating the concept of equivalent crack length to indirectly estimate crack propagation [35]. Other common test configurations are compact shear tests and block shear tests. This method offers advantages in terms of simplicity in both testing procedures and specimen preparation. However, it is advised that the compact shear method is not a practical approach due to the difficulty in achieving stable crack propagation [36, 37]. In addition, its validity is questionable when cracking occurs on only one side rather than symmetrically on both sides [27].

Comparison of G c and G f

In mode I conditions, there are studies that demonstrate the differences between Gc and Gf when the effects of FPZ are not considered [38,39,40]. Conversely, studies [21, 24] have numerically shown that the value of Gc is identical to Gf when the effect of FPZ is considered in the case of wood double cantilever beam test for mode I fracture. However, for mode II fracture in wood, there are few studies that compare these fracture properties, although the presence of FPZ is significant similar to mode I fracture. The evidence of FPZ such as “twisting, tearing, and unwinding” of cell walls is obvious during mode II crack propagation [41]. According to previous studies [23, 27, 32, 41, 42] due to the development of FPZ, it is demonstrated that mode II Gc during crack propagation can increase compared with Gc for crack initiation, but comparison with Gf obtained from the non-linear approach was beyond their scope.

Although equality is expected from a purely theoretical perspective [21, 43], the NLFM approach generally yields higher values due to energy consumption in the body of the specimen [44]. Moreover, a study asserted that specimen stiffness affects the NLFM approach more than the LEFM approach [45], which implies potential inequality between these two approaches. Therefore, this study addresses whether the LEFM and NLFM approaches can be used interchangeably for mode II fracture characterization of wood.

Materials and methods

It is well-known that the stable crack growth is essential to accurately evaluate fracture properties. When the load–displacement curve exhibits rapid crack propagation, the calculation of fracture properties may be distorted [8, 17, 33, 46]. The test setups were designed in accordance with these considerations. To ensure a stable mode II fracture, loading should be applied using a displacement-controlled method rather than a force-controlled method for this study’s test configurations (i.e., single-lap shear and beam-shaped specimens). In displacement-controlled tests with this study’s specimen geometry, the energy release rate decreases as the crack length increases, thereby stabilizing crack propagation. Therefore, displacement-controlled testing was adopted to prevent sudden crack propagation.

Mode II fracture energy (G f)

Requirements for mode II fracture energy (G f) test

To measure mode II Gf under stable crack propagation, the stiffness of the system excluding the target specimen must be high. Let \({Q}_{s}\) represent the sum of stored energy and consumed energy for crack growth of specimen (the area represented in Fig. 1), and \({Q}_{m}\) denotes the strain energy stored in components other than the specimen, such as the testing machine. Then, the derivatives of total energy of the system can be written as below:

Fig. 1
Fig. 1
Full size image

Schematic of components’ load–displacement curves of single-lap shear test

$$\frac{d\Pi }{da} = \frac{dU}{da}-\frac{dF}{da}+\frac{dW}{da} = \frac{d{Q}_{s}}{da}+\frac{d{Q}_{m}}{da}-\frac{dF}{da}$$
(1)

In Eq. (1),\(\frac{dF}{da}\) is the incremental external work of load due to the incremental crack length, which means that \(\frac{dF}{da}\) is a positive value, because crack growth increases the compliance of specimen. Therefore

$$\text{if}, \frac{d{Q}_{s}}{da}+\frac{d{Q}_{m}}{da}\le 0$$
(2)
$$\Rightarrow \frac{d\Pi }{da} = \frac{d{Q}_{s}}{da}+\frac{d{Q}_{m}}{da}-\frac{dF}{da} \le 0$$
(3)

It means that crack growth always occurs, in other words, unstable failure happens. Therefore, \(\frac{d{Q}_{s}}{da}+\frac{d{Q}_{m}}{da}\) should be larger than zero:

$$\frac{d{Q}_{s}}{da}+\frac{d{Q}_{m}}{da}\ge 0$$
(4)

\({Q}_{s}\) and \({Q}_{m}\) are functions of P and \(\delta\), and when assumed a and \(\delta\) have functional relation, by chain rule:

$$\frac{\partial {Q}_{s}}{\partial \delta }\frac{\partial \delta }{\partial a}+ \frac{\partial {Q}_{m}}{\partial \delta }\frac{\partial \delta }{\partial a}\ge 0$$
(5)
$$\text{since}, \frac{\partial \delta }{\partial a}\ge 0$$
(6)
$$\Rightarrow \frac{\partial {Q}_{s}}{\partial \delta }+\frac{\partial {Q}_{m}}{\partial \delta }\ge 0$$
(7)

By definition:

$${Q}_{s}=\int P\left(\delta \right)d\delta$$
(8)
$${Q}_{m}=\frac{1}{2}P\left(\delta \right){\delta }_{m}=\frac{P{\left(\delta \right)}^{2}}{2{k}_{m}}$$
(9)

Using Eqs. (7–9), the following equation is derived:

$$P\left(\delta \right)+\frac{P\left(\delta \right)}{{k}_{m}}\frac{\partial P\left(\delta \right)}{\partial \delta }\ge 0$$
(10)
$$\therefore {k}_{m}\ge -\frac{\partial P(\delta )}{\partial \delta }$$
(11)

In other words, the stiffness of the system outside the specimen [left term in Eq. (11)] must be higher than the slope of the strain-softening region of the specimen [right term in Eq. (11)]. The importance of a stiff system for crack stability has also been highlighted in previous research [17, 46]. It should be noticed that the above condition is a necessary condition but not a sufficient condition for stable crack growth.

Materials for mode II fracture energy (G f) test

Japanese larch (Larix kaempferi) sawn timber was used as the test material. Prior to processing, Japanese larch was stored at 25 °C and 60% humidity. The specimens were cut into a single-lap shear specimen configuration, with nominal dimensions detailed in Fig. 2.

Fig. 2
Fig. 2
Full size image

Detailed configuration of fracture energy test specimens

As previously mentioned, achieving crack stability requires that \({k}_{m}\) should be higher than the slope of the strain-softening region of the specimen. Since reducing the specimen height (h) increases stiffness, it was fabricated as short as possible within a controllable range to satisfy Eq. (11). For mode II fracture of wood, the RL and TL fracture characteristics in timber are similar [42]; thus, no distinction was made between these orientations. A total of 30 specimens were prepared.

Methods for mode II fracture energy (G f) test

The experimental configuration is illustrated in Fig. 3. To prevent rotation caused by load eccentricity and the mode I component, a steel plate and lateral support were incorporated instead of the sliding block projection used in previous research [47]. The steel plate was secured with bolts to constrain the specimen. As demonstrated in prior study [48], which introduced a steel ball and rubber band to allow only vertical displacement while restricting horizontal movement, the rollers supporting the shear specimen were clamped in place. To prevent the fixture’s movement and rotation, the opposite side was also fixed using clamps.

Fig. 3
Fig. 3
Full size image

Test configuration of fracture energy test

A Zwick UTMZ100 model testing machine was used to apply the load. The test was conducted in a displacement-controlled manner at a loading rate of 0.5 mm/min. Compression-type shear tests were chosen not only for their ease of setup but also to eliminate peel stresses. Recent research [49, 50] asserted that, compared to tension-based single-lap shear tests, compression shear tests yield more accurate shear strength values due to the reduction of normal stresses. Displacement was measured by installing an LVDT (accuracy: ± 0.0005 mm) on the load cell to track its distance from yoke as shown in \* MERGEFORMAT Fig. 3. The range from 0 to 150 N was corrected by assuming the slope between 0.1 and 0.4 times the maximum load. The fracture energy was calculated as below:

$${G}_{\text{f}}=\frac{1}{A}\int Pd\delta =\int \tau d\delta$$
(12)

Finite-element calculations

To check the effect of mixed-mode failure (mode I + mode II), 2D finite-element analyses were conducted using ANSYS 2019 R2 Fig. 4 shows the model configuration and finite-element meshes. The basic mesh size was set to 0.2 mm, refined to 0.1 mm around the crack region. Eight-node brick elements were utilized. The boundary conditions were set, such that horizontal movements of the right and left sides were restricted, and vertical movements of the top and bottom sides of the lower block were constrained. The load was applied to the top side of the specimen. Material properties were assigned as follows: longitudinal modulus of elasticity (MOE) of 10,000 MPa according to korean standard [51], radial and tangential MOE of 0.03 times the longitudinal MOE, and shear MOE of 0.06 times the longitudinal MOE.

Fig. 4
Fig. 4
Full size image

Schematic of finite-element analysis model and meshes

The mode I and II energy release rates were calculated by virtual crack closure technique (VCCT). Figure 5 describes the VCCT method and the equations are written as below [35]:

Fig. 5
Fig. 5
Full size image

Description of nodal properties used for VCCT

$${G}_{\text{I}}^{\text{VCCT}}=\frac{{\sigma }_{x}({u}_{x2}-{u}_{x1})}{2}$$
(13)
$${G}_{\text{II}}^{\text{VCCT}}=\frac{{\tau }_{x}({v}_{y1}-{v}_{y2})}{2}$$
(14)

The degree of mixed-mode failure was evaluated by the ratio of mode II energy release rate to the sum of mode I and mode II energy release rates as follows:

$${G}_{\text{ratio}}=\frac{{G}_{\text{II}}^{\text{VCCT}}}{{G}_{\text{I}}^{\text{VCCT}}+{G}_{\text{II}}^{\text{VCCT}}}$$
(15)

Mode II critical strain energy release rate (G c)

Since the standard for measuring the mode II critical energy release rate of timber is not established, measuring Gc was conducted using two different methodologies. One method is based on ASTM D7905 [52], which is the standard for measuring mode II fracture toughness of unidirectional fiber-reinforced polymer matrix composites. The other method was CBBM for measuring timber’s mode II critical strain energy release rate.

Requirements for mode II G c test

In the case of beam tests measuring Gc, crack length should exceed a certain threshold for stable fracture. The equilibrium state for stable crack growth can be written as follows [44]:

$$\frac{dG}{da}<\frac{dR}{da}$$
(16)

For the test configuration shown in Fig. 6, assuming \(\frac{dR}{da}\) equals zero, which is a conservative condition for stable crack propagation in timber, Eq. (16) can be expressed as follows:

$$\frac{dG}{{da}} = \frac{{6a\delta^{2} }}{{EIC^{2} b\left( {3a^{3} + 2b^{2} L^{3} } \right)\left( {1 + b} \right)^{2} }}\left( { - 3a^{3} + b^{3} L^{3} } \right) < 0$$
(17)
Fig. 6
Fig. 6
Full size image

Schematic of end-notched flexure test

$$\Rightarrow a\ge \sqrt[3]{\frac{{b}^{2}}{3}}L$$
(18)

If load is applied in the middle of the beam (b = 1), stability is secured when \(a\ge 0.7L\), which is generally known as the stable condition for ENF fracture tests [32, 53].

ASTM D7905-based test

ASTM D7905 [52] requires specimens with a 100 mm span and a 40 mm initial crack length. In this configuration, the Gc cannot be evaluated under conditions with more than 10 mm of crack growth. However, in other research, timber seems to have increasing mode II Gc due to FPZ developments, and a plateau state is observed when crack growth reaches about 20–30 mm [23, 27, 42]. Therefore, when specimen dimensions are unmodified, the test results might not fully reflect the fracture characteristics of timber. To address this problem, specimens with longer spans and initial crack lengths were used for tests (twice the ASTM D7905 [52] specimen dimensions). The specimen configuration and size are demonstrated in Fig. 7. A total of 10 specimens were produced with 23 × 23 mm cross sections, 200 mm length, and 80 mm initial crack. The span-to-depth ratio was less than 9 to ensure failure would not occur due to bending loading. To prevent friction between wood pieces during the loading process, PET films were inserted in the crack.

Fig. 7
Fig. 7
Full size image

Test configuration of ASTM D7905 [52]-based method

The load was applied at the middle of the span according to ASTM D7905 [52]. The displacement of the specimens was obtained through a linear variable differential transformer (LVDT) as shown in Fig. 7. The compliance-calibration coefficient was determined by linear least squares regression analysis utilizing data from 40, 60, and 80 mm initial crack lengths, which are twice the ASTM D7905 [52] crack lengths (20, 30, and 40 mm). The loading rate of 0.5 mm/min was selected for tests. Fracture tests were carried out at crack length 60 mm. In this setup, the condition for stable crack growth is satisfied, i.e., Eq. (18) is fulfilled. Critical strain energy release rates and crack length were calculated as below:

$${G}_{c}=\frac{3m{P}_{\text{max}}^{2} {a}_{0}^{2}}{2B}$$
(19)
$$m=\text{Compliance calibration coefficient}$$
$${a}_{0}=\text{initial crack length}(30\text{ mm})$$
$$B=\text{specimen width}\left(23\text{ mm}\right)$$

CBBM method

The ENF specimens with 23 × 23 mm cross sections and 500 mm length with an initial crack length of 182 mm were used based on existing literature [27, 42]. The roller supports were positioned 20 mm in from the ends of the specimens, so the specimens were loaded with a 460 mm span and 162 mm initial crack length, as shown in Fig. 8.

Fig. 8
Fig. 8
Full size image

Test configuration of CBBM method

The loading point was the center of the span, so stable crack growth requirements were satisfied. The loads were applied at a 2 mm/min loading rate. The equivalent crack length (aeq) and flexural modulus (Ef) were obtained as follows:

$${a}_{eq}={\left(\frac{C}{{C}_{0}}{a}_{0}^{3}+\frac{2}{3}\left(\frac{C}{{C}_{0}}-1\right){L}^{3}\right)}^{1/3}$$
(20)
$${E}_{f}=\frac{3{a}_{0}^{3}+2{L}^{3}}{12I{C}_{0}}$$
(21)

\({C, C}_{0}=Compliance and initial compliance\) Based on equivalent crack length and flexural modulus, critical strain energy release rates were determined using the following equation:

$${G}_{c}=\frac{3{P}^{2}{a}_{eq}^{2}}{8B{E}_{f}I}$$
(22)

Results and discussion

Result of NLFM approach (G f)

Finite-element analysis results

Figure 9 shows the ratio of the mode II energy release rate to the sum of mode I and mode II energy release rates (Gratio) for various crack lengths. At a normalized crack length of 0.1, the value of Gratio was about 0.986, and as the crack length increases, the effect of mode I diminishes. This trivial mode I effect can also be found in other research [47, 54].

Fig. 9
Fig. 9
Full size image

Gratio values for different normalized crack lengths

When shear stress is induced by a compression force, the mode I component becomes insignificant because of the reduction of normal stresses [49, 50]. The compression force tends to close the crack, while tensile force tends to open the crack. This is one of the reasons why compression-type shear tests were adopted. According to the finite-element analysis, the effect of mode I fracture for this study's test configuration appears to be minor compared to mode II fracture; therefore, pure mode II fracture was assumed.

Fracture energy test results

Crack propagation behavior

Figure 10 shows an example of the crack propagating gradually. The crack initiated at either the top or bottom of the shear region and expanded as the displacement applied by the machine increased. In some cases, crack propagation was visually observable during testing. However, in most specimens, measuring the crack length was highly challenging, highlighting the inherent difficulty of methods requiring crack length estimation.

Fig. 10
Fig. 10
Full size image

Crack propagation of fracture energy test (a Crack initiation, b, c crack propagation, d completely fractured specimen)

The tested specimens failed in the manner shown in Fig. 11. Most specimens exhibited non-planar shear failure surfaces. The non-planar fracture surfaces caused interference between the two separated pieces, which was a key factor enabling the specimen to continue resisting the load even after significant shear cracking had occurred.

Fig. 11
Fig. 11
Full size image

Examples of fractured specimens (a, c Front view of specimens, b, d top view of specimens)

Unstable crack propagation

Among the 30 specimens used in this experiment, 2 specimens exhibited rapid failure, resulting in unstable load–displacement curves. This unstable crack growth is caused by the material’s quasi-brittle characteristics, which make it difficult to analyze fracture properties. Rapid failures during tests were also reported in other research [28, 29]. In these cases, no strain-softening region was observed after the maximum load. The shear failure behavior of rapid crack growth is characterized by the load–displacement curve shown in Fig. 12. When failure occurs in this manner, calculating fracture energy based on the area under the load–displacement curve introduces a high degree of uncertainty; therefore, these specimens were excluded from the analysis.

Fig. 12
Fig. 12
Full size image

Load–displacement curve of unstable failure test

Unintended crack propagation

Among the 30 specimens, 15 specimens experienced failure in areas other than the intended shear plane (mainly due to bending), as shown in Fig. 13. A representative load–displacement curve of these specimens is presented in Fig. 14. This type of failure could also be seen in other literature [47] with similar phenomena showing a decrease of load prior to shear failure in the load–displacement curve. For these specimens, distinguishing the shear area was impossible, so they were excluded from the analysis.

Fig. 13
Fig. 13
Full size image

Examples of unintended failures (a Failure of top side, b failure of top and bottom sides)

Fig. 14
Fig. 14
Full size image

Load–displacement curve of unintended failure

This phenomenon is believed to have occurred, because the target shear plane was not clearly defined as the weakest point. By adjusting the dimensions of the shear plane (e.g., introducing single or double notches and grooves into the target area [23, 26]), it is expected that the frequency of unintended failures could be reduced. Using specimens with shorter length (“w” in Fig. 2) can be another option to prevent bending failures, but it should be cautious, because this can lead to smaller values of fracture properties [25].

Stable crack propagation

Out of 30 specimens, 13 specimens exhibited stable shear behavior that allows for the calculation of mode II fracture energy. A load–displacement curve for the shear loading of Japanese larch wood specimens is shown in Fig. 15. After shear failure occurred in the specimens, continuous load resistance was observed. The reasons for the residual strength are believed to be: (1) friction at the fractured shear surface and (2) compression between the specimen pieces due to the non-planar shear surface, as shown in Fig. 16, interference of two separate pieces.

Fig. 15
Fig. 15
Full size image

Load–displacement curve of stable failure

Fig. 16
Fig. 16
Full size image

Interference of two separate pieces

Residual strength can introduce distortion in the calculation of the mode II fracture energy of wood and should be eliminated. The ideal method would be to correct the load starting from the point at which the residual strength begins to manifest, but it is difficult to clearly define this point. In this study, the minimum load of the descending curve was defined as the residual strength. To ensure a conservative approach to the calculation of fracture energy, all residual loads occurring after the maximum load were eliminated, as depicted in Fig. 17.

Fig. 17
Fig. 17
Full size image

Elimination of residual resistance in fracture energy test curve

The measured displacement includes not only the deformation caused by the shear of the wood but also the displacement due to the compression of the wood and fixture hardware. To accurately calculate the fracture energy, all displacements other than the shear displacement of the wood must be eliminated. The displacement due to the compression of the wood was excluded from the load–displacement curve, as shown in Fig. 18. The compression displacement caused by the fixture hardware was considered negligible compared to the deformation of the wood; thus, it was ignored.

Fig. 18
Fig. 18
Full size image

Elimination of compression deformation in fracture energy test curve

The calculation of fracture energy was based on the area under the load–displacement curve, excluding compression deformation and residual strength, as shown in Fig. 18. The average fracture energy (Gf) calculated in this experiment is shown in Table 1.

Table 1 Result of fracture energy tests
Full size table

Dissipated energy in the body of specimen

Given that the NLFM result includes the energy dissipated in the body of specimen [44], the mode II Gf may be overestimated. The energy consumed other than the target fracture (shear fracture) was measured through compression loading–unloading tests of specimens with dimensions identical to half of the single-lap shear specimen (10 mm \(\times\) 15 mm \(\times\) 15 mm). Figure 19 shows one example of loading–unloading curve and the dissipated energy was calculated as 2 times the area of curve. A total of 10 specimens were tested. The corrected value (Gf,cor) is shown in Table 1. Note that the higher COV for Gf,cor is a statistical artifact of the correction process and does not indicate reduced measurement reliability.

Fig. 19
Fig. 19
Full size image

Compression loading–unloading test curve of wood specimens (10 mm × 15 mm × 15 mm)

The fracture energy of wood reported in existing literature varies by species but is generally within the range of 0.5–2.0 N/mm [10, 17, 25, 28, 37]. The average fracture energy measured for Japanese larch in this experiment is approximately 1.4 N/mm, which is within the range reported in the previous literature.

Result of LEFM approach (G c)

ASTM D7905-based method

In preliminary tests according to ASTM D7905 [52] (2L = 80 mm, a0 = 30 mm for fracture test), significant local crushing at the support and loading point occurred before mode II failure. Therefore, test span and initial crack length were doubled to facilitate crack propagation by shear. Despite the efforts to avoid local bearing fracture before shear cracking, some degree of crushing was inevitable. The local deformation of specimens due to the relatively high compression force applied perpendicular to the grain implies that compliance might be influenced. It means that a longer initial crack length is needed to exclude the effect of bearing.

The specimens in this study can be classified as deep beams with a span-to-depth ratio of about 9, so the effect of shear deformation on compliance was considerably high. Therefore, the displacement due to shear was excluded from the compliance as below:

$$C={C}_{\text{test}}-\frac{3{b}^{2}L}{5(1+b)A{G}_{xy}}$$
(23)
$${G}_{xy}=\text{The shear modulus of timber}$$

For fracture tests (a0 = 60 mm), all specimens exhibited crack propagation or cracking noise prior to reaching the maximum load, indicating that the calculated critical energy release rates represent crack propagating Gc rather than crack initiation Gc [41]. The load–displacement data at crack lengths of 40 mm and 80 mm were used for compliance calibration. According to ASTM D7905 [52], data points above 90 N are required. However, significant initial nonlinearity was observed above 90 N. Therefore, compliance calibration was performed using load–displacement data above 150 N (approximately 0.1Pmax). Figure 20 illustrates one of the Gc test results. According to ASTM D7905 [52], the compliance-calibration (CC) method was utilized for calculating Gc. The value of m in Eq. (19) was determined by linear least squares regression analysis using data below 0.75, 0.5 and 0.375 Pmax for crack lengths of 40 mm, 60 mm, and 80 mm, respectively, that coincide with a half of the maximum load for the given initial crack length. The results are presented in Table 2, following the procedure outlined in Eq. (19).

Fig. 20
Fig. 20
Full size image

Load–displacement curves of ASTM D7905 [52] tests (a0 = 40,80 mm: non-fracture tests, a0 = 60 mm: fracture test)

Table 2 Result of critical strain energy release rate tests based on ASTM D7905 [52]
Full size table

The measured Gc of wood based on ASTM D7905 [52] ranges within 0.5–2.0 N/mm [23, 27, 32, 34, 42]. The above Gc value is not in accordance with other research, and it seems that Gc is not determined correctly. One reason for the inaccuracy might be the reinforcing effect of crack resistance due to localized compression around the loading point [22, 23, 42]. For fracture tests with a 160 mm span and a 60 mm initial crack length, the newly formed crack and FPZ are very close to the applied force. Moreover, given that a plate was used for the loading procedure (i.e., it is not perfectly point-loading), the tip of the crack might be more influenced by the loading force than expected. Another reason is the local compression deformation of the specimens. In this study’s test setup, the LVDT measured not only the bending displacement of the beam but also the local compression displacement of the loading point. The total displacement of this test was only about 2–3 mm, which means that even small deformation might affect the compliance significantly.

These results emphasize the need for specimens with longer available crack propagation and lower expected maximum load for fracture. It can be achieved by lengthening the specimens’ span and initial crack length. This is the motivation for conducting another test to measure mode II critical strain energy release rate, which is discussed in the next section.

CBBM method

Beams with 460 mm span and 162 mm end notch were utilized for determining mode II Gc of timber. The CBBM method was selected for its simplicity that does not require monitoring crack length and its reliability that can produce trustworthy Gc values when self-similar FPZ is developed [22, 27, 41, 42]. The total number of tested specimens was 24; however, 11 specimens exhibited rapid crack extension during crack propagation. These specimens were excluded from the analysis. The detailed analysis procedure is described below:

  1. 1.

    Initial compliance(C0) was calculated by utilizing data from 0.1Pmax to 0.4Pmax.

  2. 2.

    At each point of the loading curve, compliance(C) was recorded.

  3. 3.

    Initial compliance and compliance were corrected considering the effect of shear deformation. (Eq. 23)

  4. 4.

    From Eqs. (20) and (21), the equivalent crack length and flexural modulus were determined.

  5. 5.

    The strain energy release rate was evaluated using Eq. (22).

Figure 21a shows the average loading curves and equivalent crack lengths of the test specimens, and Fig. 21b demonstrates the average energy release rate curve. At the early stage of loading, the propagation crack length (aeq–a0) shows high fluctuation. It is caused by the extremely sensitive characteristic of compliance in the low displacement region. Even though the same variation has occurred in both the initial and latter stages of loading, the compliance is more influenced by the initial region’s fluctuation. Because equivalent crack length is calculated based on compliance, the variation in propagation crack length becomes stabilized as displacement during the test increases. The R-curve (Fig. 21b) indicates that as the crack extends, the specimens’ crack resistance also increases. This is the nature of timber; it retards cracking through mechanisms, such as microcracks and fiber-bridging [8, 55, 56].

Fig. 21
Fig. 21
Full size image

CBBM method test result (a Load–displacement curve and propagation crack length, b R-curve)

Figure 22 shows one example of the loading curve and propagation crack length of specimens with unstable and stable crack growth. In Fig. 22a, the equivalent crack length jumps from 30.1 mm to 49.8 mm, which means that rapid fracture occurred. On the other hand, Fig. 22b shows the case of slow but continuous crack extension. At about 50–60 mm crack growth, the loading curve starts to rise. As the crack tip gets close to the middle of the specimen span, crack growth becomes restrained. It is thought to be an indication of the confinement of the FPZ due to the loading force. The enhancement effect is remarkable in Fig. 22c, d which show the strain energy release rate for each specimen. The strain energy release rate increases rapidly at about propagation crack length 50–60 mm. This phenomenon is also reported in other research [22, 23]. This is one of the reasons why the Gc value is determined as the average value for propagation crack length under 50 mm.

Fig. 22
Fig. 22
Full size image

Typical test curves of CBBM method. (a, c Test result of unstable crack growth, b, d test result of stable crack growth)

The critical strain energy release rate from the CBBM test results is shown in Table 3. The Gc is determined as the average value of the strain energy release rate for propagation crack length 30–50 mm, which is a similar region with existing literature [23, 27, 42]. The result is about 1.37 N/mm, which is within the range of previous research [23, 27, 32, 34, 42].

Table 3 Result of critical strain energy release rate test-based CBBM
Full size table

Comparison of LEFM and NLFM approaches

The average values of the LEFM approach Gc and NLFM approach Gf showed approximately 11.5% difference; however, when the dissipated energy in the bulk of the specimen is deducted, the difference becomes 3.7%. To statistically verify the difference between the results of the two approaches, t tests were performed using experimental data. Since the ratio of the standard deviation (S.D) of Gf to Gc fell within the range of 0.5 to 2, equal population variances were assumed, and pooled estimators were used for the t tests. At a significance level of 0.05, the null hypothesis (H0: average of Gc = average of Gf or Gf,cor) was not rejected (p value > 0.05, Table 3).

In addition, the two-sample Kolmogorov–Smirnov (KS) test was conducted to determine whether the data sets of Gf (or Gf,cor) and Gc originated from the same population. Dmax is determined from the maximum difference between the empirical cumulative distribution function (CDF) of LEFM and NLFM results, Table 4. In both cases, Dmax was less than the critical D value of the two-sample KS test for a significance level (\(\alpha\)) of 0.05. Thus, at a 5% significance level, the hypothesis that the two data sets originate from the same population could not be rejected (Dmax < Dcritical).

Table 4 Statistical comparision of fracture energy and critical strain energy release rate
Full size table

These findings suggest that when the effects of the FPZ, such as fiber bridging and microcracking, are adequately accounted for, the critical strain energy release rate (Gc) aligns well with the fracture energy (Gf). Although Gf did not show a significant difference from Gc in a statistical sense, the energy consumed in the bulk of the specimen seems to affect the result of the NLFM approach to some extent. It indicates that when the FPZ and consumed energy in the body of the specimen are considered, NLFM and LEFM-based results can be thought to be equal reasonably for wood's mode II fracture energy.

Limitations and future research

Mode II critical strain energy release rates and fracture energy of wood were compared in this study, but several limitations exist. First, the measurements are not entirely accurate. For fracture energy Gf, the magnitude and the starting point of the residual force were unclear. The perfect elimination of the residual force effect was not feasible in this study. Regarding the critical strain energy release rate (Gc), the measurement of both the actual and equivalent crack length was limited. Instead, the equivalent crack length was used for the calculation of Gc, which may have led to misestimation.

The ratio of stable curves in fracture energy tests is low, which is partly due to the presence of unstable specimens but also the occurrence of unintended fractures. Adjusting the specimen configuration may help improve the yield. There is also a need to test with other configurations.

A fundamental issue is that there is no global consensus on methods for measuring Gc and Gf of wood. ASTM D7905 [52], a method for measuring the Gc of unidirectional fiber-reinforced polymer matrix materials, seems to be applicable to wood, since wood also has unidirectional fiber characteristics. However, it is necessary to be cautious when asserting that the results of this testing method represent the Gc of wood. Therefore, finding the most appropriate testing method for wood through comparisons of different measurement techniques is necessary.

Conclusions

This study compared the mode II critical strain energy release rate (Gc) and mode II fracture energy (Gf) of Japanese larch using two methodologies for measuring Gc: a modified ASTM D7905 [52] approach and the compliance-based beam method (CBBM), and a single-lap shear test for measuring Gf. Experimental conditions were designed to induce stable fracture behavior under shear loading and obtain reliable load–displacement curves. However, the results of ASTM D7905 [52]-based test seem to be not reliable due to short available crack growth length. To address the inherent limitation of NLFM approaches that include energy dissipated in the bulk of the specimen, a correction was done using compression loading–unloading tests. The results are as follows:

  1. 1.

    Out of 30 specimens, 13 showed stable curves for shear failure, and the average mode II fracture energy (Gf) for the Japanese Larch specimens was observed to be 1.53 N/mm. After correction for bulk energy dissipation, the corrected fracture energy (Gf,cor) was 1.42 N/mm.

  2. 2.

    The average Gc result was 1.37 N/mm, and 13 specimens showed stable fracture among a total of 24 samples prepared.

  3. 3.

    Statistical analysis using t tests and Kolmogorov–Smirnov tests showed no significant difference between Gc and both Gf and Gf,cor values (significance level 0.05). The difference between Gc and Gf was 11.5%, but this reduced to only 3.7% when comparing Gc with Gf,cor.

The critical strain energy release rate (Gc) from the linear elastic fracture mechanics (LEFM) approach aligns well with the fracture energy (Gf) from the non-linear fracture mechanics (NLFM) approach when both the fracture process zone (FPZ) effects and bulk energy dissipation are properly considered. This study demonstrates that LEFM and NLFM approaches can be used interchangeably for mode II fracture characterization of wood when appropriate corrections are applied, thereby enabling more flexible experimental design choices.