Introduction

For structural applications in Europe, and more specifically the Nordic countries, Norway spruce (Picea abies) is the dominant wood species utilized. However, in Europe, there exists a significant amount of hardwood. For example, in Sweden and Finland, birch (Betula pendula) makes up 13% of the forest stock (SLU 2022), whereas in central Europe, beech represents a substantial amount of the forest stock. Currently, birch is primarily used in the pulpwood industry and for manufacturing furniture, but in recent years, the interest of extending the fields of application to also include structural use has increased. Due to global warming and the altered growth climate that follows, only relying on a single wood species in the construction industry might not be feasible long-term. However, to enable a potential diversification in the construction industry by utilizing birch, a better understanding of the mechanical behavior and the fracture properties of birch is required.

Elastic behavior of wood

A significant portion of research dealing with structural applications reports on the elastic mechanical behavior of softwoods in general, and Norway spruce in particular. The literature is however scarse when it comes to hardwoods for structural applications, including silver birch. Large and extensive research on the mechanical behavior of wood and wood-based materials has been carried out by e.g., Kollman et al. (1968, 1975) and Niemz et al. (2023). Some parts of the literature that are oriented towards hardwoods focus on North American wood species. For example, Bodig and Jayne (1982), present various mechanical properties of both hardwoods and softwoods, however, all are species native to North America. Some mechanical properties for both silver birch and Norway spruce can be found in the work of Hearmon (1961) in addition to other hardwood and softwood species. The elastic parameters of different wood species were studied by Dahl (2009), where a complete set of elastic parameters for Norway spruce is presented, based on the findings from different sources. Heräjärvi (2004) studied the static bending properties of Finnish birch wood, which were examined together with their dependency on the density. In addition, the possibilities of utilizing birch in engineered wood products, such as glued- and cross laminated timber, have been evaluated by e.g., Jeitler et al. (2016) and Obernosterer and Jeitler (2020), and the results were found promising. Determination of mechanical parameters for birch plywood has been carried out by Wang et al. (2022a, 2022b). In addition, an extensive experimental study on bending strength and stiffness of birch timber boards has been carried out by Lemke et al. (2023).

Collins and Fink (2022) studied the tensile strength parallel to grain of birch on small-scale specimens. Sixty planks were initially sawn from 30 different trees, from which 70 smaller timber boards were acquired. Defect-free parts could then be identified from the 70 timber boards. These defect-free parts were then used for manufacturing small-scale specimens, which were designed as dog bone shaped specimens for determining the ultimate tensile strength.

Fracture behavior of wood

In addition to the elastic behavior of wood-based structures, the behavior at failure, often characterized by fracture in wood, is of particular importance to ensure structural safety. When considering the fracture characteristics of materials, these are often evaluated in terms of the specific fracture energy \(G_{\text {f}}\), i.e., the energy required to create a unit area of traction-free surface. Although the specific fracture energy is very influential on the fracture behavior of solids, it is not the only governing factor. When speaking of fracture, the material brittleness is a commonly used term. The material brittleness is not only governed by the specific fracture energy, \(G_{\text {f}}\), but also by the stiffness, E, and the ultimate strength, f, of the material. The material brittleness is often characterized with the so called characteristic length. The characteristic length, denoted \(l_{\text {ch}}\), is determined according to \(l_{\text {ch}} = \dfrac{EG_{\text {f}}}{f^2}\).

Evaluating the fracture energy is not a new endeavour, and different experimental methods have been developed for this very purpose. Such methods were originally developed and utilized for determining the fracture energy of steel and concrete but have with time been adopted and modified for testing wood. Some examples of such methods are the double cantilever beam (DCB), the tapered double cantilever beam (TDCB), compact tension (CT) specimens and the single-edge-notched beam (SENB) loaded in three-point-bending. All methods have their respective strengths and weaknesses, but in general, the SENB is simple and convenient to use, since it does not require any monitoring of the crack-opening width (de Moura and Dourado 2018).

Wood is often considered a quasi-brittle material, in contrast to perfectly brittle materials, such as glass (Smith et al. 2003). Usually, the peak load for quasi-brittle materials is preceded by a non-linear behavior. However, unlike brittle materials, which usually experience sudden catastrophic failure, quasi-brittle materials experience toughening mechanisms in the fracture process zone (FPZ), e.g., micro-cracking and fiber-bridging in wood. As a consequence, quasi-brittle materials usually experience strain-softening behavior. As such, to accurately evaluate the fracture energy of wood using non-linear fracture mechanics, it is important that stable crack propagation is achieved during experimental testing. Here, stable crack propagation refers to the ability to accurately capture the strain-softening response of the specimens used in the previously mentioned experimental methods (DCB, TDCB, CT and SENB specimens), without experiencing sudden large load-drops. A method suitable for this is described in the standard NT BUILD 422 (Nordtest 1993). The standard, which specifies a method for determining the specific fracture energy in wood in tension perpendicular to grain, is named the Nordtest method, and was originally proposed by Gustafsson (1988). It is a powerful method in the sense that only the applied force and displacement of a SENB specimen have to be recorded. In addition, the Nordtest method works well for sufficiently small wooden specimens, since this ensures that the only source of energy dissipation is the creation of a crack. For smaller specimens, the fracture process zone can be quite large in comparisons to the absolute specimen size. Under such conditions, methods based on linear elastic fracture mechanics, such as the R-curve concept, break down, and are not feasible for evaluation of the specific fracture energy.

Wood, which usually is considered an orthotropic material, is often characterized by its three principal directions; \(\text {L}\), the longitudinal direction of the fibers, \(\text {R}\), the radial direction and \(\text {T}\), the tangential direction, with respect to the growth rings. Consequently, six different crack propagation systems can be identified. These systems are denoted LR, RL, TL, LT, RT and TR, where the first letter denotes the direction normal to the crack plane and the second letter denotes the crack propagation direction, see Fig. 1. A crack in any of the first four systems is generally governed by cell fracture, whereas the latter two are governed by cell separation (Conrad et al. 2003).

Fig. 1
figure 1

Different crack propagation systems in wood. The first letter denotes the direction normal to the crack plane, and the second the direction of crack propagation

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In practice, wooden structural elements often have to be modified with respect to their geometry, by for example introducing holes or notches. Such geometrical modifications typically introduce stresses that act perpendicular to the fiber direction (Thelandersson and Larsen 2003), creating a risk of fracture occurring along the grain, i.e., fracture in the TL or RL-propagation system. To better understand the effect of utilizing birch in such applications, it is important to characterize the fracture behavior in these crack propagation systems. Previous research indicates that the TL-direction generally seems to have a lower fracture energy for several wood species (Reiterer et al. 2002), and as such, the TL-direction is the focus of the present work.

The fracture behavior of Norway spruce has previously been evaluated by Ostapska and Malo (2020) and Reiterer et al. (2002). In both works, the fracture behavior was evaluated by means of a wedge splitting test, from which the fracture energy and the stress intensity factor were determined. The fracture energy in the TL-direction was determined by Reiterer et al. (2002), Reiterer and Stanzl-Tschegg (2002) and Dourado et al. (2015), using SENB specimens. In addition, the fracture behavior of both Norway spruce and birch was evaluated by Tukiainen and Hughes (2013, 2016). Forsman et al. (2021) evaluated the fracture energy of birch and spruce in the TL-direction by means of the Nordtest method. In the Nordtest method, a rectangular fracture area is prescribed, but Forsman et al. (2021) used a triangular fracture area to achieve sufficient stability of the load–displacement curve during experimental testing. Numerous studies have been conducted with regards to how a triangular fracture area affects the fracture toughness in testing of materials such as metals, composites and ceramics (Dlouhý and Boccacini 2001; Sheikh et al. 2015). However, to the author’s best knowledge, comparisons between a rectangular and triangular fracture area have not been carried out for the materials in the present work; spruce and birch.

In Eurocode 5 (EN 2004), a few design criteria are based on concepts from fracture mechanics, although this is not explicitly stated in the code. One example is the design criterion with respect to splitting for beams with connections loaded perpendicular to grain. The design criterion employs a single material characteristic value regardless of strength class and wood species. Consequently, if birch would be used to a greater extent for structural applications, its fracture behavior and fracture energy should be considered, and different standardized values should potentially be adopted to ensure a material-efficient design practice.

Establishing mechanical parameters, such as the fracture energy, is also important in the field of numerical simulations. It is well-known that accurate material characteristics are crucial to be able to achieve accurate numerical models which can produce reliable results. Determining mechanical parameters on the clearwood or small-scale specimen level, and then using said parameters for simulations at the structural level is a common approach in engineering science. As such, if the fracture behavior could be established on the clearwood level, for birch, this behavior could then be utilized at a structural level in numerical models of load-bearing applications, to evaluate the effect of using birch. Such numerical models often, in turn, serve as a base for development of design criteria, and it is thus of great importance to establish representative input parameters to said models.

Aim and objective

Accelerating the use of hardwoods, particularly birch, in the construction industry can provide both structural and environmental benefits, such as enhanced biodiversity in forestry. However, to increase the use of birch in structural applications, its mechanical properties must be well understood.

The primary aim of the present paper is to further develop the understanding of the fracture behavior of birch, using the Nordtest method to evaluate the specific fracture energy. A secondary aim is to assess the difference in evaluated fracture energy when using different geometries for the fracture surfaces—herein a rectangular and a triangular fracture area. With previous work indicating increased material brittleness in birch, compared to spruce, modifications to the experimental method might be needed to achieve a stable post-peak behavior of the load–displacement response, which is crucial for reliable evaluation of test results. In turn, the present work could lead to a better understanding of the fracture behavior of birch, which could aid in optimizing design processes where birch is utilized.

The remainder of the paper is structured as follows. Section “Materials and methods” introduces the experimental method used for evaluating the fracture energy, the tested materials, a description of the post-processing of the experimental data, and lastly a brief description of the numerical modeling approach applied. Section “Results and discussion” presents the results, together with a discussion. Lastly, in Section “Conclusions and outlook”, conclusions and suggestions for further studies are presented.

Materials and methods

Evaluation of specific fracture energy

In the Nordtest method, a wood beam is loaded in three-point-bending until failure, see Figs. 2 and 3. The beam is assembled from three parts; two identical lateral pieces and one middle piece. In the middle piece, a notch is sawn just before testing, to ensure stability and crack-initiation at the mid-section of the beam. The lateral pieces are oriented with their fiber direction in the beam’s longitudinal direction, and the middle piece is oriented to achieve the desired crack propagation system. In the present work, this corresponds to the TL-propagation system, see Fig. 1. The beam is loaded in a displacement controlled setting and the load, P, and the corresponding displacement, u, are recorded. The aim is to capture the complete load versus displacement response, including the descending part after the peak load has been reached, until zero load at complete failure of the specimen. The specific fracture energy of the specimen can then be calculated as follows (Nordtest 1993):

$$\begin{aligned} G_{\text {f}} =\frac{W + mgu_\text {0}}{A_\text {c}} \end{aligned}$$
(1)

where

$$\begin{aligned} m = 5/6 \cdot m_{\text {tot}} + 2 \cdot m_{\text {prism}} \end{aligned}$$
(2)

and \(m_{\text {tot}}\) is the mass of the test specimen and \(m_{\text {prism}}\) is the mass of the steel prism on top of the beam, see Fig. 3. W equals the total work done to fracture the specimen, i.e., the area under the load–displacement curve. \(A_c\) is the fracture area of the specimen, which is different for the rectangular and triangular fracture surfaces, respectively, see Fig. 4. \(u_0\) is the load-point displacement at complete failure, and g is the gravitational acceleration.

Fig. 2
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Schematic two-dimensional illustration of the Nordtest set-up. The beam depth is equal to d and \(h_c\) denotes the ligament length

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Fig. 3
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Exploded view of the SENB specimen. Dimensions according to Fig. 2

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Fig. 4
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The two different fracture surfaces used in the present work

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Experimental testing

SENB specimens

Two wood species, spruce and birch, and two different geometries of the fracture area were used in the four test series of this study, see Tables 1 and 2. The different geometries of the fracture surfaces are illustrated in Fig. 4. The triangular fracture area has a ligament length, \(h_{\text {c}}\), of 10 mm, whereas the rectangular fracture area has a ligament length of 8 mm. In the Nordtest standard, the beam height, d, is prescribed to 60 mm. However, in the present work a beam height of \(d = 20\) mm was adopted for two reasons: (i) the raw material used for creating the birch specimen did not allow for larger dimensions and (ii) to improve post-peak stability and thus ensuring proper evaluation of the fracture energy. A smaller specimen entails a larger relative size of the FPZ in comparison to the specimen size, in turn leading to a more ductile behavior during testing. In addition, it has previously been shown by Persson et al. (1993) that, for larger specimen sizes, the effect of plastic dissipation affects the evaluated fracture energy. However, for a sufficiently small specimen size, the plastic dissipation becomes negligible.

Table 1 Spruce test series
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Table 2 No. of birch specimens from each birch timber piece
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The specimens made of Norway spruce were manufactured from structural timber of strength class C24. Timber boards of nominal size \(45 \times 90~\hbox {mm}^2\) were initially planed and sawn length-wise into sticks with a cross-sectional area of \(20 \times 20~\hbox {mm}^2\). The sticks with the growth-ring orientations most appropriate for evaluating the fracture energy in the TL-direction were further used for creating the middle pieces for the SENB specimens, whereas the remaining sticks were utilized to create the lateral pieces. Lateral pieces of spruce were used for the spruce series as well as for the birch series. The lateral pieces were sawn to nominal lengths of 60 mm and the middle pieces were sawn into cubes, with nominal dimensions of \(20 \times 20\times 20~\hbox {mm}^3\).

In Table 1, S and TS denote spruce specimens with a rectangular and triangular fracture area, respectively. Initially, four different spruce timber boards were planed and split into nine sticks in total. The four boards were identified by the numbers 1–4, and the sticks within each board by the numbers 1–3. As such, all spruce specimens were extracted from the same stick, from the same timber board—in this case the third stick from the fourth timber board (thus the names S43 and TS43).

The material used for the birch specimens originated from Finland, where its tensile strength in the direction of the grain previously had been experimentally tested using dog bone specimens (Collins and Fink 2022). The broken specimens were acquired from Aalto University where the pieces in the best condition after testing were chosen. The pieces with the most optimal fiber orientation for testing in a TL-orientation, see Fig. 1, were selected for the Nordtest specimens in the present work. It should be noted that the specimens were extracted from the outer parts of the dog bone specimens, see Fig. 5. These outer parts have been clamped, but free from large stresses in comparison to the middle-section of the dog bone specimens.

Fig. 5
figure 5

Schematic illustration of the dog bone specimens and where the material tested in the present work was extracted from. After sawing/planing, birch specimens were only created from the areas marked by dashed red squares (color figure online)

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Due to dimensional limitations, 20 specimens could not be acquired from a single dog bone specimen. Instead, between two and seven middle pieces for the SENB specimens according to Fig. 3 could be extracted from each dog bone specimen. Consequently, one series of birch specimens included birch wood from four to five different dog bone specimens. The number of specimens acquired from each dog bone specimen is shown in Table 2. In this table, TB denotes a triangular fracture area and B a rectangular. Due to manufacturing problems, specimens B01 and B07 were unfit for further processing, and they were excluded from the study. In addition, one middle piece was excluded from each of the test series TS43 and TB10 due to erroneous notch preparation. In total, 82 specimens were created and testing was carried out on 80 specimens. In all other aspects, the approach for manufacturing the birch middle pieces was the same as for spruce; the pieces were planed into sticks with a nominal cross-sectional dimension of \(20 \times 20 \ \text {mm}^2\) and then sawn into cubes with nominal dimensions of \(20 \times 20 \times 20 \ \text {mm}^3\), cf. Figure 3.

All pieces, spruce and birch, were conditioned in a climate chamber with a relative humidity of 60% and a temperature of \(20 \pm 0.2\) degrees Celsius until the change in weight was less than 0.1 g over 24 h. The parts were glued together with a commercial PVAc wood adhesive, and again conditioned in the same climate chamber. The dimensions and the weight of each birch specimen were determined to the nearest 0.1 mm and 0.1 g, respectively, and again conditioned until less than 0.1 g change in weight was observed over 24 h. Before testing, the positions for the supports were marked on all specimens to ensure repeatability of the test set-up to the greatest extent possible. When the specimens had been marked, they were placed in a ziplock bag in the climate room and taken to a small band-saw, where the notches were sawn. The band-saw had a blade with a measured thickness of 0.54 mm. Directly after sawing the notch, each specimen was placed back inside the ziplock bag.

To ensure the nominal notch geometries, special jigs were used. For the triangular notch, a jig which enabled the correct orientation of the specimen during cutting was used. As such, the first cut was made in this jig, followed by a 90-degree rotation of the specimen, followed by a second cut, creating the triangular notch geometry. In addition, an attached wood piece behind the saw blade was used, ensuring the intended nominal saw cut depth into the specimens.

Density and moisture content

The density was evaluated by measuring the dimensions and the weight of all specimens. All birch pieces were weighed before, during and after conditioning in the climate chamber, whereas the complete set of spruce pieces was weighed only before conditioning. A smaller set of the spruce pieces was then weighed during and after conditioning, to ensure moisture content equilibrium.

The moisture content (MC) was determined on specimens cut from the middle pieces, after testing. The oven drying method was used, with a temperature of \(105\,^{\circ }\hbox {C}\). The mean moisture content for each series is shown in Table 3.

Table 3 Mean measured density and moisture content of the four test series
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Testing procedure

The experimental testing of the 80 specimens was carried out in batches of five to ten specimens. The specimens were loaded in a displacement controlled manner in a MTS-machine, see Fig. 6.

Fig. 6
figure 6

Experimental test set-up

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During the testing of the specimens, a load cell with a capacity of 500 N was utilized. A total displacement of 5 mm was applied to the spruce specimens, whereas 7 mm was applied to the birch specimens. The rate of loading was set to a constant 0.75 mm/min, which resulted in the peak load being reached after roughly 90 s, and a total test duration of 400–600 s.

Due to the horizontal displacement of the beam at the roller support, which increases with the vertical displacement, some specimens experienced downfall from the test set-up before reaching the full prescribed displacement. To ensure that they were still comparable with respect to their fracture energy, the difference in fracture energy between the downfall specimens and the fully displaced specimens was examined; the difference was found to be negligible.

Post-processing of experimental data

Load–displacement curves were recorded with a sampling frequency of 128 Hz. Due to the high sampling rate, a sensitivity study on the influence of sampling frequency on the estimated fracture energy was carried out. This included the use of sampling frequencies from 0.25 to 128 Hz in the evaluation of fracture energy. It was found that a sampling frequency of 1 Hz was sufficient, and thus this frequency of included data points was used for the further post-processing of the recorded data.

Determination of fracture surface area

From Eq. 1, it is obvious that the area of the fracture surface is decisive in calculating the fracture energy. As such, it is of interest to determine the fracture area of each specimen with a sufficiently accurate approach. The area of the fracture surface was measured after the tests. With respect to the specimens that experienced downfall, explained in the previous section, these specimens had cracks that had propagated enough for the specimens to still be considered completely fractured, having a negligible ligament length left, and a low remaining load left (1–2% of the peak load). In the present work, the fracture area was measured with the program GNU Image Processing Program (GIMP) (The GIMP Development Team 2019). Initially, photos were captured of all fracture surfaces together with a ruler, for scale. The pictures were then imported into GIMP, and with the histogram function, the number of pixels in a certain selected area could be determined. By scaling the number of pixels to the ruler, the specimen fracture area could be determined. In Table 4, the nominal and mean measured areas are shown for all four series. As can be seen, the average area is fairly close to the nominal area for the two spruce series, however, the deviation is larger for the two birch series. Due to this, all analyses were carried out with respect to the measured area, \(A_{\text {real}}\), to allow for proper evaluation of the fracture energy.

Table 4 Nominal and mean measured fracture area (measured with GIMP) for the different test series
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Evaluation of stability

Currently, there is no method in the Nordtest standard for evaluating the stability. In this paper, the method proposed by Forsman et al. (2021) has been utilized. The method defined a load drop parameter, \(\text {LC}\), based on the magnitude of the largest load drop in relation to the maximum load:

$$\begin{aligned} \text {LC} = |P_i - P_{i+1}|/P_{\text {max}} \times 100 \end{aligned}$$
(3)

In Eq. 3, \(P_i\) and \(P_{i+1}\) denote two subsequently recorded load data points, and \(P_{\text {max}}\) denotes the maximum load recorded. An arbitrary limit level of load-drop, LC, can be decided (e.g., 5, 10, 15, 20%) defining all curves exceeding this limit as being unstable. It is important to note that this method does not necessarily eliminate the arbitrariness regarding the definition of stability, however, to some extent it makes it quantifiable. Table 5 presents an overview of the distribution of load-drops found.

Table 5 No. of specimens experiencing differently sized load-drops
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Load–displacement curves

An example of a recorded load–displacement curve is shown in Fig. 7. From the unprocessed load–displacement curves, two problems are evident. Firstly, the curve does not start with an initial load and displacement equal to zero, due to the fact that the recording of data is initiated before the beam actually is in contact with the loading equipment that applies the force. Secondly, it is apparent that some non-linearities occur at the onset of loading when the beam is establishing contact with the prism and the supports. To simplify the comparison between different tests, and between tests and numerical simulations, the initial non-linear part was removed and all curves were modified such that both the initial load and displacement were equal to 0.

Fig. 7
figure 7

Schematic example of a recorded load–displacement curve with illustration of how the displacement increases with zero load until contact is initiated (1) and nonlinearities that occur after initial contact (2)

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For each curve, the non-linearity was dealt with by conducting successive linear regressions between two points. The distance between the two points was set to 10% of the maximum load, for each curve. For each such 10% interval, the coefficient of determination, \(R^2\), was determined. The interval with the maximum \(R^2\)-value was chosen to represent the actual elastic stiffness of the specimen. Any point on the curve prior to this interval was replaced with a point coinciding with this stiffness. The value of the displacement at the intersection of this straight line and the x-axis could then be determined and subtracted from all recorded displacement values. With this methodology, the initial non-linear part of the curves becomes linear and all curves originate at (0,0). The effect on the evaluated fracture energy by using this approach to modify the curves was in most cases less than 1%, compared to evaluating the fracture energy, \(G_{\text {f}}\), with the unprocessed data.

Numerical modeling

The calculation model used for the numerical simulations of the experimental Nordtest set-up is schematically shown in Fig. 8. Two-dimensional plane stress conditions were assumed and symmetry was utilized in the beams’ middle-section, i.e., at the pre-defined crack path. The pre-defined crack path was modeled as a cohesive zone with non-linear springs, where the non-linear springs were modelled with a strain-softening behavior. The non-linear springs had an initial length of zero and were in their left-most degree of freedom (dof) connected to the beam, with the other node connected to the ground, using ’SPRING1’ in the commercial FE-software ABAQUS (Dassault 2019). The springs could only transfer force in the horizontal direction. Two-dimensional, linear, plane stress elements (CPS3) were used, with an element size of 0.25 mm for the middle piece, and 1 mm for the lateral piece.

Fig. 8
figure 8

Schematic illustration of the calculation model of the symmetric half of the SENB specimen (left) and close-up of the non-linear springs (right), with an initial length of zero

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Mechanical parameters

The elastic parameters of spruce and birch have not been tested experimentally in the present work, and instead values were collected from the literature. The elastic parameters collected for birch were acquired from the works of Dinwoodie (2002) and are presented in Table 6 together with the corresponding values for spruce. The values used for spruce are from Dahl (2009), where mechanical parameters for Norway spruce were compiled from a number of sources. For the tensile strength perpendicular to the grain, \(f_{\text {t}}\), a value of 7 MPa was used for birch, acquired from Kollman et al. (1968), and a value of 3 MPa for spruce (Danielsson 2013). These values were used as a starting point for the numerical simulations, but modified as needed to fit the elastic part of the load displacement curve, see Section “Results and discussion”.

Table 6 Elastic parameters and tensile strength perpendicular to grain for spruce and birch, respectively
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A generic linear strain-softening behavior is shown in Fig. 9. It should be noted that the displacement value \(\delta _1\) should theoretically equal 0, i.e., no deformation in the FPZ occurs until the tensile strength, \(f_{\text {t}}\), has been reached. However, this is complicated in practice due to issues with convergence, and instead, a high initial stiffness, \(k_{\text {init}}\), resulting in negligible elastic deformation at maximum stress is chosen. For a sufficiently high initial stiffness, the global load–displacement behavior is not affected. Here, a stiffness of \(5 \cdot 10^{11} \text { Pa/m}\) was chosen after carrying out a sensitivity study with respect to the initial stiffness, see Section “Effect of initial stiffness and element size”. The springs were modelled as ten times stiffer in compression compared to tension. The fracture energy for both wood species was acquired from the present work. Based on these parameters, linear strain-softening relations could be established for spruce and birch. The parameter \(\delta _1\), was determined according to

$$\begin{aligned} \delta _1 = \frac{f_{\text {t}}}{k_{\text {init}}} \end{aligned}$$
(4)

and the parameter \(\delta _2\) according to

$$\begin{aligned} \delta _2 = \frac{2G_{\text {f}}}{f_{\text {t}}} \end{aligned}$$
(5)
Fig. 9
figure 9

Schematic illustration of a cohesive linear softening law used for simulating quasi-brittle fracture. Note that the figure is not drawn to scale

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The strain-softening behavior of spruce has been determined before to some extent (Dourado et al. 2015, 2008; Stanzl-Tschegg et al. 1995). However, to the authors best knowledge, the literature does not contain any work that has determined the strain-softening behavior of birch. Consequently, the same shape of the strain-softening behavior of spruce has been utilized, but instead scaled with the tensile strength and fracture energy of birch.

The strain-softening behavior is usually expressed as a stress-displacement relation, see Fig. 9. As such, since the springs of the FE-model carry a force, and not a stress, the stress-displacement relation has to be converted to a force-displacement relation. Conveniently, this can be carried out by the relation

$$\begin{aligned} F_{\text {spring}} = \sigma A_{\text {spring}} \end{aligned}$$
(6)

where \(\sigma\) is the tensile stress perpendicular to the grain, \(F_{\text {spring}}\) is the force in the spring, and \(A_{\text {spring}}\) its tributary area. For the model with a rectangular fracture area, all springs had the same tributary area, except the top- and bottom-springs, which had half the tributary area of the other springs. For the specimens with a triangular fracture area, the tributary area was different for each spring, due to the geometry of the notch, resulting in a linear variation of tributary area along the height of the fracture plane. The initial stiffness, \(k_{\text {init}}\), was not modified to account for the varying width of the triangular fracture surface. This is motivated since \(k_{\text {init}}\) anyhow should be regarded as infinitely stiff.

Effect of initial stiffness and element size

To evaluate proper initial stiffness, \(k_{\text {init}}\) and element size, a convergence study was carried out with regard to these parameters. The element sizes were 0.5 mm (large), 0.25 mm (medium) and 0.125 mm (small). The initial stiffness was tested for values of \(5 \times 10^{11}\) Pa/m, \(5 \times 10^{12}\) Pa/m, and \(5 \times 10^{13}\) Pa/m. The convergence study was carried out using the mechanical parameters for spruce presented in Table 6. The effect of the element size is shown in Fig. 10a, where the relative difference between the input fracture energy and the output fracture energy evaluated from the external work was determined for three different levels of mesh refinement and \(k_{\text {init}}\). As is shown, the error reduces drastically from the large to medium element size, but next to nothing between the medium and small element size. As such, the medium mesh was deemed sufficient. As can be seen in Fig. 10b, the different values of initial stiffness, with the medium element size, did not markedly affect the same relative difference between input fracture energy and external work. Consequently, a stiffness of \(5 \cdot 10^{11}\) Pa/m was deemed sufficient for subsequent analyses.

Fig. 10
figure 10

Relative error between input fracture energy and output fracture energy evaluated from external work in the FE-model for a different element sizes in the middle piece, and, b different values of the initial stiffness \(k_{\text {init}}\) for the medium element size

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Results and discussion

Experimental results

In the present work, the Nordtest method was modified by introducing a triangular fracture area, in addition to the normally employed rectangular fracture area. The reason for this was due to previous work implying increased material brittleness in birch (Forsman et al. 2021), in turn resulting in reduced stability of the post-peak behavior. The load–displacement curves acquired from the experimental testing are shown in Fig. 11, together with a load–displacement curve from the numerical simulations. It is evident that the birch specimens showed a more stable post-peak behavior than the spruce specimens, see Table 5. Out of the 39 spruce specimens, 27 specimens had a maximum load-drop larger than 5%, whereas for the birch specimens the same number was only seven. No birch specimens, and only four spruce specimens, had a maximum load-drop larger than 10%. In addition, no apparent difference in stability could be identified between the rectangular and triangular fracture area, for either species. However, this might be a direct consequence of the size of the specimen, see e.g., Karihaloo (1995), since for sufficiently small specimens, post-peak instability should not be an issue.

Fig. 11
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Numerical load–displacement curves plotted in conjunction to experimental load–displacement curves. Legend valid for all plots

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The fracture energy of the spruce specimens in the present work was evaluated to 188 J/\(\hbox {m}^2\) for the S43-series and 257 J/\(\hbox {m}^2\) for the TS43-series, see Table 7. The individual results for each specimen are shown in Online Resource 1. The results are in accordance with previous findings in the literature. For example, the fracture energy of spruce, conditioned to a moisture content of 11–13%, was evaluated by means of linear elastic fracture mechanics to 150 J/\(\hbox {m}^2\) by Dourado et al. (2015). Similiary, it was evaluated to 150 J/\(\hbox {m}^2\), at a moisture content of 12–13%, by Stanzl-Tschegg et al. (1995), by means of a wedge splitting method. In another study (Reiterer et al. 2002), the fracture energy for spruce was evaluated to a slightly higher value; 213 J/\(\hbox {m}^2\) with a moisture content of around 12%. Slightly higher values of the fracture energy for spruce has been established by e.g., Riberholt et al. (1992) and Stefansson (2001), where a fracture energy of 298 and 283 J/\(\hbox {m}^2\) was acquired, respectively.

Table 7 Evaluated fracture energies for the different test series
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A clear outlier can be identified for the S43 series. This outlier corresponds to specimen S4320, (see Online Resource 1), and did have a slight deviation in fiber orientation compared to the other specimens. It is however difficult to assess the exact reason for this deviation, but one explanation could be that there was a nearby knot in the timber board, in turn causing deviation in the fibers.

The fracture energies for the birch series, B and TB, were evaluated to 700 and 611 J/\(\hbox {m}^2\), respectively, see Table 7. Forsman et al. (2021) determined the fracture energy of birch for five different series, each conditioned in different relative humidity. Four out of five series were either very wet or very dry, and as such not comparable to the present work. However, one series had a moisture content of 13.6%, which is close to the MC in the present work. For this moisture content, the fracture energy was evaluated to 460 J/\(\hbox {m}^2\). Compared to this, the values acquired in the present work are markedly higher. It should however be noted, a larger quantity of specimens has been evaluated in the present work. In addition, the discrepancy between these values could in part be explained by the natural variation of mechanical properties in wood.

Effect of density and moisture content

The density in relation to the fracture energy for all specimens, both spruce and birch, is shown in Fig. 12. The density of the spruce specimens showed a small variation, see Table 3. The coefficient of variation in specific fracture energy for the S43- and TS43-series was 3.3% and 1.5%, respectively. This can most likely be explained by the fact that all spruce specimens originated from the same timber board. As such, all spruce specimens were extracted in a perfectly sequential order from the same timber piece, with the exception of the exclusion of specimens containing knots and/or other imperfections. For the two spruce series, the only large deviation in density was from the previously mentioned outlier, S4320, which did have a quite high density compared to the other specimens (see Fig. 12).

Fig. 12
figure 12

Fracture energy in relation to density for all specimens

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The birch specimens were, in contrast to the spruce specimens, created from nine different timber pieces. The B-series was created from four timber pieces, and the TB-series from five timber pieces. However, the B-series still does display a very low CoV in density, only 1.9%. In comparison, the TB-series displays the largest CoV in density of all series, at 8.9%. This makes it difficult to draw any conclusions with regards to how the density and the fracture energy are correlated, since only one series had a larger variation in density. As a comparison, the work by Forsman et al. (2021) evaluated the fracture energy with the same method as in the present work. In comparison, the mean density of birch was evaluated to 647 kg/\(\hbox {m}^3\), which is similar to the density for all specimens in the present work (660 kg/\(\hbox {m}^3\)).

In addition to density, the moisture content (MC) was evaluated for all specimens, since previous work (Reiterer et al. 2002; Tukiainen and Hughes 2016; Forsman et al. 2021) indicates that there is a dependency between the MC and the fracture energy. The relation between MC and fracture energy from the specimens in the present work is shown in Fig. 13. The mean values of MC of the S43 and TS43-series were 14.2% and 14.5%, respectively. The previously mentioned outlier with regard to density, S4320, was also a slight outlier with regard to moisture content. It did have the lowest moisture content of all specimens in the series S43. Both the S43 and TS43-series showed a low variation in moisture content, as shown in Table 3. The B-series had a mean moisture content of 11.4% and the TB-series a mean moisture content of 12.5%. The coefficient of variation for the B-series was almost as low as for the spruce series—3.4%, whereas for the TB-series it was lower than both of the two spruce series: 1.2%.

Fig. 13
figure 13

Fracture energy in relation to the moisture content for all specimens

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The dependency between moisture content and fracture energy seems in general to be more pronounced for larger differences in moisture content. For a moisture content in common structural applications, i.e., 8–16%, the change in fracture energy seems to be relatively small, based on previous research (Tukiainen and Hughes 2016; Forsman et al. 2021; Reiterer and Stanzl-Tschegg 2002). Hence, the small difference in moisture content between species should not affect the comparison in fracture energy between species.

Effect of notch geometry and fracture area

As is evident from Fig. 14 and Table 7, the effect of the notch geometry on the evaluated fracture energy is not systematic between spruce and birch specimens. For the spruce specimens, the triangular fracture area yields an increase in the mean value of the evaluated fracture energy, whereas for the birch specimens, the triangular fracture area instead yields a reduction. A factor that could influence this is potentially the annual ring orientation.

Fig. 14
figure 14

Evaluated fracture energy for the different test series

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The annual ring orientations for the two spruce series were not perfect. For the spruce specimens with a rectangular fracture area, an orientation very similar to a perfect TL-system was acquired. However, for the spruce specimens with a triangular fracture area, the deviation from a pure TL-system was roughly 17–\(18^{\circ }\). For the birch specimens, the deviation from a pure TL-system seemed to be smaller than for the spruce specimens, and no consistent effect on the fracture energy could be identified. However, identifying the annual ring orientation was somewhat more difficult for the birch specimens.

The fracture energy in the present work was not evaluated with the nominal area, but instead with the measured fracture area, \(A_{\text {real}}\), as measured with GIMP, cf. Section 2. As seen in Table 4, \(A_{\text {real}}\), for the spruce specimens was fairly close to the nominal area. However, for the birch specimens, the deviation in mean of \(A_{\text {real}}\), compared to the nominal area, was much larger. In addition, the variation in \(A_{\text {real}}\) was higher for the birch specimens compared to the spruce specimens. This is likely an artefact from the manufacturing process. In addition, it should be noted, that the variation in \(A_{\text {real}}\) for the triangular fracture area (see Table 4), is higher for both the spruce and birch specimens, compared to the specimens with a rectangular fracture area, which might imply additional uncertainties when manufacturing the specimens with the triangular fracture area.

When manufacturing specimens with the triangular fracture area, the sawing of the triangular notch is the most crucial moment. Since two cuts are needed to create the triangular notch, several difficulties are introduced. After the first cut is made, the specimen is rotated \(90^{\circ }\) around its longitudinal axis, and cut a second time. If alignment, orientation and rotation is not executed with high precision, the second cut might not enter the specimen in the same plane as the first cut. As a consequence, multiple cracks could propagate instead of the intended single crack, possibly resulting in a greater value of the evaluated fracture energy. However, this does not seem to be a systematic error occurring in the present work; if it was, a systematic increase in evaluated fracture energy would likely have been acquired for the specimens with a triangular fracture area.

For both the spruce and birch specimens, a t-test showed a significant difference between the means of the fracture energy between the series with a triangular vs. rectangular fracture area. The p-value for the t-test between the two spruce series was \(1.06 \cdot 10^{-8}\), implying a significant difference between the two series. For the birch-series the p-value was 0.02, still implying a difference in mean between the two series.

Experimental versus numerical load displacement curves

In addition to the experimental testing, a small set of simplified numerical simulations was carried out in the present work, where the Young’s modulus in the tangential direction (\(E_{\text {T}}\)) of the middle piece was manually fitted to ensure that the elastic region of the numerical load–displacement curves matched the experimental elastic region well. \(E_{\text {T}}\) was manually fitted to 1000, 620, 350 and 200 MPa, for the B-, TB, S-, and TS-series, respectively. The numerical load–displacement curves corresponding to these values of \(E_{\text {T}}\) combined with the other mechanical parameters in Table 6 and the fracture energy in Table 7 are shown in Fig. 11. As can be seen, the numerical model is in reasonable agreement with the experimental load–displacement curves, capturing also the influence of the shape of the fracture area. To achieve a well fitting elastic region for the specimens with a triangular fracture area, the tangential Young’s modulus has to be markedly lower compared to the specimens with a rectangular fracture area. This is explained by the fact that the numerical model for the specimens with a triangular fracture area is based on the same 2d-geometry as the specimens with a rectangular fracture area. Consequently, without lowering the value of \(E_{\text {T}}\), the stiffness of the model with triangular fracture area is too high, especially close to the lower edge of the fracture area. As such, it is important to note that the simulations carried out in the present work, need further work, potentially by creating three dimensional models, or by modifying the two-dimensional models in such a way that the stiffness of the specimens with a triangular fracture area is properly accounted for.

In addition, the peak load is slightly too low for all series except for the spruce specimens with a rectangular fracture area. This implies that the tensile strength values that have been adopted from the literature might be too conservative in comparison to the tensile strengths of the tested material, both for spruce and birch.

Conclusion and outlook

In the present work, the specific fracture energy for birch and spruce was evaluated by means of the Nordtest method. In addition, the effect of different notch geometries on the evaluation of the specific fracture energy was studied. Following the previous discussion, the main conclusions of the work are:

  • The specific fracture energy was evaluated to 221 \(\text {J/m}^2\) for the spruce specimens, and to 656 \(\text {J/m}^2\) for the birch specimens, averaged over all specimens of the respective species.

  • When evaluating the stability of the tests in terms of the suggested load drop criterion, the spruce specimens seem to be markedly more unstable compared to the birch specimens.

  • An influence on evaluated fracture energy of the shape of the fracture area was found.

  • With a low variation in density for three out of four test series, it is difficult to draw any conclusions with regards to how the density and fracture energy are correlated.

  • For the specimen size adopted in the present work, the triangular fracture area was not necessary, sufficient stability could be achieved with a rectangular fracture area.

  • With what seems to be a marked increase in fracture energy for birch, compared to spruce, current design codes might need revision and rework to account for this. In addition, the increase in fracture energy can potentially indicate higher load-carrying capacity in e.g., connections, or beams with notches or holes.

Since determining the specific fracture energy of birch still is a novel area, it would be beneficial to evaluate the fracture energy of birch grown in different geographical locations, but also in different parts of a single tree, to properly assess how much the fracture energy varies. In addition, studying the effect on the fracture energy from e.g., distance to pith and annual ring orientation would be interesting.

Future numerical work could also use the results of the experimental tests to determine the softening behavior of birch, with an optimization method. This could extend the use of the SENB method to not only determine the specific fracture energy, but also to determine the softening behavior. Since the softening behavior is partly governed by the tensile strength, a calibration of such models would also yield an estimation of the tensile strength of birch in tension perpendicular to grain.

In Section “Materials and methods”, it was mentioned that the effect of plastic dissipation on the evaluation of fracture energy is negligible for sufficiently small specimens. However, this effect has previously only been evaluated for specimens with a rectangular fracture area. It would be of interest to quantify this effect in the specimens with a triangular fracture area as well, since the plastic effects might be different due to the different specimen geometry.