Examination of the failure behavior of wood with a short crack in the radial–longitudinal system by single-edge-notched bending test

Abstract

The failure behavior of wood with a short crack was examined by conducting the single-edge-notched bending tests of a radial-longitudinal system on Agathis specimens. In the test, the mode I critical stress intensity factor was measured, and its validity was checked by the result from double cantilever beam testing method. The mode I critical stress intensity factor decreased when the crack length approached zero. With crack length correction, a constant critical stress intensity factor was obtained over a wide range of crack length including crack-free specimen.

Introduction

In actual cases as timber engineering and wood drying, the crack lengths in wood are often short. In a previous study [1], the single-edge-notched bending (SENB) tests of tangential–radial (TR) system were conducted using specimens of Agathis (Agathis sp.) with varying the crack lengths that were smaller than those adopted in several existing studies involving the single-edge-notched tension or bending tests [2–8]. When considering an additional crack length ahead of the crack tip, the mode I critical stress intensity factor was effectively predicted independently from the initial crack length. In addition, the nominal bending strength could also be predicted in both cases of with and without a crack based on the concept for the additional crack length.

In practical operations such as timber engineering and drying, a crack induced in solid wood often propagates along the fiber, so it is important to investigate whether the concept described above is valid for the system including the fiber direction. In this study, the radial–longitudinal (RL) system SENB tests were conducted, and the validity of the results was examined by comparing test data with those obtained from double cantilever beam (DCB) tests.

Theory

Three-point SENB test analyses

Figure 1 shows a schematic diagram of the three-point SENB test. The specimen which had a crack of length a at its center was supported with a span of S, and the load P was applied at the mid-span. The initial crack length is defined as a. In this loading condition, the nominal bending stress σ _n is derived from the elementary beam theory as follows:

$$ \sigma_{\text{n}} = \frac{3SP}{{2BW^{2} }}, $$

(1)

where B and W are the beam width and depth, respectively. This notation is applicable to crack-free specimens. When a crack-free specimen is bent, the failure-by-bending moment is induced when σ _n reaches its critical value σ _nc, which is usually defined as the bending strength of the material.

Fig. 1

Schematic diagram of the three-point single-edge-notched bending (SENB) test

The fracture behavior of a material with a long crack should be analyzed using the mode I stress intensity factor K _I or the energy release rate G _I, each of which is derived from fracture mechanics theory. In SENB testing of specimens with a short crack, however, it is difficult to measure G _I appropriately, because the loading-line compliance does not vary with the crack length [1]. In contrast, K _I is more easily measured using an approximating equation.

The mode I stress intensity factor K _I is derived using the following equation:

$$ K_{\text{I}} = \sigma_{\text{n}} \sqrt {\pi a} f\left( \frac{a}{W} \right), $$

(2)

where f(a/W) is the crack geometry factor. The fracture is initiated when K _I reaches the critical stress intensity factor, defined as K _Ic, which is obtained by substituting σ _nc for σ _n in Eq. (2). Gross and Srawley derived a f(a/W)–a/W relationship for isotropic material as follows [9]:

$$ f\left( \frac{a}{W} \right) = 1.09 - 1.735\left( \frac{a}{W} \right) + 8.20\left( \frac{a}{W} \right)^{2} - 14.18\left( \frac{a}{W} \right)^{3} + 14.57\left( \frac{a}{W} \right)^{4} . $$

(3)

There was a concern that the orthotropy of the wood might influence the crack geometry factor. However, Eq. (3) was appropriate as the crack geometry factor for the TR system SENB specimens with the configuration similar to this study according to the virtual crack closure technique (VCCT) applied to the results obtained from the finite element analyses [1]. Moreover, the validity of Eq. (3) was verified for wood with a strong orthotropy, such as TL system fracture mechanics testing analyses [10]. Hereafter, the mode I stress intensity factor is calculated using the crack geometry factor represented by Eq. (3).

Double cantilever beam test analyses

Double cantilever beam test in the RL system (Fig. 2) was conducted to examine the validity of SENB testing results. According to beam theory and fracture mechanics, the mode I energy release rate can be obtained as follows [11]:

$$ G_{\text{I}} = \frac{{3P^{2} C_{\text{L}} }}{2B}\left( {\frac{3Hl}{4}.\frac{{C_{\text{L}} }}{{C_{\text{S}} }}} \right)^{{ - \frac{1}{3}}} $$

(4)

where C _L is the compliance obtained from the load–deflection relationship, and C _s is the compliance obtained from the load–longitudinal strain relationship, which was measured at l = a/2.

Fig. 2

Schematic diagram of double cantilever bending (DCB) test

Here, the energy release rate was converted to the stress intensity factors as follows:

$$ K_{\text{I}} = \sqrt {\frac{{E_{x} G_{\text{I}} }}{{c_{\text{I}} }}} , $$

(5)

where

$$ c_{\text{I}} = \frac{1}{\sqrt 2 }\sqrt {\frac{{E_{x} }}{{E_{y} }}} \sqrt {\sqrt {\frac{{E_{x} }}{{E_{y} }}} + \frac{1}{2}\left( {\frac{{E_{x} }}{{G_{xy} }} - 2v_{xy} } \right)} $$

(6)

in which E _x and E _y are Young’s moduli in the x and y directions, respectively, and G _xy and ν _xy are the shear modulus and Poisson’s ratio in the xy plane, respectively [12]. In this study, xy plane denotes RL plane. The elastic modulus values were obtained by compression tests and vibration tests [1], and are listed in Table 1.

Table 1 Elastic constants used for the conversion to stress intensity factor

Experimental

Materials

Agathis (Agathis sp.) lumber, with a density of 480 ± 10 kg/m³ at 12 % moisture content (MC), was used for the tests. When examining the fracture mechanics properties of the radial–longitudinal system, the influence of the annual rings is often significant [8]. Agathis is a tropical wood species with no significant growth rings; hence, the influence of annual rings could be ignored in the analysis. The lumber had no defects such as knots or grain distortions, so that the specimens cut from it could be regarded as small and clear. The lumber was stored for several months in a room at a constant temperature of 20 °C and a relative humidity of 65 % before the test and was confirmed to be in an air-dried condition. These conditions were maintained throughout the tests. The equilibrium MC was approximately 12 %.

Single-edge-notched bending tests

All of the specimens were cut from the lumber described above so that they were long-matched to the dimensions shown in Table 2. The numbers of specimens were 50, 25, and 15 for specimen A, B, and C, respectively. For the cracked specimens, a crack was produced along the radial direction in the RL system. The crack was first cut with a band saw (thickness = 1 mm), it was then extended ahead of the crack tip using a razor blade. The specimen was supported by the span shown in Table 2, and a load was applied to the specimen at a crosshead speed of 1 mm/min for the test with span lengths of 60 and 120 mm, a crosshead speed of 2 mm/min was used for the test with a span length of 240 mm. The test was conducted until the load markedly decreased.

Table 2 Specimen configurations used for the SENB tests

Figure 3 shows a typical load–loading-line deflection relation obtained by the SENB test. The load increases linearly until it is maximum, and then drops immediately without displaying a nonlinear phase. This trend is similar to that obtained from the SENB tests of the specimen with the TR system [1]. In this research, the critical load for crack propagation P _c was determined as the maximum load, as the same manner with that in the previous study [1]. By substituting P _c into Eq. (1), the nominal bending strength σ _nc corresponding to initial crack length a was obtained [1]. Then, the mode I critical stress intensity factor K _Ic corresponding to the initial crack length was obtained by substituting σ _nc and f(a/W) represented as Eq. (3) into Eq. (2).

Fig. 3

Typical example of the load–loading-line deflection relation by the SENB test

Double cantilever beam test

All of the specimens were cut from the lumber mentioned above so that they were side matched into the initial dimensions of 30 mm (radial direction) × 15 mm (tangential direction) × 315 mm (longitudinal direction). The number of specimens was six. The crack was first cut along the longitudinal direction in the longitudinal–radial plane with a band saw (thickness = 1 mm), and then it was extended 1 mm ahead of the crack tip using a razor blade so that the crack length would be as that mentioned below. After cutting the crack, the loading blocks of MDF with the dimensions of 30 mm in length, 30 mm in height, and 15 mm in thickness were bonded by epoxy resin on the upper and lower cantilever portions symmetrically as shown in Fig. 2. Crack length, which was defined as the distance from the line of load application to the crack tip, was 150 mm. Load P was applied to the specimen by pins through universal joints at a crosshead speed of 5 mm/min until the load markedly decreased. The total testing time was about 10 min.

The loading-line displacement δ was measured by the crosshead travel since it was confirmed that the machine compliance was small enough to be ignored [7], whereas the longitudinal strain ε was measured by a strain gauge (gauge length = 2 mm; FLA-2-11, Tokyo Sokki Co., Tokyo) bonded at the midpoint between the loading line and crack tip on the top cantilever portion (l = a/2). The C _L and C _S values were obtained from the linear portions of P–δ and P–ε relationships, respectively [11]. The critical load for crack propagation P _c was defined as that at the onset of nonlinearity in the load–loading-line displacement curve [7, 13, 14].

Results and discussion

Three-point SENB test

Figure 4 shows the relationship between the nominal bending strength σ _nc and initial crack length a. According to the results for metals [15, 16], the nominal bending strength of a cracked specimen will approach the bending strength of a crack-free specimen when the crack length is decreased. From the results obtained here, however, the value of σ _nc is markedly smaller than that of the crack-free specimen, and it decreases as the crack length increases. This tendency is similar to that for the results obtained using the specimens with the TR system [1], and it suggests that fracture mechanics theory is essential for analyzing the failure behavior of solid wood with a crack even when the crack length is short.

Fig. 4

Relationship between the nominal bending strength σ _nc and crack length a by the SENB test

Figure 5 shows the relationship between the mode I critical stress intensity factor K _Ic and the initial crack length a. The K _Ic value obtained by the DCB test, the details of which are mentioned below, is also shown in Fig. 5. The K _Ic value can be regarded to be a material parameter independently determined from the initial crack length. Nevertheless, the K _Ic value is not constant and it significantly increases as the initial crack length increases.

Fig. 5

Relationship between the mode I critical stress intensity factor K _Ic and crack length a

The dependence of K _Ic value on the initial crack length a can be reduced by introducing an additional crack length Δ into Eq. (2) [1, 17]:

$$ K_{\text{I}} = \sigma_{\text{n}} \sqrt {\pi \left( {a + \Updelta } \right)} f\left( {\frac{a + \Updelta }{W}} \right) = \sigma_{\text{n}} \sqrt {\pi a^{\prime}} f\left( {\frac{{a^{\prime}}}{W}} \right), $$

(7)

where a′ is defined as the corrected crack length. Table 3 shows the value of the additional crack length Δ corresponding to each specimen type. The value of Δ was determined as follows: (1) the values of K _Ic were calculated under various values of Δ in Eq. (7), (2) the standard deviations of K _Ic corresponding to each crack length were obtained, (3) the sums of the standard deviations, defined as s _D, corresponding to each Δ were obtained and compared. The value of Δ in Table 3 was determined as that from which the smallest value of s _D was derived. Figure 6 shows the relationship between the mode I critical stress intensity factor K _Ic and the corrected crack length a′. By conducting the crack length correction, the dependence of K _Ic on the crack length is effectively reduced. The concept for introducing the additional crack length is also applicable for the crack-free specimen. Although the dimensions of the specimens vary significantly, the variation of the additional crack length values is relatively small. These additional crack length values are similar to those obtained from the tests of the specimens with the TR system [1]. In a previous study, we examined the mode II fracture behavior of Agathis by asymmetric four-point bending tests of a specimen with a short crack in the TL system, which were cut from the same lumber used in this investigation [17]. The test results were analyzed with the same manner conducted here, and we found that the additional crack length values obtained in this study are shorter than that obtained from the mode II tests [17]. Table 4 shows the average values of K _Ic obtained with and without correcting the crack length.

Table 3 Additional crack length corresponding to each specimen type

Fig. 6

Relationship between the mode I critical stress intensity factor K _Ic and corrected crack length a′

Table 4 Mode I critical stress intensity factor K _Ic obtained with and without correcting the crack length

The relationships between σ _nc and a can be conventionally predicted by substituting the values of K _Ic without the crack length correction in Table 4 and f(a/W) into Eq. (2) [15]. In addition, the σ _nc–a relationships are also predicted by substituting the values of K _Ic with the crack length correction in Table 4, Δ in Table 3, and f((a + Δ)/W) into Eq. (7). This method is our proposal. Figure 7 compares the predicted and experimentally obtained σ _nc–a relationships. When the crack length is not corrected, the predicted strength increases markedly when the crack length approaches zero. When the crack length is corrected, however, the σ _nc–a relationship is effectively predicted over the full range of crack lengths, including when a = 0. This is the same result as those in the previous works [1, 17]. Therefore, the failure behavior is probably described by fracture mechanics theory even when the specimen has no crack as shown in Fig. 8 in comparison with the failure behavior of metal [15, 16].

Fig. 7

Comparison of nominal bending strengths experimentally obtained and predicted on the basis of fracture mechanics theory

Fig. 8

A relationship between strength and crack length for wood

For wood as quasi-brittle material, so-called fracture process zone influence [18–24] might not be so significant because the nonlinearity in the load–loading-line deflection relation is not significant as Fig. 3 shows. Natural cracks inherently contained in wood might be significant to the phenomenon obtained in this study. Further researches including the microscopic observation will reveal this phenomenon.

Comparison with DCB test results

The average K _Ic value by DCB test was 0.42 ± 0.022 MPa m^1/2. The average K _Ic values by SENB are 0.35 ± 0.058 MPa m^1/2 for 0.5–4 mm crack length specimens before correction and 0.39 ± 0.044 MPa m^1/2 for all specimens including crack-free specimens after additional crack length correction. The results obtained from the SENB tests approached to DCB results after the correction. By the t test between the results obtained from the DCB and SENB tests, the probability value (P value) was 0.011 before correction and the difference was significant at 5 % level. In contrast, the P value was 0.19 after correction, so the difference was not significant at 5 % level. Although the DCB specimens used here were not longitudinally matched to the SENB specimens but were only in the same lumber, the results may show the validity of introducing the additional crack length.

There have been several approaches on the application of fracture mechanics to the crack free specimens [25, 26]. Further researches are required for revealing the failure behavior of solid wood based on our proposal as well as the examinations described above.