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Mode II wood fracture characterization using the ELS test
Engineering Fracture Mechanics 74 (2007) 2133–2147 www.elsevier.com/locate/engfracmech
Mode II wood fracture characterization using the ELS test M.A.L. Silva a, J.J.L. Morais a, M.F.S.F. de Moura
b,* ,
J.L. Lousada
c
a
b
CETAV/UTAD, Departamento de Engenharias, Quinta de Prados, 5001-801 Vila Real, Portugal Faculdade de Engenharia da Universidade do Porto, Departamento de Engenharia Mecaˆnica e Gesta˜o Industrial, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal c CEGE/UTAD, Departamento de Florestal, Quinta de Prados, 5001-801 Vila Real, Portugal Received 20 June 2006; received in revised form 28 September 2006; accepted 20 October 2006 Available online 5 December 2006
Abstract This paper describes experimental and numerical studies on the application of the end loaded split test to mode II wood fracture characterization. A new data reduction scheme, based on the specimen compliance and on the equivalent crack concept, is proposed. The method presents three main advantages relatively to the classical methodologies: it does not require crack measurement during propagation; it accounts for the root rotation at the clamping point and includes the effect of the fracture process zone at the crack tip. The new procedure was numerically validated using a two-dimensional finite element analysis including a cohesive damage model, which allows the simulation of crack initiation and growth. The results demonstrated the good performance of the model and the applicability of the end loaded split test for mode II wood fracture characterization. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Wood; Mode II; End loaded split; R-Curve; Data reduction scheme
1. Introduction Recently, due to ecological and economical reasons, the applications of wood in building and in engineering works are increasing worldwide. However, these applications are still limited due to lack of adequate failure criterion. It is also known that fracture mechanics allows better mechanical rupture description relatively to classical strength of material approaches. In this context, it is fundamental to provide accurate measurements of fracture properties such as the critical strain energy release rates. The mode I wood fracture characterization has received the attention of many researchers. The double cantilever beam [1], the single edged notched loaded in three-point bending [2] and the wedge splitting [3], are the most common used tests. The mode II fracture characterization is not so well studied, which can be explained by some difficulties associated to experimental tests. In fact, problems related to unstable crack growth and to crack monitoring during propagation avoid a rigorous measurement of GIIc. There are three fundamental tests presented in the *
Corresponding author. Tel.: +351 225081727; fax:+351 225081584. E-mail address: [email protected] (M.F.S.F. de Moura).
0013-7944/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2006.10.012
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Nomenclature CBBM compliance based beam method CBT corrected beam theory CCM compliance calibration method ELS end loaded split ENF end notched flexure FPZ fracture process zone L, R, T longitudinal, radial and tangential wood directions SENF stabilised end notched flexure 4ENF four point end notched flexure SEM scanning electronic microscope a crack length aeq equivalent crack length a0 initial crack length DaFPZ crack length correction Di (i = I, II) crack length corrections C0 initial compliance C compliance ri (i = I, II) stress component on i direction ru,i (i = I, II) material strength on i direction T shear stresses Em (m = L, R, T) Young’s modulus E1 longitudinal Young’s modulus E2 transversal Young’s modulus G13 shear modulus Gm,n (m,n = L, R, T) shear modulus mm,n (m,n = L, R, T) Poisson’s ratios d vector of relative displacements di (i = I, II) current relative displacement du,i (i = I, II) maximum relative displacement do,i (i = I, II) onset relative displacement dom,i (i = I, II) onset relative displacements in mixed-mode dum,i (i = I, II) ultimate relative displacements in mixed-mode d loading displacement P applied load U strain energy Gi (i = I, II) strain energy release rates Gic (i = I, II) critical strain energy release rates Mf bending moment I second moment of area Aj cross-section area Cj half-thickness of the beam Vj transverse load L free-length of the specimen Lef effective beam length L1 specimen length B specimen width H half-height of the specimen
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literature. The most popular one is the end notched flexure (ENF), which was developed for wood fracture characterization [4]. Several other authors [5–8] suggested different approaches in order to enhance the performance of the referred test. Recently, de Moura et al. [9] and Silva et al. [10] proposed a new data reduction scheme based on specimen compliance to overcome the problems that hinder a correct measurement of GIIc of wood using the ENF. Yoshihara [11] studied the applicability of the four point end notched flexure test (4ENF) to determine the mode II R-curve of wood. The advantage of the test method proposed by this author is related to unnecessary crack measurement during propagation. Kageyama [12] proposed a stabilised end notched flexure (SENF) test for experimental characterization of mode II crack growth. A special displacement gage was developed for direct measurement of the relative shear slip between crack surfaces of the ENF specimen. The test was performed under constant crack shear displacement rate, which guarantees stable crack propagation. In the context of polymer composites, the end loaded split (ELS) test was also used [13,14] to measure the interlaminar critical strain energy release rate in mode II. Wang et al. [13] argue that this method is the most suitable for measuring the R-curve, which is justified by the longer beam length for crack extension that provides more stable fracture condition. Blackman et al. [14] applied the ELS test to mode II fracture characterization of adhesively bonded carbon–fibre-reinforced composite adherends. The authors affirm that ELS test induces more stable crack propagation and that it is more adequate for toughened adherends concerning their deformations, which must remain elastic during the test. In conclusion, although the ELS test involves more complexities during experiments relatively to the ENF test, it provides a larger range of crack length where the crack propagates stably. In fact, the ENF test requires that a0/L > 0.7 to obtain stable crack propagation [6], whereas in the ELS test a0/L > 0.55 [13] is sufficient. On the other hand, the ELS experimental setup can be considered simpler than the 4ENF and SENF tests. The objective of this work is to verify the adequacy of the ELS test to mode II fracture characterization of wood. In this context, ELS specimens were used to determine the GIIc of clear Pinus pinaster wood in the RL system, which is the most frequent crack propagation system. The numerical model is based on a two-dimensional (2D) finite element analysis considering the ABAQUSÒ software and a cohesive mixed-mode damage model implemented via user defined interface finite elements [15,16]. In order to avoid crack length measurement during propagation and to account for the effects of beam root rotation at the clamping point and for the effects of fracture process zone (FPZ) at the crack tip, a new data reduction scheme based on specimen compliance and named compliance based beam method (CBBM), is proposed. This method was applied to obtain the mode II R-curves from the experimental P–d curves. Numerical simulations of damage initiation and propagation were also performed in order to simulate the real test conditions. The application of the referred new data reduction scheme to the P–d curves obtained numerically evidenced an excellent agreement between numerical and experimental GIIc predictions.
2. Numerical analysis 2.1. Cohesive damage model In order to simulate damage initiation and propagation a cohesive mixed-mode damage model based on interface finite elements was considered. A constitutive relationship between the vectors of stresses (r) and relative displacements (d) is established (Fig. 1). The method requires the local strengths (ru,i, i = I, II) and the critical strain energies release rates (Gic) as inputted data parameters. Damage onset is predicted using a quadratic stress criterion
rI ru;I
2
þ
rII ¼ ru;II
rII ru;II
2
¼1
if rI P 0
ð1Þ
if rI 6 0
where ri, (i = I, II) represent the stresses in each mode. Crack propagation was simulated by a linear energetic criterion
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σu,i
Pure mode model Gic i = I, II
σum,i
Mixed-mode model Gi
δi δ om,i
δ um,i
δo,i
δ u,i
Fig. 1. Pure and mixed-mode damage model.
GI GII þ ¼ 1: GIc GIIc
ð2Þ
Basically, it is assumed that the area under the minor triangle of Fig. 1 represents the energy released in each mode, which is compared to the respective critical fracture energy represented by the bigger triangle. The subscripts ‘‘o’’ and ‘‘u’’ refer to onset and ultimate relative displacement, respectively, and the subscript ‘‘m’’ states for mixed-mode case. More details about the used model are presented in [9]. 2.2. Model description The ELS specimen geometry used in this work is represented in Fig. 2. The specimen axis 1, 2 and 3 coincide with the wood principal material directions L (longitudinal direction of tracheid cells), T (tangential direction to the annual rings) and R (radial direction of parenchyma cells), respectively. Hence, R is normal to the crack plane and L is the direction of crack extension, corresponding to the so called RL crack propagation system of wood. The chosen geometry (see Table 1) was based on the previous authors experience with the ENF specimen [9,10]. The initial crack length, a0 = 105 mm, was chosen in order to provide a stable crack propagation, which according to Wang et al. [13], happens when a0/L P 0.55. Table 2 presents the nominal elastic properties of the Pinus pinaster wood.
2h
3,R
1,L d
a0 L L1
B
2,T
1,L Fig. 2. The ELS specimen geometry.
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Table 1 Nominal specimen geometry L1 (mm)
L (mm)
2h (mm)
d (mm)
B (mm)
a0 (mm)
230
175
20
25
20
105
Table 2 Elastic properties of the Pinus pinaster wood [9] EL (GPa)
ER (GPa)
ET (GPa)
tLR
tTL
tRT
GLR (GPa)
GTL (GPa)
GRT (GPa)
15.13
1.91
1.01
0.47
0.51
0.59
1.12
1.04
0.17
A preliminary 3D numerical analysis was done using the commercial code ABAQUSÒ considering 35,250 three-dimensional 8-node brick elements and 4890 interface finite elements (Fig. 3). The Virtual Crack Closure Technique was used to verify the Gi (i = I, II, III) distributions at the crack front. Fig. 4 shows the clear predominance of mode II along the specimen width. In fact, the average of mode II component is above of 99.5% of the total strain energy released rate. A geometrically non-linear crack growth simulation was subsequently performed. It was verified that the crack front is linear and no difference was detected between crack length at the edges and at the centre of the specimen, which validates a 2D approach.
Fig. 3. Mesh of the three-dimensional finite element model.
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G II/(GI+GII+GIII) (%)
100.5 100.0 99.5 99.0 98.5 0.0
0.2
0.4
0.6
0.8
1.0
y /B Fig. 4. Distribution of GII along of the specimen width.
Fig. 5. Mesh of the two-dimensional finite element model.
The 2D numerical model includes 1211 8-node isoparametric plane stress elements and 130 compatible sixnode interface finite elements along the crack path (see detail 2 in Fig. 5). At the initial crack, ‘‘opened’’ interface finite elements were considered in order to allow free relative sliding of the crack surfaces (friction effects were neglected [10]) and prevent interpenetration of the specimen arms (see detail 1 in Fig. 5). Contact conditions were imposed between the actuator, considered as a rigid body, and the specimen. A geometrically non-linear analysis considering small increment values (0.5% of the applied displacement) was performed to obtain smooth crack propagation. 3. Data reduction schemes 3.1. Classical methods There are two main classical methodologies proposed in the literature to obtain the GIIc value from the ELS test: the compliance calibration method (CCM) and the corrected beam theory (CBT). The CCM is based on the Irwin–Kies equation GIIc ¼
P 2 dC : 2B da
ð3Þ
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Assuming a cubic relationship between the compliance (C) and the measured crack length a [17], C ¼ D þ ma3 ;
ð4Þ
where D and m are constants. GIIc can now be obtained from GIIc ¼
3P 2 ma2 : 2B
ð5Þ
The CBT proposed by Wang et al. [17] leads to, 2
GIIc ¼
9P 2 ða þ jDII jÞ ; 4B2 h3 E1
ð6Þ
where DII is a crack length correction to account for the transverse shear and beam rotation effects. Following a finite element calibration procedure, Wang et al. [18] found that DII = 0.49DI, DI being the correction for mode I obtained for the double cantilever beam test vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " u 2 # u E1 C t ; ð7Þ 32 DI ¼ h 1þC 11G13 where pffiffiffiffiffiffiffiffiffiffi E1 E2 C ¼ 1:18 : G13
ð8Þ
Both of these methods are dependent on the crack length measurements during propagation which are very difficult to carry out experimentally. In fact, in the mode II fracture characterization tests, the crack tends to close which hinders a clear visualization of crack tip (see Fig. 6). 3.2. Compliance based beam method The strain energy of the specimen due to bending and shear effects is Z LZ h 2 Z L M 2f s dx þ B dy dx; U¼ 2E I 2G 1 13 0 0 h
ð9Þ
where Mf is the bending moment, I the second moment of area and
Fig. 6. Microscope image of a crack tip during the propagation process; detail 1: initial crack length; detail 2: undefined crack tip.
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3 Vj s¼ 2 Aj
! y2 1 2 ; cj
ð10Þ
where Aj, cj and Vj represent, respectively, the cross-section area, the half-thickness of the beam and the transverse load of the j segment (0 6 x 6 a and a 6 x 6 L). From Castigliano theorem, the displacement at the loading point for a crack length a is dU P 3a3 þ L3 3PL d¼ ¼ þ : ð11Þ 3 dP 5BhG13 2Bh E1 In order to include the root rotation effects at clamping and several effects not incorporated in the beam theory, e.g., crack root rotation, contact stresses and friction effects between specimen arms at the pre-crack and stress concentration at the crack tip, an effective beam length (Lef) can be achieved. In fact, from Eq. (11), considering the initial crack length (a0) and the initial compliance (C0) measured during the experimental tests, it can be written L3ef 3Lef 3a30 þ þ C 0 ¼ 0: 3 2Bh E1 5BhG13 2Bh3 E1
ð12Þ
The solution of this equation is presented in Appendix A The numerical simulations were performed considering a perfect clamping and this effective length. Previous studies considering the ENF test [9] showed that the FPZ has a non negligible influence on the measured GIIc. To take account for its influence a correction of the real crack length (DaFPZ) should be considered. From Eq. (11) the compliance (C) during crack propagation can be expressed as C¼
3ða þ DaFPZ Þ3 þ L3ef 3Lef þ : 5BhG13 2Bh3 E1
Combining Eqs. (13) and (12) we obtain the equivalent crack length,
1=3 2Bh3 E1 aeq ¼ a þ DaFPZ ¼ ðC C 0 Þ þ a30 : 3 Substituting, in Eq. (6), a + DII by the result of Eq. (14),GIIc can be obtained from
2=3 9P 2 2Bh3 E1 3 þ a0 GIIc ¼ 2 3 ðC C 0 Þ : 3 4B h E1
ð13Þ
ð14Þ
ð15Þ
This method allows the attainment of GIIc without crack measurement during propagation which can be considered an important advantage. In fact, Eq. (15) only depends on applied load and displacement during crack growth. Additionally, the influences of root rotation at the clamping point and the effect of FPZ on the compliance are both included by using this methodology. It can also be observed that GIIc depends on the Young’s modulus (E1 = EL), which can vary between different specimens. Consequently, it is necessary to provide a previous measurement of the longitudinal modulus (EL) for each specimen. 4. Experimental tests Nineteen specimens of Pinus pinaster wood with 12% moisture content and average oven dry specific density equal to 0.57 were machined. The nominal specimen geometry is presented in Table 1. Three-point bending tests were executed to measure the longitudinal modulus. These tests were performed on specimens with a length of 500 mm and a span equal to 460 mm. Subsequently, these specimens were cut and used to carry out the ELS tests. The initial crack was introduced in two steps, just before the fracture test. First, a starter notch was machined using a band saw with 1 mm thickness. Then, the crack was extended between 2 and 5 mm, by applying a low impact load on a cutting blade. The initial crack length was measured with a resolution of
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0.001 mm using an optical microscope. The measurement was performed before the fracture tests and at the both sides of the specimen in order to minimize the misalignments caused during execution. The mechanical tests were conducted in a screw-driven universal testing machine (Instron 1125). The ELS test fixture included a linear guidance system allowing horizontal movement of translation of the clamping grip during loading (see Fig. 7). The tests were conducted in a displacement control mode with a crosshead speed of 5 mm/min. Two sheets of Teflon film were inserted between the initial crack surfaces in order to minimize the friction effects. During the tests, the applied load and displacement values were recorded (frequency 5 Hz). Fracture surfaces were observed using a scanning electronic microscope (SEM) Philips-FEI Quanta 400. Fig. 8 is a photomicrograph of the initial crack surface, where three different zones are visible. Region A is
Fig. 7. The end loaded split (ELS) test fixture.
Fig. 8. Initial crack sections: (A) sawn section; (B) sliced section; (C) Mode I pre-cracked section.
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the tip of the starter notch machined with the band saw, whereas region B is the portion of the initial crack cut by the blade. Zone C of the initial crack corresponds to the mode I fracture surface created ahead of cutting blade, due to its penetration into wood under an impact loading. Indeed, this section exhibits the characteristic appearance of fracture in perpendicular-to-grain tension, through the middle lamella of tracheids [19]. The transition between the surface of mode I pre-crack and the surface of mode II (parallel to grain shear loading) crack can be clearly seen on Fig. 9. The mode II crack surface is characterized by severe twisting, tearing, and unwinding of tracheid walls [19]. However, a significant difference exists between the morphology of fracture surfaces of earlywood and latewood, due to the greater thickness of the tracheid walls of latewood (Fig. 10). These findings in SEM investigations allows fracture surfaces mode II characterization and gives additional confidence on the application of ELS test to determine the mode II fracture resistance of wood in the RL fracture system.
Fig. 9. Transition between Mode I pre-crack and Mode II crack.
Fig. 10. Fracture surface obtained by Mode II: (a) latewood and (b) earlywood.
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5. Results Fig. 11 presents the experimental P–d curves for all tested specimens. Some scatter on the initial specimen stiffness show the necessity of previous specimen modulus measurement. The R-curves of all tested specimens (Fig. 12) were determined from the experimental P–d data and using the CBBM (Eq. (15)). The values of mode II energy release rates at initiation (GIIini) and during stable crack propagation (GIIc) were obtained. The GIIini values were determined using the load and deflection measured at the point of deviation from linearity in the P–d curve (Fig. 13). These initiation values correspond to the onset of FPZ development at the pre-crack tip. The critical strain energy release rates (GIIc) were obtained from the plateau of the R-curves (Fig. 14) and characterize the self-similar stable crack propagation process. Table 3 summarises the experimental results for all specimens. Standard deviations of approximately 31.8% and 16.8% were obtained for GIIini and GIIc, respectively, which can be considered acceptable in wood. The average value for GIIc (0.939 N/mm) is similar to the one obtained by the ENF test in a previous study (0.909 N/mm) for matched specimens obtained from the same trunk [20]. In order to validate the CBBM, numerical simulations based on the proposed cohesive damage model were performed for each specimen considering their measured dimensions of cross section, initial crack length, longitudinal modulus and GIIc . The effective length (Lef) of each specimen, given by Eq. (12), was used in the 500 450 400
P (N)
350 300 250 200 150 100 50 0 0
5
10 δ (mm)
15
20
Fig. 11. The ELS experimental P–d curves.
1.4 1.2
G II (N/mm)
1 0.8 0.6 0.4 0.2 0 105
115
125
135
145
155
a (mm) Fig. 12. The ELS experimental R-curves.
165
175
185
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500
400
GIIc
300
P (N)
G II ini
200
Experimental curve
100
Numerical curve 0 0
2
4
6
8
δ (mm)
10
12
14
Fig. 13. Comparison between the numerical and experimental P–d curves (specimen 13). 1.25
1
G II (N/mm)
GIIc
0.75
Experimental curve Numerical curve
0.5
GII ini 0.25
0 110
115
120
125
130
135
140
145
150
a (mm) Fig. 14. Comparison between the numerical and experimental R-curves (specimen 13).
numerical model instead of the real length of the beam outside the clamping fixture (dimension L in Fig. 2). This allows the consideration of a perfect clamping condition on the numerical model. The numerical P–d results were used to obtain the mode II R-curve considering the CBBM. Figs. 13 and 14 show an excellent agreement between numerical and experimental referred curves for a given specimen. As it can be seen in Table 3, globally, an excellent agreement between the numerical and experimental results was obtained. An important aspect concerning the numerical simulations is related to the influence of the limit stress ru,II, which in this case corresponds to su,LR, on the maximum load. This property was not experimentally measured and some variability can be expected in wood. To overcome this difficulty, fine–tune iterative process was performed for each specimen in order to minimize the difference between the numerical and experimental peak load values. This methodology originated a mean value of 9.27 MPa for su,LR with a standard deviation of 2.82 MPa.
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Table 3 Synopsis of the experimental and numerical results for all ELS specimens tested Specimen
Experimental results
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
387.84 338.28 394.92 344.76 449.76 349.44 362.16 349.56 390.72 342.24 350.52 352.20 387.96 292.92 381.24 394.44 368.52 329.64 349.32
0.359 0.184 0.108 0.429 0.332 0.278 0.335 0.417 0.385 0.372 0.441 0.510 0.309 0.193 0.194 0.252 0.333 0.350 0.281
1.041 0.763 1.080 0.822 1.148 1.163 1.108 0.936 1.150 0.855 0.815 0.911 0.922 0.733 1.092 1.031 0.843 0.754 0.680
390.09 337.29 392.17 337.94 461.05 354.60 354.57 348.12 384.35 344.39 354.09 352.21 389.11 298.41 385.27 391.66 361.67 331.22 342.63
0.366 0.184 0.106 0.449 0.335 0.267 0.341 0.420 0.412 0.376 0.459 0.502 0.313 0.200 0.197 0.252 0.333 0.372 0.279
1.030 0.745 1.081 0.813 1.142 1.151 1.090 0.918 1.140 0.846 0.810 0.904 0.913 0.734 1.084 1.025 0.849 0.747 0.669
Average
364.023
0.319
0.939
363.728
0.324
0.931
9.182
31.779
16.847
9.488
32.366
16.905
P max (N)
Std. dev. (%)
GIIini (N/mm)
Numerical results GIIc (N/mm)
P max (N)
GIIini (N/mm)
Comparison numerical/ experimental GIIc (N/mm)
P max (%)
GIIini (%)
GIIc (%)
0.579 0.293 0.696 1.980 2.511 1.475 2.095 0.412 1.631 0.629 1.019 0.004 0.296 1.873 1.056 0.706 1.859 0.480 1.915
1.845 0.101 2.269 4.397 0.822 4.151 1.740 0.858 6.365 1.187 4.025 1.643 1.255 3.798 1.431 0.022 0.002 3.346 0.565
1.046 2.302 0.106 1.044 0.526 1.007 1.657 1.978 0.905 1.079 0.577 0.675 0.976 0.186 0.778 0.655 0.656 0.885 1.687
Table 4 Alteration of GIIc (%) for a variation of 1% in each parameter a0
B
h
E1
C0
C
0.931
0.495
1.392
0.469
1.155
1.473
Table 5 Errors obtained by the three different methods between numerical and experimental results for GIIc Experimental results
Comparison numerical/experimental
Specimen
GIIc (N/mm)
CBBM, error (%)
CCM, error (%)
CBT, error (%)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1.041 0.763 1.080 0.822 1.148 1.163 1.108 0.936 1.150 0.855 0.815 0.911 0.922 0.733 1.092 1.031 0.843 0.754 0.680
1.05 2.30 0.11 2.13 0.53 1.01 1.66 0.18 0.91 1.08 0.58 0.67 1.19 0.19 0.78 0.65 0.66 0.89 1.69
0.25 1.63 0.52 2.38 0.15 3.64 3.07 0.24 2.37 1.25 1.55 1.63 0.49 0.05 1.57 1.64 1.12 3.45 2.22
8.34 6.69 6.79 12.43 9.92 14.68 14.53 13.64 6.04 18.34 12.26 16.19 15.31 12.09 6.70 6.04 17.94 17.61 14.63
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The sensibility of GIIc relatively to the parameters included in Eq. (15) was also verified. Table 4 presents the alteration in GIIc for a variation of 1% in each parameter. It can be observed that compliances, beam height and initial crack length are the most influent parameters and they should be carefully measured. The performance of the CBBM was also numerically evaluated comparing the obtained results with the ones given by the CCM and CBT (Eqs. (5) and (6), respectively). Both of these methods depend on rigorous crack monitoring during propagation, which is very difficult to perform experimentally. However, numerically it is easy to accomplish crack evolution owing to the cohesive damage model. Table 5 presents the errors obtained from the three methods for all tested specimens. The CBBM and CCM shows good agreement relatively to the inputted values and the CBBM presents the advantage of being independent of crack monitoring during propagation. 6. Conclusions A new data reduction scheme based on the specimen compliance is proposed for the ELS test. The method does not require crack length measurement during the experimental tests which was verified to be very difficult to perform accurately in these tests. On the other hand, the proposed methodology accounts for the effects of beam root rotation at the clamping point and for the FPZ size at the crack extremity. The method was applied to the experimental results to obtain the mode II energy release rates at initiation and during stable crack propagation. The experimental tests were individually simulated considering a two-dimensional analysis based on a cohesive damage model, in order to numerically validate the suggested data reduction scheme. Excellent agreement was obtained between simulation and experimental results which confirms the validity of the proposed method and the applicability of the ELS test to fracture characterization of wood. Acknowledgements The authors thank Professor Alfredo B. de Morais of University of Aveiro, Portugal, for his valuable advices and discussion about the subject. The authors would also to thank the Portuguese Foundation for Science and Technology (FCT, Research Project POCTI/EME/45573/2002) for supporting the work here presented. Appendix A Eq. (12) can be expressed as, ax3 þ bx þ c ¼ 0;
ðA1Þ
where the coefficients a, b and c are, respectively a¼
1 ; 2Bh3 E1
b¼
3 ; 5BhG13
c¼
3a30 C0: 2Bh3 E1
ðA2Þ
Using the MatlabÒ software and only keeping the real solution we have, x¼
1 2b A ; 6a A
ðA3Þ
being 00
1 113 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ffi 2a 4b þ 27c A a2 A : A ¼ @@108c þ 12 3 a
ðA4Þ
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References [1] Yoshihara H, Kawamura T. Mode I fracture toughness estimation of wood by DCB test. Compos Part A: Appl Sci Manufact 2006;37:2105–13. [2] Morel S, Dourado N, Valentin G, Morais J. Wood: a quasibrittle material R-curve behavior and peak load evaluation. Int J Fract 2005;131:385–400. [3] Reiterer A, Burgert I, Sinn G, Tschegg S. The radial reinforcement of the wood structure and its implication on mechanical and fracture mechanical properties – a comparison between two tree species. J Mater Sci 2002;37:935–40. [4] Barrett JD, Foschi RO. Mode II stress-intensity factors for cracked wood beams. Engng Fract Mech 1977;9:371–8. [5] Russell AJ, Street KN. Moisture and temperature effects on the mixed-mode delamination fracture of unidirectional graphite/epoxy. ASTM STP 1985;876:349–70. [6] Carlsson LA, Gillespie JW, Pipes RB. On the analysis and design of the end notched flexure (ENF) specimen for mode II testing. J Compos Mater 1986;20:594–604. [7] Whitney JM, Gillespie JW, Carlsson LA. Singularity approach to the analysis of the end notched flexure specimen. Proceedings of the American society for composites 2nd technical conference. Lancaster; 1987. p. 391–8. [8] Wang J, Qiao P. Novel beam analysis of the end notched flexure specimen for mode II fracture. Engng Fract Mech 2004;71:219–31. [9] de Moura MFSF, Silva MAL, de Morais AB, Morais JJL. Equivalent crack based mode II fracture characterization of wood. Engng Fract Mech 2006;73:978–93. [10] Silva MAL, de Moura MFSF, Morais JJL. Numerical analysis of the ENF test for mode II wood fracture. Compos Part A: Appl Sci Manufact 2006;37:1334–44. [11] Yoshihara H. Mode II R-curve of wood measured by 4-ENF test. Engng Fract Mech 2004;71:2065–77. [12] Kageyama K, Kikuchi M, Yanagisawa N. Stabilized end notched flexure test: characterization of mode II interlaminar crack growth. In: O’Brien TK, editor. Compos Mater Fatigue Fract. ASTM STP 1110, Vol. 3. Philadelphia: American Society for Testing and Materials; 1991. p. 210–25. [13] Wang H, Vu-Khanh T. Use of end-loaded-split (ELS) test to study stable fracture behaviour of composites under mode II loading. Compos Struct 1996;36:71–9. [14] Blackman BRK, Kinloch AJ, Paraschi M. The determination of the mode II adhesive fracture resistance, GIIc, of structural adhesive joints: an effective crack length approach. Engng Fract Mech 2005;72:877–97. [15] de Moura MFSF, Gonc¸alves JPM, Marques AT, Castro PMST. Modeling compression failure after low velocity impact on laminated composites using interface elements. J Compos Mater 1997;31:1462–79. [16] Gonc¸alves JPM, de Moura MFSF, Castro PMST, Marques AT. Interface element including point-to-surface constraints for three dimensional problems with damage propagation. Engng Comput 2000;17:28–47. [17] Davies P, Blackman BRK, Brunner AJ. Mode II delamination. In: Moore DR, Pavan A, Williams JG, editors. Fracture mechanics testing methods for polymers adhesives and composites. Amsterdam, London, New York: Elsevier; 2001. p. 307–34. [18] Wang Y, Williams JG. Corrections for mode II fracture toughness specimens of composite materials. Compos Sci Technol 1992;43:251–6. [19] Zink AG, Pellicane J, Shuler CE. Ultrastructural analysis of softwood fracture surfaces. Wood Sci Technol 1994;28:329–38. [20] de Moura MFSF, Silva MAL, Morais JJL, de Morais AB, Lousada JL. A new methodology for the characterization of the mode II fracture of Pinus pinaster wood. Mech Adv Mater Struct, submitted for publication.
Mode II wood fracture characterization using the ELS test M.A.L. Silva a, J.J.L. Morais a, M.F.S.F. de Moura
b,* ,
J.L. Lousada
c
a
b
CETAV/UTAD, Departamento de Engenharias, Quinta de Prados, 5001-801 Vila Real, Portugal Faculdade de Engenharia da Universidade do Porto, Departamento de Engenharia Mecaˆnica e Gesta˜o Industrial, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal c CEGE/UTAD, Departamento de Florestal, Quinta de Prados, 5001-801 Vila Real, Portugal Received 20 June 2006; received in revised form 28 September 2006; accepted 20 October 2006 Available online 5 December 2006
Abstract This paper describes experimental and numerical studies on the application of the end loaded split test to mode II wood fracture characterization. A new data reduction scheme, based on the specimen compliance and on the equivalent crack concept, is proposed. The method presents three main advantages relatively to the classical methodologies: it does not require crack measurement during propagation; it accounts for the root rotation at the clamping point and includes the effect of the fracture process zone at the crack tip. The new procedure was numerically validated using a two-dimensional finite element analysis including a cohesive damage model, which allows the simulation of crack initiation and growth. The results demonstrated the good performance of the model and the applicability of the end loaded split test for mode II wood fracture characterization. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Wood; Mode II; End loaded split; R-Curve; Data reduction scheme
1. Introduction Recently, due to ecological and economical reasons, the applications of wood in building and in engineering works are increasing worldwide. However, these applications are still limited due to lack of adequate failure criterion. It is also known that fracture mechanics allows better mechanical rupture description relatively to classical strength of material approaches. In this context, it is fundamental to provide accurate measurements of fracture properties such as the critical strain energy release rates. The mode I wood fracture characterization has received the attention of many researchers. The double cantilever beam [1], the single edged notched loaded in three-point bending [2] and the wedge splitting [3], are the most common used tests. The mode II fracture characterization is not so well studied, which can be explained by some difficulties associated to experimental tests. In fact, problems related to unstable crack growth and to crack monitoring during propagation avoid a rigorous measurement of GIIc. There are three fundamental tests presented in the *
Corresponding author. Tel.: +351 225081727; fax:+351 225081584. E-mail address: [email protected] (M.F.S.F. de Moura).
0013-7944/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2006.10.012
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Nomenclature CBBM compliance based beam method CBT corrected beam theory CCM compliance calibration method ELS end loaded split ENF end notched flexure FPZ fracture process zone L, R, T longitudinal, radial and tangential wood directions SENF stabilised end notched flexure 4ENF four point end notched flexure SEM scanning electronic microscope a crack length aeq equivalent crack length a0 initial crack length DaFPZ crack length correction Di (i = I, II) crack length corrections C0 initial compliance C compliance ri (i = I, II) stress component on i direction ru,i (i = I, II) material strength on i direction T shear stresses Em (m = L, R, T) Young’s modulus E1 longitudinal Young’s modulus E2 transversal Young’s modulus G13 shear modulus Gm,n (m,n = L, R, T) shear modulus mm,n (m,n = L, R, T) Poisson’s ratios d vector of relative displacements di (i = I, II) current relative displacement du,i (i = I, II) maximum relative displacement do,i (i = I, II) onset relative displacement dom,i (i = I, II) onset relative displacements in mixed-mode dum,i (i = I, II) ultimate relative displacements in mixed-mode d loading displacement P applied load U strain energy Gi (i = I, II) strain energy release rates Gic (i = I, II) critical strain energy release rates Mf bending moment I second moment of area Aj cross-section area Cj half-thickness of the beam Vj transverse load L free-length of the specimen Lef effective beam length L1 specimen length B specimen width H half-height of the specimen
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literature. The most popular one is the end notched flexure (ENF), which was developed for wood fracture characterization [4]. Several other authors [5–8] suggested different approaches in order to enhance the performance of the referred test. Recently, de Moura et al. [9] and Silva et al. [10] proposed a new data reduction scheme based on specimen compliance to overcome the problems that hinder a correct measurement of GIIc of wood using the ENF. Yoshihara [11] studied the applicability of the four point end notched flexure test (4ENF) to determine the mode II R-curve of wood. The advantage of the test method proposed by this author is related to unnecessary crack measurement during propagation. Kageyama [12] proposed a stabilised end notched flexure (SENF) test for experimental characterization of mode II crack growth. A special displacement gage was developed for direct measurement of the relative shear slip between crack surfaces of the ENF specimen. The test was performed under constant crack shear displacement rate, which guarantees stable crack propagation. In the context of polymer composites, the end loaded split (ELS) test was also used [13,14] to measure the interlaminar critical strain energy release rate in mode II. Wang et al. [13] argue that this method is the most suitable for measuring the R-curve, which is justified by the longer beam length for crack extension that provides more stable fracture condition. Blackman et al. [14] applied the ELS test to mode II fracture characterization of adhesively bonded carbon–fibre-reinforced composite adherends. The authors affirm that ELS test induces more stable crack propagation and that it is more adequate for toughened adherends concerning their deformations, which must remain elastic during the test. In conclusion, although the ELS test involves more complexities during experiments relatively to the ENF test, it provides a larger range of crack length where the crack propagates stably. In fact, the ENF test requires that a0/L > 0.7 to obtain stable crack propagation [6], whereas in the ELS test a0/L > 0.55 [13] is sufficient. On the other hand, the ELS experimental setup can be considered simpler than the 4ENF and SENF tests. The objective of this work is to verify the adequacy of the ELS test to mode II fracture characterization of wood. In this context, ELS specimens were used to determine the GIIc of clear Pinus pinaster wood in the RL system, which is the most frequent crack propagation system. The numerical model is based on a two-dimensional (2D) finite element analysis considering the ABAQUSÒ software and a cohesive mixed-mode damage model implemented via user defined interface finite elements [15,16]. In order to avoid crack length measurement during propagation and to account for the effects of beam root rotation at the clamping point and for the effects of fracture process zone (FPZ) at the crack tip, a new data reduction scheme based on specimen compliance and named compliance based beam method (CBBM), is proposed. This method was applied to obtain the mode II R-curves from the experimental P–d curves. Numerical simulations of damage initiation and propagation were also performed in order to simulate the real test conditions. The application of the referred new data reduction scheme to the P–d curves obtained numerically evidenced an excellent agreement between numerical and experimental GIIc predictions.
2. Numerical analysis 2.1. Cohesive damage model In order to simulate damage initiation and propagation a cohesive mixed-mode damage model based on interface finite elements was considered. A constitutive relationship between the vectors of stresses (r) and relative displacements (d) is established (Fig. 1). The method requires the local strengths (ru,i, i = I, II) and the critical strain energies release rates (Gic) as inputted data parameters. Damage onset is predicted using a quadratic stress criterion
rI ru;I
2
þ
rII ¼ ru;II
rII ru;II
2
¼1
if rI P 0
ð1Þ
if rI 6 0
where ri, (i = I, II) represent the stresses in each mode. Crack propagation was simulated by a linear energetic criterion
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σu,i
Pure mode model Gic i = I, II
σum,i
Mixed-mode model Gi
δi δ om,i
δ um,i
δo,i
δ u,i
Fig. 1. Pure and mixed-mode damage model.
GI GII þ ¼ 1: GIc GIIc
ð2Þ
Basically, it is assumed that the area under the minor triangle of Fig. 1 represents the energy released in each mode, which is compared to the respective critical fracture energy represented by the bigger triangle. The subscripts ‘‘o’’ and ‘‘u’’ refer to onset and ultimate relative displacement, respectively, and the subscript ‘‘m’’ states for mixed-mode case. More details about the used model are presented in [9]. 2.2. Model description The ELS specimen geometry used in this work is represented in Fig. 2. The specimen axis 1, 2 and 3 coincide with the wood principal material directions L (longitudinal direction of tracheid cells), T (tangential direction to the annual rings) and R (radial direction of parenchyma cells), respectively. Hence, R is normal to the crack plane and L is the direction of crack extension, corresponding to the so called RL crack propagation system of wood. The chosen geometry (see Table 1) was based on the previous authors experience with the ENF specimen [9,10]. The initial crack length, a0 = 105 mm, was chosen in order to provide a stable crack propagation, which according to Wang et al. [13], happens when a0/L P 0.55. Table 2 presents the nominal elastic properties of the Pinus pinaster wood.
2h
3,R
1,L d
a0 L L1
B
2,T
1,L Fig. 2. The ELS specimen geometry.
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Table 1 Nominal specimen geometry L1 (mm)
L (mm)
2h (mm)
d (mm)
B (mm)
a0 (mm)
230
175
20
25
20
105
Table 2 Elastic properties of the Pinus pinaster wood [9] EL (GPa)
ER (GPa)
ET (GPa)
tLR
tTL
tRT
GLR (GPa)
GTL (GPa)
GRT (GPa)
15.13
1.91
1.01
0.47
0.51
0.59
1.12
1.04
0.17
A preliminary 3D numerical analysis was done using the commercial code ABAQUSÒ considering 35,250 three-dimensional 8-node brick elements and 4890 interface finite elements (Fig. 3). The Virtual Crack Closure Technique was used to verify the Gi (i = I, II, III) distributions at the crack front. Fig. 4 shows the clear predominance of mode II along the specimen width. In fact, the average of mode II component is above of 99.5% of the total strain energy released rate. A geometrically non-linear crack growth simulation was subsequently performed. It was verified that the crack front is linear and no difference was detected between crack length at the edges and at the centre of the specimen, which validates a 2D approach.
Fig. 3. Mesh of the three-dimensional finite element model.
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G II/(GI+GII+GIII) (%)
100.5 100.0 99.5 99.0 98.5 0.0
0.2
0.4
0.6
0.8
1.0
y /B Fig. 4. Distribution of GII along of the specimen width.
Fig. 5. Mesh of the two-dimensional finite element model.
The 2D numerical model includes 1211 8-node isoparametric plane stress elements and 130 compatible sixnode interface finite elements along the crack path (see detail 2 in Fig. 5). At the initial crack, ‘‘opened’’ interface finite elements were considered in order to allow free relative sliding of the crack surfaces (friction effects were neglected [10]) and prevent interpenetration of the specimen arms (see detail 1 in Fig. 5). Contact conditions were imposed between the actuator, considered as a rigid body, and the specimen. A geometrically non-linear analysis considering small increment values (0.5% of the applied displacement) was performed to obtain smooth crack propagation. 3. Data reduction schemes 3.1. Classical methods There are two main classical methodologies proposed in the literature to obtain the GIIc value from the ELS test: the compliance calibration method (CCM) and the corrected beam theory (CBT). The CCM is based on the Irwin–Kies equation GIIc ¼
P 2 dC : 2B da
ð3Þ
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Assuming a cubic relationship between the compliance (C) and the measured crack length a [17], C ¼ D þ ma3 ;
ð4Þ
where D and m are constants. GIIc can now be obtained from GIIc ¼
3P 2 ma2 : 2B
ð5Þ
The CBT proposed by Wang et al. [17] leads to, 2
GIIc ¼
9P 2 ða þ jDII jÞ ; 4B2 h3 E1
ð6Þ
where DII is a crack length correction to account for the transverse shear and beam rotation effects. Following a finite element calibration procedure, Wang et al. [18] found that DII = 0.49DI, DI being the correction for mode I obtained for the double cantilever beam test vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " u 2 # u E1 C t ; ð7Þ 32 DI ¼ h 1þC 11G13 where pffiffiffiffiffiffiffiffiffiffi E1 E2 C ¼ 1:18 : G13
ð8Þ
Both of these methods are dependent on the crack length measurements during propagation which are very difficult to carry out experimentally. In fact, in the mode II fracture characterization tests, the crack tends to close which hinders a clear visualization of crack tip (see Fig. 6). 3.2. Compliance based beam method The strain energy of the specimen due to bending and shear effects is Z LZ h 2 Z L M 2f s dx þ B dy dx; U¼ 2E I 2G 1 13 0 0 h
ð9Þ
where Mf is the bending moment, I the second moment of area and
Fig. 6. Microscope image of a crack tip during the propagation process; detail 1: initial crack length; detail 2: undefined crack tip.
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3 Vj s¼ 2 Aj
! y2 1 2 ; cj
ð10Þ
where Aj, cj and Vj represent, respectively, the cross-section area, the half-thickness of the beam and the transverse load of the j segment (0 6 x 6 a and a 6 x 6 L). From Castigliano theorem, the displacement at the loading point for a crack length a is dU P 3a3 þ L3 3PL d¼ ¼ þ : ð11Þ 3 dP 5BhG13 2Bh E1 In order to include the root rotation effects at clamping and several effects not incorporated in the beam theory, e.g., crack root rotation, contact stresses and friction effects between specimen arms at the pre-crack and stress concentration at the crack tip, an effective beam length (Lef) can be achieved. In fact, from Eq. (11), considering the initial crack length (a0) and the initial compliance (C0) measured during the experimental tests, it can be written L3ef 3Lef 3a30 þ þ C 0 ¼ 0: 3 2Bh E1 5BhG13 2Bh3 E1
ð12Þ
The solution of this equation is presented in Appendix A The numerical simulations were performed considering a perfect clamping and this effective length. Previous studies considering the ENF test [9] showed that the FPZ has a non negligible influence on the measured GIIc. To take account for its influence a correction of the real crack length (DaFPZ) should be considered. From Eq. (11) the compliance (C) during crack propagation can be expressed as C¼
3ða þ DaFPZ Þ3 þ L3ef 3Lef þ : 5BhG13 2Bh3 E1
Combining Eqs. (13) and (12) we obtain the equivalent crack length,
1=3 2Bh3 E1 aeq ¼ a þ DaFPZ ¼ ðC C 0 Þ þ a30 : 3 Substituting, in Eq. (6), a + DII by the result of Eq. (14),GIIc can be obtained from
2=3 9P 2 2Bh3 E1 3 þ a0 GIIc ¼ 2 3 ðC C 0 Þ : 3 4B h E1
ð13Þ
ð14Þ
ð15Þ
This method allows the attainment of GIIc without crack measurement during propagation which can be considered an important advantage. In fact, Eq. (15) only depends on applied load and displacement during crack growth. Additionally, the influences of root rotation at the clamping point and the effect of FPZ on the compliance are both included by using this methodology. It can also be observed that GIIc depends on the Young’s modulus (E1 = EL), which can vary between different specimens. Consequently, it is necessary to provide a previous measurement of the longitudinal modulus (EL) for each specimen. 4. Experimental tests Nineteen specimens of Pinus pinaster wood with 12% moisture content and average oven dry specific density equal to 0.57 were machined. The nominal specimen geometry is presented in Table 1. Three-point bending tests were executed to measure the longitudinal modulus. These tests were performed on specimens with a length of 500 mm and a span equal to 460 mm. Subsequently, these specimens were cut and used to carry out the ELS tests. The initial crack was introduced in two steps, just before the fracture test. First, a starter notch was machined using a band saw with 1 mm thickness. Then, the crack was extended between 2 and 5 mm, by applying a low impact load on a cutting blade. The initial crack length was measured with a resolution of
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0.001 mm using an optical microscope. The measurement was performed before the fracture tests and at the both sides of the specimen in order to minimize the misalignments caused during execution. The mechanical tests were conducted in a screw-driven universal testing machine (Instron 1125). The ELS test fixture included a linear guidance system allowing horizontal movement of translation of the clamping grip during loading (see Fig. 7). The tests were conducted in a displacement control mode with a crosshead speed of 5 mm/min. Two sheets of Teflon film were inserted between the initial crack surfaces in order to minimize the friction effects. During the tests, the applied load and displacement values were recorded (frequency 5 Hz). Fracture surfaces were observed using a scanning electronic microscope (SEM) Philips-FEI Quanta 400. Fig. 8 is a photomicrograph of the initial crack surface, where three different zones are visible. Region A is
Fig. 7. The end loaded split (ELS) test fixture.
Fig. 8. Initial crack sections: (A) sawn section; (B) sliced section; (C) Mode I pre-cracked section.
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the tip of the starter notch machined with the band saw, whereas region B is the portion of the initial crack cut by the blade. Zone C of the initial crack corresponds to the mode I fracture surface created ahead of cutting blade, due to its penetration into wood under an impact loading. Indeed, this section exhibits the characteristic appearance of fracture in perpendicular-to-grain tension, through the middle lamella of tracheids [19]. The transition between the surface of mode I pre-crack and the surface of mode II (parallel to grain shear loading) crack can be clearly seen on Fig. 9. The mode II crack surface is characterized by severe twisting, tearing, and unwinding of tracheid walls [19]. However, a significant difference exists between the morphology of fracture surfaces of earlywood and latewood, due to the greater thickness of the tracheid walls of latewood (Fig. 10). These findings in SEM investigations allows fracture surfaces mode II characterization and gives additional confidence on the application of ELS test to determine the mode II fracture resistance of wood in the RL fracture system.
Fig. 9. Transition between Mode I pre-crack and Mode II crack.
Fig. 10. Fracture surface obtained by Mode II: (a) latewood and (b) earlywood.
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5. Results Fig. 11 presents the experimental P–d curves for all tested specimens. Some scatter on the initial specimen stiffness show the necessity of previous specimen modulus measurement. The R-curves of all tested specimens (Fig. 12) were determined from the experimental P–d data and using the CBBM (Eq. (15)). The values of mode II energy release rates at initiation (GIIini) and during stable crack propagation (GIIc) were obtained. The GIIini values were determined using the load and deflection measured at the point of deviation from linearity in the P–d curve (Fig. 13). These initiation values correspond to the onset of FPZ development at the pre-crack tip. The critical strain energy release rates (GIIc) were obtained from the plateau of the R-curves (Fig. 14) and characterize the self-similar stable crack propagation process. Table 3 summarises the experimental results for all specimens. Standard deviations of approximately 31.8% and 16.8% were obtained for GIIini and GIIc, respectively, which can be considered acceptable in wood. The average value for GIIc (0.939 N/mm) is similar to the one obtained by the ENF test in a previous study (0.909 N/mm) for matched specimens obtained from the same trunk [20]. In order to validate the CBBM, numerical simulations based on the proposed cohesive damage model were performed for each specimen considering their measured dimensions of cross section, initial crack length, longitudinal modulus and GIIc . The effective length (Lef) of each specimen, given by Eq. (12), was used in the 500 450 400
P (N)
350 300 250 200 150 100 50 0 0
5
10 δ (mm)
15
20
Fig. 11. The ELS experimental P–d curves.
1.4 1.2
G II (N/mm)
1 0.8 0.6 0.4 0.2 0 105
115
125
135
145
155
a (mm) Fig. 12. The ELS experimental R-curves.
165
175
185
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500
400
GIIc
300
P (N)
G II ini
200
Experimental curve
100
Numerical curve 0 0
2
4
6
8
δ (mm)
10
12
14
Fig. 13. Comparison between the numerical and experimental P–d curves (specimen 13). 1.25
1
G II (N/mm)
GIIc
0.75
Experimental curve Numerical curve
0.5
GII ini 0.25
0 110
115
120
125
130
135
140
145
150
a (mm) Fig. 14. Comparison between the numerical and experimental R-curves (specimen 13).
numerical model instead of the real length of the beam outside the clamping fixture (dimension L in Fig. 2). This allows the consideration of a perfect clamping condition on the numerical model. The numerical P–d results were used to obtain the mode II R-curve considering the CBBM. Figs. 13 and 14 show an excellent agreement between numerical and experimental referred curves for a given specimen. As it can be seen in Table 3, globally, an excellent agreement between the numerical and experimental results was obtained. An important aspect concerning the numerical simulations is related to the influence of the limit stress ru,II, which in this case corresponds to su,LR, on the maximum load. This property was not experimentally measured and some variability can be expected in wood. To overcome this difficulty, fine–tune iterative process was performed for each specimen in order to minimize the difference between the numerical and experimental peak load values. This methodology originated a mean value of 9.27 MPa for su,LR with a standard deviation of 2.82 MPa.
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Table 3 Synopsis of the experimental and numerical results for all ELS specimens tested Specimen
Experimental results
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
387.84 338.28 394.92 344.76 449.76 349.44 362.16 349.56 390.72 342.24 350.52 352.20 387.96 292.92 381.24 394.44 368.52 329.64 349.32
0.359 0.184 0.108 0.429 0.332 0.278 0.335 0.417 0.385 0.372 0.441 0.510 0.309 0.193 0.194 0.252 0.333 0.350 0.281
1.041 0.763 1.080 0.822 1.148 1.163 1.108 0.936 1.150 0.855 0.815 0.911 0.922 0.733 1.092 1.031 0.843 0.754 0.680
390.09 337.29 392.17 337.94 461.05 354.60 354.57 348.12 384.35 344.39 354.09 352.21 389.11 298.41 385.27 391.66 361.67 331.22 342.63
0.366 0.184 0.106 0.449 0.335 0.267 0.341 0.420 0.412 0.376 0.459 0.502 0.313 0.200 0.197 0.252 0.333 0.372 0.279
1.030 0.745 1.081 0.813 1.142 1.151 1.090 0.918 1.140 0.846 0.810 0.904 0.913 0.734 1.084 1.025 0.849 0.747 0.669
Average
364.023
0.319
0.939
363.728
0.324
0.931
9.182
31.779
16.847
9.488
32.366
16.905
P max (N)
Std. dev. (%)
GIIini (N/mm)
Numerical results GIIc (N/mm)
P max (N)
GIIini (N/mm)
Comparison numerical/ experimental GIIc (N/mm)
P max (%)
GIIini (%)
GIIc (%)
0.579 0.293 0.696 1.980 2.511 1.475 2.095 0.412 1.631 0.629 1.019 0.004 0.296 1.873 1.056 0.706 1.859 0.480 1.915
1.845 0.101 2.269 4.397 0.822 4.151 1.740 0.858 6.365 1.187 4.025 1.643 1.255 3.798 1.431 0.022 0.002 3.346 0.565
1.046 2.302 0.106 1.044 0.526 1.007 1.657 1.978 0.905 1.079 0.577 0.675 0.976 0.186 0.778 0.655 0.656 0.885 1.687
Table 4 Alteration of GIIc (%) for a variation of 1% in each parameter a0
B
h
E1
C0
C
0.931
0.495
1.392
0.469
1.155
1.473
Table 5 Errors obtained by the three different methods between numerical and experimental results for GIIc Experimental results
Comparison numerical/experimental
Specimen
GIIc (N/mm)
CBBM, error (%)
CCM, error (%)
CBT, error (%)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1.041 0.763 1.080 0.822 1.148 1.163 1.108 0.936 1.150 0.855 0.815 0.911 0.922 0.733 1.092 1.031 0.843 0.754 0.680
1.05 2.30 0.11 2.13 0.53 1.01 1.66 0.18 0.91 1.08 0.58 0.67 1.19 0.19 0.78 0.65 0.66 0.89 1.69
0.25 1.63 0.52 2.38 0.15 3.64 3.07 0.24 2.37 1.25 1.55 1.63 0.49 0.05 1.57 1.64 1.12 3.45 2.22
8.34 6.69 6.79 12.43 9.92 14.68 14.53 13.64 6.04 18.34 12.26 16.19 15.31 12.09 6.70 6.04 17.94 17.61 14.63
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The sensibility of GIIc relatively to the parameters included in Eq. (15) was also verified. Table 4 presents the alteration in GIIc for a variation of 1% in each parameter. It can be observed that compliances, beam height and initial crack length are the most influent parameters and they should be carefully measured. The performance of the CBBM was also numerically evaluated comparing the obtained results with the ones given by the CCM and CBT (Eqs. (5) and (6), respectively). Both of these methods depend on rigorous crack monitoring during propagation, which is very difficult to perform experimentally. However, numerically it is easy to accomplish crack evolution owing to the cohesive damage model. Table 5 presents the errors obtained from the three methods for all tested specimens. The CBBM and CCM shows good agreement relatively to the inputted values and the CBBM presents the advantage of being independent of crack monitoring during propagation. 6. Conclusions A new data reduction scheme based on the specimen compliance is proposed for the ELS test. The method does not require crack length measurement during the experimental tests which was verified to be very difficult to perform accurately in these tests. On the other hand, the proposed methodology accounts for the effects of beam root rotation at the clamping point and for the FPZ size at the crack extremity. The method was applied to the experimental results to obtain the mode II energy release rates at initiation and during stable crack propagation. The experimental tests were individually simulated considering a two-dimensional analysis based on a cohesive damage model, in order to numerically validate the suggested data reduction scheme. Excellent agreement was obtained between simulation and experimental results which confirms the validity of the proposed method and the applicability of the ELS test to fracture characterization of wood. Acknowledgements The authors thank Professor Alfredo B. de Morais of University of Aveiro, Portugal, for his valuable advices and discussion about the subject. The authors would also to thank the Portuguese Foundation for Science and Technology (FCT, Research Project POCTI/EME/45573/2002) for supporting the work here presented. Appendix A Eq. (12) can be expressed as, ax3 þ bx þ c ¼ 0;
ðA1Þ
where the coefficients a, b and c are, respectively a¼
1 ; 2Bh3 E1
b¼
3 ; 5BhG13
c¼
3a30 C0: 2Bh3 E1
ðA2Þ
Using the MatlabÒ software and only keeping the real solution we have, x¼
1 2b A ; 6a A
ðA3Þ
being 00
1 113 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ffi 2a 4b þ 27c A a2 A : A ¼ @@108c þ 12 3 a
ðA4Þ
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