Holzforschung, Vol. 6, pp. 8 89, 007 Copright b Walter de Gruter Berlin New York. DO 0.55/HF.007.0 Simple estimation of critical stress intensit factors of wood b tests with double cantilever beam and three-point end-notched fleure Hiroshi Yoshihara* Facult of Science and Engineering, Shimane Universit, Matsue, Shimane, Japan *Corresponding author. Facult of Science and Engineering, Shimane Universit, Nishikawazu-cho 060, Matsue, Shimane 690-8504, Japan E-mail: osihara@riko.shimane-u.ac.jp Abstract Simple equations are proposed for calculation of critical stress intensit factors b tests using double cantilever beam (DCB) and three-point end-notched fleure (ENF). The calculation modes are named here as modes and and are based on the beam theor and 95 previousl published data on the elasticit properties of woods. The validit of the data was eamined on specimens of western hemlock wood with various crack lengths. The influence of the elastic properties is more significant on the stress intensit factor calculated in mode than that calculated in mode. Further work is needed, particularl for measuring the mode stress intensit factor. However, it is obvious from the eperiments with western hemlock that the critical stress intensit factors can be determined b the equations proposed here. Kewords: critical stress intensit factor; DCB test; ENF test; mode ; mode ; wood. ntroduction Recentl, it has become common practice to investigate cracks based on the energ release rate G rather than stress intensit factor K, because the former is mathematicall well defined and eperimentall measurable (Adams et al. 00). The calculations of the mode and mode fracture characteristics rel on tests with double cantilever beam (DCB) and end-notched fleure (ENF) approaches, respectivel. Here, the fracture toughness G c and G c can be defined mathematicall according to the beam theor. Fracture toughness measurement with the DCB and ENF tests has been increasingl applied (Morel et al. 00, 00, 005; Jensen 005; Yoshihara 005, 006a,b; de Moura et al. 006; Silva et al. 006; Yoshihara and Kawamura 006). Nevertheless, both techniques have a drawback: the load-deflection relation, which is mandator for determining the fracture toughness, often deviates from the elementar beam theor because the deformation around the crack tip cannot be predicted. There are three was to correct the deformation around the crack tip. ) A factor for correcting the crack length to the solution (Williams 989; Chatterjee 99; Wang and Williams 99; Corleto and Hogan 995; Williams and Hadavinia 00) can be introduced. The elastic constant of the beam is needed for this approach, and it must be measured in separate tests. ) The equivalent crack length corrected b the loading-line compliance can be used instead of the actual crack length (Blackman et al. 005; de Moura et al. 006; Silva et al. 006). Separate tests are not necessar for measuring this parameter. ) The strain at a certain point of the specimen can be measured (Yoshihara 005, 006a,b; Yoshihara and Kawamura 006). The deflection and strain should be measured simultaneousl, along with the applied load. Besides the tests discussed above, there are more possibilities for measuring fracture properties. Compact tension (CT), single-edge notched tension (SENT), double-edge notched tension (DENT), and single-edge notched bending (SENB) tests have been described for measuring the fracture properties in mode (Schniewind and Pozniak 97; Mall et al. 98; Triboulot et al. 984; Kretschmann and Green 996; King et al. 999), whereas compact shear (CS), and double-edge notched shear (DENS) tests have been developed for measuring the fracture properties in mode (Mall et al. 98; Cramer and Pugel 987; Prokopski 995; Stanzl-Tschegg et al. 996; Xu et al. 996). n these methods, the critical stress intensit factor K c is usuall calculated b approimating equations instead of deriving the fracture toughness G c. These equations often depend on the critical load for crack propagation and specimen configuration alone, and the do not contain an load-deflection relation or elastic constants of the material. n terms of simplicit, measurement of K c using an approimating equation is more advantageous than that of G c. This idea ma be applicable to DCB and ENF tests, but it would be desirable to determine the critical stress intensit factors K c and K c in the course of the tests. Nevertheless, publications are rare in this field (Barrett and Foschi 977). n the present work, DCB and three-point ENF (ENF) tests were conducted and the critical stress intensit factors were analzed b simple approimating equations derived from the beam theor. The validit of the equations was eamined b comparing the results with eperimental data. Theories Figure shows diagrams of the DCB and ENF tests, for which specimens were prepared b cutting a crack along the neutral ais of a rectangular bar. As shown in Figure a, the load was applied to the upper and lower cantilever portion of the specimen in opposite directions to each other in the DCB test, whereas in the ENF test the load was applied to the midspan (Figure b). When neglecting the transverse shear deflection and considering the deformation around the crack tip, the deflection
Simple estimation of mode and mode critical stress intensit factors 8 Figure Schematic diagram of DCB and ENF tests. at the loading point d is derived b the modified beam theor, as follows (Adams et al. 00): DCB: 8PŽ aq H. ds () EBH ENF: PwL qž aqh. ds, () 8EBH where P is the applied load, a is the crack length, E is Young s modulus in the longitudinal ais, B is the width of the specimen, H is the depth of the specimen, L is the span length in the ENF test, and and are factors for correcting the crack length in the DCB and ENF tests, respectivel. These factors are ignored in the elementar beam theor. There are various theoretical models for deriving these correction factors and (Williams 989; Chatterjee 99; Wang and Williams 99; Corleto and Hogan 995; Williams and Hadavinia 00). The question as to which model is most applicable cannot be answered et. According to the Williams and Hadavinia model, these factors are given b the elastic constants of the beam (Williams and Hadavinia 00; Adams et al. 00) as: D G E B E E s 0.4 q0.c -n F () E G D G E B E E s0.4 0.4 q0.c -n F, (4) E G where E is Young s modulus in the depth direction, and G and n are the shear modulus and Poisson s ratio in the length-depth plane, respectivel. The loading-line compliance C L is derived b d/p, so: DCB: 8Ž aq H. CLs (5) EBH ENF: L qž aqh. C s. L (6) 8E BH The energ release rates in modes and, G and G, respectivel, are given as: P C L P Ž aqh. G s Ø s (7) B a E B H 9P Ž aqh. Gs. (8) 6E BH t is time-consuming to determine the energ release rates b Eqs. (7) and (8) because the elastic constants E, E, G, and n that include and should be measured b separate tests independentl of the fracture test. Nevertheless, this obstacle can be overcome when the longitudinal strain is measured simultaneousl at a certain point of the beam, together with the loading point deflection. When the longitudinal strain on the bottom surface of a specimen is measured at a point located at sl (0-l-a in the DCB test and a-l-l in the ENF test) as shown in Figure the load-strain compliance C S, which is defined as /P, is derived b the beam theor independentl of the crack length as follows: DCB: 6l CSs (9) EBH ENF: l C s. S (0) 4E BH From Eqs. (5), (6), (9), and (0), aq H and aq H are written as follows: BHl C L E C F aqhs Ø () D 4 C S G B C E L C F aqhs HlØ - L. () D CS G B substituting Eqs. (9) and () into Eq. (7), E and are eliminated and G is derived as follows: B E L L - C F D G P C Hl C Gs Ø. () B 4 C S Similarl, G is derived b substituting Eqs. (0) and () into Eq. (8) as follows: B E S L C F D S G P C C Gs HlØ - L. (4) 4BLH C
84 H. Yoshihara Article in press - uncorrected proof Equations () and (4) signif that the energ release rates G and G can be independentl obtained directl b the DCB and ENF fracture tests, respectivel, while correcting the influences of transverse shear force and crack tip deformation. The author previousl called this approach the compliance combination method, and verified that the fracture toughness can be obtained appropriatel b modes and (Yoshihara 005, 006a,b; Yoshihara and Kawamura 006). The energ release rates G and G can be transformed into the stress intensit factors K and K, respectivel, when using the following relations (Sih et al. 965): EG K s (5) a EG K s, (6) a where E E B E E a s q C -n F E E DG G (7) E B E E a s q C -n F. E DG G (8) B substituting Eqs. (7) and (8) into Eqs. (5) and (6), respectivel, the stress intensit factors are derived as follows: DCB: P B a E K s Cp Ø qq F (9) BH D H G ENF: P B a E K s Cp Ø qq F, (0) B H D H G be obtained simpl b the specimen configuration, crack length, and applied load. For eample, Murph (979, 988) proposed a center slit fleure test for measuring the mode stress intensit factor, and derived p and q as.4 and 0.6, respectivel, b averaging the values of several wood species. From 95 data on the elastic constants for the longitudinal-tangential and longitudinal-radial planes of 5 hardwood and 0 softwood species collected b Hearmon (948), Kollmann and Côte (968), and the Forestr and Forest Products Research nstitute Japan (004), the values of E /E and E /G n were found to be in the range of 8 8 for hardwoods and for softwoods. Material and methods Specimens Western hemlock (Tsuga heterophlla Sarg.) lumber, with a densit of 0.48"0.0 g cm - at % moisture content (MC), was tested. As shown in Figure, the annual rings were flat enough to ignore their curvature. This lumber had no defects (knots or grain distortions) so that the specimens cut from it could be regarded as small and clear. The lumber was stored before the test for approimatel ear in a room at 08C and 65% relative humidit. The equilibrium MC condition was approimatel %. Compression tests Young s moduli E and E, shear modulus G, and Poisson s ratio n are required for transforming the energ release rates obtained b the beam theor and compliance combination methods. With this aim, the stress intensit factor was obtained using Eqs. (5) and (6). These elastic constants were determined b compression tests. A short-column specimen with the dimensions of 40 mm=0 mm=0 mm was prepared from the lumber described above. When measuring E and n, the long ais of the specimen was coincident with the longitudinal direction of wood, whereas when measuring E, the long ais of the specimen was aligned with the tangential direction. When measuring G, the long ais was inclined at 458 to the grain. Strain gauges where l p s a () qs a l ps a () q s a As represented b Eqs. (7), (8), (), and (), the parameters p, q, p, and q include the elastic constants E, E, G, and n,sok and K are dependent on the elastic properties of the test material. When these parameters are approimated using constants, K and K can Figure mage of a cross-section of the material used in this eperiment.
Simple estimation of mode and mode critical stress intensit factors 85 were bonded at the centers of longitudinal-tangential planes, and compression load was applied along the long ais of the specimen at a crosshead speed of mm min -. Young s moduli E and E were obtained from the stress-strain relation in the loading direction. Poisson s ratio n was obtained b the relation between the longitudinal and tangential strains b appling the compression load in the longitudinal direction. For the 458 inclined specimen, Young s modulus in the loading direction E 45 was obtained from the stress-strain relation. The strain in the direction perpendicular to the loading ais was simultaneousl measured, and Poisson s ratio n 45 was obtained, as well as Young s modulus E 45. The shear modulus G was determined from the following equation: E 45 G s. () qn Ž. 45 The averages of E, E, G, and n were substituted into Eqs. (), (4), (7), and (8), and the fracture toughness G c and G c (obtained b the beam theor and b the compliance combination methods) were transformed into the critical stress intensit factors K c and K c, respectivel. DCB tests The dimensions of the specimens were 5 mm=5 mm= 5 mm (R=T=L). A crack was produced along the longitudinal direction in the longitudinal-tangential plane, which is the socalled TL sstem. The crack was first cut with a band saw (thickness mm), and then etended mm ahead of the crack tip using a razor blade. Loading blocks of western hemlock with dimensions of 0 mm in length, 0 mm in height, and 5 mm in thickness were bonded b epo resin on the upper and lower cantilever portions opposite each other (Figure ). Crack length, which was defined as the distance from the line of load application to the crack tip, varied from 60 to 60 mm in intervals of 0 mm. Load was applied to the specimen b pins through universal joints at a crosshead speed of 5 mm min - until the load markedl decreased. The total testing time was approimatel 0 min (five repetitions). The loading-line displacement d was measured b the crosshead travel, since it was confirmed that the machine compliance was small enough to be ignored (Yoshihara and Kawamura 006), whereas the longitudinal strain was measured using a strain gauge (gauge length mm; FLA--, Toko Sokki Co., Toko) bonded at the midpoint between the loading line and crack tip on the top cantilever portion (lsa/), in a location similar to that in previous work (Yoshihara and Kawamura 006). The loading-line compliance C L and the load-longitudinal strain compliance C S were obtained from the linear portions of P-d and P- relations, respectivel. As shown in Figure, the critical load for crack propagation P c was defined as the load at the intersection point between the load-loading line displacement curve and straight line with a 5% increase in compliance (Yoshihara 005, 006a; Yoshihara and Kawamura 006). B substituting P c, C L, and C S into Eq. (), the mode fracture toughness G c was obtained. Then the value of G c was transformed into the critical stress intensit factor K c using Eq. (5). ENF tests Similar to the DCB tests, all of the specimens were cut from the lumber described above. The initial dimensions were 5 mm=5 mm=450 mm (R=T=L). The crack was cut in the same manner as for the DCB tests, and the crack length varied from 60 to 60 mm in intervals of 0 mm. Two sheets of 0.5- Figure Definition of critical load P c. mm-thick Teflon film were inserted between the crack surfaces to reduce the friction between the upper and lower cantilever beams. These specimens were supported b 400-mm spans, and a load was applied to the midspan at a crosshead speed of mm min - until significant non-linearit in the load-loading line displacement relation was induced. The total testing time was approimatel 0 min. Five specimens were used for one testing condition. To avoid the influence of indentation of the loading nose, the loading-line deflection d was measured b an LVDT set below the specimen, whereas the longitudinal strain was measured using a strain gauge (gauge length mm; FLA--) bonded to the bottom surface of the midspan (lsl). The loading-line compliance C L and load-longitudinal strain compliance C S were obtained from the linear portions of P-d and P- relations, respectivel. According to the elementar beam theor, a fracture propagates unstabl when the crack length is smaller than 0.7 times the half-span length (Carlsson et al. 986). n the ENF tests conducted here, however, the crack propagated stabl in ever specimen because of the large value of additional crack length H (Yoshihara 005). Hence, the critical load for crack propagation was determined as the load at the intersection point between the load-loading line displacement and a straight line with a 5% increase in compliance, similar to the DCB test. B substituting P c, C L, and C S into Eq. (4), the mode fracture toughness G c was obtained. Then the value of G c was transformed into the critical stress intensit factor K c using Eq. (6). Analsis methods The critical stress intensit factors were analzed b the following si methods: modified beam theor, compliance combination, and linear approimations A, B, C, and D. The details are described below. n appling the modified beam theor, the fracture toughness G c was calculated b substituting Young s modulus E, critical load P c, initial crack length a 0, and correction factor into Eq. (7), whereas G c was calculated b substituting E, P c, a 0, and into Eq. (8). Then the values of G c and G c were transformed into K c and K c, respectivel, b substituting these values and E, E, G, and n into Eqs. (5) and (6), respectivel. n the compliance combination method, G c and G c were obtained b substituting P c, C L, and C S into Eqs. () and (4), respectivel, then transformed into the critical stress intensit factors in the same manner as for the modified beam theor method.
86 H. Yoshihara Article in press - uncorrected proof n the linear approimation methods, the values of K c and K c were obtained b substituting P c into Eqs. (9) and (0), respectivel, but the values of the parameters were different in methods A D. n methods A and B, the parameters were derived from the data for compression tests on western hemlock and 95 data from the literature (Hearmon 948; Kollmann and Côte 968; Forestr and Forest Products Research nstitute Japan 004), respectivel. When comparing for western hemlock obtained from Eqs. () and (), however, obtained from Eq. () was.45"0.66, approimatel.5-fold greater than the value obtained from Eq. (), which was.6. Similarl, obtained from Eq. () was.67"., approimatel five-fold greater than the value obtained from Eq. (4), which was 0.68. Several researchers have suggested that Eqs. () and (4) are no longer valid when a fracture process zone, which is the region where the material progressivel softens, develops at the crack tip (de Moura et al. 006; Silva et al. 006). Considering the eaggerated values of and, the values of q and q were thus corrected in methods C and D. n method C, the elastic constants obtained from compression tests on western hemlock were used, but the values of q and q were corrected as follows: q s.5 (4) a q s7.5. (5) a n these equations, and were derived b Eqs. () and (4), respectivel. n method D, the elastic constants were from 95 data obtained from the literature, and the values of q and q were derived b Eqs. (4) and (5), respectivel. Results and discussion Figure 4 shows the dependence of the parameters p, q, p, and q on the values of E /E and E /G n. The Table Elastic properties used for the transformation of energ release rates G and G into stress intensit factors K and K, respectivel. E (GPa) E (GPa) G (GPa) n 4.9"0.4 0.7"0.0 0.90"0. 0.5"0.0 Results are mean"sd. influence of variation in the elastic constants is rather significant on the values of p and q. This indicates that K might not be properl obtained when some constant values are derived for p and q. When the value of E /E is restricted to the range from to 4, however, variations in p and q are approimatel within 0%. n contrast, the influence of variation in the elastic constants is rather small on the values of p and q, so that determination of K b this method is promising. K is probabl more correct than K in view of the wide range of possible elastic properties. Table lists Young s moduli in the longitudinal and tangential directions, E and E, respectivel, and the shear modulus and Poisson s ratio on the longitudinal-tangential plane, G and n, respectivel, for western hemlock obtained from compression tests. From the averages of these elastic constants, the values of E /E and E /G n used in linear approimation methods A and C were derived as 0.4 and 5.5, respectivel. B substituting these values into Eqs. (7), (8), (), (), (4), and (5), the parameters p, q, p, and q were obtained as listed in Table. The values of p, q, p, and q (Table ) were obtained from a similar procedure applied to 95 data taken from the literature. Figure 4 Dependence of the parameters p, q, p, and q on the elastic properties E /E and E /G n.
Simple estimation of mode and mode critical stress intensit factors 87 Table Parameters p, q, p, and q for deriving stress intensit factors in the linear approimation methods. Method Mode Mode p q p q A.04.68.5 0.46 B."0..78"0..6"0.0 0.46"0.0 C.04.5.5.0 D."0..6"0.0.6"0.0.5"0.8 The parameters in methods A and C were obtained from compression test data for western hemlock, whereas those in methods B and D were given b the 95 data reported b Hearmon (948), Kollmann and Côte (968), and the Forestr and Forest Products Research nstitute Japan (004). Data for B and D are mean"sd. Figure 5 Comparison of the relations between the mode critical stress intensit factor K c and crack length a obtained b each analtical method. Figure 5 shows comparisons of K c corresponding to the crack length a obtained b each analtical method. The influence of crack length is not significant on K c, and statistical analses reveal that there are no differences between the values of K c obtained b the si methods at the significance level of 0.0. As mentioned above, variations in p and q are approimatel within 0% for E /E in the range from to 4. Since the value of E /E for western hemlock used here was 0.4, K c could be obtained appropriatel. n methods A and C, the parameters obtained for a specific species are needed, so there is a concern that K c for another species cannot be measured b these methods. Among the 95 data analzed, 5 data satisf E /E in the range 4, so it is feasible to obtain K c for rather various species b methods B and D. As described in previous work, the mode fracture toughness can be obtained b the compliance combination method more appropriatel than deriving the correction factor from the elastic constants (Yoshihara and Kawamura 006). When comparing methods B and D, method D in which the compliance combination method is used to determine the value of q is more appropriate than method B, in which q is obtained using consisting of Figure 6 Comparison of the relations between the mode critical stress intensit factor K c and crack length a obtained b each analtical method.
88 H. Yoshihara Article in press - uncorrected proof the elastic constants. Therefore, the following relation is proposed for obtaining the mode stress intensit factor from DCB tests: P B a E K s C. q.6 F. (6) BHD H G Figure 6 shows a comparison of the mode critical stress intensit factor K c corresponding to crack length a obtained b each analtical method. The influence of crack length is not significant for values of K c obtained b the compliance combination and b linear approimation methods C and D. Nevertheless, statistical analses reveal that K c values obtained b the modified beam theor and b linear approimation methods A and B were smaller than those obtained b the other methods at significance level 0.0 when the crack length was smaller than 0 mm. Murph (988) conducted an asmmetric four-point bending test on a specimen with a center slit to measure K c, and derived values of.4 for p and 0.6 for q. The value of K c for clear Douglas fir tended to decrease when the crack length was decreased. This tendenc is similar to the K c -a relations obtained b the modified beam theor and linear approimations A and B. This can be eplained b the small value of q used in these methods. n addition, the influence of variations in the elastic constants is rather small on the data for p and q. Accordingl, linear approimation D, which is based on data for various species, is recommended for deriving the K c -a relation, and the following relation is proposed for obtaining the mode stress intensit factor from ENF tests: P B a E K s C.6 q.5 F. (7) BH D H G As mentioned above, there is a concern that Eq. (6) is not effective when measuring the mode stress intensit factor of a material if E /E is outside the range 4. On the other hand, Eq. (7) is applicable for measuring the mode stress intensit factor. To verif the broad validit range for the proposed method, further research is needed on DCB and ENF tests and various wood species should be included in the test program. Conclusions To obtain critical stress intensit factors according to modes and b DCB and ENF tests, equations are proposed based on the beam theor. Data on elastic properties previousl published were used to validate the data obtained from western hemlock with various crack lengths. The influence of elastic properties is more significant for measuring the mode stress intensit factor than that of mode. The mode stress intensit factor needs more attention in the future. As for the specimens of western hemlock eamined here, however, both stress intensit factors could be properl obtained, and it appears that the critical stress intensit factors can be determined b the equations proposed here. Acknowledgements The author would like to thank Prof. Tohru Uehara of Shimane Universit for his help in conducting the eperiment. 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