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250200Applied load F [N]200150100501501005000 10 20 30 40Cross-head displacement Δ DCB [mm](a)40008000 2.5 5 7.5 10Load-point deflection Δ MMB [mm](b)Applied load F [N]30020010060040020000 2 4 6 800 2 4 6 8Load-point deflection Δ MMB [mm]Load-point deflection Δ MMB [mm](c)(d)Experiment FEA G c = G c,nlin FEA G =c ,G cmax FEA G =c G cfit ,Figure 13: Load-deformation curves of the adhesive fracture tests for mode mixing ratios of β = 0(a),β = 0.25 (b), β = 0.5 (c),andβ = 0.75 (d).18
4. Fracture Toughness CriteriaThis section deals with the evaluation and validation of suitable fracture toughness criteria available from literature.The basis for fitting these criteria are the CERR values for specific mode mixing ratios which have been obtaineddirectly via experiments for β = 0 or indirect by the semi-numerical estimate as described previously for β = 0.25,β = 0.5, and β = 0.75. The fracture toughness for the single mode II case, G IIc ,isusedasafitting parameter. In thissection we first review some of the available criteria for mixed mode I/II scenarios, then present the results of leastsquares fits against the basic values deduced in section 3.3, and then present a simple approach to include mode IIIbehavior which is based on some basic assumptions.4.1. Review of Available CriteriaSeveral criteria have been applied in the past, ranging from power laws with uniform or different exponents formode I and II over polynomial criteria to the one proposed by Benzeggagh and Kenane [60], which is referred toas the BK-criterion in the following. Surveys on available criteria can be found e.g. in [6, 61]. An extract of theseare reviewed in the sequel. Let us first consider that the most general fracture-mechanics-based criterion for crackpropagation is that the total energy release rate G T , which is defined as the sum of the single mode energy releaserates, is equal to the mixed mode CERR. This criterion readsG c = G I + G II = G T ⇔ G T= 1 . (9)G cThe most easy imaginable mode interaction criterion is that there is no interaction at all. In this case the fracturetoughness remains unchanged when altering the mode mixing ratio. However, such a criterion is evidently wrong andis not considered here. Another simple interaction criterion is a power law. The most general power law from whichsome other criteria can be deduced is defined by( ) α1( ) α2 GI GII+ = 1 , (10)G Ic G IIcsee [62]. A wide range of material response can be displayed with this criterion by choosing any combination of α 1and α 2 [63, 64]. If equal values α 1 = α 2 = 1 are utilized we obtain the linear interactionG I+ G II= 1 . (11)G Ic G IIcSubstitution of (1)and(9) and some basic mathematical manipulations yield an expression for the mixed mode CERRin terms of the mode mixing ratio:( 1 − βG c = + β ) −1(12)G Ic G IIcA linear interaction of the different failure modes may not be sufficient for an accurate representation of the mixedmode CERR. For a quadratic interaction we have to set α 1 = α 2 = 2 so we obtain( ) 2 ( ) 2 GI GII+ = 1 . (13)G Ic G IIcAgain substituting (1) and(9) leads to an expression for the mixed mode CERR in terms of the mode mixing ratiowhich is given by⎡( ) 2 1 − βG c = ⎢⎣ +G Ic( ) 2 −1/2βG⎤⎥⎦IIc. (14)The same methodology can be used for deducing a more general power law which includes (12) and(14). For thispurpose we set α 1 = α 2 = α and we obtain[( ) α ( ) α ] −1/α 1 − β βG c = +. (15)G Ic G IIc19
Page 1 and 2: Karlsruher Institut für Technologi
Page 3 and 4: Adhesive Joints in Composite Lamina
Page 5 and 6: 1.41.41.21.2Applied load F/F max [-
Page 7 and 8: Fw 2tadhesive0°pre-crackDCBFl adha
Page 9 and 10: (a)(b)Figure 5: Photographs of the
Page 11 and 12: 600200450150Applied Load [N]300150A
Page 13 and 14: (a) (b) (c)Figure 9: Photographs of
Page 15 and 16: 3.1.1. Shell ElementsThe shell elem
Page 17 and 18: In the MMB case, see Fig. 11 (b), r
Page 19: 3.3. Adhesive Fracture ToughnessFor
Page 23 and 24: 5000500040004000CERR Gc [J/m 2 ]300
Page 25 and 26: total shear energy release rate G s
Page 27 and 28: Havre de Grace, Maryland, USA (Mar.

Karlsruher Institut fÃ¼r Technologie Institut fÃ¼r Baustatik Adhesive ...

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