Corrected beam theory

Thus, corrected beam theory can be used to deduce the value of Ef from the mode II fracture test, and this allows an important cross-eheck on the results to be performed, as the value can be compared to the known, or independently measured value, , as used in eqn (1). Also the value of Ef so deduced should be independent of crack length, a. 

If one assumes that the measured crack length is likely to be only approximate in the presence of such extensive microcracking, the use of beam theory expressions requiring this parameter will also be subject to this uncertainty. However, corrected beam theory can be used to determine an effective crack length, a, via the measured compliance value and eqn (3) may be reananged to give an effective crack length thus fa)... 

Note G c values deduced via corrected beam theory 2,3, Cjic values deduced via corrected beam theory using eftective crack lengths, 0. ... 

Corrected Beam Theory (CBT) values of initiation from the Mode-1 pre-crack according to [1] are summarised in Tables 2-4 for the two laboratories. The values in Table 2 can be compared with the average values for the unidirectional lay-up reported by a third participating laboratory in [4] for initiation from the insert 273 J/m (standard deviation 2.1%), initiation from the Mode-I pre-crack 304 J/m (2.8%), and maximum propagation values of 369 JW (1.9%). The sign convention for the A-value is explained in the notes to the tables. 

According to the requirements specified in the ISO standard [1] deviation of the delamination propagation from mid-plane invalidates the test. In that sense, any data analysis of the cross-ply laminates, therefore, will yield invalid results. This point was discussed extensively in [4] where the authors also used the Finite-Element method to supplement their analysis and concluded that the corrected beam theory data reduction scheme seemed to remain applicable and that the non-symmetric cross-ply material yielded apparently valid fracture toughness data, even though these were probably affected by transverse cracking. 

Notes ( ) Analysis using the 5% offset value and the corrected beam theory method. ( ) Visual initiation rather than 5% offset value. ( ) If specimen 3/1 is considered an outlier and eliminated, the average is 342+49 J/m. C ) From compliance measurement. 

Notes ( ) Analysis using the 5% offset value and the corrected beam theory method. ( ) From compliance measurement. 

The mode I fracture resistance of adhesive joints is most commonly determined using the double cantilever beam (DCB) test. This test was initially described in the ASTM standard (ASTM 1990) and has been developed more recently in the British standard (BSI2001) and the international standard (ISO 2009). The original ASTM test standard specified metallic substrates and the critical strain energy release rate in mode I, Gic, was determined for repeated crack initiations using a version of simple, shear corrected beam theory. The later standards additionally accommodate nonmetaUic substrates and employ corrected beam theory to determine values of Qc 4t both crack initiation and during steady-state crack propagation. 

The analysis methods developed assume that the adhesive layer is thin and that the adhesive has a low value of Young s modulus compared to the substrate material. Thus, the contribution of the adhesive layer to the overall beam compliance is assumed to be negligible. In the ASTM method for mode I (ASTM 1990), G[c is determined using simple, shear corrected beam theory ... 

In the BS and ISO methods, additionally an experimental compliance and a corrected beam theory analysis are specified for the determination of Qc- The experimental compliance method is based on Berry s method (Berry 1960) and is given by ... 

Gie values have been deduced using simple beam theory (SBT), corrected beam theory (CBT), and the experimental compliance method (ECM))... 

The data reduction for the tapered ENF (TENF) specimen can take the form of an experimental compliance or a simple beam theory approach, as reported by Edde and Verrenman (1995) or a corrected beam theory approach following the work of Qiao et al. (2003), using a beam on an elastic foundation model. These authors showed that substrates with a linear taper gave an almost linear relationship between compliance and crack length, within a certain range of crack lengths. 

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