l184 L.J. Hart-Smith shear-failure cut-offs, the validity of the earlier 45deg approximation for carbon/epoxy laminates is clearly d (11) confirmed. Conversely, the earlier appro be significantly conservative for glass as the author had suspected without actually ble to for round fibres in precisely quantify the effect until now. (These expressions are derived from the solutions for Point(4)in Fig. 3 follows from point (3), the zero the fibre volume fraction as a function of each array axial stress point for the fibre, by retaining the same Setting the value of the array coefficient K at unity, axial strain and multiplying the transverse strain by Re eqn(2)would then predict the following strain-amplifi- from eqn(2). The line(14)in Fig 3 then defines the cation factors for the composite materials used in failure locus of shear failures in the fibres in terms of strains the lamina. It will be apparent that point (4) lies off the (0%)T300/914C carbon/epoxy, Re=1. 517 zero longitudinal stress line for the lamina, being asso- (0%)E-glass/LY556-epoxy, Re=5. 257 ciated with an effective transverse poisson ratio of (0%)E-glass/MY750-epoxy, RE= 5.159 UTL =VTL/Re (12) (0%)AS4/3501-6 carbon/epoxy, Re= 1. 488. Given that these amplification factors are effectively instead of the unrelated VlamtL for the laminate as a reduced in the ratio (UnT/vLT), or roughly 0.2/0.3 for whole. The reason for this is that, while the fibres have carbon/epoxy, in establishing the final slope of these no axial stress at point(4), the matrix does d Square fibres in squre array d Circular fibres in square array Circular fibres in hexagonal arrayand d ˆ  4  Vf r ˆ 1
128pVf …11† for round ®bres in square arrays. (These expressions are derived from the solutions for the ®bre volume fraction as a function of each array.) Setting the value of the array coecient K at unity, eqn (2) would then predict the following strain-ampli®- cation factors for the composite materials used in failure exercise.4 (0) T300/914C carbon/epoxy, R" ˆ 1
517 (0) E-glass/LY556-epoxy, R" ˆ 5
257 (0) E-glass/ MY750-epoxy, R" ˆ 5
159 (0) AS4/3501-6 carbon/epoxy, R" ˆ 1
488. Given that these ampli®cation factors are e€ectively reduced in the ratio (fLT=LT), or roughly 0.2/0.3 for carbon/epoxy, in establishing the ®nal slope of these shear-failure cut-o€s, the validity of the earlier 45deg; approximation for carbon/epoxy laminates is clearly con®rmed. Conversely, the earlier approximation would be signi®cantly conservative for glass ®bres, as the author had suspected without actually being able to precisely quantify the e€ect until now. Point (4) in Fig. 3 follows from point (3), the zero axial stress point for the ®bre, by retaining the same axial strain and multiplying the transverse strain by R" from eqn (2). The line (1)±(4) in Fig. 3 then de®nes the locus of shear failures in the ®bres in terms of strains in the lamina. It will be apparent that point (4) lies o€ the zero longitudinal stress line for the lamina, being asso￾ciated with an e€ective transverse Poisson ratio of 0 TL ˆ f TL=R" …12† instead of the unrelated lamTL for the laminate as a whole. The reason for this is that, while the ®bres have no axial stress at point (4), the matrix does. Fig. 4. Fibre volumes for various arrays. 1184 L. J. Hart-Smith