Stiffness transverse shear modulus

For two-dimensional randomly oriented fibers in a composite, approximating theory of elasticity equations with experimental results yielded this equation for the planar isotropic composite stiffness and shear modulus in terms of the longitudinal and transverse moduli of an identical but aligned composite system with fibers of the same aspect ratio ... 

A simplified performance index for stiffness is readily obtained from the essentials of micromechanics theory (see, for example. Chapter 3). The fundamental engineering constants for a unidirectionally reinforced lamina, ., 2, v.,2, and G.,2, are easily analyzed with simple back-of-the-envelope calculations that reveal which engineering constants are dominated by the fiber properties, which by the matrix properties, and which are not dominated by either fiber or matrix properties. Recall that the fiber-direction modulus, is fiber-dominated. Moreover, both the modulus transverse to the fibers, 2, and the shear modulus, G12. are matrix-dominated. Finally, the Poisson s ratio, v.,2, is neither fiber-dominated nor matrix-dominated. Accordingly, if for design purposes the matrix has been selected but the value of 1 is insufficient, then another more-capable fiber system is necessary. Flowever, if 2 and/or G12 are insufficient, then selection of a different fiber system will do no practical good. The actual problem is the matrix systemi The same arguments apply to variations in the relative percentages of fiber and matrix for a fixed material system. 

The moduli of elasticity, G for shear and E for tension, are ratios of stress to strain as measured within the proportional limits of the material. Thus the modulus is really a measure of the rigidity for shear of a material or its stiffness in tension and compression. For shear or torsion, the modulus analogous to that for tension is called the shear modulus or the modulus of rigidity, or sometimes the transverse modulus. 

Experimental load deflection curves (Fig. 3.) illustrate the large difference in crack propagation observed in each case. A difference in stiffness between both bonded specimens is observed and results from either a difference in the bond line quality or from interfacial conditions. For both specimens, adherends were made from the same sample of wood. Both wood substrates contained no apparent defects and had the same longitudinal Young s modulus (14500 MPa). Both also had the same growth characteristics (oven dry specific density, annual growth rings), and as a consequence very close values of transverse and shear modulus adjacent to the bond line. Thus, any difference in stiffness is likely to be due to... 

As shown in Figures 14.5 and 14.6, the axial extensional stiffness C33 and axial Young s modulus 3 rise sharply with increasing k but become saturated above k — A, closely following the behavior of P and P. The transverse extensional stiffness transverse Young s modulus and the axial (C ) and transverse (C ) shear moduli show moderate decreases while the cross-plane stiffness C 3 exhibits a slight increase. The anisotropy patterns of C33 3 and... 

Coupling terms of laminate stiffness matrix Bending terms of laminate stiffness matrix Longitudinal Young s modulus of the lamina Transverse Young s modulus of the lamina In-plane shear modulus of the lamina Out-of-plane shear modulus of lamina (in the 1-3 plane) Out-of-plane shear modulus of lamina (in 2-3 plane) Moment stress resultants per unit width Force stress resultants per unit width Laminate reduced stiffness terms Transformed reduced stiffness terms... 

It may be seen from Fig. 3 that, as transverse crack density increases, all stiffness properties of the laminate are significantly reduced. Longitudinal and transverse moduli of the undamaged laminate, calculated from the classical lamination theory, are 166.5 GPa, shear modulus 44 GPa, Poisson s ratio 0.19. When transverse cracking in the 90° layer reaches saturation, the laminate longitudinal and shear moduli are predicted to lose more than 45% of their value. Inclusion of tensile residual stresses into the analysis would lead to even more significant reduction in the longitudinal modulus and Poisson s ratio, but reduction in shear modulus would remain the same. Predictions for a [O/OOi], SiC/CAS laminate are shown in Fig. 4. 

Numerical results for SiC/CAS cross-ply laminates of different lay-ups have shown that transverse and longitudinal cracks cause significant reduction of all laminate stiffness properties. Reduction in the longitudinal modulus occurs mainly due to transverse macrocracks, while the shear modulus appears to be the most affected by the presence of longitudinal macrocracks. 

The techniques of analysis are essentially the same as when isotropic adherends are used, although due attention must be paid to the low longitudinal shear stiffness of unidirectional composites. As Demarkles (1955) showed, even with metallic adherends in which the shear modulus is of the order of 25-30% of Young s modulus, it is necessary to take account of adherend shears. With unidirectional composites, this modulus ratio may be as low as 2%, and so the adherend shears become extremely important. The use of lamination techniques in which fibres are placed at different angles to the plate axis leads to reduced longitudinal and increased shear moduli. However, the transverse modulus (i.e. through the thickness of the adherend) remains... 

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