Stiffness unidirectional composites

Figures 9.19 and 9.20 present a survey of the mechanical properties of some (unidirectional) composites, in comparison with some other materials. In Figure 9.19 the values of modulus and strength are plotted as such, while in Figure 9.20 these values have been divided by the specific mass. From Figure 9.20 the enormous advantage of composites with respect to stiffness and strength per unit weight, in comparison to metals, is clearly visible. The modem carbon and aramide composites are superior to those based on glass fibres, for the specific stiffness even by a factor between 4 and 5.

Hsiao HM, Daniel IM. Effect of fibre waviness on stiffness and strength reduction of unidirectional composites under compressive loading. Compos Sci Technol 1996 56 581-93. 

We now turn our attention from considerations of stiffness to stress-strain and tensile behavior. Variations in strength and modulus as a function of direction (Broutman and Krock, 1967, Chapter 12) have been treated by several investigators for example, Tsai (1965) and Brody and Ward (1971). Even though the polymer matrix typically has such a low modulus that it does not contribute much overall to the composite modulus, the matrix can by no means be neglected, because failure often involves catastrophic crack growth in the matrix (see below). Stress-strain curves for unidirectional composites are typically fairly linear up to failure for loading in the direction of the fibers (Broutman and Krock, 1967, p. 370), but quite nonlinear transverse to the fiber direction. The stress to rupture is also very low in the latter case, presumably due to a high concentration of stress in the matrix. 

The Halpin-Tsai model is a well-known composite theory to predict the stiffness of unidirectional composites as a functional of aspect ratio. In this model, the lorrgitudi-nal Ejj and transverse engineering modtrli are expressed in the following general form ... 

For most applications, resistance to sfiesses in more than one direction is essential, and in these cases, a unidirectional composite is not acceptable. Strength and stiffness in both dimensions of a plane, as in a car body panel, or isotropically in all three dimensions is normally required. Consequently, the orientation or the fibres must be modified to provide resultants in the required directions. This may be achieved in two directions by the use of woven or knitted cloths or with non-woven, random felts or mats. In three dimensions, random orientation of the reinforcement is usually the only possibility. However, the maximum strength attainable drops sharply in multidirectional composites, not only because of the smaller fraction of fibres contributing to resisting the stress but also becanse the maximum packing density of the reinforcing fabric decreases and hence the overall volume fraction of reinforcement decreases. Despite this fall-off, the strength of the fibres is snch that a very substantial reinforcement may still result. 

This chapter began by describing briehy the elasticity of anisotropic materials, providing the fundamental relationships and the allowed simplihcations by the existence of material planes of symmetry. The current unidirectional composites are usually classihed as transversely isotropic materials. In this case, only hve independent elastic constants are necessary to fully characterize unidirectional composites. The micromechanics provides the analytical and numerical approaches to predict the elastic constants based on the elastic properties of the composite constituents. Several classical closed formulas are revisited and compared with experimental data. Finally, stiffness and compliance transformations are given in the context of unidirectional composites. Experimental data are used to assess theoretical predictions and illustrate the off-axis in-plane elastic properties. 

Fig. 11 Effect of consolidation temperature on the flexural stiffness of unidirectional composites produced by using coextruded PP tape technology. Over the range of consolidation temperature considered here, there is no significant effect on the mechanical properties of the final composite, indicating that this large temperature window would make composites based on coextruded tapes less susceptible to thermal relaxation during consolidation than systems with a smaller temperature processing window. Reproduced with kind permission from Sage Publieatioiis frran [159]...

The tensile stiffness of LCP foils reported by Lefevre [217] was 71 GPa, which is directly comparable to other unidirectional composite laminates described in this analysis. Since the mechanical properties of a two-dimensional foil are shown in Fig. 14, the volume fraction of reinforcement is described as 100%, and the stiffness of the composite is not shown. The LCP fibre-based unidirectional composite with the highest mechanical properties included in this review was reported by Pegoretti et al. [188] and was based on LCP fibres with a tensile stiffness of 72-84 GPa (depending on gauge length). However, due to the final composite fibre volume fraction of approximately 50%, the composites had a tensile stiffness of approximately 58 GPa. As before, although the precursor fibres appeared to be stiffer than the PE fibres reported by Marais and Feillard, the final composites were less stiff, which is probably at least partially due to the lower fibre volume fraction of this LCP-based self-reinforced polymer composite. 

Fig. 14 Stiffness of unidirectional composites compared to the stiffness of the precursor fibres. Source data by polymer system PE [31], LCP [217], PA (aramid) [10], PP [161], PET [135], PMMA [25], Note that the PET and PMMA composite samples shown here were tested in flexure, rather than in tension...

The 1, 2, and 3 axes in Fig. 4.29 are special and are called the ply axes, or material axes. The 1 axis is in the direction of the fibers and is called the longitudinal axis or the fiber axis. The longitudinal axis typically has the highest stiffness and strength of any direction. Any direction perpendicular to the fibers (in the 2, 3 plane) is called a transverse direction. Sometimes, to simplify analysis and test requirements, ply properties are assumed to be the same in any transverse direction. This is the transverse isotropy assumption it is approximately satisfied for most unidirectional composite plies. 

Micromechanical models have been widely used to estimate the mechanical and transport properties of composite materials. For nanocomposites, such analytical models are still preferred due to their predictive power, low computational cost, and reasonable accuracy for some simplified stmctures. Recenfly, these analytical models have been extended to estimate the mechanical and physical properties of nanocomposites. Among them, the rule of mixtures is the simplest and most intuitive approach to estimate approximately the properties of composite materials. The Halpin-Tsai model is a well-known analytical model for predicting the stiffness of unidirectional composites as a function of filler aspect ratio. The Mori-Tanaka model is based on the principles of the Eshelby s inclusion model for predicting the elastic stress field in and around the eflipsoidal filler in an infinite matrix. 

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... 

The experimental procedures for determining engineering elastic constants for unidirectional composite lamina will be discussed in Section 8.3. The stiffness matrix can be expressed in terms of engineering elastic constants by taking the inverse of Equation 8.52 ... 

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