Unidirectionally reinforced lamina

For a unidirectionally reinforced lamina in the 1-2 plane as shown in Figure 2-7 or a woven lamina as in Figure 2-1, a plane stress state is defined by setting... 

Figure 2-8 Physical Symmetry of a Unidirectionally Reinforced Lamina...

First, consider uniaxial tension loading in the 1-direction on a flat piece of unidirectionally reinforced lamina where only the gage section is shown in Figure 2-20. The specimen thickness is not just one lamina, but several laminae all of which are at the same orientation (a single lamina would be too fragile to handle). The strains and E2 are measured so, by definition,... 

For each of the failure criteria, we will generate biaxial stresses by off-axis loading of a unidirectionally reinforced lamina. That is, the uniaxial off-axis stress at 0 to the fibers is transformed into biaxial stresses in the principal material coordinates as shown in Figure 2-35. From the stress-transformation equations in Figure 2-35, a uniaxial loading obviously cannot produce a state of mixed tension and compression in principal material coordinates. Thus, some other loading state must be applied to test any failure criterion against a condition of mixed tension and compression. 

The foregoing are but examples of the types of mechanics of materials approaches that can be used. Other assumptions of physical behavior lead to different expressions for the four elastic moduli for a unidirectionally reinforced lamina. For example, Ekvall [3-2] obtained a modification of the rule-of-mixtures expression for and of the expression for E2 in which the triaxial stress state in the matrix due to fiber restraint is accounted for ... 

A cross-ply laminate in this section has N unidirectionally reinforced thotropic) layers of the same material with principal material directions srnatingly oriented at 0° and 90° to the laminate coordinate axes. The sr direction of odd-numbered layers is the x-direction of the laminate, e fiber direction of even-numbered layers is then the y-direction of the linate. Consider the special case of odd-numbered layers with equal kness and even-numbered layers also with equal thickness, but not essarily the same thickness as that of the odd-numbered layers, te that we have imposed very special requirements on how the fiber sntations change from layer to layer and on the thicknesses of the ers to define a special subclass of cross-ply laminates. Thus, these linates are termed special cross-ply laminates and will be explored his subsection. More general cross-ply laminates have no such con-ons on fiber orientation and laminae thicknesses. For example, a neral) cross-ply laminate could be described with the specification t/90° 2t/90° 2t/0° t] wherein the fiber orientations do not alter-e and the thicknesses of the odd- or even-numbered layers are not same however, this laminate is clearly a symmetric cross-ply lami-e. 

Fibers are often regarded as the dominant constituents in a fiber-reinforced composite material. However, simple micromechanics analysis described in Section 7.3.5, Importance of Constituents, leads to the conclusion that fibers dominate only the fiber-direction modulus of a unidirectionally reinforced lamina. Of course, lamina properties in that direction have the potential to contribute the most to the strength and stiffness of a laminate. Thus, the fibers do play the dominant role in a properly designed laminate. Such a laminate must have fibers oriented in the various directions necessary to resist all possible loads. 

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. 

In the case of unidirectionally reinforced layers, or laminae (not to be confused with the similar-looking lamellae, cf. Section 1.3.6.1), a composite is formed by laminating the layers together. If all the layers were to have the same fiber orientation (see Figure 5.117a), the material would still be weak in the transverse direction. Therefore,... 

The characteristic features of a cord—rubber composite have produced the netting theory (67—70), the cord—inextensible theory (71—80), the classical lamination theory, and the three-dimensional theory (67,81—83). From structural considerations, the fundamental element of cord—rubber composite is unidirectionally reinforced cord—mbber lamina as shown in Figure 5. From the principles of micromechanics and orthotropic elasticity laws, engineering constants of tire T cord composites in terms of constitutive material properties have been expressed (72—79,84). The most commonly used Halpin-Tsai equations (75,76) for cord—mbber single-ply lamina L are expressed in equation 5 ... 

Fig. 5. Calendered unidirectionally reinforced single-ply cord—mbber lamina where 0 is...

This chapter is devoted to the analysis of the elastic properties and their characterization for laminated advanced composites. It starts with a general overview of composite stiffness and then moves to lamina analysis focused on unidirectional reinforced composites. The analysis of laminated composites is addressed through the classical lamination theory (CLT). The last section describes full-field techniques coupled with inverse identification methods that can be employed to measure the elastic constants. 

The typical building block of a composite structure is the lamina, with a typical thickness of 0.125 mm. The lamina stress-strain relationships are described for orthotropic, transverse isotropic and isotropic materials. When a lamina is reinforced with unidirectional fibres it can be assumed to be a transversely isotropic material. In this chapter, theoretical determination of lamina elastic properties, assumed to be a transversely isotropic material, using micromechanics approaches is presented and illustrated with experimental data. 

Figure 5.120 Elastic constants for unidirectional (a) glass fiber reinforced epoxy and (b) graphite fiber-reinforced epoxy laminae. Reprinled, by permission, from P. C. Powell and A. J. I. Housz, Engineering with Polymers, pp. 222, 223. Copyrighl 1998 by Stanley Thorned Publishers.

Consider the same unidirectional lamina with the stresses now applied perpendicular to the fiber axis as shown in Fig. 12. The local stress at the fiber matrix interface can be calculated and compared to the nominally applied stress on the whole lamina to give K, the stress concentration factor. The plot of the results of this analysis shows that the interfacial stresses at the point of maximum principal stress can range up to 2.6 times the applied stress depending on the moduli of the constituents and the volume fraction of the reinforcement. For a typical graphite-epoxy composite, with a modulus ratio of 70 and a volume fraction of 70 % the stress concentration factor at the interface is about 2.4. That is, the local stresses at the interface are a factor of 2.4 times greater than the applied stress. 

This chapter has focused on the elastic qualities of advanced fibre-reinforced composites, in terms of characterization, measurement and prediction from the basic constituents, i.e. the fibre and matrix. Unidirectional fibre-reinforced polymers were the material selected for these brief elastic analyses. These comprised the micromechanics approaches which were applied to predicting the lamina elastic properties from the basic constituents and the classical lamination theory which was used to predict the elastic properties of composites materials composed of several laminae stacked at different orientations. The theoretical predictions were compared against available experimental data, illustrating the predictive capability of the theoretical analysis. Finally, a brief overview was delivered on the identification methods for elastic properties based on full-field measurements. This approach proved to be suitable for anisotropic and heterogeneous materials. 

Micromechanics of a Unidirectional Fiber-Reinforced Composite Layer (Lamina)... 

The prediction of macromechanical strength properties of a unidirectional fiber-reinforced composite lamina using micromechanics models has met with less success than the elastic moduli predictions of the earlier sections. The structural designer will most likely rely primarily on results from mechanical tests that measure the macromechanical strength properties of the composite lamina directly. Nevertheless, it is instructive to look at the micromechanics model for tensile strength in the fiber direction of a lamina to gain a better understanding of how the composite functions. 

---