Understanding lamina stiffness

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. 

The most general linear elastic relationship between the stresses and strains is given as 

3 Example of randomly distributed fibres computationally generated over a lamina cross-section. 

4 Example of a representative volume element (adapted from [14]). 

When the stresses and strains are symmetric the nnmber of independent constants is rednced to 36. Hooke s law can be written in a contracted notation  

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Tsai and Pagano [2-7] ingeniously recast the stiffness transformation equations to enable ready understanding of the consequences of rotating a lamina in a laminate. By use of various trigonometric identities between sin and cos to powers and sin and cos of multiples of the angle, the transformed reduced stiffnesses. Equation (2.85), can be written as... 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

This section is devoted to those special cases of laminates for which the stiffnesses take on certain simplified values as opposed to the general form in Equation (4.24). The general force-moment-strain-curvature relations in Equations (4.22) and (4,23) are far too comprehensive to easily understand. Thus, we build up our understanding of laminate behavior from the simplest cases to more complicated cases. Some of the cases are almost trivial, others are more specialized, some do not occur often in practice, but the point is that all are contributions to the understanding of the concept of laminate stiffnesses. Many of the cases result from the common practice of constructing laminates from laminae that have the same material properties and thickness, but have different orientations of the principal material directions relative to one another and relative to the laminate axes. Other more general cases are examined as well. 

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