Anisotropic material engineering constants

Compare the transformed orthotropic compliances in Equation (2.88) with the anisotropic compliances in terms of engineering constants in Equation (2.91). Obviously an apparenf shear-extension coupling coefficient results when an orthotropic lamina is stressed in non-principal material coordinates. Redesignate the coordinates 1 and 2 in Equation (2.90) as X and y because, by definition, an anisotropic material has no principal material directions. Then, substitute the redesignated Sy from Equation (2.91) in Equation (2.88) along with the orthotropic compliances in Equation (2.62). Finally, the apparent engineering constants for an orthotropic iamina that is stressed in non-principal x-y coordinates are... 

In summary, the engineering constants for anisotropic materials and orthotropic materials loaded in non-principal material coordinates can be most effectively thought of In strictly functional terms ... 

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... 

As with other anisotropic materials, Hooke s Law for cubic crystals may be expressed in terms of the engineering elastic constants. Equation (2.57) can be written as... 

Again each term on the right hand side of Elq. 2.40 represents a double summation and each coefficient of strain is an independent set of material parameters. Thus, many more than 81 parameters may be required to represent a nonlinear heterogeneous and anisotropic material. Further, for viscoelastic materials, these material parameters are time dependent. The introduction of the assumption of linearity reduces the number of parameters to 81 while homogeneity removes their spatial variation (i.e., the parameters are now constants). Symmetry of the stress and strain tensors (matrices) reduces the number of constants to 36. The existence of a strain energy potential reduces the number of constants to 21. Material symmetry reduces the number of constants further. For example, an orthotropic material, one with three planes of material symmetry, has only 9 constants and an isotropic material, one with a center of symmetry, has only two independent constants (and Eq. 2.39 reduces to Eq. 2.28). Now it is easy to see why the assumptions of linearity, homogeneity and isotropy are used for most engineering analyses. 

The magnitude of the elastic moduli obtained for an anisotropic material will depend on the orientation of the coordinates used to describe the material elastic response. However, if the material elastic moduli are known for coordinates aligned with the principal material directions, then the elastic moduli for any other orientation can be determined through appropriate transformation equations. Thus, only four elastic constants are needed in order to fully characterize the in-plane maaoscopic elastic response of an orthotropic lamina. The reference coordinates in the plane of the lamina are aligned with longitudinal axis (L) parallel to the fibers, and the transverse axis (7) perpendicnlar to the fibers. The engineering orthotropic elastic moduli of the lamina defined earher are... 

These three engineering constants are sufficient to define the stress-strain relationships of an isotropic material. Hence, one simple test provides all material properties needed to completely define the mechanical response of a linear elastic isotropic material. A larger number of tests will be needed to obtain the engineering constants required to define the macroscopic elastic response of an anisotropic lamina. 

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