Orthotropic material definition

Identity Equation (2.97) by interpreting Equation (2.88) using Equation (2.90) as well as Equations (2.91) and (2.62). Explain tbe key logical step that enables you to use both Equations (2.90) and (2.91) for anisotropic materials and Equations (2.62) and (2.88) for orthotropic materials in this problem. That Is, in what way can we interpret a material as satistying both definitions ot a material ... 

Qi are for anisotropic materials. Qy for orthotropic materials are obtained by deleting Ug and Uy from the definitions of Qy. 

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... 

What has been accomplished in preceding sections on stiffness relationships serves as the basis for determination of the actual stress field what remains is the definition of the allowable stress field. The first step in such a definition is the establishment of allowable stresses or strengths in the principal material directions. Such information is basic to the study of strength of an orthotropic lamina. 

The relationships between elastic constants which must be satisfied for an isotropic material impose restrictions on the possible range of values for the Poisson s ratio of -1 < v <. In a similar manner, there are restrictions in orthotropic and transversely isotropic materials. These constraints are based on considerations of the first law of thermodynamics [15]. Moreover, these constraints imply that both the stiffness and compliance matrices must be positive-definite, i.e. each major diagonal term of both matrices must be greater than 0. 

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