Stiffness and compliance transformations

Stress and strain are second-order tensors while stiffness and compliance are fonrth-order tensors [14]. Hence these entities are mled by the tensor transformation laws that establish the relationships between the components 

25 Transverse Poisson s ratio of unidirectional carbon/epoxy composite as a function of fibre volume fraction (E = 5.28 GPa, 

Therefore it becomes necessary to use the transformation laws which relate the tensors in one coordinate system to another in a rotated coordinate system. Briefly, the relationship between the stresses in the principal material and global coordinates [14] is given by 

26 Schematic representation of (a) principai materiai coordinate system (1-2-3) and (b) iamina coordinated system (x-y-z). 

Therefore the transformed stiffness [C] may be determined using the relationships obtained for the stress and strain transformation. 

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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. 

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