3.2 COMPOSITE NOTATION

Hooke's law relationships, starting with the proportionality between stress and strain, govern the theory of the mechanical properties of composite materials. Their inherent anisotropy means that their mathematical description requires reference frames (coordinate systems), and transformations between these reference frames. Physical entities that obey certain transformation relations are called tensors. Tensors come in different ranks, depending on their complexity. A scalar is a zero-rank tensor since it has only magnitude and no direction. A vector is a tensor of the first rank. The various coefficients of expansion (e.g., thermal, moisture) for a composite are second rank tensors, and so are stresses, strains, curvatures, moments and loads. Tensor notation requires two subscripts; i and j, each of which can take on three values. The first subscript denotes a direction normal to the area or face on which that parameter acts. The second subscript denotes the actual direction in which that parameter (e.g., stress) is acting. For example, normal stresses give repeated subscripts; thus ? _ii means a stress normal to the number one face. Shear stresses have mixed subscripts, viz., i ? j, such as ? ₁₂. This means a shear stress in the 1-2 directions plane. Since i and j can each assume 3 values it would seem that there are nine independent parameters (e.g., stresses) which can act on any material element. Considerations of symmetry (or rotation of the element) indicate that parameters with reversed subscripts are equal, e.g.,