Contracted notation stresses

The generalized Hooke s law relating stresses to strains can be written in contracted notation as... 

Table 2-1 Tensor versus Contracted Notation for Stresses and Strains...

Contracted notation is a rearrangement of terms such that the number of indices is reduced although their range increases. For second-order tensors, the number of indices is reduced from 2 to 1 and the range increased from 3 to 9. The stresses and strains, for example, are contracted as in Table A-1. Similarly, the fourth-order tensors for stiffnesses and compliances in Equations (A.42) and (A.43) have 2 instead of 4 free indices with a new range of 9. The number of components remains 81 (3 = 9 ). 

In contracted notation, the stress-strain and strain-stress relations. Equations (A.42) and (A.43), are written as... 

The stress-strain relations in this book are typically expressed in matrix form by use of contracted notation. Both the stresses and strains as well as the stress-strain relations must be transformed. First, the stresses transform for a rotation about the z-axis as in Figure A-1 according to... 

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 ... 

A three-dimensional representation of the stresses in contract notation is presented in Fig. 11.5. 

Contracted notation (CN) has been introduced in the equations, where it is convenient for computer solution to use single-digit subscripts to designate stress and strain terms. The relationships between (1) tensor or elasticity notation, (2) engineering notation (EN), and (3) CN are defined in Table 8.3. 

Let us first consider the case of an isotropic material, then simplify it for the case of an orthotropic material (same properties in the two directions orthogonal to the fiber axis—in this case, directions 2 and 3), snch as a nnidirectionally reinforced composite lamina. Eqnation (5.128) can be written in terms of the strain and stress components, which are conpled dne to the anisotropy of the material. In order to describe the behavior in a manageable way, it is cnstomary to introdnce a reduced set of nomenclature. Direct stresses and strains have two snbscripts—for example, an, 22, ti2, and Y2i, depending on whether the stresses and strains are tensile (a and s) or shear (t and y) in natnre. The modnli should therefore also have two subscripts En, E22, and G 2, and so on. By convention, engineers nse a contracted form of notation, where possible, so that repeated snbscripts are reduced to just one an becomes a, En becomes En but Gn stays the same. The convention is fnrther extended for stresses and strains, such that distinctions between tensile and shear stresses and strains are... 

To avoid cumbersome notation, the symbol for the conventional mobilities has been contracted to u. The absence of a bar above the u stresses that it is not the absolute mobility... 

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