Contracted notation strains

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... 

In crystals with the LI2 structure (the fcc-based ordered structure), there exist three independent elastic constants-in the contracted notation, Cn, C12 and 044. A set of three independent ab initio total-energy calculations (i.e. total energy as a function of strain) is required to determine these elastic constants. We have determined the bulk modulus, Cii, and C44 from distortion energies associated with uniform hydrostatic pressure, uniaxial strain and pure shear strain, respectively. The shear moduli for the 001 plane along the [100] direction and for the 110 plane along the [110] direction, are G ooi = G44 and G no = (Cu — G12), respectively. The shear anisotropy factor, A = provides a measure of the degree of anisotropy of the electronic charge... 

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 ... 

Eor this reason and the confusion that can arise in defining the momentum and strain-displacement equations, one must use care in solving problems with the contracted notation. The value of p in terms of i,j varies notoriously between authors, but the IEEE standard [4] is the most commonly accepted version here p= (i +f) (1 - sgn(i -j) I) -b sgn(j -j) (9 - i -j) for i,j = 1,2,3 here the vertical bars represent an absolute value while sgn(-) is the signum, zero when x is zero and the sign of x otherwise. 

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... 

The BOLS notation indicates that the contraction of the mean lattice constant of the entire solid originates from the CN imperfection-induced bond contraction of surface atoms and the fraction of the surface atoms of the entire solid. The following expressions formulate the surface strain and nanosolid densihcation [82, 83] ... 

---