Identity tensor

The identity tensor by is zero for i J and unity for i =J. The coefficient X is a material property related to the bulk viscosity, K = X + 2 l/3. There is considerable uncertainty about the value of K. Traditionally, Stokes hypothesis, K = 0, has been invoked, but the vahdity of this hypothesis is doubtful (Slattery, ibid.). For incompressible flow, the value of bulk viscosity is immaterial as Eq. (6-23) reduces to... 

These are covariant and contravariant representations of the Cartesian identity tensor, and inverses of each other. 

We next present a relationship between the inverses of the projected tensors S and T within the soft and hard subspaces, respectively. By requiring that the matrix product of the block matrices (A.l) and (A.2) yield the identity tensor, it is straightforward to show that the elements of the matrix... 

In this expression, I is the identity tensor, W and (VV)T are, respectively, the velocity-gradient tensor and its transpose (Appendix B.2). 

The macroscopic drained stiffness tensor Chom(f) and Biot s coefficient B( ) can then be expressed as a function of the average of A over the fluid saturated pore space (I=fourth identity tensor) ... 

Here, e. and Epj are homogeneous inelastic strains defined in the respective phase domains L and are dimensionless constants satisfying + /ps Lj =1(1 the identity tensor of rank 4) M, and Mj are constants having the same dimension as the elastic modulus and satisfying +/pszM2 = 0 N, and Nj are also dimensionless constants relating to... 

Cartesian component notation offers an extremely convenient shorthand representation for vectors, tensors, and vector calculus operations. In this formalism, we represent vectors or tensors in terms of their typical components. For example, we can represent a vector A in terms of its typical component Ait where the index i has possible values 1, 2, or 3. Hence we represent the position vector x as x, and the (vector) gradient operator V as Note that there is nothing special about the letter that is chosen to represent the index. We could equally well write x . x/ . or x , as long as we remember that, whatever letter we choose, its possible values are 1, 2, or 3. The second-order identity tensor I is represented by its components %, defined to be equal to 1 when i = j and to be equal to 0 if z j. The third-order alternating tensor e is represented by its components ,< , defined as... 

Evaluation involving the combination of the local g-tensors into the molecular-state g-tensor, and that combining the local and pair-interaction D-tensors into the molecular-state / -tensor, are satisfied even when the tensors are neither diagonal nor collinear care must be taken, however, to refer all the tensors to the same reference system. Therefore the difference in centres means not only that SA Sb> the systems SA = SB can be taken as different when otherwise identical tensors are differently oriented. 

It can be proved that U is a symmetric and positive definite tensor, which is a measure of the local stretching (or contraction) of material at X. V is also is a symmetric and positive definite second-order tensor called the left stretch tensor, which is a measure of the local stretching (or contraction) of the material in the deformed configuration at x. R is a proper orthogonal tensor, that is, R R = I or detR = 1, where T means transpose, I is the identity tensor, and det is the determinant. 

In the above equations, 7 is the identity tensor and T is the Maxwell stress tensor Ty is the representation of this tensor with Einstein notation. 

The metric tensor of Euclidean space 3 is the identity tensor I. Its componental representation is gij, K or 5/, depending on basis, that is. 

---