L&L - Elasticity - couple questions on notation

[osnarf]

Edit - maybe I have the notaton figured out now and am just confused.

Homework Statement

The below relevant quotations come from Landau and Lifgarbagez, volume 7 : The theory of elasticity, chapter 1.

2. Relevant quotations

Page 2:

equation 1.2
dl'² = dl² + 2u_ikdx_idx_k

where u_ik is the strain tensor, dl is the original distance between two points, and dl' is the deformed distance between the two points. x_i are co-ordinates.

Like any symetrical tensor, u_ik can be diagonalised at any given point.
...
If the strain tensor is diagonalised at any given point, the element of length (1.2) near it becomes:

dl'² = (D_ik + 2u_ik)dx_idx_k\
= (1 + 2u⁽¹⁾)dx₁² + (1 + 2u⁽²⁾)dx₂² + (1 + 2u⁽³⁾)dx₃²

^^^Where, in the book, D is a squigly d (lower case delta?). Looks like the d used in variations.

Page 5 (last paragraph):

Let us determine the moment of the forces on a portion of the body. The moment of the force F can be written as an antisymmetrical tensor of rank two, whose components are F_ix_k - F_kx_i, where x_i are the co-ordinates of the point where the force is applied.

The Attempt at a Solution

In quotation 2 - where did D come from? What is it?

In quotation 3 - Is he using Einstiein summation notation still, because I don't understand why there would only be two components of the force, or two co-ordinates, because everything so far as been 3 dimensional. I don't understand how this is the moment tensor (it does make sense if its done for all 3 2d planes (xy, yz, zx), and F_i is a component of force in the i direction, then it's a scalar returned for the norm of the moment, directed in the direction normal to the plane - but then you get either a 2nd order diagonal tensor, or a first order tensor, neither of which is an antisymmetrical tensor of rank 2)

Thanks for your help.

 

[osnarf]

Just got to this part:

o_ik = -pD_ik
It's (the stress tensor o_ik's) non-zero components are simply equal to the pressure.

Where D is the same squigly D from quotation 2 above, and p is pressure.

So I take it D is the identity tensor? Could somebody please explain quotation 2 to me. I can post more if you don't have the book.

 

[osnarf]

Another quick question - why are all the tensors (which seem to be 3x3 matrices) written as u_ik (only two subscripts)?

EDIT - one last thing (to someone who has an older version of the book). Mine was printed kind of sloppily and I can't make out what it says above equation 2.9 (page 8 in my book). I'm trying to read the sentence that's starts out Substituting (2.8) in the first integral, we find... what are the terms on the side of the equation opposite the surface integral? Thanks again.