L&L - Elasticity - couple questions on notation

In summary, the author is discussing the theory of elasticity and how strains in a body can be calculated. The author states that the strain tensor, uik, can be diagonalized at any given point and that the moment of the force on a portion of the body can be determined.
  • #1
osnarf
209
0
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 + 2uikdxidxk
where uik is the strain tensor, dl is the original distance between two points, and dl' is the deformed distance between the two points. xi are co-ordinates.

Like any symetrical tensor, uik 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 = (Dik + 2uik)dxidxk\
= (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 Fixk - Fkxi, where xi 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 Fi 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.
 
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  • #2
Just got to this part:

oik = -pDik
It's (the stress tensor oik'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.
 
  • #3
Another quick question - why are all the tensors (which seem to be 3x3 matrices) written as uik (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.
 
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Related to L&L - Elasticity - couple questions on notation

1. What is elasticity?

Elasticity is a measure of how responsive a variable is to changes in another variable. In economics, it specifically refers to the responsiveness of quantity demanded or supplied to changes in price.

2. What is the difference between price elasticity of demand and price elasticity of supply?

Price elasticity of demand measures the change in quantity demanded in response to a change in price, while price elasticity of supply measures the change in quantity supplied in response to a change in price. They both use the same formula, but the difference lies in the perspective - demand or supply.

3. How is elasticity measured?

Elasticity is measured using the formula: (Percentage change in quantity / Percentage change in price). The result is a unitless number, and the magnitude of the number indicates the degree of elasticity.

4. What are some factors that affect elasticity?

Some factors that affect elasticity include the availability of substitutes, the necessity of the good or service, and the proportion of the consumer's income spent on the good or service. Generally, goods or services with more substitutes, less necessity, and a smaller proportion of income spent will have a higher elasticity.

5. How does elasticity impact pricing decisions?

Elasticity plays a crucial role in pricing decisions. If demand for a good is highly elastic, a price increase will likely lead to a significant decrease in quantity demanded, and therefore, a decrease in revenue. On the other hand, if demand is inelastic, a price increase may result in a smaller decrease in quantity demanded, and possibly an increase in revenue. This information helps businesses set prices that maximize their profits.

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