Interrelationships among elastic constants

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Problem 2.2

The entries in Table 2.2a express, for isotropic media, the quantities at the heads of the columns in terms of the pairs of elastic constants or velocities at the left ends of the rows. The first three entries in the ninth row are equations (2.1j), (2.1a), and (2.1k) and the next two entries in the same row are formulas for the P- and S- wave velocities and (see problem 2.5). Starting from these five relations, derive the other relations in the table.

Background

For isotropic media, any two of the elastic constants can be considered as independent and the others can be expressed in terms of these two. The P- and S-wave velocities, and , given by the equations


(2.2a)

[see Sheriff and Geldart, 1995, equations (2.28) and (2.29)] can also be expressed in terms of any two elastic constants (plus the density ).

Solution

Denoting the equations by row and column (as for matrix elements) and using a comma instead of a period, we use equations (9,1) to (9,3) to derive the equations that do not involve and , then we use equations (9,6) and (9,7) (see equation (2.2a)) to derive the rest. From equation (9,1),


(3,5)

From equation (9,2),


(5,5)

Solving equation (9,2) for , we have ,


(6,4)

From equation (9,3),


(7,5)

Equating from equations (9,2) and (9,3) [or from equations (6,4) and (7,5)] gives


(8,4)

thus

(6,3)

and

(4,5)

Solving equation (4,5) for gives


(8,2)

Substituting equation (4,5) into equation (8,4), we get,


(4,4)

We use equation (7,5) to eliminate from equation (6,3):

that is,

so


(5,3)

Solving for , we get


(7,2)
Figure 2.2a)  Relations between elastic constants and velocities for isotropic media.

We use equation (8,4) to express in equation (9,1) in terms of . Thus


(8,1)

Solving equation (8,1) for gives


(2,5)

We now eliminate from equation (9,1) using equation (7,5)


(7,1)

Solving equation (7,1) for and gives


(3,3)


(2,4)

Next we use equation (6,4) to eliminate from equation (9,1):


(6,1)

Using equations (9,1) and (5,5) we get


(5,1)

Solving equation (5,1) for and gives


(3,2)


(1,4)

Using equations (4,4) and (4,5) to replace in equation (9,1) by gives


(4,1)

Solving equation (4,1) for and , we get


(1,3)


(2,2)

The last equation (of this group) can be obtained by substituting equation (1,3) into equation (4,5):


(1,5)

Equations (10,1) to (10,3) express , , and in terms of the P- and S-wave velocities, and (see equation 2.2a). To introduce and , we write equations (10,1) to (10,3) in terms of and . Thus,


(10,1)


(10,2)


(10,3)

Equation (10,4) is the second equation in equation (2.2a). To get equation (10,5), we write


(10,5)

To verify column 6, we start with equation (9,6) and express and in terms of the required pair of constants. Thus, using equations (1,4) and (1,5) we get


(1,6)

Following the same procedure, using equations (2,4) and (2,5), we get


(2,6)

In the same way, we get the following results:


(3,6)


(4,6)


(5,6)


(6,6)


(7,6)


(8,6)

Column 7 is merely the square root of column 4 after dividing by . Column 8 is obtained by dividing column 7 by column 6.

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