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 $ {\alpha } $ and $ {\beta } $ (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, $ {\mathrm {\alpha } } $ and $ {\mathrm {\beta } } $, given by the equations


$ {\begin{aligned}{\mathrm {\rho } }{\mathrm {\alpha } }^{2}=\left({\mathrm {\lambda } }+2{\mathrm {\mu } }\right),{\quad }{\mathrm {\rho } }\beta ^{2}=\mu \end{aligned}} $ (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 $ {\mathrm {\rho } } $).

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 $ {\mathrm {\alpha } } $ and $ {\mathrm {\beta } } $, then we use equations (9,6) and (9,7) (see equation (2.2a)) to derive the rest. From equation (9,1),


$ {\begin{aligned}{{\mathrm {\lambda } }\left(E-3{\mathrm {\mu } }\right)=2{\mathrm {\mu } }^{2}-{\mathrm {\mu } }E,}\\{{\mathrm {\lambda } }={\mathrm {\mu } }\left(E-2{\mathrm {\mu } }\right)/\left(3{\mathrm {\mu } }-E\right).}\end{aligned}} $ (3,5)

From equation (9,2),


$ {\begin{aligned}{{\mathrm {\lambda } }\left(2{\mathrm {\sigma } }-1\right)+2{\mathrm {\mu } }{\mathrm {\sigma } }=0,}\\{{\mathrm {\lambda } }=2{\mathrm {\mu } }{\mathrm {\sigma } }/\left(1-2{\mathrm {\sigma } }\right).}\end{aligned}} $ (5,5)

Solving equation (9,2) for $ {\mathrm {\mu } } $, we have $ 2{\mathrm {\mu } }{\mathrm {\sigma } }={\mathrm {\lambda } }\left(1-2{\mathrm {\sigma } }\right) $,


$ {\begin{aligned}{\mathrm {\mu } }={\mathrm {\lambda } }\left(1-2{\mathrm {\sigma } }\right)/2{\mathrm {\sigma } }.\end{aligned}} $ (6,4)

From equation (9,3),


$ {\begin{aligned}{\mathrm {\lambda } }=k-{\frac {2}{3}}{\mathrm {\mu } }.\end{aligned}} $ (7,5)

Equating $ {\mathrm {\mu } } $ from equations (9,2) and (9,3) [or from equations (6,4) and (7,5)] gives


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\mu}} ={\mathrm{\lambda}} \left(1-2{\mathrm{\sigma}} \right)/2{\mathrm{\sigma}} =\frac{3}{2} \left(k-{\mathrm{\lambda}} \right), \end{align} (8,4)

thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} k={\mathrm{\lambda}} \left(1-2{\mathrm{\sigma}} \right)/3{\mathrm{\sigma}} +{\mathrm{\lambda}} ={\mathrm{\lambda}} \left(1+{\mathrm{\sigma}} \right)/3\sigma, \end{align} (6,3)

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\lambda}} =\frac{3{\mathrm{\sigma}} k}{\left(1+{\mathrm{\sigma}} \right)}. \end{align} (4,5)

Solving equation (4,5) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}} gives


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\sigma}} =\frac{{\mathrm{\lambda}} }{3k-{\mathrm{\lambda}} }. \end{align} (8,2)

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


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\mu}} =\frac{3}{2} \left(k-\frac{3{\mathrm{\sigma}} k}{\left(1+{\mathrm{\sigma}} \right)} \right)=\frac{3k\left(1-2{\mathrm{\sigma}} \right)}{2\left(1+{\mathrm{\sigma}} \right)}. \end{align} (4,4)

We use equation (7,5) to eliminate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} from equation (6,3):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} k=\left(k-\frac{2}{3} {\mathrm{\mu}} \right)\left(\frac{1+{\mathrm{\sigma}} }{3{\mathrm{\sigma}} } \right)=\frac{k\left(1+{\mathrm{\sigma}} \right)}{3{\mathrm{\sigma}} } -\frac{2{\mathrm{\mu}} \left(1+{\mathrm{\sigma}} \right)}{9{\mathrm{\sigma}} }, \end{align}

that is,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} k\left(\frac{1+{\mathrm{\sigma}} }{3{\mathrm{\sigma}} } -1\right)=\frac{2{\mathrm{\mu}} \left(1+{\mathrm{\sigma}} \right)}{9{\mathrm{\sigma}} }, \end{align}

so


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} k=\frac{2{\mathrm{\mu}} \left(1+{\mathrm{\sigma}} \right)}{3\left(1-2{\mathrm{\sigma}} \right)}. \end{align} (5,3)

Solving for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}} , we get


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\sigma}} =\frac{\left(3k-2{\mathrm{\mu}} \right)}{2\left(3k+{\mathrm{\mu}} \right)}. \end{align} (7,2)
Figure 2.2a)  Relations between elastic constants and velocities for isotropic media.

We use equation (8,4) to express Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} in equation (9,1) in terms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(k,\; {\mathrm{\lambda}} \right) . Thus


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E=\frac{\frac{3}{2}\left(k-{\mathrm{\lambda}} \right)\left[3{\mathrm{\lambda}} +3\left(k-{\mathrm{\lambda}} \right)\right]}{{\mathrm{\lambda}} +\frac{3}{2} \left(k-{\mathrm{\lambda}} \right)} =\frac{9k\left(k-{\mathrm{\lambda}} \right)}{\left(3k-{\mathrm{\lambda}} \right)}. \end{align} (8,1)

Solving equation (8,1) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} gives


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\lambda}} =3k\left(3k-E\right)/\left(9k-E\right). \end{align} (2,5)

We now eliminate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} from equation (9,1) using equation (7,5)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {E = \frac{{\mu \left( {3k - 2\mu + 2\mu } \right)}}{{\left( {k - 2\mu /3 + \mu } \right)}}}\\ { = 9k\mu /\left( {3k + \mu } \right).} \end{align} (7,1)

Solving equation (7,1) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} gives


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} k ={\mathrm{\mu}} E/3\left(3{\mathrm{\mu}} -E\right), \end{align} (3,3)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\mu}} =3kE/\left(9k-E\right). \end{align} (2,4)

Next we use equation (6,4) to eliminate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} from equation (9,1):


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E={\mathrm{\mu}} \left(\frac{6{\mathrm{\mu}} {\mathrm{\sigma}} +2{\mathrm{\mu}} -4{\mathrm{\mu}} {\mathrm{\sigma}} }{2{\mathrm{\mu}} {\mathrm{\sigma}} +{\mathrm{\mu}} -2{\mathrm{\mu}} {\mathrm{\sigma}} } \right)={\mathrm{\lambda}} \left(1-2{\mathrm{\sigma}} \right)\left(1+{\mathrm{\sigma}} \right)/{\mathrm{\sigma}}. \end{align} (6,1)

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


$ {\begin{aligned}{E={\mathrm {\mu } }\left(6{\mathrm {\mu } }{\mathrm {\sigma } }\right)/\left(2{\mathrm {\mu } }+{\mathrm {\mu } }-2{\mathrm {\mu } }{\mathrm {\sigma } }\right)}\\{{\quad }=2{\mathrm {\mu } }^{2}\left(1+{\mathrm {\sigma } }\right)/{\mathrm {\mu } }=2{\mathrm {\mu } }\left(1+{\mathrm {\sigma } }\right).}\end{aligned}} $ (5,1)

Solving equation (5,1) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} gives


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\sigma}} =E/2{\mathrm{\mu}} -1=\left(E-2{\mathrm{\mu}} \right)/2\mu, \end{align} (3,2)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\mu}} =E/2\left(1+{\mathrm{\sigma}} \right). \end{align} (1,4)

Using equations (4,4) and (4,5) to replace Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left({\mathrm{\lambda}}, \; {\mathrm{\mu}} \right) in equation (9,1) by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left({\mathrm{\sigma}}, \; k\right) gives


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} E=\frac{3k\left(1-2{\mathrm{\sigma}} \right)}{2\left(1+{\mathrm{\sigma}} \right)} \times \frac{\left(9{\mathrm{\sigma}} k+3k-6{\mathrm{\sigma}} k\right)}{\left[3{\mathrm{\sigma}} k+\left(3k-6{\mathrm{\sigma}} k\right)/2\right]} =3k\left(1-2{\mathrm{\sigma}} \right). \end{align} (4,1)

Solving equation (4,1) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}} , we get


$ {\begin{aligned}k=E/3\left(1-2{\mathrm {\sigma } }\right),\end{aligned}} $ (1,3)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\sigma}} =\left(3k-E\right)/6k. \end{align} (2,2)

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


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\lambda}} =\frac{3{\mathrm{\sigma}} k}{\left(1+{\mathrm{\sigma}} \right)} =\frac{3{\mathrm{\sigma}} E}{3\left(1-2{\mathrm{\sigma}} \right)\left(1+{\mathrm{\sigma}} \right)} =\frac{{\mathrm{\sigma}} E}{\left(1+{\mathrm{\sigma}} \right)\left(1-2{\mathrm{\sigma}} \right)}. \end{align} (1,5)

Equations (10,1) to (10,3) express Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\sigma}} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): k in terms of the P- and S-wave velocities, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\alpha}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\beta}} (see equation 2.2a). To introduce Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\alpha}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\beta}} , we write equations (10,1) to (10,3) in terms of $ \left({\mathrm {\lambda } }+2{\mathrm {\mu } }\right) $ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} . Thus,


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {E ={\mathrm{\mu}} \left(\frac{3{\mathrm{\lambda}} +2{\mathrm{\mu}} }{{\mathrm{\lambda}} +{\mathrm{\mu}} } \right)={\mathrm{\mu}} \left[\frac{3\left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right)-4{\mathrm{\mu}} }{\left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right)-{\mathrm{\mu}} } \right] }\\ { {\qquad}{\quad}=\frac{{\mathrm{\rho}}\beta ^{2} \left(3{\mathrm{\rho}}{\mathrm{\alpha}} ^{2} -4{\mathrm{\rho}}\beta ^{2} \right)}{{\mathrm{\rho}}\left({\mathrm{\alpha}} ^{2} -\beta ^{2} \right)} =\frac{{\mathrm{\rho}}\beta ^{2} \left(3{\mathrm{\alpha}} ^{2} -4\beta ^{2} \right)}{{\mathrm{\alpha}} ^{2} -\beta ^{2} }}, \end{align} (10,1)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\sigma}} =\frac{{\mathrm{\lambda}} }{2\left({\mathrm{\lambda}} +{\mathrm{\mu}} \right)} =\frac{\left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right)-2{\mathrm{\mu}} }{2\left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right)-2{\mathrm{\mu}} } =\frac{{\mathrm{\alpha}} ^{2} -2\beta ^{2} }{2\left({\mathrm{\alpha}} ^{2} -\beta ^{2} \right)}, \end{align} (10,2)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} k={\mathrm{\lambda}} +\frac{2{\mathrm{\mu}} }{3} =\left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right)-\frac{4}{3} {\mathrm{\mu}} ={\mathrm{\rho}}\left({\mathrm{\alpha}} ^{2} -\frac{4}{3} \beta ^{2} \right). \end{align} (10,3)

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


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\lambda}} =\left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right)-2{\mathrm{\mu}} ={\mathrm{\rho}}\left({\mathrm{\alpha}} ^{2} -2\beta ^{2} \right). \end{align} (10,5)

To verify column 6, we start with equation (9,6) and express Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\mu}} in terms of the required pair of constants. Thus, using equations (1,4) and (1,5) we get


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {{\mathrm{\rho}}{\mathrm{\alpha}} ^{2} =\left({\mathrm{\lambda}} +2{\mathrm{\mu}} \right)=\frac{E{\mathrm{\sigma}} }{\left(1+{\mathrm{\sigma}} \right)\left(1-2{\mathrm{\sigma}} \right)} +\frac{2E}{2\left(1+{\mathrm{\sigma}} \right)}}\\ {{\qquad}=\left(\frac{E}{1+{\mathrm{\sigma}} } \right)\left(\frac{{\mathrm{\sigma}} }{1-2{\mathrm{\sigma}} } +1\right)=\frac{E\left(1-{\mathrm{\sigma}} \right)}{\left(1+{\mathrm{\sigma}} \right)\left(1-2{\mathrm{\sigma}} \right)}}. \end{align} (1,6)

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


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {{\mathrm{\rho}}{\mathrm{\alpha}} ^{2} =3k\left(\frac{3k-E}{9k-E} \right)+\left(\frac{6kE}{9k-E} \right) }\\ {{\qquad}=\left(\frac{3k}{9k-E} \right)\left(3k-E+2E\right)=\frac{3k\left(3k+E\right)}{9k-E}.} \end{align} (2,6)

In the same way, we get the following results:


$ {\begin{aligned}{{\mathrm {\rho } }{\mathrm {\alpha } }^{2}={\mathrm {\mu } }\left({\frac {E-2{\mathrm {\mu } }}{3{\mathrm {\mu } }-E}}\right)+2{\mathrm {\mu } }=\left({\frac {\mathrm {\mu } }{3{\mathrm {\mu } }-E}}\right)\left(E-2{\mathrm {\mu } }+6{\mathrm {\mu } }-2E\right)}\\{{\qquad }=\left({\frac {\mathrm {\mu } }{3{\mathrm {\mu } }-E}}\right)\left(4{\mathrm {\mu } }-E\right),}\end{aligned}} $ (3,6)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {{\mathrm{\rho}}{\mathrm{\alpha}} ^{2} =\left(\frac{3k{\mathrm{\sigma}} }{1+{\mathrm{\sigma}} } \right)+3k\left(\frac{1-2{\mathrm{\sigma}} }{1+{\mathrm{\sigma}} } \right)=\left(\frac{3k}{1+{\mathrm{\sigma}} } \right)\left({\mathrm{\sigma}} +1-2{\mathrm{\sigma}} \right)} \\ {{\qquad} =\frac{3k\left(1-{\mathrm{\sigma}} \right)}{\left(1+{\mathrm{\sigma}} \right)},} \end{align} (4,6)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\rho}}{\mathrm{\alpha}} ^{2} =\left(\frac{2{\mathrm{\mu}} {\mathrm{\sigma}} }{1-2{\mathrm{\sigma}} } \right)+2{\mathrm{\mu}} =2{\mathrm{\mu}} \left(\frac{{\mathrm{\sigma}} }{1-2{\mathrm{\sigma}} } +1\right)=2{\mathrm{\mu}} \left(\frac{1-{\mathrm{\sigma}} }{1-2{\mathrm{\sigma}} } \right), \end{align} (5,6)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\rho}}{\mathrm{\alpha}} ^{2} ={\mathrm{\lambda}} +\frac{{\mathrm{\lambda}} \left(1-2{\mathrm{\sigma}} \right)}{{\mathrm{\sigma}} } =\frac{{\mathrm{\lambda}} }{{\mathrm{\sigma}} } \left(1-{\mathrm{\sigma}} \right), \end{align} (6,6)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\rho}}{\mathrm{\alpha}} ^{2} =\left(k-\frac{2}{3} {\mathrm{\mu}} \right)+2{\mathrm{\mu}} =k+\frac{4}{3} {\mathrm{\mu}}, \end{align} (7,6)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\mathrm{\rho}}{\mathrm{\alpha}} ^{2} ={\mathrm{\lambda}} +3\left(k-{\mathrm{\lambda}} \right)=3k-2\lambda. \end{align} (8,6)

Column 7 is merely the square root of column 4 after dividing by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\rho}} . Column 8 is obtained by dividing column 7 by column 6.

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