Magnitudes of elastic constants

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Problem 2.4

To illustrate the relationships and magnitudes of the elastic constants, complete Table 2.4a. Note that these values apply to specific specimens; the elastic constants for rocks range considerably, especially as porosity and pressure change.

Solution

We use the row-column notation to designate equations from Table 2.2a.

Water

Since water is a fluid we know that $ {\mathrm {\mu } }=0=\beta $. Equation (4,1) shows that $ E=0 $ also. From equation (4,5) we get $ {\mathrm {\lambda } }=k=2.1\times 10^{9} $ Pa.

Table 2.4a. Magnitudes of elastic constants and velocities.
Constant Water Stiff mud Shale Sandstone Limestone Granite
$ E\,\,\,({\times }10^{9}{\hbox{Pa}}) $ 16 54 50
$ k\,\,\,({\times }10^{9}{\hbox{Pa}}) $ 2.1
$ {\mathrm {\mu } }\,\,\,({\times }10^{9}{\hbox{Pa}}) $
$ {\mathrm {\lambda } }\,\,\,({\times }10^{9}{\hbox{Pa}}) $
$ {\mathrm {\sigma } } $ 0.50 0.43 0.38 0.34 0.25 0.20
$ {\mathrm {\rho } }\,\,\,({\hbox{g/cm}}^{3}) $ 1.0 1.5 1.8 1.9 2.5 2.7
$ {\alpha }\,\,\,({\hbox{km/s}}) $ 1.5 1.6 3.2
$ {\beta }\,\,\,({\hbox{km/s}}) $

Stiff mud

Because $ {\mathrm {\sigma } }<0.5 $, stiff mud is equivalent to a solid, hence $ {\mathrm {\mu } }\neq 0 $. From $ {\mathrm {\rho } } $ and $ {\mathrm {\alpha } } $ we get $ \left({\mathrm {\lambda } }+2{\mathrm {\mu } }\right) $ using equation (9,6), while equation (9,2) expresses $ {\mathrm {\sigma } } $ in terms of $ {\mathrm {\lambda } } $ and $ {\mathrm {\mu } } $, thus enabling us to find both $ {\mathrm {\lambda } } $ and $ {\mathrm {\mu } } $.

We have:


$ {\begin{aligned}{{\rho }{{\alpha }^{2}}=\left({{\lambda }+2{\mu }}\right)=\left({1.5\;{\rm {g/c}}{{\rm {m}}^{3}}}\right){\times }{{(1.6{\times }{{10}^{5}}\;{\rm {cm/s}})}^{2}}}\\{{\qquad }=3.8{\times }{{10}^{10}}\;{\rm {dynes/c}}{{\rm {m}}^{2}}=3.8{\times }{{10}^{9}}\;{\rm {Pa}},}\\{{\quad }{\sigma }=0.43={\lambda }/2\left({{\lambda }+{\mu }}\right),\;{\rm {i}}.{\rm {e}}.,\;0.86{\mu }=0.14{\lambda }.}\end{aligned}} $

Solving the two equations, we get $ {\mathrm {\lambda } }=2.9{\times }10^{9} $ Pa, $ {\mathrm {\mu } }=0.47{\times }10^{9} $ Pa. Using equation (6,1),


$ {\begin{aligned}{E={\mathrm {\lambda } }{\frac {\left(1+{\mathrm {\sigma } }\right)\left(1-2{\mathrm {\sigma } }\right)}{\mathrm {\sigma } }}=2.9{\times }10^{9}{\times }1.43{\times }0.14/0.43}\\{{\qquad }{\qquad }{\qquad }{\qquad }{\quad }=1.4{\times }10^{9}\ \mathrm {Pa} .}\end{aligned}} $

Equation (6,3) gives


$ {\begin{aligned}k={\mathrm {\lambda } }{\frac {\left(1+{\mathrm {\sigma } }\right)}{3{\mathrm {\sigma } }}}=2.9{\times }10^{9}{\times }1.43/1.29=3.2{\times }10^{9}\ \mathrm {Pa} .\end{aligned}} $

Finally, to get $ {\mathrm {\beta } } $ we note that $ {\mathrm {\lambda } } $, $ {\mathrm {\mu } } $, and $ k $ are in $ N/m^{2} $, and so we must express $ {\mathrm {\rho } } $ in appropriate units of $ kg/m^{3} $, i.e., $ {\mathrm {\rho } }=1.5{\times }10^{3} $ $ kg/m^{3} $. We now have


$ \beta =({\mathrm {\mu } }/{\mathrm {\rho } })^{1/2}=\left(0.47{\times }10^{9}/1.5{\times }10^{3}\right)=0.56\ \mathrm {\hbox{km/s}} . $

Shale

As with stiff mud, we have been given $ {\mathrm {\rho } } $, $ \alpha $, and $ {\mathrm {\sigma } } $, so that again $ {\mathrm {\rho } } $ and $ {\mathrm {\alpha } } $ give us $ \left({\mathrm {\lambda } }+2{\mathrm {\mu } }\right) $ and $ {\mathrm {\sigma } } $ gives us $ {\mathrm {\lambda } }/2\left({\mathrm {\lambda } }+{\mathrm {\mu } }\right) $ so that we can solve these equations for $ {\mathrm {\lambda } } $ and $ {\mathrm {\mu } } $, then find $ k $ and $ E $ using equations (9,3) and (9,1). Thus,


$ {\begin{aligned}\left(\lambda +2\mu \right)=\rho \alpha ^{2}=\left(1.8\ \mathrm {g/cm} ^{3}\right)\times (3.2\times 10^{5}\ \mathrm {cm/s} )^{2}=18.4\times 10^{9}\ \mathrm {Pa} ,\end{aligned}} $

$ {\begin{aligned}\sigma =0.38=\lambda /2\left(\lambda +\mu \right),\quad \mathrm {i.e.} ,0.76\mu =0.24\lambda .\end{aligned}} $

Solving the two equations gives $ {\mathrm {\lambda } }=11{\times }10^{9} $ Pa, $ {\mathrm {\mu } }=3.6{\times }10^{9} $ Pa.

From equation (9,3), $ k={\mathrm {\lambda } }+{\frac {2}{3}}{\mathrm {\mu } }=14{\times }10^{9} $ Pa, and equation (9,1) gives


$ {\begin{aligned}E={\mathrm {\mu } }{\frac {\left(3{\mathrm {\lambda } }+2{\mathrm {\mu } }\right)}{\left({\mathrm {\lambda } }+{\mathrm {\mu } }\right)}}=9.9{\times }10^{9}\ \mathrm {Pa} .\end{aligned}} $

Finally,


$ {\begin{aligned}\beta =({\mathrm {\mu } }/{\mathrm {\rho } })^{1/2}=(3.6{\times }10^{9}/1.8{\times }10^{3})^{1/2}=1.4\ \mathrm {\hbox{km/s}} .\end{aligned}} $

Sandstone

We are given the elastic constants $ E $ and $ {\mathrm {\sigma } } $ (plus $ {\mathrm {\rho } } $), so we get $ k $, $ {\mathrm {\mu } } $, $ {\mathrm {\lambda } } $, $ \alpha $, $ {\mathrm {\beta } } $ using equations (1,3) to (1,7) in Table 2.2a. Thus


$ {\begin{aligned}{k=16{\times }10^{9}/3{\times }0.32=17{\times }10^{9}\ \mathrm {Pa} ,}\\{{\mathrm {\mu } }=16{\times }10^{9}/2{\times }1.34=6.0{\times }10^{9}\ \mathrm {Pa} ,}\\{{\mathrm {\lambda } }=16{\times }10^{9}{\times }0.34/1.34{\times }0.32=13{\times }10^{9}\ \mathrm {Pa} ,}\\{{\mathrm {\alpha } }=[16{\times }10^{9}{\times }0.66/1.34{\times }0.32{\times }1.9{\times }10^{3}]^{1/2}=3.6\ \mathrm {\hbox{km/s}} ,}\\{{\mathrm {\beta } }=[16{\times }10^{9}/2{\times }1.34{\times }1.9{\times }10^{3}]^{1/2}=1.8\ \mathrm {\hbox{km/s}} .}\end{aligned}} $

[We could also have obtained $ {\mathrm {\alpha } } $ and $ {\mathrm {\beta } } $ by using equations (9,6) and (9,7).]

Limestone

We solve in the same way as with sandstone since we are given the same constants:


$ {\begin{aligned}{k=54{\times }10^{9}/3{\times }0.50=36{\times }10^{9}\ \mathrm {Pa} ,}\\{{\mathrm {\mu } }=54{\times }10^{9}/2{\times }1.25=22{\times }10^{9}\ \mathrm {Pa} ,}\\{{\mathrm {\lambda } }=54{\times }10^{9}{\times }0.25/1.25{\times }0.50=22{\times }10^{9}\ \mathrm {Pa} ,}\\{{\mathrm {\alpha } }=[54{\times }10^{9}{\times }0.75/1.25{\times }0.50{\times }2.5{\times }10^{3}]^{1/2}=5.1\ \mathrm {\hbox{km/s}} ,}\\{{\mathrm {\beta } }=[54{\times }10^{9}/2{\times }1.25{\times }2.5{\times }10^{3}]^{1/2}=2.9\ \mathrm {\hbox{km/s}} .}\end{aligned}} $

Granite

Again the solution is the same as for sandstone.


$ {\begin{aligned}{k=50{\times }10^{9}/3{\times }0.60=28{\times }10^{9}\ \mathrm {Pa} ,}\\{{\mathrm {\mu } }=50{\times }10^{9}/2{\times }1.2=21{\times }10^{9}\ \mathrm {Pa} ,}\\{{\mathrm {\lambda } }=50{\times }10^{9}{\times }0.20/1.2{\times }0.60=14{\times }10^{9}\ \mathrm {Pa} ,}\\{{\mathrm {\alpha } }=[50{\times }10^{9}{\times }0.80/1.2{\times }0.60{\times }2.7{\times }10^{3}]^{1/2}=4.5\ \mathrm {\hbox{km/s}} ,}\\{{\mathrm {\beta } }=[50{\times }10^{9}/2{\times }1.2{\times }2.7{\times }10^{3}]^{1/2}=2.8\ \mathrm {\hbox{km/s}} .}\end{aligned}} $

Table 2.4b summarizes the results.

Table 2.4b Magnitudes of elastic constants.
Constant Water Stiffmud Shale Sandstone Limestone Granite
$ E\,\,\,({\times }10^{9}{\hbox{Pa}}) $ 0 1.4 9.9 16 54 50
$ K\,\,\,({\times }10^{9}{\hbox{Pa}}) $ 2.1 3.2 13 17 36 28
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}}\,\,\,({\times} 10^{9} \hbox{Pa}) 0 0.47 3.6 6.0 22 21
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}}\,\,\,({\times} 10^{9} \hbox{Pa}) 2.1 2.9 11 13 22 14
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}} 0.50 0.43 0.38 0.34 0.25 0.20
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}}\,\,\,(\hbox{g/cm}^{3}) 1.0 1.5 1.8 1.9 2.5 2.7
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}}\,\,\,(\hbox{km/s}) 1.5 1.6 3.2 3.6 5.1 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/":): {\mathrm{\beta}}\,\,\,(\hbox{km/s}) 0 0.56 1.4 1.8 2.9 2.8

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