Gate_2020
5)
A sample of Granite is observed to have a P- wave velocity of 5
km/sec and density of 2600 kg/m3.
The Bulk modules of the granite, assuming it to be a poissions solid
is ____________ kilo-Pascal(Kpa).
(Thanks to Chandrasekhar, ANU)
Solution:
Given
that V_{p}= 5 km/sec =5000m/sec
Density(ρ) = 2600 kg/m3
Bulk
Modules(K)= ?
Poisson solid (\nu) =0.25
\frac{V_{p}}{V_{s}} =\sqrt{\frac{2(1-\upsilon)}{(1-2\upsilon)}}
\frac{V_{p}}{V_{s}} =\sqrt{\frac{2(1-0.25)}{(1-2*0.25)}}
\frac{V_{p}}{V_{s}} =\sqrt{\frac{2(0.75)}{(1-0.5)}}
\frac{V_{p}}{V_{s}} =\sqrt{\frac{1.5}{0.5}}
\frac{V_{p}}{V_{s}} =\sqrt{\frac{1.5}{0.5}}
\frac{V_{p}}{V_{s}} =\sqrt{3}
\frac{V_{p}}{V_{s}} =\sqrt{\frac{2(1-0.25)}{(1-2*0.25)}}
\frac{V_{p}}{V_{s}} =\sqrt{\frac{2(0.75)}{(1-0.5)}}
\frac{V_{p}}{V_{s}} =\sqrt{\frac{1.5}{0.5}}
\frac{V_{p}}{V_{s}} =\sqrt{\frac{1.5}{0.5}}
\frac{V_{p}}{V_{s}} =\sqrt{3}
V_{s}=
\frac{V_{p}}{\sqrt{3}}
V_{s}=
\frac{5000}{1.732}
V_{s}=
2886.83 m/sec^2
We
know that velocity of S-wave can be defined as follws
V_{s}^{2}=\frac{\mu}{\rho}
\mu=\rho\times
V_{s}^{2}
\mu=(2600)\times
(2886.83)^{2}
\mu
= 2.16\times 10^{10}
Velocity
of the P-wave can be defined as follows
V_{p}^{2}=
\frac{K + \frac{4}{3}\mu}{\rho}
(5000)^{2}=
\frac{K + \frac{4}{3}( 2.16\times 10^{10})}{2600}
6.5\times
10^{10} = K + 2.88 \times 10^{10}
K
= 3.62\times 10^{10} pascal
K= 3.62 \times 10^{7} kilo - pascal
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