GATE-2006
Q. The Young's modulus (E), density of rock(ρ) and Poisson's ratio (σ) are used to determine Shear wave velocity(β) in a rock.
84. Which one of the following expression is correct?
Solution:
Solution:
Young's modulus : a
measure of elasticity, equal to
the ratio of the stress acting on a substance to the
strain produced.
the ratio of the stress acting on a substance to the
strain produced.
The relation between
Young's modulus (E) and
Lame's constants (λ, μ)
Lame's constants (λ, μ)
E=
μ ((3 λ +2μ)/ (λ+ μ)) -------------------------(1)
Shear
wave velocity (Vs)
Vs=
√ (μ/ρ)
----------------------------------------(2)
Poisson's
ratio (σ):
Is the ratio of transverse
contraction strain to longitudinal extension strain in
the direction of stretching force.
contraction strain to longitudinal extension strain in
the direction of stretching force.
σ=(λ/2(λ
+ μ)) ----------------------(3)
E=
μ (( 3μ+2 λ)/ (λ+ μ))
=
μ ( ((2λ+2μ)/ (λ+μ)) + (λ /(λ+μ)) )
=
μ ( (2(λ+μ)/ (λ+μ)) + (λ /(λ+μ)) )
=
μ ( 2
+ (λ /(λ+μ)) ) (
σ=(λ/2(λ+
μ))
=
μ ( 2
+ (2λ
/2(λ+μ))
)
=
μ ( 2
+ 2(σ)
)
E
= 2 μ
( 1
+ (σ) ) -----------------------(4)
μ
= ( E / (
2
(1+
σ)
) )
--------------------(5)
Vs
(or) β = √ (μ/ρ) =
√ (E
/ 2(1+σ
) ρ)
β=
√ (E
/ 2ρ
(1+σ
) )
-------------(6)
ρ=
density and σ =
Poisson's ratio
Formula written wrongly please check it..
ReplyDeleteE= μ (( 3λ+2μ )/ (λ+ μ))
fermoranma Chris Lavin click here
ReplyDeletedownload
click
click
wealthrockmencong