GATE-2018 (24) :Wyllie – Time average equation

GATE-2008

61. The Lame's coefficient (λ) can be written in terms of compressibility of the material (β) and possion's ratio (σ) as

Solution:

The Bulk modulus (k) is defined in terms of Lame's 

constants λ and µ is

 

k=λ + (2/3)* µ –----------------------------(1)

Compressibility (β) = 1 / Bulk modulus 

 

                     β = 1 / k

Poisson's ratio (υ) in terms of Lame's constants λ and µ is

Poisson's ration (υ)= λ / 2 ( λ + µ ) 

----------------------------(2)

  

The relation between υ , λ and β is

1/ k = 1 / ( λ + (2/3) * µ) ---------(3)

β = 3 / (3λ + 2µ) ----------(4)

β = 3 / (2λ + 2µ + λ)

1 / β = (2λ + 2µ + λ) / 3

multiplying by λ with both sides 

 

1 / (β* λ)= ((2λ + 2µ) / 3 λ) + (λ / 3 λ) ------------( 5)

Substitute eq(2) value in eq (5)

 

1 / (β* λ)= ( 1/ (3*υ)) + ( 1/ 3 )

1 / (β* λ)= (1 + υ) / 3 υ

(β* λ)= 3 υ / (1 + υ) 

 

Therefore λ = 3 υ / ( β*(1 + υ))