Correct Answer - Option 4 : 1, 2 and 3
Statement I: True and Statement II: True
The term ‘dilation’ means a change in the volume of material due to the application of load or we can say that dilation is the volumetric strain in the material.
The ‘invariant’ means constant (does not change).
The volumetric strain is given by:
\({ \in _v} = \;{ \in _x} + \;{ \in _y} + \;{ \in _z} + \;{ \in _x}{ \in _y} + \;{ \in _z}{ \in _{y\;}} + \;{ \in _x}{ \in _z} + { \in _x}{ \in _y}{ \in _z}\;\)
In infinitesimal strain theory, displacement of material is much smaller than any relevant dimension of the body so that its geometry and properties of material can be assumed to be unchanged.
\({ \in _{\rm{x}}}{ \in _{\rm{y}}} + {\rm{\;}}{ \in _{\rm{z}}}{ \in _{{\rm{y\;}}}} + {\rm{\;}}{ \in _{\rm{x}}}{ \in _{\rm{z}}} + { \in _{\rm{x}}}{ \in _{\rm{y}}}{ \in _{\rm{z}}}{\rm{\;can\;be\;negelcted}}.\)
\({\epsilon_v} = \frac{{\left( {1 - 2\mu } \right)\left( {\;{\sigma _x} + \;{\sigma _y} + \;{\sigma _Z}} \right)}}{E}{\rm{\;}}\) (If \({ \in _x}{ \in _y} + \;{ \in _z}{ \in _{y\;}} + \;{ \in _x}{ \in _z} + { \in _x}{ \in _y}{ \in _z}\) is neglected)
i.e. Dilation or volumetric strain is proportional to the algebraic sum of all normal stresses only in case of infinitesimal strain otherwise not.
a) For a given material, E and μ are constant.
b) In infinitesimal strain theory, follows the stress invariant concept i.e. sum of the normal stresses acting in three mutually perpendicular directions( x,y,z) is constant.
⇒ \({\sigma _x} + {\sigma _y} + {\sigma _Z} = constant\)
From a) and b) ⇒ ϵv or dilation is constant.
Statement III: True
The relation between Modulus of Elasticity (E) and Shear Modulus (G) is given by:
E = 2G (1 + μ)
E/G = 2(1+μ )
Now, we know that
0 ≤ μ ≤ 0.5 then E/G ratio is,
⇒ 2 ≤ 2(1 + μ ) ≤ 3 i.e. denominator is greater than 1 in the above equation, the shear modulus ends up being less than the Young’s modulus
