Q.State the relation connecting Young's modulus ($Y$), bulk modulus ($K$), the shear modulus of rigidity ($\eta$), and Poisson's ratio ($\sigma$) for a homogeneous, isotropic elastic solid (no derivation is required). Using this relation, explain why an isotropic elastic solid is fully characterised by only two independent elastic constants even though four elastic quantities ($Y$, $K$, $\eta$, $\sigma$) are commonly used to describe it.

For a homogeneous, isotropic elastic solid, Young's modulus $Y$, the bulk modulus $K$, the shear modulus of rigidity $\eta$, and Poisson's ratio $\sigma$ are related by

(WBCHSE's syllabus states that these relations are to be used directly, without derivation.)

Because these relations hold for any isotropic elastic material, they mean that the four quantities $Y$, $K$, $\eta$, $\sigma$ are not four separate, freely-choosable material properties -- they are four different "views" of just two genuinely independent underlying elastic constants. Concretely: if any two of the four quantities are measured or known for a particular material, the remaining two are then completely determined -- they can be calculated directly from the relations above, without needing any further independent experiment. …

State the standard relation connecting $Y$, $K$, $\eta$, $\sigma$ (used as given, per WBCHSE's syllabus note, not derived); explain that since the relation always holds, fixing any two of the four quantit …

• Attempting to derive the relation from scratch, which is unnecessary and not required by the syllabus -- it is meant to be applied directly. …