The three-dimensional stress-strain relationship for an isotropic material is given as
{
$\sigma$xx
$\sigma$yy
$\sigma$zz
$\tau$yz
$\tau$xz
$\tau$xy}
=
[
P Q Q
Q P Q
Q Q P
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
R 0 0
0 R 0
0 0 R]
{
𝜀xx
𝜀yy
𝜀zz
$\gamma$yz
$\gamma$xz
$\gamma$xy}
where, $P, Q$ and $R$ are the three elastic constants, $\sigma$ and $\tau$ represent normal and shear
stresses and 𝜀 and $\gamma$ represent normal and engineering shear strains. Which one of
the following options is correct?
Options:
(A) $R = \frac{P - Q}{2}$
(B) $R = \frac{Q - P}{2}$
(C) $Q = \frac{P - R}{2}$
(D) $Q = \frac{R - P}{2}$
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2024 Q.13)**
**1. Constitutive Relations for Isotropic Linear Elastic Material**:
For an isotropic material, Hooke's law relates normal stresses to normal strains and shear stresses to engineering shear strains:
$$\sigma_{xx} = P \varepsilon_{xx} + Q \varepsilon_{yy} + Q \varepsilon_{zz}$$
$$\tau_{xy} = R \gamma_{xy}$$
In terms of Lamé constants $\lambda$ and $\mu = G$:
$$P = \lambda + 2G, \quad Q = \lambda, \quad R = G$$
**2. Relationship Between $P, Q,$ and $R$**:
Subtracting $Q$ from $P$:
$$P - Q = (\lambda + 2G) - \lambda = 2G = 2R$$
Dividing by $2$:
$$R = \frac{P - Q}{2}$$
**3. Options Analysis**:
- **(A) $R = \frac{P - Q}{2}$**: ✅ **CORRECT**. Directly derived from isotropic symmetry relations.
- **(B) $R = \frac{Q - P}{2}$**: ❌ **INCORRECT**. Sign error ($P > Q$ for real elastic materials).
- **(C) $Q = \frac{P - R}{2}$**: ❌ **INCORRECT**. Violates Lamé identity.
- **(D) $Q = \frac{R - P}{2}$**: ❌ **INCORRECT**.
**4. Official Master Key**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: For an isotropic elastic solid, shear modulus $G$ relates Lamé constants by $G = \frac{C_{11} - C_{12}}{2}$. Here $C_{11}=P, C_{12}=Q, G=R \implies R = \frac{P - Q}{2}$.