Question Statement:
A stress field is given by $\sigma_{xx} = \sigma_{zz} = C_1y$; $\sigma_{yy} = C_2y$; $\tau_{xy} = \tau_{yz} = \tau_{zx} = 0$, where $C_1$ and $C_2$ are non-zero constants. If the stress field satisfies equilibrium, which one of the following options is correct?
Options:
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(A) There is no body force per unit volume.
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(B) There is a constant body force per unit volume in the y-direction.
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(C) The body force per unit volume varies linearly in the y-direction.
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(D) The direction of the body force per unit volume depends on the value of C1.
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.28)**
**1. Intuitive Physical Setup & Given Parameters**:
Given stress field in a 3D Cartesian continuum:
- Normal stresses: $\sigma_{xx} = C_1 y$, $\sigma_{yy} = C_2 y$, $\sigma_{zz} = C_1 y$
- Shear stresses: $\tau_{xy} = 0$, $\tau_{yz} = 0$, $\tau_{zx} = 0$
- $C_1, C_2 \neq 0$ are non-zero constants.
We are given that the stress field satisfies the static equilibrium equations:
\frac{\partial \sigma_{ji}}{\partial x_j} + B_i = 0
We need to determine the characteristics of the body force per unit volume $\vec{B} = (B_x, B_y, B_z)$.
**2. Step-by-Step Continuum Mechanics Derivation**:
1. **Equations of Equilibrium in Cartesian Coordinates**:
- Along $x$-direction:
\frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} + B_x = 0
\frac{\partial (C_1 y)}{\partial x} + 0 + 0 + B_x = 0 \implies 0 + B_x = 0 \implies B_x = 0
- Along $z$-direction:
\frac{\partial \tau_{xz}}{\partial x} + \frac{\partial \tau_{yz}}{\partial y} + \frac{\partial \sigma_{zz}}{\partial z} + B_z = 0
0 + 0 + \frac{\partial (C_1 y)}{\partial z} + B_z = 0 \implies 0 + B_z = 0 \implies B_z = 0
- Along $y$-direction:
\frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_{yy}}{\partial y} + \frac{\partial \tau_{zy}}{\partial z} + B_y = 0
0 + \frac{\partial (C_2 y)}{\partial y} + 0 + B_y = 0
C_2 + B_y = 0 \implies B_y = -C_2
2. **Analysis of the Body Force Vector**:
- $B_x = 0$, $B_z = 0$, and $B_y = -C_2 = \text{constant}$.
- Because $C_2$ is a non-zero constant, there is a **constant body force per unit volume in the $y$-direction**.
- Note that $B_y$ is completely independent of $C_1$ and spatial coordinates $(x, y, z)$.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) There is no body force per unit volume**: ❌ **INCORRECT**. Requires $\frac{\partial \sigma_{yy}}{\partial y} = 0$, but $\frac{\partial (C_2 y)}{\partial y} = C_2 \neq 0$.
- **(B) There is a constant body force per unit volume in the y-direction**: ✅ **CORRECT**. $B_y = -C_2 = \text{const}$, $B_x = B_z = 0$.
- **(C) The body force per unit volume varies linearly in the y-direction**: ❌ **INCORRECT**. The derivative of a linear function is constant.
- **(D) The direction of the body force per unit volume depends on the value of C1**: ❌ **INCORRECT**. Depends solely on $C_2$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B**
- **Exam Takeaway**: Differential equilibrium requires $B_i = -\partial \sigma_{ji}/\partial x_j$. Differentiating $\sigma_{yy} = C_2 y$ yields constant body force $B_y = -C_2$.