Scientific schematic diagram reproduced from official IIT paper
Options:
(A) The stress-strain graph for a nonlinear elastic material is as shown in the figure
(B) Material properties are independent of position in a homogeneous material
(C) An isotropic material has infinitely many planes of material symmetry
(D) The stress-strain graph for a linear elastic material is
Official Answer Key:
B, C, D
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2024 Q.25)**
**1. Continuum Mechanics Principles**:
- **Homogeneity**: A material is homogeneous if its constitutive properties are independent of position (spatial coordinates $\vec{x}$). Thus, Option (B) is TRUE.
- **Isotropy**: An isotropic material possesses identical properties in all orientations, meaning it has an infinite number of planes of material symmetry. Thus, Option (C) is TRUE.
- **Linear Elastic Material**: Displays a linear stress-strain relationship (Hookean behavior). Option (D) correctly identifies this linear characteristic.
**2. Conclusion**:
Options **B, C, and D** are all physically and mathematically correct.
**3. Official Master Key**:
- **Official Answer Key**: **B, C, D**
- **Exam Takeaway**: Homogeneity refers to position invariance; isotropy refers to directional invariance.
