A cylindrical rod is subjected to an uniaxial tensile load. Which of the following statements regarding the rod's material behavior is correct if it is made of a linearly elastic, isotropic material?
- AThe ratio of lateral strain to longitudinal strain depends only on the applied stress.
- BThe Poisson's ratio is independent of the material properties.
- CThe modulus of elasticity and shear modulus are related by the Poisson's ratio.Correct
- DThe strain energy density depends only on the longitudinal strain.
Solution
Concept
For a linearly elastic isotropic material, E, G and Poisson's ratio are related.
Formula
E = 2G(1 + μ)
Step-by-step solution
1. The material is linear elastic and isotropic. 2. Check each option. 3. (A) Lateral strain / longitudinal strain = Poisson's ratio. It is a material constant, not dependent on stress. 4. (B) Poisson's ratio is a material property, so (B) is wrong. 5. (C) E = 2G(1 + μ). So E and G are related by μ. This is correct. 6. (D) Strain energy density = σ²/2E = ½ σ ε. It depends on stress and strain, and on E. It is not only on longitudinal strain. 7. Answer: (C).
Explanation
In an isotropic elastic material only two constants are independent. E, G and μ are linked by E = 2G(1 + μ). Poisson's ratio is a fixed material property.
Why the other options are wrong
(A) The ratio is Poisson's ratio, a material constant, not dependent on stress. (B) Poisson's ratio is a material property. (D) Energy density also depends on E (U = ½ E ε²), so it is not independent of the material.
Shortcut
Remember E = 2G(1 + μ) = 3K(1 − 2μ). Any option linking E, G and μ is true.
Common mistake
Thinking Poisson's ratio changes with load. It is constant in the elastic range.
Correct answer: C) The modulus of elasticity and shear modulus are related by the Poisson's ratio.