Relationship between modulus of elasticity E, modulus of rigidity C (G), and Poisson’s ratio μ: select the correct formula.
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AE = 2 C (1 + μ)
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BE = 3 C (1 − 2 μ)
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CC = E (1 − 2 μ) / 2 (1 + μ)
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DE = C / (1 − μ)
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EE = 2 C (1 − μ)
Answer
Correct Answer: E = 2 C (1 + μ)
Explanation
Introduction / Context:In isotropic linear elasticity, only two independent elastic constants are required. The commonly used relations connect Young’s modulus E, shear modulus C (also denoted G), bulk modulus K, and Poisson’s ratio μ.
Given Data / Assumptions:
- Material is homogeneous and isotropic.
- Linear elastic response.
- Standard definitions of E, C (G), μ.
Concept / Approach:The fundamental isotropic elasticity relations include E = 2 C (1 + μ) and E = 3 K (1 − 2 μ). These allow conversion between sets of elastic constants based on measured data.
Step-by-Step Solution:
Start from generalized Hooke’s law for isotropic solids.Relate normal and shear strains to stresses using E, μ and C.Derive E = 2 C (1 + μ).Verification / Alternative check:Cross-check with E = 3 K (1 − 2 μ) and the identity K = E / 3 (1 − 2 μ) to ensure consistency.
Why Other Options Are Wrong:E = 3 C (1 − 2 μ) confuses K with C.C = E (1 − 2 μ) / 2 (1 + μ) is algebraically equivalent to the correct formula, but the question asks for E as a function of C and μ; the simplest correct expression is E = 2 C (1 + μ).Other forms given are incorrect dimensionally or miss the 1 + μ factor.
Common Pitfalls:Interchanging K and C; omitting μ; using plastic-range values where these linear relations no longer hold.
Final Answer:
E = 2 C (1 + μ)