---
title: "Derive Lame's constants"
subject: "Computational Techniques in Civil Engineering"
chapter: 3
key: "semester_8/computational_techniques_in_civil_engineering:derive-lame-constants"
source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/derive-lame-constants
site: BCT Engineering
---

# Derive Lame's constants

Computational Techniques in Civil Engineering · Chapter 3: Elasticity in solids

- Asked in: 2077 Chaitra, 5 marks; 2070 Bhadra, 5 marks (asked 2 times)
- Video: [Elastic Constants and their Relationship | Lec-5](https://www.youtube.com/watch?v=sZi0W3pr8Jg) — Unacademy GATE & ESE - Civil & ES
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/derive-lame-constants
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-3

## Question

Derive the expression for Lame's constants.

## Short answer

Lame's constants are λ = νE/[(1+ν)(1-2ν)] and μ = E/[2(1+ν)] = G. They appear in the stress-strain law σ_ij = λ e δ_ij + 2μ ε_ij, derived from Hooke's law by adding normal strains and solving for stresses.

## Answer

Lame's constants $\lambda$ and $\mu$ express the isotropic stress-strain law in the compact form
$$
\sigma_{ij}=\lambda\,e\,\delta_{ij}+2\mu\,\varepsilon_{ij},\qquad e=\varepsilon_x+\varepsilon_y+\varepsilon_z
$$
that is, $\sigma_x=\lambda e+2\mu\varepsilon_x$, $\sigma_y=\lambda e+2\mu\varepsilon_y$, $\sigma_z=\lambda e+2\mu\varepsilon_z$, $\tau_{xy}=\mu\gamma_{xy}$ etc.

#### Derivation
Hooke's law: $\varepsilon_x=\dfrac{1}{E}[(1+\nu)\sigma_x-\nu\Theta]$, with $\Theta=\sigma_x+\sigma_y+\sigma_z$.

Adding the three normal equations: $e=\dfrac{1-2\nu}{E}\Theta$, so $\Theta=\dfrac{E}{1-2\nu}e$.

Solve for $\sigma_x$:
$$
\sigma_x=\frac{E}{1+\nu}\varepsilon_x+\frac{\nu}{1+\nu}\Theta=\frac{E}{1+\nu}\varepsilon_x+\frac{\nu E}{(1+\nu)(1-2\nu)}\,e
$$
Compare with $\sigma_x=\lambda e+2\mu\varepsilon_x$:
$$
\boxed{\lambda=\frac{\nu E}{(1+\nu)(1-2\nu)}},\qquad \boxed{\mu=\frac{E}{2(1+\nu)}=G}
$$
For shear, $\tau_{xy}=G\gamma_{xy}$ with $G=\dfrac{E}{2(1+\nu)}$, which confirms $\mu=G$ (the shear modulus).

Inverse relations: $E=\dfrac{\mu(3\lambda+2\mu)}{\lambda+\mu}$, $\nu=\dfrac{\lambda}{2(\lambda+\mu)}$. The bulk modulus is $K=\lambda+\tfrac23\mu$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In Lame's form sigma_x = lambda e + 2 mu epsilon_x, what is e?
   A. The Young's modulus
   B. The shear strain
   C. The sum of the normal stresses
   D. The volumetric strain epsilon_x + epsilon_y + epsilon_z
   Answer: D. e = epsilon_x + epsilon_y + epsilon_z is the volumetric strain. (Key point: Lame form)

2. Which expression gives lambda?
   A. nu E/((1 + nu)(1 - 2 nu))
   B. E/(2(1 + nu))
   C. E/((1 + nu)(1 - 2 nu))
   D. nu E/(1 - nu^2)
   Answer: A. Comparing sigma_x = E eps_x/(1 + nu) + nu E e/((1 + nu)(1 - 2 nu)) with lambda e + 2 mu eps_x gives lambda = nu E/((1 + nu)(1 - 2 nu)). (Key point: Result for lambda)

3. Why is mu identified with G?
   A. Because tau_xy = mu gamma_xy and tau = G gamma with G = E/(2(1 + nu)), the same as mu
   B. Because mu = lambda
   C. Because mu = E/(1 - 2 nu)
   D. Because mu is the bulk modulus
   Answer: A. The 2 mu coefficient of epsilon_x is E/(1 + nu), so mu = E/(2(1 + nu)), and the shear law confirms mu = G. (Key point: mu is shear modulus)

## Related in Computational Techniques in Civil Engineering

- [Isotropic vs anisotropic; Lame's constants](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Aisotropic-anisotropic-lame) — Computational Techniques in Civil Engineering, chapter 3

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