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

# Isotropic vs anisotropic; Lame's constants

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

- Asked in: 2074 Bhadra, 2+2 marks
- Video: [Generalised hooke's law | Strength of material](https://www.youtube.com/watch?v=IPzkPvS_MoY) — STUDY WITH FAIZ
- 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/isotropic-anisotropic-lame
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-3

## Question

Differentiate between isotropic and anisotropic material body. Derive the expressions for Lame's constants for a linearly elastic isotropic material body.

## Short answer

Isotropic: two constants (E, ν), properties same in all directions. Anisotropic: up to 21 constants, properties direction-dependent. Lame's constants: λ=νE/[(1+ν)(1-2ν)], μ=E/[2(1+ν)]=G, giving σ_ij=λ e δ_ij+2μ ε_ij.

## Answer

#### Isotropic and anisotropic bodies

| Isotropic body | Anisotropic body |
|---|---|
| Elastic properties are the same in all directions at a point | Properties change with direction |
| Two independent constants, $E$ and $\nu$ | Up to 21 independent constants (general); 9 for orthotropic |
| Example: steel, concrete (usually), homogeneous soil | Timber, rolled plates, layered rock and soil, fibre composites |
| Normal stress produces only normal strain | Normal stress can also produce shear strain |

#### Lame's constants
For a linearly elastic isotropic body, Hooke's law is $\varepsilon_x=\dfrac{1}{E}[(1+\nu)\sigma_x-\nu\Theta]$ with $\Theta=\sigma_x+\sigma_y+\sigma_z$. Adding the normal equations gives $e=\varepsilon_x+\varepsilon_y+\varepsilon_z=\dfrac{1-2\nu}{E}\Theta$. Therefore
$$
\sigma_x=\frac{E}{1+\nu}\varepsilon_x+\frac{\nu E}{(1+\nu)(1-2\nu)}\,e=2\mu\,\varepsilon_x+\lambda\,e
$$
so that
$$
\lambda=\frac{\nu E}{(1+\nu)(1-2\nu)},\qquad \mu=\frac{E}{2(1+\nu)}=G
$$
and $\tau_{xy}=\mu\gamma_{xy}$. These are Lame's constants; $\mu$ is the shear modulus.

## Check yourself

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

1. Which statement distinguishes isotropic from anisotropic bodies?
   A. Isotropic: properties depend on direction; anisotropic: the same in all directions
   B. Isotropic: same properties in all directions with two constants E and nu; anisotropic: properties vary with direction (up to 21 constants)
   C. Both have exactly two constants
   D. Anisotropic bodies have no shear modulus
   Answer: B. An isotropic body has the same properties in every direction and two constants; an anisotropic one, like timber or layered rock, depends on direction and in general needs 21. (Key point: Isotropic vs anisotropic)

2. What is mu in terms of E and nu?
   A. E/(1 - 2 nu)
   B. nu E/((1 + nu)(1 - 2 nu))
   C. E/(1 - nu^2)
   D. E/(2(1 + nu)), the shear modulus G
   Answer: D. The coefficient of epsilon_x is E/(1 + nu) = 2 mu, so mu = E/(2(1 + nu)) = G. (Key point: Lame constant mu)

3. What is the key step before solving for sigma_x in the Lame derivation?
   A. Assuming nu = 0.5
   B. Adding the normal strain equations to get e = ((1 - 2 nu)/E) Theta
   C. Setting all shear strains to zero
   D. Differentiating Hooke's law
   Answer: B. Adding the normal equations gives the relation between volumetric strain and Theta, allowing Theta to be eliminated. (Key point: Derivation step)

## Related in Computational Techniques in Civil Engineering

- [Derive Lame's constants](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Aderive-lame-constants) — Computational Techniques in Civil Engineering, chapter 3, asked 2×
- [Constitutive relation for 2D isotropic material](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Aconstitutive-2d-isotropic) — Computational Techniques in Civil Engineering, chapter 3
- [Constitutive relations for 3D elastic solid](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Aconstitutive-3d-solid) — Computational Techniques in Civil Engineering, chapter 3, asked 2×

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