Table of Contents
Elastic Constants
Elastic constants describe how a material deforms under stress and are fundamental to structural engineering, materials science, and geophysics. For isotropic materials (same properties in all directions), only two independent elastic constants are needed to fully characterize the elastic behavior. The most commonly used pair is Young's modulus (E) and Poisson's ratio (ν), from which all other elastic constants can be derived.
Young's modulus measures stiffness (resistance to stretching), Poisson's ratio describes lateral contraction when stretched, shear modulus measures resistance to shearing, and bulk modulus measures resistance to uniform compression. These constants are interconnected through well-defined mathematical relationships that must be satisfied for any real isotropic material.
Relationships
Common Material Values
| Material | E (GPa) | ν | G (GPa) | K (GPa) |
|---|---|---|---|---|
| Steel | 200 | 0.30 | 76.9 | 166.7 |
| Aluminum | 69 | 0.33 | 25.9 | 67.6 |
| Copper | 117 | 0.34 | 43.7 | 121.9 |
| Rubber | 0.01-0.1 | ~0.49 | ~E/3 | ~1000*E |
| Cork | ~0.03 | ~0 | ~E/2 | ~E/3 |
Physical Meaning
- Young's Modulus (E): Stiffness under uniaxial tension/compression
- Poisson's Ratio (ν): Ratio of lateral strain to axial strain (0 to 0.5)
- Shear Modulus (G): Resistance to shape change at constant volume
- Bulk Modulus (K): Resistance to volume change under hydrostatic pressure
- Lamé lambda: Used in wave propagation and continuum mechanics
FAQ
What are the physical limits on Poisson's ratio?
For stable isotropic materials, Poisson's ratio must satisfy -1 < ν < 0.5. Most materials have positive ν between 0.2 and 0.5. Incompressible materials (rubber) approach 0.5. Cork has ν near 0, which is why it seals bottles well without bulging outward. Auxetic materials have negative ν, expanding laterally when stretched. The limits ensure positive-definite strain energy.
Why does rubber have ν close to 0.5?
Rubber is nearly incompressible, meaning its volume barely changes under stress. When stretched, it must contract laterally by almost the same amount to maintain constant volume. The theoretical limit of 0.5 corresponds to perfect incompressibility (infinite bulk modulus). Liquids are perfectly incompressible and would have ν = 0.5 if they could sustain shear stress.
How are elastic constants measured?
Young's modulus is measured via tensile testing (stress-strain curve slope). Poisson's ratio requires measuring both axial and lateral strain simultaneously, typically using strain gauges or extensometers. Shear modulus can be measured by torsion testing. Ultrasonic methods measure elastic wave velocities, from which all elastic constants can be calculated non-destructively.