On the physical interpretation of elastic constants

1. Introduction

This very brief write up attempts to explain the physical interpretation of elastic constants: elastic modulus, poisson's ratio, bulk modulus, shear modulus, thermal expansion coefficient.

 

2. Physical meaning of elastic constants

It is important to have a feel for the physical significance of the  elastic constants: elastic modulus, poisson's ratio, bulk modulus, shear modulus, thermal expansion coefficient.

 

 Young’s modulus (E) is the slope of the stress—strain curve in uniaxial tension.  It has dimensions of stress ( N/m2 ) and is usually large – for steel, E=210×10^9 N/m2. You can think of E as a measure of the stiffness of the solid. The larger the value of E, the stiffer the solid.  For a stable material, E>0.

Poisson’s ratio ν is the ratio of lateral to longitudinal strain in uniaxial tensile stress. It is dimensionless and typically ranges from 0.2—0.49, and is around 0.3 for most metals.  For a stable material, −1<ν<0.5. It is a measure of the compressibility of the solid.  If ν=0.5, the solid is incompressible – its volume remains constant, no matter how it is deformed.  If ν=0, then stretching a specimen causes no lateral contraction.  Some bizarre materials have ν<0 --  if you stretch a round bar of such a material, the bar increases in diameter!!

Thermal expansion coefficient quantifies the change in volume of a material if it is heated in the absence of stress.  It has dimensions of (degrees Kelvin)^-1 and is usually very small.  For steel, α≈6^-10×10^-6 K^-1

Bulk modulus quantifies the resistance of the solid to volume changes.  It has a large value (usually bigger than E).

Shear modulus quantifies its resistance to volume preserving shear deformations.  Its value is usually somewhat smaller than E.