Principles of Dairy Machine Design

Module 3. Stress

Lesson 11
POISSON’S RATIO

11.1 Introduction

When a material is compressed in one direction, it usually tends to expand in the other two directions perpendicular to the direction of compression. This phenomenon is called the Poisson effect. Poisson's ratio m (mu) is a measure of the Poisson effect. On the molecular level, Poisson’s effect is caused by slight movements between molecules and the stretching of molecular bonds within the material lattice to accommodate the stress. When the bonds elongate in the direction of load, they shorten in the other directions. This behaviour multiplied millions of times throughout the material lattice is what drives the phenomenon.

11.2 Poisson's Ratio

Whenever a bar is subjected to tensile load, its length will increase but its lateral dimension will decrease. Thus changes in longitudinal and lateral dimensions are of opposite nature.

The transverse dimension ‘b’ changes to and the deformation has the opposite signs. The ratio of transverse strain to the longitudinal strain is called Poisson’s ratio and is designated as

The Poisson’s ratio is dimensionless parameter.

For most of engineering materials its value is in between 0 to 0.5.

For most of the metals its value is in the range of 0.2 to 0.3.

The Modulus of Elasticity (E), the Modulus of Rigidity (G) and the Poisson’s ratio, m are related by the relation,

When same stress is applied on all the six faces of a cube then the ratio of normal stress to the volumetric strain (change of volume to original volume) is known as bulk modulus of elasticity, denoted by K. The relation between E and K is

11.3 Shear Stress

When the section is subjected to two equal and opposite forces P acting tangentially across the resisting section, the resistance set up is called Shear Stress.

Fig. 11.1 Shear stress

11.4 Volumetric Strain of Rectangular Bar

V = l.b.t

11.5 Volumetric Strain of Rod

11.6 Bars of Varying Sections

Total change in length will be equal to an algebra is sum of deformation in all (individual) sections.

Fig. 11.2 Bars of varying section

dl₁= e₁l₁; dl₂ = e₂ l₂; dl₃ = e₃ l₃