Chapter-3 Poisson - S Ratio PDF
Chapter-3 Poisson - S Ratio PDF
Chapter-3 Poisson - S Ratio PDF
Solid Mechanics-I
Stress-strain Relationship
Solid Mechanics-I
Stress-strain Relationship
7/17/2014
Solid Mechanics-I
Stress-strain Relationship
POISSONS RATIO
In all engineering materials, the elongation produced by an axial
tensile force P in the direction of the force is accompanied by a
contraction in transverse direction
All materials considered will be assumed to be both homogeneous
and isotropic, i.e., their mechanical properties will be assumed
independent of both position and direction
It follows that the strain must have the same value for any
transverse direction, referred to as the lateral strain
Poissons ratio is an important constant for a given material and is
defined as:
lateral strain
=
axial strain
=
Solid Mechanics-I
Stress-strain Relationship
7/17/2014
POISSONS RATIO
Consider an element of material subjected to uniaxial stress x the
corresponding strain system is shown.
Stress-strain Relationship
POISSONS RATIO
Element subjected to triaxial stresses x, y, and z, total strain in xdirection is therefore composed of a strain due to x, and lateral
strains due to y and z.
Similarly,
=
=
Solid Mechanics-I
Stress-strain Relationship
7/17/2014
POISSONS RATIO
Equations giving strains can be solved to determine stress
components:
= 1+ 12 + + + 1+
+ + + 1+
1+ 12
= 1+ 12 + + + 1+
Lame constant, = 1+ 12 , = 2 1+
Similarly, =
= 3(12)
Plane stress
In many practical situations stress
component in z-direction is zero
and is referred as plane stress
= ,
=
Solid Mechanics-I
Stress-strain Relationship
POISSONS RATIO
Plane strain
If the strain in z-direction is zero ( = 0), this referred to a plane
strain condition
Stress-strain Relationship
7/17/2014
POISSONS RATIO
Solid Mechanics-I
Stress-strain Relationship
POISSONS RATIO
Solid Mechanics-I
10
Stress-strain Relationship
7/17/2014
POISSONS RATIO
Problem 3-33
Plug has diameter of 30mm and fits within a rigid sleeve having an
inner diameter of 32mm. Both are 50mm long. Determine the axial
pressure that must be applied at the top of plug to cause it contact the
sides of sleeve. Also, how far must the plug be compressed downward
to do this? Youngs modulus is 5MPa and Poissons ratio is 0.45.
Solution:
Plug should be compressed with the
particular amount of load in order to
touch the sleeve. That load is used to
determine the axial pressure.
Solid Mechanics-I
PLUG
Sleeve
Stress-strain Relationship
11
POISSONS RATIO
That load will do the compression and resulted in lateral strain, which
can be calculated using,
32 30
=
=
= 0.06667
30
=
= 0.1481
= =
= 741
Answer
= L
= 0.1481 50
= 7.41
Solid Mechanics-I
12
Answer
Stress-strain Relationship