Equation of State for Solids For use in Manufacturing

1
Equation of State for Solids For use in
Manufacturing

• P M V Subbarao
• Professor
• Mechanical Engineering Department
• I I T Delhi

Usable form of EOS..
2
The Need

• Knowledge of the equation of state P V T -
  relation, is of primary importance in
  Manufacturing .
• It provides insight into the nature of metal
  working/cutting theories, and determines the
  values of fundamental thermodynamic parameters.

3
Universal Equation of State for Solids
where
and
V0 is the volume of solid and B0 is bulk modulus
at reference pressure .
4
Constants of EoS
Parameter Gold Nacl Xenon
B0 (1010 Pa) 16.6 2.35 0.302
?0(10-5 K-1) 4.25 12.0 60.0
(?B/?p)0 5.5 6.5 5.35 7.8
TR, K 300 298 60
5
A common equation of state for Solid
Vm molar volume T temperature p pressure
C1, C2, C3, C4, C5 empirical constants The
empirical constants are all positive and specific
to each substance. For constant pressure
processes, this equation is often shortened to
Vmo molar volume at 00C A, B empirical
constants
6
p-V-T Diagram of crystalline solid Phase
Pressure
Temperature
Volume
7
Thermodynamic Property Models for Manufacturing
Solids

• Manufacturing systems deal with three fundamental
  properties, namely
• Stress
• Strain
• Temperature
• Unlike general thermodynamic Scalar EoS,
  manufacturing systems demands Tensor EoS.
• Basic definition of stress is

8
Tensor Nature of Stress
In 1823, the French mathematician, Augustin Baron
Cauchy (17891857) introduced the concept of
stress by eliminating the difficulty that .s is a
function of two vectors, at the price that stress
became a second-order tensor .
9
Stress Tensor
10
Definition of strain

• The strain can be specified by the displacement
  of each point in the solid from its position in
  some reference configuration.
• We treat the solid as a continuum, and consider
  only strains which are effectively uniform over
  distances of several atomic spacings.
• One Dimensional Strain

11
Deformation of Solid
12
Three Dimensional Strain
13
Pure Translation of A Solid Projection
14
Translation Rotation of A Solid Projection
15
Translation Deformation of A Solid Projection
16
Translation, Rotation Deformation of a Solid
Projection
17
The strain Vs Rotation