equation of state for solids for use in manufacturing usable form of eos….. p m v subbarao...
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Equation of State for Solids For use in Manufacturing
Usable form of EOS…..
P M V SubbaraoProfessor
Mechanical Engineering Department
I I T Delhi
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.
Universal Equation of State for Solids
whereand
V0 is the volume of solid and B0 is bulk modulus at reference pressure .
RX TTBeX
X
BXTp
001
20 01
2,
3
1
0
RTV
VX
12
3
0
0 p
B
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
A common equation of state for Solid
TpCpCTCTCCVm 542
321Vm = 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
20 1 TBTAVV mm
Vmo = molar volume at 00CA, B = empirical constants
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:
Ad
Fd
Tensor Nature of Stress
dA
Fd
In 1823, the French mathematician, Augustin Baron Cauchy (1789–1857) introduced the concept of stress by eliminating the difficulty that .σ is a function of two vectors, at the price that stress became a second-order tensor .
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:AB
ABBAex
''
dx
dxdxxu
dxex
x
uex
xeu x