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

p-V-T Diagram of crystalline solid Phase

Volume

Pressure

Temperature

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 .

Stress 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

Deformation of Solid

Three Dimensional Strain

zeyexeu xzxyxx

zeyexev yzyyyx

zeyexew zzzyzx

xeU

jiji xeu

Pure Translation of A Solid Projection

xdx

A

B C

Dxd

Translation & Rotation of A Solid Projection

xdx

A

B

C

D

xd

Translation & Deformation of A Solid Projection

xdx

xd

A

B C

D

Translation, Rotation & Deformation of a Solid Projection

The strain Vs Rotation

22jiijjiij

ij

eeeee

ijijije

i

j

j

iij x

u

x

u

2

1