1. Equation of State for Solids For use in Manufacturing Usable form of EOS….. P M V Subbarao Professor Mechanical Engineering Department I I T Delhi

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 V 0 is the volume of solid and B 0 is bulk modulus at reference pressure.

4. Constants of EoS ParameterGoldNaclXenon B 0 (10 10 Pa)16.62.350.302  0 (10 -5 K -1 ) 4.2512.060.0 (  B/  p) 0 5.5 – 6.55.357.8 T R, K30029860

5. A common equation of state for Solid V m = molar volume T = temperature p = pressure C 1, C 2, C 3, C 4, C 5 = empirical constants The empirical constants are all positive and specific to each substance. For constant pressure processes, this equation is often shortened to V mo = molar volume at 0 0 C A, B = empirical constants

6. p-V-T Diagram of crystalline solid Phase Volume Pressure Temperature

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 (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.

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