M. Ali Etaati Eindhoven University of Technology Math. & Computer Science Dept. CASA Apr. 12 2006

1. M. Ali EtaatiEindhoven University of TechnologyMath. & Computer Science Dept.CASAApr. 12 2006 Continuum Mechanics General Principles

2. Presentation Layout Introduction Conservation of mass Conservation of Momentum The moment of momentum principles Conservation of energy; First law of Thermodynamics First law of Thermodynamics (including couple stress) Internal energy and Entropy production; second law of Thermodynamics Summary and conclusion as an example

3. Divergence theorem: Integral Transformation; Divergence (Gauss’s) Theorem Green’s theorem: Stokes theorem:

4. V Vn dt v dt n dS Flux across a surface

5. V Vn dt v dt n dS Mass Flux: Flux across a surface Volume Flux: Momentum Flux: (a vector) Kinetic Energy Flux: (a scalar)

6. vn n P v dS V S Conservation of mass; the continuity equation

7. Continuity equation Incompressible material

8. Reynolds transport theorem Rate of increase of the total amount of A possessed by the material instantaneously inside the control surface Net rate of outward flux of A carried by mass transport through the control surface “S” - = “A” is any property of the material Rate of increase of the total amount of A inside the control surface “S” Then it will result in Reynolds theorem: Material form of mass:

9. tdS bdV dS V dV S “t” is external surface force “b” is external body force Momentum principles; equation of motion and equilibrium Momentum balance

10. “t” External surface force, “T” Stress tensor Cauchy’s equations of Motions Equilibrium equations (no acceleration)

11. x2 x1 x3 The moment of momentum principles or (Symmetrical Stress Tensor)

12. x2 x1 x3 “ m ” Average couple traction,(per unit area) “ M ” couple tensor , “ c ” Average total body couple (per unit mass) Momentum equation; Couple stress

13. Which “ l ” spin angular momentum (per unit mass) Momentum equation; Rotational momentum principle Whichresults in (Non-symmetrical Stress Tensor)

14. Thermodynamic system ( closed system for continuous matter ) • Power input Conservation of energy • Heat input “ q ” heat flux vector “ r ” distributed internal heat source per unit mass (possibly from a radiation field)

15. First law of Thermodynamics “ u” specific internal energy and , the rate of deformation Finally results in ( the nonpolar case ): • Remark on internal energy

16. Power of couple stress Such that First law of Thermodynamics (including couple stress) Energy equation with couple stresses

17. Reversible and irreversible processes Second law of Thermodynamics

18. Entropy in classical thermodynamics • Ideal gas Second law of Thermodynamics (entropy) (Constant volume) (Entropy as a state function)

19. Gibbs relation • Enthalpy Second law of Thermodynamics (entropy) Then,

20. “ ” the rate of increase of the system’s entropy “ r ” distributed internal heat source per unit mass (possibly from a radiation field) “ ” entropy production rates due to internal irreversible processes “ q ” the outward heat flux vector Or better to say: Second law of Thermodynamics (entropy production) “ v “ is a set of “ n “ variables including all the mechanical and electrical state variables for continuum thermodynamics

21. Summary as an example