Continuum Mechanics General Principles M. Ali Etaati Eindhoven University of Technology Math. & Computer Science Dept. CASA Apr

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

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

Flux across a surface V v dt dS n V n dt

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

Conservation of mass; the continuity equation V v vnvn n P dS S

Continuity equation Incompressible material

Rate of increase of the total amount of A inside the control surface “S” “A” is any property of the material 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” = - Reynolds transport theorem Then it will result in Reynolds theorem: Material form of mass:

Momentum principles; equation of motion and equilibrium b dV V t dS dS SdV Momentum balance “t” is external surface force “b” is external body force

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

The moment of momentum principles x2x2 x3x3 x1x1 or (Symmetrical Stress Tensor)

Momentum equation; Couple stress x2x2 x3x3 x1x1 “ m ” Average couple traction,(per unit area) “ M ” couple tensor, “ c ” Average total body couple (per unit mass)

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

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

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

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

Second law of Thermodynamics Reversible and irreversible processes

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

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

Second law of Thermodynamics (entropy production) “ ” 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 “ v “ is a set of “ n “ variables including all the mechanical and electrical state variables for continuum thermodynamics Or better to say:

Summary as an example