M. Ali Etaati Eindhoven University of Technology Math. & Computer Science Dept. CASA Apr. 12 2006. Continuum Mechanics General Principles. Presentation Layout Introduction Conservation of mass Conservation of Momentum The moment of momentum principles
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Continuum Mechanics
General Principles
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:
Stokes theorem:
Vn dt
v dt
n
dS
Mass Flux:
Flux across a surface
Volume Flux:
Momentum Flux:
(a vector)
Kinetic Energy Flux:
(a scalar)
Incompressible 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”

=
“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:
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
“t” External surface force,
“T” Stress tensor
Cauchy’s equations of Motions
Equilibrium equations (no acceleration)
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
Which “ l ” spin angular momentum (per unit mass)
Momentum equation; Rotational momentum principle
Whichresults in
(Nonsymmetrical Stress Tensor)
Conservation of energy
“ q ” heat flux vector
“ r ” distributed internal heat source per unit mass (possibly from a radiation field)
“ u” specific internal energy and
, the rate of deformation
Finally results in ( the nonpolar case ):
Such that
First law of Thermodynamics (including couple stress)
Energy equation with couple stresses
Second law of Thermodynamics
Second law of Thermodynamics
(entropy)
(Constant volume)
(Entropy as a state function)
“ ” 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