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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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M. Ali EtaatiEindhoven University of TechnologyMath. & Computer Science Dept.CASAApr. 12 2006
Continuum Mechanics
General Principles
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
Divergence theorem:
Integral Transformation; Divergence (Gauss’s) Theorem
Green’s theorem:
Stokes theorem:
V
Vn dt
v dt
n
dS
Flux across a surface
V
Vn dt
v dt
n
dS
Mass Flux:
Flux across a surface
Volume Flux:
Momentum Flux:
(a vector)
Kinetic Energy Flux:
(a scalar)
vn
n
P
v
dS
V
S
Conservation of mass; the continuity equation
Continuity equation
Incompressible material
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:
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
The moment of momentum principles
or
(Symmetrical Stress Tensor)
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)
First law of Thermodynamics
“ 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)
Second law of Thermodynamics
(entropy)
Then,
“ ” 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
Summary as an example