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

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M. Ali Etaati Eindhoven University of Technology Math. &amp; 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

(Non-symmetrical Stress Tensor)

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)

First law of Thermodynamics

“ u” specific internal energy and

, the rate of deformation

Finally results in ( the nonpolar case ):

• Remark on internal energy

Power of couple stress

Such that

First law of Thermodynamics (including couple stress)

Energy equation with couple stresses

Reversible and irreversible processes

Second law of Thermodynamics

Entropy in classical thermodynamics

• Ideal gas

Second law of Thermodynamics

(entropy)

(Constant volume)

(Entropy as a state function)

Gibbs relation

• Enthalpy

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