Continuum Mechanics

General Principles

M. Ali EtaatiEindhoven University of Technology

Math. & Computer Science Dept.CASA

Apr. 12 2006

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

Vn dt

Flux across a surface

Volume Flux:

Mass Flux:

Momentum Flux: (a vector)

Kinetic Energy Flux: (a scalar)

V

v dt

dS

n

Vn dt

Conservation of mass; the continuity equation

V

v

vn

nP

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 dSdS

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 jiji nTt

Equilibrium equations (no acceleration)

The moment of momentum principles

n̂

x2

x3

x1

1)1( * St

3)3( * St

St *

Vb **2

)2( * St

0mnrmnTe

0

0

0

2112

1331

3223

TT

TT

TT

or

3)3(

2)2(

1)1( ntntntt

(Symmetrical Stress Tensor)

Momentum equation; Couple stress

n̂

x2

x3

x1

1)1( * Sm

3)3( * Sm

Sm *

Vc **2

)2( * Sm

3)3(

2)2(

1)1( nmnmnmm

“ m ” Average couple traction,(per unit area)

“ M ” couple tensor ,

“ c ” Average total body couple (per unit mass)

jjii nm

jijiM )(

Momentum equation; Rotational momentum principle

Which “ l ” spin angular momentum (per unit mass)

Which results in

]12[3

]31[2

]23[1

2

2

2

TTe

TTe

TTe

mnmn

mnmn

mnmn

(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

j

ijiij x

vvD

,

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

yield

elastic

Second law of Thermodynamics(entropy)

• Entropy in classical thermodynamics

rev

dqds )(

• Ideal gas

mRpv

pdvdqdu

v

dvRdcdq v )(

dcdu v )( (Constant volume)

(Entropy as a state function)

Second law of Thermodynamics(entropy)

• Gibbs relation

• Enthalpy

Then,

)(, pj

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

s

“ 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