Conservation Laws for Continua e 3 e 1 e 2 Original Configuration Deform ed Configuration S R R 0 S 0 b t n V T(n) y x u(x) or ij j j i b a y y σ b a e 3 e 1 e 2 Original Configuration Deform ed Configuration S R R 0 S 0 n V y x u(x) V 0 Mass Conservation Linear Momentum Conservation 0 0 i i i i const const V v v dV t y t y x x 0 ( ) 0 i i const const v t y t y y y v Angular Momentum Conservation ij ji
Conservation Laws for Continua. Mass Conservation. Linear Momentum Conservation. Angular Momentum Conservation. Work-Energy Relations. Rate of mechanical work done on a material volume. Conservation laws in terms of other stresses. Mechanical work in terms of other stresses. - PowerPoint PPT Presentation
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Conservation Laws for Continua
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
S
R
R0S0b
t
n
V T(n)yx
u(x)
or ijj j
ib a
y
y σ b a
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
S
R
R0S0n
V
yx
u(x)V0
Mass Conservation
Linear Momentum Conservation
0 0i i
i iconst constV
v vdV
t y t y
x x
0 ( ) 0i
iconst const
vt y t
yy y
v
Angular Momentum Conservation ij ji
( ) 12i i i ij ij i ii
A V V V
dr T v dA b v dV D dV v v dVdt
n
Rate of mechanical work done ona material volume
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
S
R
R0S0b
t
n
V T(n)yx
u(x)
Conservation laws in terms of other stresses
0 0 0 0ij
j ji
Sb a
x
S b a
0 0 0 0ik jk
j ji
Fb a
x
TF b a
Mechanical work in terms of other stresses
0 0
( )0 0 0
12i i i ij ji i ii
A V V V
dr T v dA b v dV S F dV v v dVdt
n
0 0
( )0 0 0
12i i i ij ij i ii
A V V V
dr T v dA b v dV E dV v v dVdt
n
Work-Energy Relations
2
0iij ij i i i i i
V
dvD dV dA
dtV V S
dV v b v dV t v
Principle of Virtual Work (alternative statement of BLM)
12
ji iij ij
j j i
vv vL Dy y y
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
S2R0S0b
tVyx
u(x) S1
ji ii
j
dvby dt
If for all iv
Then
i ij jn t 2Son
Thermodynamics
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
S
R
R0
S0 b
tSpecific Internal EnergySpecific Helmholtz free energy s
Temperature
Heat flux vector qExternal heat flux q
First Law of Thermodynamics ( )d KE Q Wdt
iij ij
iconst
qD q
t y
x
Second Law of Thermodynamics 0dS ddt dt
Specific entropy s
( / )0i
i
qs qt y
1 0ij ij ii
D q sy t t
Transformations under observer changesTransformation of space under a change of observer
e3
e1
e2
DeformedConfiguration
b
n
y
DeformedConfiguration
b*
n*
y*
e2*
e3*
e2*
Inertial frame
Observer frame
* *0 0( ) ( )( )t t y y Q y y
All physically measurable vectors can be regarded as connecting two points in the inertial frame
These must therefore transform like vectors connecting two points under a change of observer
* * * * b Qb n Qn v Qv a Qa
Note that time derivatives in the observer’s reference frame have to account for rotation of the reference frame
*** * * * *0
0 0
2 * *2 2 2 * ** * * 2 * *0 0
0 02 2 2 2
( ( )) ( ( ))
( )( ( )) ( ( )) 2 ( )
T
T
dd d dt tdt dt dt dt
d d td d d d dt tdt dt dtdt dt dt dt
yy yv Qv Q Q Q y y Ω y y
y yy y Ω ya Qa Q Q Q y y Ω y y Ω
Tddt
QΩ Q
The deformation mapping transforms as * *0 0( , ) ( ) ( ) ( , )t t t t y X y Q y X y
The deformation gradient transforms as *
*
y yF Q QFX X
The right Cauchy Green strain Lagrange strain, the right stretch tensor are invariant * * * * *T T T C F F F Q QF C E E U U
The left Cauchy Green strain, Eulerian strain, left stretch tensor are frame indifferent * * * *T T T T T B F F QFF Q QCQ V QVQ
The velocity gradient and spin tensor transform as
* * * 1 1
* * *( ) / 2
T T
T T
L F F QF QF F Q QLQ Ω
W L L QWQ Ω
The velocity and acceleration vectors transform as **
* * * * *00 0
2 * *2 2 2 * ** * * 2 * *0 0
0 02 2 2 2
( ( )) ( ( ))
( )( ( )) ( ( )) 2 ( )
T
T
dd d dt tdt dt dt dt
d d td d d d dt tdt dt dtdt dt dt dt
yy yv Qv Q Q Q y y Ω y y
y yy y Ω ya Qa Q Q Q y y Ω y y Ω
(the additional terms in the acceleration can be interpreted as the centripetal and coriolis accelerations) The Cauchy stress is frame indifferent * Tσ QσQ (you can see this from the formal definition, or use
the fact that the virtual power must be invariant under a frame change) The material stress is frame invariant * Σ Σ The nominal stress transforms as * 1 1( ) T T TJ J S QF Q Q F Q SQσ σ (note that this
transformation rule will differ if the nominal stress is defined as the transpose of the measure used here…)
Some Transformations under observer changes
e3
e1
e2
DeformedConfiguration
b
n
y
DeformedConfiguration
b*
n*
y*
e2*
e3*
e2*
Inertial frame
Observer frame
Objective (frame indifferent) tensors: map a vector from the observed (inertial) frame back onto the inertial frame
t n σ
* *T T σ QσQ D QDQ
Invariant tensors: map a vector from the reference configuration back onto the reference configuration
0 T m Σ
* Σ Σ
Mixed tensors: map a vector from the reference configuration onto the inertial frame
d dy F x* F QF
Some Transformations under observer changes
Constitutive Laws
General Assumptions:1.Local homogeneity of deformation (a deformation gradient can always be calculated)2.Principle of local action (stress at a point depends on deformation in a vanishingly small material element surrounding the point)
Restrictions on constitutive relations: 1. Material Frame Indifference – stress-strain relations must transform consistently under a change of observer 2. Constitutive law must always satisfy the second law of thermodynamics for any possible deformation/temperature history.
Equations relating internal force measures to deformation measures are knownas Constitutive Relations
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
1 0ij ij ii
D q sy t t
Fluids
Properties of fluids• No natural reference configuration• Support no shear stress when at rest
Kinematics• Only need variables that don’t depend
on ref. config
Conservation Laws
e3
e1
e2
DeformedConfiguration
S
R
b
t
y
iij
j
vL
y
( ) / 2 ( ) / 2ij ij ij ij ij ji ij ij jiL D W D L L W L L ki ijk ijk ij
j
vW
y
1 1( ) 2 ( )2 2
k i i i
k
i i k i i ii ik k ik ik k
kx const y const y const y const
ik k ik k k k ijk j k
i iy const
v v y v v va L v D W v
t y t t t t
vv v W v v v v
y t y
0 or 0ikk
iconst const
vD
t t y
x y
i
ji i ii k ij ji
j k y const
v vb v
y y t
iij ij
iconst
qD q
t y
x
1 0ij ij ii
D q sy t t
General Constitutive Models for Fluids
Objectivity and dissipation inequality show that constitutive relations must have form
Internal EnergyEntropyFree EnergyStress response functionHeat flux response function
ˆ( , ) ˆ( , )s s
ˆ ( , ) s
ˆ ˆ ˆ( , , ) ( , ) ( , , )visij ij ij eq ij ij ijD D
ˆ , , ,i i iji
q q Dy
In addition, the constitutive relations must satisfy
2
2 2
22
2 2 2
ˆˆ ˆ
ˆ ˆ ˆˆ ˆ
ˆˆ
eq
eq eqeq
eqvv
s
s
cc
( , , ) 0 , , 0vis
ij ij ij ii i
D D qy y
ˆ( , )vc
where
Constitutive Models for Fluids
ˆ ( ) ( )ij eq ij Elastic Fluid
0log log( 1)v v v ij ij ij
pc c c R s p R
Ideal Gas
ˆ ( , ) ( ( , ) ( , ) ) 2 ( , )( / 3)ij eq kk ij ij kk ijD D D Newtonian Viscous
1 1 2 3 2 1 2 3 3 1 2 3
ˆ ( , )( , ) ( , , , , ) ( , , , , ) ( , , , , )ij eq ij ij ij ik kjI I I I I I D I I I D D
Non-Newtonian
ˆ ˆ ˆ( , , ) ( , ) ( , , )visij ij ij eq ij ij ijD D ˆ ( , ) s
Derived Field Equations for Newtonian Fluids
1 ( )2
i k
ji i i ii i i k k k ijk j k
j k iy const y const
v v vb a a v v v v
y y t t y
Combine BLM
With constitutive law. Also recall
Compressible Navier-Stokes 2 ( , ) / 3 ( , ) ( , )ij kk ij i i eq kki j
p D D b a p Dy y
221 23
eq jii i
i j j j i
vvb a
y y y y y
With density indep viscosity
1 ( )2
k
eq ii k k ijk j k
i iy const
vb v v v
y t y
For an elastic fluid (Euler eq)
21 ii i
i j j
vp b ay y y
For an incompressible Newtonianviscous fluid
12
jiij
j i
vvD
y y
0i
i
vy
Incompressibility reduces mass balance to
0 or 0ikk
iconst const
vD
t t y
x yMust always satisfy mass conservation
Unknowns: , ip v
Derived Field Equations for Fluids
2 2 2
21 2 ( )
3i k l k i
ijk ijk k ij j ij j j l l l k j k const
v v vb D
y y y y y y y y y t
x
Vorticity transport equation (constant temperature, density independent viscosity)
For an elastic fluid
ki ijk ijk ij
j
vW
y
Recall vorticity vector
( ) k iijk k ij j i
j k const
vb D
x y t
x
2( )i i
ijk k ij jj j j const
b Dy y x t
xFor an incompressible fluid
k i kijk ij j i
j kconst
a vD
y t y
x
If flow of an ideal fluid is irrotational at t=0 and body forces are curl free, then flow remains irrotational for all time (Potential flow)