Conservation Laws for Continua
Mass Conservation
 
vi



t x const
yi
V


vi

dV

0



0

t x const
yi

 vi


0
t y const yi
e2

  y  (  v)  0
t y const
or
 ij
 yi
  bj   a j
e2
Angular Momentum Conservation
x
 ij   ji
V
R
Deformed
Configuration
Original
Configuration
n
S
R0
e1
e3
u(x)
y
e3
S0
n
S
V0
x
e1
Linear Momentum Conservation
y  σ   b   a
R0
S0
u(x)
y
Original
Configuration
V
R
b
T(n)
t
Deformed
Configuration
Work-Energy Relations
Rate of mechanical work done on
a material volume
e2

d  1
(n )
r   Ti vi dA   bi vi dV    ij Dij dV     vi vi dV 
dt  2

A
V
V
V
u(x)
y
x
e1
e3
n
S
R0
S0
V
R
b
T(n)
t
Deformed
Configuration
Original
Configuration
Conservation laws in terms of other stresses
  S  0 b  0 a
Sij
xi
 0 b j  0 a j
     F T    0b   0a


Mechanical work in terms of other stresses


d 1

r

b
v
dV

S
F
dV


v
v
dV

 ii
 ij ji 0 dt  2 0 i i 0 


A
V
V0
V0



d 1

(n )
r   Ti vi dA   bivi dV   ij Eij dV0    0vivi dV0 
dt  2

A
V
V0
V0

Ti(n)vi dA 


 ik F jk
 0b j  0 a j
xi
Principle of Virtual Work (alternative statement of BLM)
1   vi  v j 
 Dij  


2  y j
yi 
 vi
 Lij 
y j
R0
S0
e2
e3
If
  ij Dij dV   
V
V
dvi
 vi dV 
dt
Then
 bi vi dV   ti vi dA  0
V
S2
 ji
y j
  bi  
ni ij  t j
on
dvi
dt
S2
y
x
e1
u(x)
S2
V
t
Deformed
Configuration
Original
Configuration
for all
b
 vi
S1
Thermodynamics
S
S0
Temperature 
Specific Internal Energy 
Specific Helmholtz free energy      s
Heat flux vector q
External heat flux q
Specific entropy s
First Law of Thermodynamics
Second Law of Thermodynamics
e2
R
e1
e3
t
Original
Configuration
d
(  KE)  Q  W
dt

b
R0
q

  ij Dij  i  q
t xconst
yi
dS d 

0
dt dt
s (qi /  ) q
 
 0
t
yi

1 
 
 
 ij Dij  qi

s
0
 yi
t 
 t
Deformed
Configuration
Transformations under observer changes
Transformation of space under
a change of observer
y*  y*0 (t )  Q(t )(y  y0 )
n
Inertial frame
e2
b
y
e1
dQ T
Ω
Q
dt
e3
All physically measurable vectors can be
regarded as connecting two points in the
inertial frame
Deformed
Configuration
e2*
n*
e3*
e2*
These must therefore transform like
vectors connecting two points under a
change of observer
Observer frame
b*
y*
Deformed
Configuration
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
dy
d T * *
dy* dy*0
v  Qv  Q
 Q Q (y  y 0 (t )) 

 Ω(y*  y*0 (t ))
dt
dt
dt
dt
*
*
a  Qa  Q
d 2y
dt
2
Q
d2
dt
2
T
Q (y
*
 y*0 (t )) 
dy* dy*0 (t )
 2 dΩ  * *

 Ω 

)
 (y  y 0 (t ))  2Ω(
2
2
dt
dt
dt


dt
dt
d 2 y*
d 2 y*0
Some Transformations under observer changes
The deformation mapping transforms as y* (X, t )  y*0 (t )  Q(t )  y(X, t )  y0 
y*
y
The deformation gradient transforms as F 
Q
 QF
X
X
The right Cauchy Green strain Lagrange strain, the right stretch tensor are invariant
*
C*  F*T F*  FT QT QF  C E*  E U*  U
The left Cauchy Green strain, Eulerian strain, left stretch tensor are frame indifferent
B*  F*F*T  QFFT QT  QCQT
The velocity gradient and spin tensor transform as

V*  QVQT

L*  F*F*1  QF  QF F 1QT  QLQT  Ω
W*  (L*  L*T ) / 2  QWQT  Ω
The velocity and acceleration vectors transform as
dy
d T *
dy* dy*0
*
v  Qv  Q
 Q Q (y  y 0 (t )) 

 Ω(y*  y*0 (t ))
dt
dt
dt
dt
*
dy* dy*0 (t )
 2 dΩ  *
*
a  Qa  Q
Q
Q (y

 Ω 

)
 (y  y 0 (t ))  2Ω(
2
2
2
2
dt
dt
dt


dt
dt
dt
dt
(the additional terms in the acceleration can be interpreted as the centripetal and coriolis accelerations)
*
d 2y
d2
T
*
 y*0 (t )) 
d 2 y*
d 2 y*0
The Cauchy stress is frame indifferent σ*  QσQT (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 S*  J (QF)1  QσQT  JF1  σQT  SQT (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
Objective (frame indifferent)
tensors: map a vector from the
observed (inertial) frame back
onto the inertial frame
t  nσ
n
Inertial frame
e2
σ*  QσQT
D*  QDQT
b
y
e1
e3
Invariant tensors: map a
vector from the reference
configuration back onto the
reference configuration
e2*
T0  m  Σ
Σ*  Σ
Deformed
Configuration
n*
e3*
e2*
Observer frame
Mixed tensors: map a vector
from the reference
configuration onto the inertial
frame
dy  Fdx
F*  QF
b*
y*
Deformed
Configuration
Constitutive Laws
Equations relating internal force measures to deformation measures are known
as Constitutive Relations
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)
e2
e1
e3
Original
Configuration
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.
1 
 
 
 ij Dij  qi

s
0
 yi

t

t


Deformed
Configuration
Fluids
Properties of fluids
•
•
No natural reference configuration
Support no shear stress when at rest
S
e2
y
b
R
t
Kinematics
•
e1
Only need variables that don’t depend
on ref. config
Lij 
ai 

vi
y j
Lij  Dij  Wij
e3
Dij  (Lij  L ji ) / 2 Wij  (Lij  L ji ) / 2
i ijk
Deformed
Configuration
vk
  ijk Wij
y j
vi
v y
v
v
v
 i k  i
 Lik vk  i
  Dik  Wik  vk  i
t x const yk t
t y const
t y const
t y const
k
i
i
i
v
1 
1 
(vk vk )  2Wik vk  i

(vk vk ) ijk  j vk
2 yi
t y const 2 yi
k
Conservation Laws
 vi


  Dkk  0 or

0
t xconst
t y const yi
 v

v

  bi    i vk  i
 yk
y j
t y const 
i


 ji

q

  ij Dij  i  q
t xconst
yi
 ij   ji
1 
 
 
 ij Dij  qi

s
0
 yi
t 
 t
General Constitutive Models for Fluids
Objectivity and dissipation inequality show that constitutive relations must have
form
Internal Energy   ˆ(  , )
s  sˆ(  , )
Entropy
  ˆ (  , )     s
Free Energy
Stress response function  ij  ˆij ( ,  , Dij )  ˆeq ( , )ij  ˆijvis ( , , Dij )
Heat flux response function qi  qˆi  ,  ,  , Dij 

yi

In addition, the constitutive relations must satisfy
ˆeq   2
ˆeq
ˆ

sˆ
 


2
cv  
where
cv ( ,  ) 
 2ˆ
 2
ˆ

sˆ  


ˆeq  
ˆeq

 2
2
cv
  ˆeq


 2  2
ˆ

 ijvis (  , , Dij ) Dij  0


qi   , ,
yi

 
0

 yi
Constitutive Models for Fluids
 ij  ˆij ( ,  , Dij )  ˆeq (  , )ij  ˆijvis (  , , Dij )
  ˆ (  , )     s
Elastic Fluid
 ˆ ( )
p
(  1) 
Ideal Gas
  cv 
Newtonian Viscous
 ˆ ( , )
Non-Newtonian
 ij   eq ( )ij
  cv    cv log  R log   s0 
 ij   p ij    R ij
 ij  ( eq ( , )   ( , ) Dkk )ij  2 ( , )( Dij  Dkkij / 3)
  ˆ (  , )
 ij   eq (  , )ij  1( I1, I 2 , I3 ,  , )ij  2 ( I1, I 2 , I3 ,  , ) Dij  3 ( I1, I 2 , I3 ,  , ) Dik Dkj
Derived Field Equations for Newtonian Fluids
Unknowns:
p, vi
 vi


  Dkk  0 or

0
t xconst
t y const yi
Must always satisfy mass conservation
Combine BLM
 ji
y j
 bi   ai
ai 
vi
v
v
1 
vk  i
 i

(vk vk ) ijk  j vk
yk
t y const t y const 2 yi
i
k
v j 
1  v
Dij   i 

2  y j yi 
With constitutive law. Also recall
Compressible Navier-Stokes

p

2
 (  , ) Dij  Dkk  ij / 3   bi   ai
yi
y j


2
  2   v j
1  eq   2 vi




b a
 yi
 y j y j   3  y j yi i i
With density indep viscosity
For an incompressible Newtonian
viscous fluid

1 p   2vi

b a
 yi  y j y j i i
Incompressibility reduces mass balance to
For an elastic fluid (Euler eq)
p   eq (  , )   (  , ) Dkk

 eq
yi
 bi  
vi
0
yi
vi
1 
 
(vk vk )   ijk  j vk
t y const 2 yi
k
Derived Field Equations for Fluids
i ijk
Recall vorticity vector
vk
  ijk Wij
y j
ijk
ak i
v

 Dij j  k i
y j
t xconst
yk
Vorticity transport equation (constant temperature, density independent viscosity)
2
v

  2i
1
 
2   2 vl 

  vk 



ijk

  
 ijk
(bk )  Dij j  k i  i



 y j y j  2
y j 
3  yl yk 
y j
yk
t
 yl yl 

For an elastic fluid
ijk
For an incompressible fluid
v


(bk )  Dij j  k i  i
x j
yk
t xconst

  2i


 ijk
(bk )  Dij j  i
 y j y j
x j
t
xconst
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)
x const
Derived field equations for fluids
For an elastic fluid
• Bernoulli
H  
For irrotational flow
For incompressible fluid
 eq
1
 vi vi    constant

2
H  
p
 eq
1
 vi vi    constant

2
1
 vi vi    constant
 2
along streamline
everywhere
Normalizing the Navier-Stokes equation
Incompressible Navier-Stokes
v
1 p   2 vi
1 


 bi  i

(vk vk ) ijk  j vk
 yi  y j y j
t y const 2 yi
k
L Characteristic length
V Characteristic velocity
f Characteristic frequency
P Characteristic pressure change
yi  Lyˆi
vi  Vvˆi
t  ftˆ
p  pˆ P
b  gbˆ
V
L
i
Normalize as
vˆi
vˆi
pˆ
1  2vˆi
1 ˆ
Eu


bi  St

vˆi
2
ˆ
yˆi Re yˆ j yˆ j Fr
t y const yˆi
k
Reynolds number
Re  VL / 
Froude number
Fr  V / gL
Eu  P / V 2
St  fL / V
Strouhal number
Euler number
Limiting cases most frequently used
vˆi
vˆi
pˆ
1  2vˆi
1 ˆ
Eu


bi  St

vˆi
2
yˆi Re yˆ j yˆ j Fr
tˆ y const yˆi
k
Ideal flow
Re  
vˆi
vˆi
pˆ
1 ˆ
Eu

bi  St

vˆi
2
ˆ
yˆi Fr
t y const yˆi
k

Stokes flow
 eq
yi
 bi  
vi
1 
 
(vk vk )   ijk  j vk
t y const 2 yi
k
V 0
vˆ
pˆ
1  2vˆi


 bˆi  i
yˆi Re yˆ j yˆ j
tˆ y const
k
v
1 p   2 vi


 bi  i
 yi  y j v j
t y const
k
Solving fluids problems: control volume approach
Governing equations for a control volume (review)
B
R
Example
v1 A
i
j
Steady 2D flow, ideal fluid
Calculate the force acting on the wall
Take surrounding pressure to be zero
1
A4
v0
A0 

A5

B
d
n  σdA  bdV 
 vdV  (  v) v  ndA
dt



R
R
B

( p0n  jdA) j  A0 0v02 sin  j 
A
v2
 ( p  p0 )dAj  0
A3
F   A0 0v02 sin  j
A2
A3
Exact solutions: potential flow
If flow irrotational at t=0, remains irrotational; Bernoulli holds everywhere
Irrotational: curl(v)=0 so

yi
vi
 2
0
0
yi
yi yi
Mass cons
1

 vi vi   
 constant
 2
t
p
Bernoulli
 
vi 
a 2V ( y  V t )
r
2
r  ( y  V t )( y  V t )
V
e2
a
e1
Exact solutions: Stokes Flow
Steady laminar viscous flow between plates
Assume constant pressure gradient in horizontal direction
V

v
1 p   vi
p
 f

 bi  i


0
2
 yi  y j v j
t y const
L

y
2
k
2
2
y2
h
y1
Solve subject to boundary conditions
p
 y

v  V 2 
y2 (h  y2 )  e1
 h 2L

V p  h

σ

  y2 
h L  2

Exact Solutions: Acoustics
Assumptions:
Small amplitude pressure and density fluctuations
Irrotational flow
Negligible heat flow
Approximate N-S as:
v
p

  bi   i
yi
t y const
k
p

 cs2
t
t
For small perturbations:
Mass conservation:
Combine:
 2vi
2 p

  bi  
yi yi
yi t
cs 
p
 s const
v

 i 0
t xconst
yi
 2 vi
 2v j
t
yi y j
 cs2
2
0
 2
t 2
(Wave equation)
 cs2

0
yi yi
yk const
 2vi
2 p


yi t
t 2
yk const
Wave speed in an ideal gas
Assume heat flow can be neglected
Entropy equation:
  cv

q
s
  i  q  s  const
t xconst
yi
  cv    cv log  R log   s0 
p   R
s   cv log  R log   s0      R/cv exp[(s  s0 ) / cv )
R / cv    1
Hence:
cs 
so
p  k 
p
p
 k  1     R
 s const

Application of continuum mechanics to elasticity
Modulus G' (N/m2)
u S
S0
e2
y
e3
Glassy
Viscoelastic
109
x R0
e1
b
Original
Configuration
R
Rubbery
t
Deformed
Configuration
Melt
5
10
Glass Transition
temperature Tg
(frequency)-1
Material characterized by
General structure of constitutive relations
Fij  ij 
u S
S0
e2
x R0
y
e1
e3
Original
Configuration
b
R
t
Deformed
Configuration
ui
x j
S  JF1  σ
Cij  Fki Fkj
Bij  Fik F jk
Qi  JFik1qk
Sij  JFik1 kj
Σ  JF1  σ  FT ij  JFik1 kl F jl1
S ji
x j
 0bi  0
vi
t x const
i
1
1 
 
 
ij Cij  Qi
 0 
s
0
2
 xi
t 
 t
F*  QF
B*  F*F*T  QBQT
C*  F*T F*  C
Σ*  Σ
Frame indifference,
dissipation inequality
ˆ
ˆ ij  2 0
Cij
ˆ ij

 2 0
sˆ
Cij
sˆ  
ˆ

Qk

0
xk
Forms of constitutive relation used in literature
I1 
I1  trace(B)  Bkk
I2 

 
1 2
1
I1  B B  I12  Bik Bki
2
2

I 3  det B  J 2
I1
J 2/3
I2
=
Bkk
J 2/3
1
B  B  1  2 Bik Bki 
I2 
  I12 
   I1  4 / 3 
J 4/3 2 
J 4/3  2 
J

J  det B
B  12b(1)  b(1)  22b(2)  b(2)  32b(3)  b(3)
• Strain energy potential
W  0
W (F)  Wˆ (C)  U (I1, I 2 , I3 )  U (I1, I 2 , J )  U (1, 2 , 3 )
1
W
Fik
J
F jk
 ij 
 ij 
 ij 
2  U
U
 I1

I 2
I3  I1


U
U
B

B
B
ij
 ij
ik kj   2 I3

I

I

2
3

 U
 U
2  1  U
U 
U  ij
1 U
 I1
B

I

2
I

B
B
ij
 2/ 3 
 ij  1

2
ik kj  
4/ 3 I
J  J

I

I

I

I
3

J
J

2
1
2
2
 1


 ij 
3 U (3) (3)
1 U (1) (1)
2 U (2) (2)
bi b j 
bi b j 
b b
123 1
123 2
123 3 i j
Specific forms for free energy function

K
U  1 ( I1  3)  1 ( J  1)2
2
2
• Neo-Hookean material
 ij 
1
1
1
  

Bij  Bkk  ij   2  Bkk Bij  [ Bkk ]2  ij  Bik Bkj  Bkn Bnk ij   K1  J  1 ij

5/ 3 
7
/
3
3
3
3
 J


J
U
N

i  j 1
• Arruda-Boyce
1

Bij  Bkk  ij   K1  J  1  ij

5/ 3 
3

J
1 
• Generalized polynomial function
U
1 


K
U  1 ( I1  3)  2 ( I 2  3)  1 ( J  1)2
2
2
2
• Mooney-Rivlin
• Ogden
 ij 
N
Cij ( I1  3)i ( I 2  3) j 
N

Ki
( J  1)2i
2
i 1
2i
K



(1 i  2 i  3 i  3)  1 ( J  1)2
2
2
i 1 i

 1
 K
1
11
2
U    ( I1  3) 
( I12  9) 
( I13  27)  ...   J  1
2
4
20
1050
 2
 2
Solving problems for elastic materials
(spherical/axial symmetry)
• Assume incompressiblility
e3
eR
• Kinematics
 rr

σ 0
 0


0

0
0 

0 
  
 Frr

F 0
 0

 Brr

B 0
 0

2
0
B
0
 dr 
r
Brr  
 B  B   
 dR 
R
dr
r
Frr 
F  F 
dR
R
2
 dr  r 

   1
 dR  R 
• Constitutive law
0 

0 
F 
0
F
0

e1
e2
2
r 3  a3  R3  A3
 U
 rr  2 
0 

0 
B 
e
r e

 I1
 I1
I1 U 2 I 2 U U 2 
U 


Brr   p
 Brr 
I 2 
3 I1
3 I 2 I 2

 U
U 
I U
2 I U
U

2
1
     2 
 I1
 2

B
 p
 B 
I 2 
3 I1
3 I 2 I 2
 I1

• Equilibrium (or use PVW)
• Boundary conditions
d rr 1
 2 rr      0br  0
dr
r

ur (a)  ga
 rr (a)  ta

ur (b)  gb
 rr (b)  tb
(gives ODE for p(r)
Linearized field equations for elastic materials
Approximations:
• Linearized kinematics
• All stress measures equal
• Linearize stress-strain relation
S
S0
e2
R
ui  ui* (t )
 ij  Cijkl ( kl   kl  )
 ij ni  t*j (t )
on 1R
Elastic constants related to strain energy/unit vol
ε  Sσ  αT
t
e1
e3
1  ui u j 
 ij  


2  x j xi 
 ij
xi
 b j  
Deformed
Configuration
Original
Configuration
 2u j
t 2
on  2 R
ˆij
 2U
Cijkl 

 kl  ij  kl
ˆij
 2U
ij 


 ij 
σ  C(ε  αT )
Isotropic materials:
ij 
1 

 ij   kk ij  Tij
E
E
b
R0
 ij 
E 

 ET
 kk  ij  
 ij
 ij 
1  
1  2
 1  2
Elastic materials with isotropy


1  
 
1 


 11 
 

1 
 
22



 0
 33 
E
0
0




 23  (1   )(1  2 ) 
 0
13 
0
0




12 
 0
0
0

1
 11 
 
 
22



  33  1  

 
2

 23  E  0
0
 213 



 212 
 0
ij 

1

0
0
0


1
0
0
0
1 

 ij   kk ij  Tij
E
E
0
0
0
1  2 
2
0
0
0
0
0
0
1  2 
2
0


  11 


   22 
0    33   ET


  2 23  1  2
0   213 


  212 
1  2  
2 
0
0
0
1 
1 
 
1 
 
0
0
 
0 
0
0
0   11 
1 

1 
0
0
0   22 


 
1 
0
0
0   33 



T


 
2 1   
0
0   23 
0 
0 
0
2 1   
0  13 


 
0
0
2 1     12 
0 
 ij 
E 

 ET
 kk  ij  
 ij
 ij 
1  
1  2
 1  2
Solving linear elasticity problems
spherical/axial symmetry
e3
 RR
0

   0 
 0
0

• Kinematics
• Constitutive law
• Equilibrium
du
 RR 
dR
0 

0 
  
 RR

  0
 0

0

0
0 

0 
 
eR

e1
u
   
R
 du 
2   dR  ET 1
 
1   u  1  2 1
 R 
 RR 
1 
E
  

   1   1  2   
d RR 1
 2 RR      0bR  0
dR
R

d 2u
dR
2
• Boundary conditions


2 du 2u
d  1 d
  1   d T 1  1  2 


R2u  

0b( R)
 2
2
R dR R
dR  R dR
E 1  
 1   dR
 
ur (a)  ga
 rr (a)  ta
ur (b)  gb
 rr (b)  tb

e
R e
e2
Some simple static linear elasticity solutions
Navier equation:
 2uk
 2ui
bi 0  2ui
1

 0 
1  2 xk xi
xk xk
  t 2
ui 
Potential Representation (statics):

2(1  ) 
1

  xk  k  
 i 
E 
4(1  ) xi

i 
Point force in an infinite solid:
Pi
4 R
 2 i
  0bi
x j x j
 0
ui 
(1   )  Pk xk xi

 (3  4 ) Pi 

8 E (1   ) R  R 2

 ij 
Pi x j  Pj xi 
 3Pk xk xi x j Pk xk  ij

 (1  2 )


R
R
8 E (1   ) R 2 
R3

 ij 
Pi x j  Pj xi   ij Pk xk
 3Pk xk xi x j

(1

2

)

R
8 (1   ) R 2 
R3
(1   )
1
Point force normal to a surface:
i 
ui 
(1  )i3
R
 



P
(1  2 )(1  )

log(R  x3 )
R
i3 (1  2 ) 
x 
(1  ) P  x3 xi

3i  i 
 3  (3  4 )

2 E  R
R
R  x3 
R 

e3

 xi x j x3 (1  2 )(2 R  x3 )
 ij 
3

xi x j   ij x32  x3  i 3 x j   j 3 xi
2
3
2

2 R 
R
R( R  x3 )
P
 2
  0bi xi
xk xk


   ( R  x )2 i3 j3  ij 
(1  2 ) R 2
3


e1
Simple linear elastic solutions
 33   0
Spherical cavity in infinite solid under remote stress:
e2

(1  ) 0
a3 (1  ) 0  (5  1)
i 
x3 i3 
xi  5 x3 i3 

(1   )
R3 (7  5 )  2(1  2 )

a
2 2
 (1  ) 0
a (1  ) 0  (7  5 ) 2
2
2
2 3x3 a

(3x3  R ) 
R a 


(1   )
R3 (7  5 )  2(1  2 )
R 2 
3
(1   ) 0  
5(5  4 )
 2 
ui 
2 E  
(7  5 )

a3
6

3 (7  5 )
R
e1
e3
a5 
 x3 i3
R5 
 2
(5  6)


 (1   ) (7  5 )
a3
3

3 (7  5 )
R
a5  5 x32   
1  2   xi
R5 
R   

3a3 xi x j 
x32 
a2 
a2

3  5  4


6

5


5

10




ij
 0 2(7  5 ) R3 
R 2 
2(7  5 ) R5 
R2
R 2 
 ij
3a3
 i3 j 3 
3

a5  15a x3 ( x j i3  xi j 3 )  a 2

 
 (7  5 )  5(1  2 ) 3  3 5  

 R2

(7  5 ) 
R
R 
(7  5 ) R5


a3
Dynamic elasticity solutions
Plane wave solution
Navier equation
ui  ai f (ct  xk pk )
 2uk
 2ui
bi 0  2ui
1

 0 
1  2 xk xi
xk xk
  t 2

Solutions:

  0c2 ak 
ai pi  0 

1  2
pi ai pk  0
c2  c22  0 / 
ai   pi  c2  cL2  2 (1  ) / 0 (1  2 )