Final Review Slides
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Transcript Final Review Slides
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 xconst
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 JF1 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 nσ
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 xconst
t y const yi
v
v
bi i vk i
yk
y j
t y const
i
ji
q
ij Dij i q
t xconst
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 ( , )
2ˆ
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 Dkkij / 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 xconst
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 xconst
yk
Vorticity transport equation (constant temperature, density independent viscosity)
2
v
2i
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 xconst
2i
ijk
(bk ) Dij j i
y j y j
x j
t
xconst
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 2L
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 xconst
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 xconst
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 JF1 σ
Cij Fki Fkj
Bij Fik F jk
Qi JFik1qk
Sij JFik1 kj
Σ JF1 σ FT ij JFik1 kl F jl1
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
123 1
123 2
123 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
2i
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 αT
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 αT )
Isotropic materials:
ij
1
ij kk ij Tij
E
E
b
R0
ij
E
ET
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
213
212
0
ij
1
0
0
0
1
0
0
0
1
ij kk ij Tij
E
E
0
0
0
1 2
2
0
0
0
0
0
0
1 2
2
0
11
22
0 33 ET
2 23 1 2
0 213
212
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
ET
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 ET 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 )