continuum
description
Transcript of continuum
Conservation Laws for Continua
e3
e1
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OriginalConfiguration
DeformedConfiguration
S
R
R0S0b
t
n
V T(n)
yx
u(x)
or ij
j ji
b ay
y σ b a
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
S
R
R0S0
n
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
v
t y t
yy y
v
Angular Momentum Conservation ij ji
( ) 1
2i i i ij ij i iiA V V V
dr T v dA b v dV D dV v v dV
dt
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
1
2i i i ij ji i iiA V V V
dr T v dA b v dV S F dV v v dV
dt
n
0 0
( )0 0 0
1
2i i i ij ij i iiA V V V
dr T v dA b v dV E dV v v dV
dt
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)
1
2ji i
ij ijj j i
vv vL D
y y y
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
S2R0S0
b
t
Vyx
u(x) S1
ji ii
j
dvb
y dt
If for all iv
Then
i ij jn t 2Son
Thermodynamics
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
S
R
R0
S0 b
t
Specific Internal EnergySpecific Helmholtz free energy s
Temperature
Heat flux vector q
External heat flux q
First Law of Thermodynamics ( )d
KE Q Wdt
iij ij
iconst
qD q
t y
x
Second Law of Thermodynamics 0dS d
dt dt
Specific entropy s
( / )0i
i
qs q
t y
10ij ij i
iD q s
y t t
Transformations under observer changes
Transformation 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 t
dt dt dt dt
d d td d d d dt t
dt 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
Td
dt
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 y
F 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 t
dt dt dt dt
d d td d d d dt t
dt 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
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OriginalConfiguration
DeformedConfiguration
10ij ij i
iD q s
y 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
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e1
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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 k
i ijk ijk ijj
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
10ij ij i
iD q s
y 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 ˆˆˆ
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( )
2i 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
pD D b a p D
y y
221 2
3eq ji
i ii j j j i
vvb a
y y y y y
With density indep viscosity
1( )
2k
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
vpb a
y y y
For an incompressible Newtonianviscous fluid
1
2ji
ijj i
vvD
y y
0i
i
v
y
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
2
1 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)
Derived field equations for fluids
• Bernoulli1
constant2
eqi iH v v
along streamline
For an elastic fluid
For irrotational flow1
constant2
eqi iH v v
everywhere
For incompressible fluid 1constant
2 i ip
v v
Normalizing the Navier-Stokes equation
21 1( )
2k
i ii k k ijk j k
i j j iy const
v vpb v v v
y y y t y
Characteristic length
Characteristic velocity
Characteristic frequency
P Characteristic pressure change
L
V
f
Normalize as
2
2
ˆ ˆ ˆˆ 1 1 ˆ ˆEu Stˆˆ ˆ ˆ ˆRe Fr
k
i i ii i
i j j iy const
v v vpb v
y y y t y
Re /VL Reynolds number
Incompressible Navier-Stokes
ˆ
ˆ
ˆ
ˆ P
ˆ
i i
i i
i
y Ly
v Vv
t ft
p p
b gb
Euler number 2Eu /P V
Froude number Fr /V gL Strouhal number St /fL V
V
L
Limiting cases most frequently used
2
ˆ ˆˆ 1 ˆ ˆEu Stˆˆ ˆFr
k
i ii i
i iy const
v vpb v
y t y
Ideal flow Re
Stokes flow
2 ˆ ˆˆ 1 ˆˆˆ ˆ ˆRe
k
i ii
i j j y const
v vpb
y y y t
2
2
ˆ ˆ ˆˆ 1 1 ˆ ˆEu Stˆˆ ˆ ˆ ˆRe Fr
k
i i ii i
i j j iy const
v v vpb v
y y y t y
0V
1( )
2k
eq ii k k ijk j k
i iy const
vb v v v
y t y
21
k
i ii
i j j y const
v vpb
y y v t
Governing equations for a control volume (review)
B
R
Solving fluids problems: control volume approach
Example
v0
v1
v2
A0
A2
A1
A3
A4
A5
ji
Steady 2D flow, ideal fluidCalculate the force acting on the wallTake surrounding pressure to be zero
( )B R R B
ddA dV dV dA
dt n σ b v v v n
3
20 0 0 0 0( ) sin ( ) 0
A A
p dA A v p p dA n j j j j
20 0 0 sinA v F j
Exact solutions: potential flow
20 0i
i i i
v
y y y
Mass cons
Bernoulli1
constant2 i i
pv v
t
If flow irrotational at t=0, remains irrotational; Bernoulli holds everywhere
Irrotational: curl(v)=0 so ii
vy
V
e1
e2
a2
2
( )( )( )
a V y V tr y V t y V t
r
Exact solutions: Stokes Flow
Vy2
y1
h
Steady laminar viscous flow between platesAssume constant pressure gradient in horizontal direction
22 2 1
2
( )2
2
y pV y h y
h L
V p hy
h L
v e
σ
2 2
22
10
k
i ii
i j j y const
v vp p fb
y y v t L y
Solve subject to boundary conditions
Exact Solutions: AcousticsAssumptions:
Small amplitude pressure and density fluctuationsIrrotational flowNegligible heat flow
Mass conservation:
ss const
pc
For small perturbations: 2
sp
ct t
k
ii
i y const
vpb
y t
Approximate N-S as:22
2k
i
i y const
vp
y t t
0i
iconst
v
t y
x
22
k
ii
i i i y const
vpb
y y y t
22 22 2
2 20 0
jis s
i j i i
vvc c
y y y yt t
Combine:
(Wave equation)
Wave speed in an ideal gas
i
iconst
qsq s const
t y
xEntropy equation:
0log logv v vc c c R s p R
/0 0log log exp[( ) / )vR c
v vs c R s s s c
Hence:
Assume heat flow can be neglected
1s
s const
p pc k R
/ 1vR c p k so
Application of continuum mechanics to elasticity
e3
e1
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OriginalConfiguration
DeformedConfiguration
S
RR0
S0 b
t
u
x
y
Material characterized by
Modulus G' (N/m2)
(frequency)-1
109
105
GlassyViscoelastic
Rubbery
MeltGlass Transitiontemperature Tg
General structure of constitutive relations
e3
e1
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OriginalConfiguration
DeformedConfiguration
S
RR0
S0 b
t
u
x
y
0
0
ˆ ˆˆ ˆ2
ˆ ˆ2
ijij
ij
ij
sC
s
C
0kk
Qx
iij ij
j
uF
x
ij ki kj
ij ik jk
C F F
B F F
1
i ik kQ JF q
1 1ij ik kjJ JFS S F σ
1 1 1Tij ik kl jlJ JF F Σ F Fσ
0 0i
ji ii
j x const
S vb
x t
01 1
02 ij ij i
iC Q s
x t t
* F QF
* Σ Σ
* * *
* * *
T T
T
B F F QBQ
C F F C
Frame indifference, dissipation inequality
Forms of constitutive relation used in literature
1
2 22 1 1
23
trace( )
1 1
2 2
det
kk
ik ki
I B
I I I B B
I J
B
B B
B
11 2 / 3 2 / 3
2 222 1 14 / 3 4 / 3 4 / 3
=
1 1
2 2
det
kk
ik ki
BII
J J
B BII I I
J J J
J
B B
B
1 2 3 1 2 1 2 3ˆ( ) ( ) ( , , ) ( , , ) ( , , )W W U I I I U I I J U F C
• Strain energy potential 0W
1ij ik
jk
WF
J F
1 31 2 2 33
22ij ij ik kj ij
U U U UI B B B I
I I I II
1 1 22 /3 4 /31 2 1 2 2
2 1 12
3ij
ij ij ik kj ijU U U U U U
I B I I B BJ I I I I I JJ J
(1) (1) (2) (2) (3) (3)31 2
1 2 3 1 1 2 3 2 1 2 3 3ij i j i j i j
U U Ub b b b b b
2 (1) (1) 2 (2) (2) 2 (3) (3)1 2 3 B b b b b b b
Specific forms for free energy function
• Neo-Hookean material21 1
1( 3) ( 1)2 2
KU I J
• Mooney-Rivlin 21 2 11 2( 3) ( 3) ( 1)
2 2 2
KU I I J
115/ 3
11
3ij ij kk ij ijB B K JJ
21 215/3 7 /3
1 1 1[ ] 1
3 3 3ij ij kk ij kk ij kk ij ik kj kn nk ij ijB B B B B B B B B K JJ J
• Generalized polynomial function 21 2
1 1
( 3) ( 3) ( 1)2
N Ni j ii
iji j i
KU C I I J
211 2 32
1
2( 3) ( 1)
2i i i
Ni
ii
KU J
• Ogden
22 31 1 12 4
1 1 11( 3) ( 9) ( 27) ... 1
2 220 1050
KU I I I J
• Arruda-Boyce
Solving problems for elastic materials (spherical/axial symmetry)
r
eR
ee
e1
e2
e3
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
rr rr rrF B
F B
F B
σ F B
2 2
rr rrdr r dr r
F F F B B BdR R dR R
• Assume incompressiblility
• Kinematics
2
1dr r
dR R
3 3 3 3r a R A
21 21
1 2 1 2 2
21 21
1 2 1 2 2
22
3 3
22
3 3
rr rr rrI IU U U U U
I B B pI I I I I
I IU U U U UI B B p
I I I I I
• Constitutive law
01
2 0rrrr r
db
dr r • Equilibrium (or use PVW)
( ) ( )r a r bu a g u b g
(gives ODE for p(r)
( ) ( )rr a rr ba t b t • Boundary conditions
Linearized field equations for elastic materials
2
2
* *1 2
1( )
2
( ) on ( ) on
j ij jiij ij ijkl kl kl j
j i i
i i ij i j
u uuC b
x x x t
u u t R n t t R
e3
e1
e2
OriginalConfiguration
DeformedConfiguration
S
R
R0
S0 b
t
2 2ˆ ˆij ijijkl ij
kl ij kl ij
U UC
Elastic constants related to strain energy/unit vol
Approximations:• Linearized kinematics• All stress measures equal• Linearize stress-strain relation
1ij ij kk ij ijT
E E
1 1 2 1 2ij ij kk ij ijE E T
Isotropic materials:
( )T σ C ε αT ε Sσ α
Elastic materials with isotropy
11 11
22 22
33 33
23 23
13 13
12 12
1 0 0 0
1 0 0 0 11 0 0 0 1
1 2 10 0 0 0 02 2 0(1 )(1 2 ) 1 2
1 22 00 0 0 0 0
22 0
1 20 0 0 0 0
2
E E T
11 11
22 22
33 33
23 23
13 13
12 12
1 0 0 0 1
1 0 0 0 1
1 0 0 0 110 0 0 2 1 0 02 0
0 0 0 0 2 1 02 0
0 0 0 0 0 2 12 0
TE
1ij ij kk ij ijT
E E
1 1 2 1 2ij ij kk ij ijE E T
Solving linear elasticity problems spherical/axial symmetry
R
eR
ee
e1
e2
e3
0 0 0 0
0 0 0 0
0 0 0 0
RR RR
1 2 1
1 11 1 2 1 2RR
duE E TdR
u
R
01
2 0RRRR R
db
dR R
• Constitutive law
22
02 2 2
1 1 1 22 2 1( )
1 1
d u du u d d d TR u b R
R dR dR dR dR EdR R R
• Equilibrium
RRdu u
dR R • Kinematics
( ) ( )r a r bu a g u b g
( ) ( )rr a rr ba t b t • Boundary conditions
Some simple static linear elasticity solutions
2
2 3
2 3
(1 )(3 4 )
8 (1 )
3(1 )(1 2 )
8 (1 )
31(1 2 )
8 (1 )
k k ii i
k k i j k k ij i j j iij
k k i j i j j i ij k kij
P x xu P
E R R
P x x x P x P x P x
R RE R R
P x x x P x P x P x
RR R
04
ii
P
R
Point force in an infinite solid:
Point force normal to a surface:
e1
P
e3
R3
3(1 ) (1 2 )(1 )
log( )ii R x
R
3 333
3
(1 ) (1 2 )(3 4 )
2i i i
i ix x xP
uE R R x RR
2
3 233 3 3 3 3 32 3 2 2
3 3
(1 2 )(2 ) (1 2 )3
2 ( ) ( )
i jij i j ij i j j i i j ij
x x x R xP Rx x x x x x
R R R R x R x
2 2 20
0 2
1
1 2k i i i
k i k k
u u b u
x x x x t
Navier equation:
2(1 ) 1
4(1 )i i k ki
u xE x
Potential Representation (statics): 2 2
0 0i
i i ij j k k
b b xx x x x
Simple linear elastic solutions
a
e2
e1
e3
30 0
3 3 3 33
3 2 22 2 2 20 0 33 3 2
(1 ) (1 ) (5 1)5
(1 ) 2(1 2 )(7 5 )
(1 ) (1 ) 3(7 5 )(3 )
(1 ) 2(1 2 )(7 5 )
i i i ia
x x xR
a x ax R R a
R R
3 50
3 33 5
23 53
3 5 2
(1 ) 5(5 4 ) 62
2 (7 5 ) (7 5 )
52 (5 6) 31
(1 ) (7 5 ) (7 5 )
i i
i
a au x
E R R
xa ax
R R R
3 23 2 23
3 2 5 2 20
33 5 23 3 3 3 3
3 5 5 2
333 5 4 6 5 5 10
2(7 5 ) 2(7 5 )
15 ( )(7 5 ) 5(1 2 ) 3
(7 5 ) (7 5 )
ij i jij
i j j i i j
a x x xa a a
R R R R R
a x x xa a a
R R R R
33 0 Spherical cavity in infinite solid under remote stress:
Dynamic elasticity solutions
( )i i k ku a f ct x p Plane wave solution
2 2 20
0 2
1
1 2k i i i
k i k k
u u b u
x x x x t
Navier equation
20 0
1 2k i i kc a p a p
2 22 00 /i ia p c c Solutions:
2 202 (1 ) / (1 2 )i i La p c c