electric charge / deformation and polarization matt pharr es 241 5/21/09
TRANSCRIPT
Electric Charge
10 protons, 10 electrons
Net charge = 10 - 10 = 0
Total charge is conserved
A B
SI Units: 1 Coulomb =
= 6.242 * 1018 elementary charges
A B
QA= 0
+QB= 0
Qnet= 0
QA= +4
+QB= -4
Qnet= 0
F’(Q’) F’’(Q’’)F’(Q’) F’’(Q’’)
Capacitor
QFF
QFF
Φ
'Q
0':
'
''
''
''''For
0''''
'''''
'
''
''''''
''''''
QF
Q
QF
QFQ
Q
QFF
QFQFF
QQconstQQ
comp
comp
SI Units: 1 Volt
= 1.602e-19 Joule
Measurement of Electric Potential
I
RV?
Current measured with galvanometer
Ohm’s Law: V = IR gives the potential
A Capacitor, a Weight, and a Battery
QlFl
l
QlFF
QlFF
,,
,
Mechanical work lPElectric work Q
In equilibrium QlPF
Q
QlF
l
QlFP
,,
,
F(l,Q) and Stress
la
Q
Experimental Relation
a
Q
l
QlFP
2
, 2
a
lQQlF
Q
QlF
2,
, 2
Recall:
oil
Φl
a
+Q
-Q
lE
aQ
D
aP
Electric field
Electric displacement
Stress field
Maxwell Stress
2
2Ez
Deformable Dielectrics
A
Reference State
P
l
a Q
Q
Current State
LE
~
A
QD ~
A
Ps
Nominal electric field
Nominal electric displacement
Nominal stress
L
lStretch
Nominal free-energy density AL
FW
QlPF
AL
Q
LA
lP
AL
F
DEsW~~
DW
s~
, D
DWE ~
~,~
L
Definition of Stress
A
Ps Nominal stress
T
No weight, no stress???
Analogous to thermal expansionStress-free deformation
Small α
Very stiff
T
σ σ
Stress generated due to constraint
How is there deformation due to voltage change?
3D Homogeneous Deformation
DEsssW~~
332211
QlPlPlPF 332211
AL
Q
LA
lP
LA
lP
LA
lP
AL
F 332211
1
3311
~,,,
DW
s
D
DWE ~
~,,,~ 321
3
3313
~,,,
DW
s
2
3312
~,,,
DW
s
DW~
,,, 321
Field Theory Recovers Maxwell Stresses in a Vacuum
3212
0111
20
321 2
1~,,,
2
EDWE
lll
F
ED 0
Electric energy per current volume
Recall21
~
D
D
210
32
111 2
~~
,,,
DDW 2
03
2021
2
12
1
E
E
ijkkijij EEEE 2
00
Q
Q
20
2E
P
E
Maxwell Stresses
Ideal Dielectric ElastomersElastomer Structure
1321 Incompressibility
DW~
,, 21
2
,~
,,2
2121
EWDW s
Stretching ization
2
212121
~
2
1,
~,,
D
WDW s
Polar-
EDE
W
EW
s
s
,,
,
2
2
21232
2
1
21131
Electrostriction
Well below extension limit
Low cross-link density
Polarization unaffected by deformation
Close to extension limit
High cross-link density
Deformation affects polarization
Deformation Affects Polarization
A model: quasi-linear dielectrics
21,
2212121 2
,,
~,, EWDW s
2
2
21232
2
1
21131
,
,
EW
EW
s
s
22
2
2
2
21232
21
1
2
1
21131
2
1,
2
1,
EEW
EEW
s
s
Ideal dielectric elastomerQuasi-linear dielectrics
Pull-in Instability
la
Q
P
l
a Q
Q
Experimental Observation for oil
Q,As l This can lead to electrical breakdown
Pull-in Instability
Exercise: Find critical electric field for instability subject to a biaxial force in the plane of membrane
2P 1P
22L
33L
11L
Q
2P 1P
22L
33L
11L
Q
Assume ideal dielectric elastomer and incompressibility
2
212121
~
2
1,
~,,
D
WDW s
Choose a free energy of stretching function: Neo-Hookean law
32
, 23
22
2121 sW
Pull-in Instability
22
31
22
23
111
211
~~,
DDWs
21
32
22
13
222
212
~~,,
DDW
s
22
21
21
~
~
~,,~
D
D
DWE
For equal biaxial stress, s1 = s2 = s and λ1 = λ2 = λ
2
~,
~~ 52
54
D
sD
E
Combining these two equations gives the following
823~
s
E
In equilibrium
Pull-in Instability
E~
reaches a peak when 0~
dEd
If s/μ = 0
mVmF
mNE
d
d
c
c
/10/10
/10~69.0
~
26.120
810
6
3/182
If s/μ = 1
56.0
~7463.1 cc E
Larger stretch
before breakdown
823~
s
E
• Multiple forces• Chemical potential• Electric current• Package Warpage• Temperature gradient
• Multiple phases
Solder: Relation to Class
Ideas from Paper Covered in Class
Kinetic laws – chemical potential, diffusion flux
Principle of virtual work – work conjugates
Traction
Deformation Rate
Eulerian vs. Lagrangian