3-9
• In piezoelectric materials, electrical
current and voltage are coupled to elastic displacement and stress:
{T} = [c] {S} - [e] {E}
{D} = [e] {S} + [Î] {E}
where {T} = stress tensor, [c] =
elastic stiffness matrix, {S} = strain tensor, [e] =
piezoelectric matrix
{E} = electric field vector, {D} =
electric displacement vector, and [Î] = is the dielectric matrix
• For a linear piezoelectric material
c11 c12 c13 c14 c15 c16 -e11 -e21 -e31
c21 c22 c23 c24 c25 c26 -e12 -e22 -e32
c31 c32 c33 c34 c35 c36 -e13 -e23 -e33
c41 c42 c43 c44 c45 c46 -e14 -e24 -e34
c51 c52 c53 c54 c55 c56 -e15 -e25 -e35
c61 c62 c63 c64 c65 c66 -e16 -e26 -e36
e11 e12 e13 e14 e15 e16 Î11 Î12 Î13
e21 e22 e23 e24 e25 e26 Î21 Î22 Î23
e31 e32 e33 e34 e35 e36 Î31 Î32 Î33
T1
T2
T3
T4
T5
T6
D1
D2
D3
=
where
T1 = T11 S1 = S11
T2 = T22 S2 = S22
T3 = T33 S3 = S33
T4 = T23 S4 = 2S23
T5 = T13 S5 = 2S13
T6 = T12 S6 = 2S12
S1
S2
S3
S4
S5
S6
E1
E2
E3
•
Elasto-electric matrix for quartz
S1
S2
S3
S4
S5
S6
-E1
-E2
-E3
et
T1
T2
T3
T4
T5
T6
D1
D2
D3
e
CE
X
S
6
2
2
10
LINES
JOIN NUMERICAL EQUALITIES
EXCEPT
FOR COMPLETE RECIPROCITY
ACROSS
PRINCIPAL DIAGONAL
INDICATES NEGATIVE OF
INDICATES TWICE THE NUMERICAL
EQUALITIES
INDICATES 1/2 (c11 - c12)
X
Î
Mathematical Description of a Quartz Resonator