Conversion of Piezoelectric coefficients

[John Kai]

Hello everyone,

I am interested on simulating the converse Piezoelectric effect of materials and I have given the mechanical (Youngs modulus and Poissoin's ratio) and electric material properties (Electric Permittivity, Piezoelectric coupling matrix etc). However from the literature I got double index notation, I am struggling to convert double index notation to triple index notation.

The double index notation from literature is as follows:

Double index notation

d11 d12 d13 d14 d15 d16 -------------- 0 0 0 0 21.3 0

d21 d22 d23 d24 d25 d26 -------------- 0 0 0 21.3 0 0

d31 d32 d33 d34 d35 d36 -------------- -2.6 -2.69 3.65 0 0 0

Triple index notation:
I would like to convert them to Triple index notation used by some commercial software providers like Abaqus and MSC Marc etc and is written as follows:

d11_1 d11_2 d11_3 -------------- 0 0 -2.6

d22_1 d22_2 d22_3 -------------- 0 0 -2.6

d33_1 d33_2 d33_3 -------------- 0 0 3.65

d12_1 d12_2 d12_3 -------------- 0 21.3 0

d23_1 d23_2 d23_3 -------------- 21.3 0 0

d31_1 d31_2 d31_3 -------------- 0 0 0

Am I doing the conversion correctly? Are the values placed in correct positions?
Double Index = Triple Index = Coupling Matrix Value
d11 = d11_1 =0
d21 = d11_2 = 0
d31 = d11_3 = -2.6
d12 = d22_1 = 0
d22 = d22_2 = 0
d32 = d22_3 = -2.6
.
...
...

d16 = d31_1 = 0
d26 = d31_2 = 0
d36 = d31_3 = 0

I am new to this filed and its kind of a big deal for me to understand all these conversions. Your feedback will be highly appreciated.
Kai

 

[Paul Colby]

I think the index pairs are

$(1,1) = 1$
$(2,2) = 2$
$(3,3) = 3$
$(2,3) = (3,2) = 4$
$(1,3) = 5$
$(1,2) = (2,1) = 6$

There are factors of two thrown in so that

$C_{nmjk}S_{jk} = C_{ab}S_b$

things like

$S_{1,2} = \frac{1}{2} S_3$

when the dust clears. $n, m, j, k$ summed over 1 - 3 and $a, b$ summed 1-6.
This may help.

https://en.wikipedia.org/wiki/Piezoelectricity

 

[John Kai]

Dear Paul Colby,

I thnak you very much for your quick reply and explanation. I have already read the Wikipedia page briefly and will go through it again.
Thanks again!

 

[dlgoff]

This Piezo Systems site has some great images that may interest you.

e.g.