Conversion of Piezoelectric coefficients

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SUMMARY

The discussion focuses on converting double index notation of piezoelectric coefficients to triple index notation, specifically for use in commercial software like Abaqus and MSC Marc. The user, Kai, presents a matrix of piezoelectric coupling values and seeks confirmation on the accuracy of their conversion process. Key values include d11, d12, and d31, with specific placements in the triple index format outlined. The conversation highlights the importance of understanding index pairs and their corresponding values in piezoelectric simulations.

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  • Understanding of piezoelectric materials and their properties
  • Familiarity with double and triple index notation in tensor mathematics
  • Experience with simulation software such as Abaqus or MSC Marc
  • Knowledge of mechanical properties like Young's modulus and Poisson's ratio
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  • Research the conversion methods between double and triple index notations in piezoelectric materials
  • Explore the piezoelectric coupling matrix and its applications in finite element analysis
  • Learn about the specific implementation of piezoelectric coefficients in Abaqus and MSC Marc
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Researchers, engineers, and students involved in material science, particularly those working with piezoelectric materials and simulations in finite element analysis software.

John Kai
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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
 
Last edited:
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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
 
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!
 
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This Piezo Systems site has some great images that may interest you.

e.g.
tech2intropiezotrans13.gif
 

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