Discussion Forum

Hi,

Although it is supposed to be a 6x6 matrix, some of the components of the elasticity matrix get cancelled out because crystal symmetries and it boils down to 21 values. Check the documentation 'Linear elastic material'. In fact since you know the Young's modulus and Poisson's ratio of Si you can calculate the eleasticity matrix and verify.

-Sankha

Hi
And its slightly tricky to regroup the terms, its either a lower or an upper triangle (forgotten wich one), probably the upper
Last time I checked, I copied the full section into EXcel and parsed into cells, then regrouped by dragging them such that my symmetry was ok

for more on the material symmetry, take a look on the recent and excellent book of E.B. Tadmor, CUP 2011 "Modelling Materials" which has also an excellent introduction to tensors and reference versus spatial frame issues. If you ewant more on that subject chek the 2nd volume, same author and editor, 2012 "Continuum Mechanics" and Thermodynamics"
The overall mathematical notation described is very close to COMSOLs (except there might be some transposed representation, normalisations is seldom unique ;)

--
Good luck
Ivar

Yes it is a lower half triangle and more or less matches up with literature values.

Thanks for your help.
Haydn

p.s. I am having to brush up my matrix mathematics a bit

Hi

That is why the books I mentionned above are interesting, they have an excellent tensor description ;)

--
Good luck
Ivar

Hi!

I read your comments and, with the risk of being redundant, I will still pose my questions.

I'm trying to build a SAW resonator in Comsol, but I'm having difficulties in the material section.
As previously stated by others, Comsol allows only 21 coefficients in the elasticity matrix.
But Silicon, for example, has a different shape for the matrix and I can't modify it properly.
Am I doing some kind of silly mistake, confusing the matrix?

Also, if I will modify the matrix in the Piezoelectric Devices Physics section (Piezoelectric Material Model), will these modifications take place also in Materials Section of the Model?

Thank you.

What shape do you have for silicon’s elasticity matrix? For anisotropic elastic materials the stress-strain matrix is symmetric and that’s why you get 21 independent coefficients instead of 36.

Nagi Elabbasi
Veryst Engineering

My problem is that I don't know how you can obtain the triangle shape matrix, in order to place it in COMSOL.
PS. It is the C-matrix the one I have to put in Comsol, right? The one that has c11,c12,c13,c33,c44 ?

Thank you!

The terminology depends on the reference you are using. It should be the tensor relating stress to strain. It is common to see this tensor referred to as the “C-Matrix” so it may be the one, but you should check.

Ok, I will try to simplify my question, by attaching a picture.
How do I go from the elasticity matrix a), which is the one I find in the articles, to matrix b), the one allowed in Comsol?
Or, is there a different way Comsol defines this matrix?

Thank you,

It seems that the difference between (a) and (b) is only due to the order of the stress and strain variables. Variable #3 in (a) is variable #1 in (b), etc. Matrix C is the elasticity matrix which you can input directly in COMSOL using the “Anisotropic” option discussed above.

Nagi Elabbasi
Veryst Engineering

SIR PLEASE I WANT GOLD ELASTICITY MATRIX PLEASE YOU CAN PROVIDE ME

It will be highly appreciated if anyone provide
1. coupling matrix, relative permittivity for silicon.
2. coupling matrix, relative permittivity, elasticity matrix for tungsten

Hi,
So just to clear it for anyone still struggling with this issue, here is some explanation:

So like most matrices, piezoresistive coefficient matrix is a 6x6 matrix, therefore it has 36 variables like this
c11 c12 c13 c14 c15 c16
c21 c22 c23 c24 c25 c26
c31 c32 c33 c34 c35 c36
c41 c42 c43 c44 c45 c46
c51 c52 c53 c54 c55 c56
c61 c62 c63 c64 c65 c66

but due to its symmetrical nature, c12=c21, c13=c31, c14=c41 and so on. Basically, the upper triangle of the matrix is equal to the lower triangle Now we get this matrix
c11 c12 c13 c14 c15 c16
c12 c22 c23 c24 c25 c26
c13 c23 c33 c34 c35 c36
c14 c24 c34 c44 c45 c46
c15 c25 c35 c45 c55 c56
c16 c26 c36 c46 c56 c66
or we can write it as
[c11 .....................................]
[c12 c22................SYM...... ]
[c13 c23 c33........................]
[c14 c24 c34 c44............... ]
[c15 c25 c35 c45 c55........]
[c16 c26 c36 c46 c56 c66]

Notice that Now we have 21 different terms instead of the original 36.
( It doesnt matter which triangle you choose, but would be same due to symmetry)
So How does COMSOL recognise these 21 terms? Here's how:

[c11........................................]
[c12 c22.................SYM..... ]
[c13 c23 c33 .....................]
[c14 c24 c34 c44.............. ]
[c15 c25 c35 c45 c55.......]
[c16 c26 c36 c46 c56 c66]
is same as:
[c01 ..................................... ]
[c02 c03...............................]
[c04 c05 c06 ......................]
[c07 c08 c09 c10 ..............]
[c11 c12 c13 c14 c15........]
[c16 c17 c18 c19 c20 c21]
so the values are numbered and added starting from c1 to c21 seperated by commas.
some materials like silicon has only 3 terms C11, C12 and C44 so if you want to write it in comsol,
C11 = c1,c2,c3
C12 = c2,c4,c5
C44 = c10,c15,c21
rest of the c terms are zeros.

Good Luck

Abdullah Siddiqui
Kind Fahd University of Petroleum and Minerals

That's god explanation

Need to know elasticity and coupling matrix for SiO2