2D Orthotropic elasticity matrix

Numerical methods and mathematical models of Elmer
pwray
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2D Orthotropic elasticity matrix

Post by pwray »

Hello
To use the 2D smitc plate solver with orthotropic materials, a 4 x 4 elasticity tensor must be entered as the Youngs Modulus, but I cant find the elements of this matrix defined anywhere in the documentation. The question has been asked a couple of times, but not definitively answered as far as I can tell.

I'm hoping that someone can explain how the matrix values are defined in terms of E11, E22, G12, G13, G23 and the Poisson ratios Nu12, Nu21

For the 6x6 case, the question is answered here. viewtopic.php?t=6410 You need to form the 6x6 compliance matrix and then invert it with a calculator or other software, is that correct?

Presumably the 4x4 case is constructed similarlarly.

Thanks,
Paul
kevinarden
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Re: 2D Orthotropic elasticity matrix

Post by kevinarden »

The smitc plate solver does not include the mid-plane stretching effect which would be associated with E11 and E22. It only includes bending, which is why there are only three terms, D1 = z deflection, D2 = Rotation in X, D3 = Rotation in Y. I believe to include the effects E11, E22, G12, G13, G23 and the Poisson ratios Nu12, Nu21 you would have to use the shell solver.
kevinarden
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Re: 2D Orthotropic elasticity matrix

Post by kevinarden »

shell solver is the more general case of a plate with six degrees of freedom, x, y, z displacement, and x y z rotation. I think it would be better suited for orthotropic materials.
plates.zip
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kevinarden
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Re: 2D Orthotropic elasticity matrix

Post by kevinarden »

The manual says that there is orthotropic capability in the plate model.
The design of the tensors E and G for orthotropic and perforated materials is discussed in
section 9.3.
pwray
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Re: 2D Orthotropic elasticity matrix

Post by pwray »

Yes I read the section, saw the equations there but dont know how to enter the values in the sif file.
Are there any examples? I believe I need a 4x4 matrix for 2D case.
kevinarden
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Re: 2D Orthotropic elasticity matrix

Post by kevinarden »

I studied the codes and the manual. The plate solver and the shell solver do not have orthotropic material inputs. The plate solver uses that term for the perforated plate case, but since hole size and spacing is the same in both directions, perforated plate isn't really orthotropic.

I am looking at at testing stresssolver, 3D elements. The previous shared topic for how to enter the 6x6 does not produce correct results.
kevinarden
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Re: 2D Orthotropic elasticity matrix

Post by kevinarden »

The orthotropic 3D case in stress solver works with
matrix.png
matrix.png (10.85 KiB) Viewed 699 times
The inverse of the above matrix is put into Elmer as
Youngs Modulus(6,6) = 8.9489E+10 3.4801E+10 3.4801E+10 0.0000E+00 0.0000E+00 0.0000E+00 \
2.981E+10 7.67E+10 2.98E+10 0.0000E+00 0.0000E+00 0.0000E+00 \
2.485E+10 2.485E+10 6.39E+10 0.0000E+00 0.0000E+00 0.0000E+00 \
0.0000E+00 0.0000E+00 0.0000E+00 2.7300E+10 0.0000E+00 0.0000E+00 \
0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 1.7300E+10 0.0000E+00 \
0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 1.7300E+10

The matrix can be assembled and inverted in a spreadsheet
ctensor.xls
(8 KiB) Downloaded 36 times
pwray
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Re: 2D Orthotropic elasticity matrix

Post by pwray »

Thanks Kevin
Yes I know about the 6x6 case, I read the posted solution which I linked to above. That's one reason Im looking at the shell solver, because it takes a 6x6. I even found a direct representation with the inversion already done:

Compliance matrix:
Capture1.PNG
Capture1.PNG (17.66 KiB) Viewed 691 times
Stiffness matrix:
Capture.PNG
Capture.PNG (33.37 KiB) Viewed 691 times
where
Capture2.PNG
Capture2.PNG (4.18 KiB) Viewed 690 times
Reference: http://homes.civil.aau.dk/lda/Continuum/material.pdf

I still dont know the layout of the reduced 4x4 matrix yet though.

Paul
kevinarden
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Re: 2D Orthotropic elasticity matrix

Post by kevinarden »

Because E11=E22=E33=70.E9 and G12=G13=G23= 27.3E9 anf v12=v13=v23=0.28
Isotrpoic Input can be a scaler
Youngs Modulus = 70.E9
and is the same as the orthotropic model input, where the values reflect the above isotopic case.
Youngs Modulus(6,6) = 8.9489E+10 3.4801E+10 3.4801E+10 0.0000E+00 0.0000E+00 0.0000E+00 \
3.4801E+10 8.9489E+10 3.4801E+10 0.0000E+00 0.0000E+00 0.0000E+00 \
3.4801E+10 3.4801E+10 8.9489E+10 0.0000E+00 0.0000E+00 0.0000E+00 \
0.0000E+00 0.0000E+00 0.0000E+00 2.7300E+10 0.0000E+00 0.0000E+00 \
0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 2.7300E+10 0.0000E+00 \
0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 2.7300E+10

I verified this in stress solver, the 3D element solver, by getting the same, correct, answer with either input.

In shellsolver and plate(smitc) I do not get the same answer. They except the input, but the correct answer is not obtained.
pwray
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Re: 2D Orthotropic elasticity matrix

Post by pwray »

Hi Kevin

Dissapointing you dont get a result in shell solver.
I believe smitc solver requires a 4x4 matrix, not 6x6.
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