Linear elasticity calculation on an Orthotropic material - writing compliance matrix in ElmerFEM

Numerical methods and mathematical models of Elmer
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Adri_Bois
Posts: 2
Joined: 19 Dec 2018, 11:21
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Linear elasticity calculation on an Orthotropic material - writing compliance matrix in ElmerFEM

Post by Adri_Bois »

Hi everyone. I'm fairly new to Elmer, and I've been so far using it for calculating the diffusion process on orthotropic material with temperature-dependent properties to great success. I found Elmer to be faster than MATLAB, the primary software I've been using since a long time, and comparable to COMSOL, which is very practical but unfortunately too expensive.

Currently I'm trying to conduct a calculation of linear elasticity on an orthotropic material. I have three moduli of elasticity on three different directions of the material as follows:
  • E_xx for the x-direction,
  • E_yy for the y-direction, and
  • E_zz for the z-direction.
Further, I also have the shear moduli as follows:
  • G_yz for the yz shear,
  • G_xz for the xz shear, and
  • G_xy for the xy shear.
And finally, I have six poisson's ratio (v_xy, v_yx, v_xz, v_zx, v_yz, v_zy) for all of the directions. Combined, they form the compliance matrix of the stress-strain equation based on Hooke's law:

Code: Select all


[ C_11 C_21 C_31 C_41 C_51 C_61 ]	[ 1/E_xx -v_yx/E_yy -v_zx/E_zz 0 0 0 ]
[ C_12 C_22 C_32 C_42 C_52 C_62 ]	[ -v_xy/E_xx 1/E_yy -v_zy/E_zz 0 0 0 ]
[ C_13 C_23 C_33 C_43 C_53 C_63 ]	[ -v_xz/E_xx -v_yz/E_yy 1/E_zz 0 0 0 ]
[ C_14 C_24 C_34 C_44 C_54 C_64 ] =	[ 0 0 0 1/G_yz 0 0		     ]
[ C_15 C_25 C_35 C_45 C_55 C_65 ]	[ 0 0 0 0 1/G_xz 0		     ]
[ C_16 C_26 C_36 C_46 C_56 C_66 ]	[ 0 0 0 0 0 1/G_xy		     ]

So my question is: how can I put all this into Elmer's code?

I've searched and researched previous questions regarding orthotropic materials posed in this forum. I've found some that advises to put the modulus elasticity as a matrix, but I don't see how I can do that with my case because not only I have three moduli of elasticity, I also have six poisson ratios and three shear moduli. The closest to my case is perhaps this one, but again the OP only uses one poisson ratio instead of six, so perhaps it's not a compliance matrix that's being used there.

Thus, does anybody have any experience in calculating the linear elasticity on orthotropic material like mine using Elmer? Could someone help me please?

Thank you.
mika
Posts: 230
Joined: 15 Sep 2009, 07:44

Re: Linear elasticity calculation on an Orthotropic material - writing compliance matrix in ElmerFEM

Post by mika »

Hi,

Might it be that the keywords of Elmer cause some semantic confusion here? If you invert your compliance matrix to obtain the elasticity tensor C in the matrix form, then it should be possible to provide the values of C as entries of the matrix defined by the keyword "Youngs Modulus". The value of the keyword "Poisson Ratio" doesn't then have any impact on the constitutive law as it is now specified completely by giving the keyword "Youngs Modulus". I think it would have been better to choose a name "Elasticity Tensor" for the keyword which is now referred to as (matrix-valued) "Youngs Modulus".

-- Mika
Adri_Bois
Posts: 2
Joined: 19 Dec 2018, 11:21
Antispam: Yes

Re: Linear elasticity calculation on an Orthotropic material - writing compliance matrix in ElmerFEM

Post by Adri_Bois »

Hi Mika,

I tried what you said (basically putting the inverse of C tensor as "Young Modulus" in Elmer) as follows:

Code: Select all

Material 1
  Name = "Wood"
  Density = 707
  Youngs modulus = 0.2912e10 0.1208e10 0.1687e10 0 0 0 \ 
                   0.1208e10 0.1712e10 0.1255e10 0 0 0 \
                   0.1687e10 0.1255e10 1.6931e10 0 0 0 \
                   0 0 0 0.1080e10 0 0 \
                   0 0 0 0 0.1387e10 0 \
                   0 0 0 0 0 0.0424e10
End
and it seems to me that it works. I have tested it on the same beam from the samples. Here are some pictures:

Image
(Deplacements)

Image
(vonMises)

I will verify this by using other software and compare it with experimental measurements afterwards. Thanks again for your help Mika. And yes, I agree with your opinion regarding renaming the "Young Modulus" in Elmer to "Elasticity Tensor".
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