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Why the lower diagonal of Elasticity matrix D in External materials equals 2G？

Posted Oct 25, 2017, 9:05 p.m. EDT 5 Replies

Wenxue Deng
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In the example of linear_elastic, D[3][3] = D[4][4] = D[5][5] = E / (1.0 + nu); // lower diagonal, note that this equals 2G For transversely isotropic material , I set the Elasticity matrix D
D[3][3] = D[4][4] = 2 * G13;
D[5][5] = E1 / (1.0 + v1);

but the result of displacement of u and v is different in quantity size from the solid mechanics

5 Replies Last Post Dec 18, 2017, 8:59 a.m. EST
Henrik Sönnerlind COMSOL Employee

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Posted: 7 years ago Oct 26, 2017, 2:11 a.m. EDT

Hi Wenxue,

It is elements of the strain tensor which are passed to the external material. This means that the shear terms are , rather than the engineering shears .

This also has the implication that the D matrix is unsymmetric (by a factor of 2) with respect to terms in the last three columns/rows. This is a fact which we have emphasized in the documentation for the upcoming 5.3a release since we have seen some cases where it has been overlooked.

Regards,

Henrik

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Henrik Sönnerlind
COMSOL
Hi Wenxue, It is elements of the strain tensor which are passed to the external material. This means that the shear terms are \epsilon_{ij}, rather than the engineering shears \gamma_{ij} = 2\epsilon_{ij}. This also has the implication that the D matrix is unsymmetric (by a factor of 2) with respect to terms in the last three columns/rows. This is a fact which we have emphasized in the documentation for the upcoming 5.3a release since we have seen some cases where it has been overlooked. Regards, Henrik
Wenxue Deng
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Posted: 7 years ago Oct 26, 2017, 10:17 a.m. EDT

Dear Henrik,

Thank you for your kind response.

I cannot understand why the D matrix is unsymmetric when the shear strain replace the engineering strain . To the best of my knowledge, I thought it's just a difference in multiple of the last three columns/rows.

Dear Henrik, Thank you for your kind response. I cannot understand why the D matrix is unsymmetric when the shear strain replace the engineering strain . To the best of my knowledge, I thought it's just a difference in multiple of the last three columns/rows.
Henrik Sönnerlind COMSOL Employee

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Posted: 7 years ago Oct 27, 2017, 1:36 a.m. EDT
Updated: 7 years ago Oct 27, 2017, 9:47 a.m. EDT

Hi Wenxue,

Think of the elements D14 and D41. If we have an elastic material and use engineering shears, , then they are equal.

But and

If we replace by in the strain representation, nothing happens to D41. D14, however, changes by a factor of 2, since it is now .

So the three last columns of D, which act as multiplier to the shear strains shift by a factor of 2.

Regards,

Henrik

-------------------
Henrik Sönnerlind
COMSOL
Hi Wenxue, Think of the elements D14 and D41. If we have an elastic material and use engineering shears, \gamma, then they are equal. But D_{14}=\frac{\partial \sigma_x}{\partial \gamma_{yz}} and D_{41}=\frac{\partial \sigma_{yz}}{\partial \epsilon_x} If we replace \gamma_{ij} by \epsilon_{ij} in the strain representation, nothing happens to D41. D14, however, changes by a factor of 2, since it is now \frac{\partial \sigma_x}{\partial \epsilon_{yz}}. So the three last columns of D, which act as multiplier to the shear strains shift by a factor of 2. Regards, Henrik
Wenxue Deng
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Posted: 7 years ago Dec 18, 2017, 8:24 a.m. EST

Dear Henrik,

Thank you for your kind response. I get it. I 'm plagued by another problem.

I have rewritten the linearelastic.c program of the example for it can be used to the transverseisotropy Material. There is five input arguments and the D matrix is different from isotropy material. See the appendix in detail.

But there are some differences in the results bettween calculated by External materials and direct calculated by orthotropic elastic. Such as mises stress and strain component. See the appendix in detail.

Dear Henrik, Thank you for your kind response. I get it. I 'm plagued by another problem. I have rewritten the linearelastic.c program of the example for it can be used to the transverseisotropy Material. There is five input arguments and the D matrix is different from isotropy material. See the appendix in detail. But there are some differences in the results bettween calculated by External materials and direct calculated by orthotropic elastic. Such as mises stress and strain component. See the appendix in detail.
Wenxue Deng
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Posted: 7 years ago Dec 18, 2017, 8:59 a.m. EST

This is the .c program appendix.

This is the .c program appendix.

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