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Hyperelastic materials in Comsol Multiphysics: why there is not a factor 2* in some of the components of the second Piola-Kirchhoff stress tensor?
Posted 2016年4月20日 GMT-4 12:17 Structural Mechanics 4 Replies
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Hi all!
When looking at the "equation view" panel of a hyperelastic material in Comsol Multiphysics it is possible to see how the program calculates the components of the second Piola-Kirchhoff stress tensor (S). Now, the formula should be:
S_ij = 2*d(Ws, C_ij)
where Ws is the strain energy and C_ij are the components of the right Cauchy-Green deformation tensor. However, when looking at the table in "equation view" in Comsol, it is possible to see that the extra-diagonal components of the stress (namely solid.Sl12, solid.Sl13 and solid.Sl23) are computed without the multiplication by 2*.
Is there anyone who noticed the same problem?
Thank you in advance for your help,
Best,
Pietro
When looking at the "equation view" panel of a hyperelastic material in Comsol Multiphysics it is possible to see how the program calculates the components of the second Piola-Kirchhoff stress tensor (S). Now, the formula should be:
S_ij = 2*d(Ws, C_ij)
where Ws is the strain energy and C_ij are the components of the right Cauchy-Green deformation tensor. However, when looking at the table in "equation view" in Comsol, it is possible to see that the extra-diagonal components of the stress (namely solid.Sl12, solid.Sl13 and solid.Sl23) are computed without the multiplication by 2*.
Is there anyone who noticed the same problem?
Thank you in advance for your help,
Best,
Pietro
4 Replies Last Post 2016年4月22日 GMT-4 03:13
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Posted:
9 years ago
Hi,
A good observation. This is actually explained in later versions of the user guide (theory section about hyperelastic materials) since the question pops up now and then.
The symmetry of the right Cauchy-Green deformation tensor will implicitly cause two equal terms to appear during the symbolic differentiation. This is an effect of some tensors being internally declared as symmetric, and cannot really be seen in Equation View.
Regards,
Henrik
A good observation. This is actually explained in later versions of the user guide (theory section about hyperelastic materials) since the question pops up now and then.
The symmetry of the right Cauchy-Green deformation tensor will implicitly cause two equal terms to appear during the symbolic differentiation. This is an effect of some tensors being internally declared as symmetric, and cannot really be seen in Equation View.
Regards,
Henrik
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Posted:
9 years ago
Thank you for your fast reply.
If I have understood correctly, the derivation of the components of S is not entirely explicit in the equation view, right?
But then, are the extra-diagonal components of the second Piola-Kirchhoff tensor already the correct ones? That is, when I use solid.Sl12 in a calculation, for example, is it the actual component that I am using? Or should I use solid.Sl12 divided by 2?
Thank you again,
Pietro
If I have understood correctly, the derivation of the components of S is not entirely explicit in the equation view, right?
But then, are the extra-diagonal components of the second Piola-Kirchhoff tensor already the correct ones? That is, when I use solid.Sl12 in a calculation, for example, is it the actual component that I am using? Or should I use solid.Sl12 divided by 2?
Thank you again,
Pietro
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Posted:
9 years ago
Hi,
All tensor components that you use are as 'expected', that is S12 is the actual shear stress value.
The expression d(solid.Ws,solid.Cl12) actually returns twice what it looks like, when solid.Cl is a symmetric tensor. If you look at the definition of the strain energy density function solid.Ws in Equation View, you will see that it is a function of ...+2*C12+... rather than ...+C12+C21+...
The C21 variable does not exist for a symmetric tensor, so instead C12 gets the factor '2'.
To sum up: Derivatives of functions of symmetric tensors return twice the 'expected' value for off-diagonal elements. The same behavior could for example appear when taking the derivative of a von Mises stress with respect to a shear stress.
Regards,
Henrik
All tensor components that you use are as 'expected', that is S12 is the actual shear stress value.
The expression d(solid.Ws,solid.Cl12) actually returns twice what it looks like, when solid.Cl is a symmetric tensor. If you look at the definition of the strain energy density function solid.Ws in Equation View, you will see that it is a function of ...+2*C12+... rather than ...+C12+C21+...
The C21 variable does not exist for a symmetric tensor, so instead C12 gets the factor '2'.
To sum up: Derivatives of functions of symmetric tensors return twice the 'expected' value for off-diagonal elements. The same behavior could for example appear when taking the derivative of a von Mises stress with respect to a shear stress.
Regards,
Henrik
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Posted:
9 years ago
Dear Henrik,
thank you for your reply and for the detailed explanation. Now I understand it!
Best,
Pietro
thank you for your reply and for the detailed explanation. Now I understand it!
Best,
Pietro