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Simple plain strain state does not produce expected deformation
Posted 10 apr 2014, 12:34 GMT-4 Structural Mechanics Version 4.4 5 Replies
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Dear all,
I am trying to simulate the scrolling of a thin, flat film/beam into a ring by using an uniaxial strain with a linear gradient. This is done in Comsol using a 2D structural mechanics model with large deformation. The resulting scrolling radius does not match the expected value given by the strain gradient.
From basic mechanical consideration the bending strain (epsb) for a freestanding beam scrolling into a ring can be given by:
epsb(y) = eps0 + (2*PI*(R-y) -2*PI*R)/(2*PI*R) = eps0-y/R
where y is the short axis of the beam, eps0 a (theoretically arbitrary) strain at y=0 and R the radius of the scrolled structure. In other words, the scrolling Radius should solely be determined by the strain gradient. This strain is implemented in Comsol using the 'initial stress and strain' node (see attached model)
solid.eil11 = Y/R
The simulation works well, the beam is bent into a perfect ring and seems to be independent of any solver settings.
However the Radius of the formed ring (as extracted from the maximum x value) is always larger by a constant value then the R used in the strain definition. E.g. if R= 60[um] then max(x) = 64.5[um],R= 70[um] then max(x) = 74.5[um].
Astonishingly, if the strain is defined as solid.eil11 = (Y-3/4*tBeam)/R the correct Radius is obtained.
This would imply that the radius is actually dependent on eps0 which doesn't make any sense to me for this very simple model.
Any Ideas as to where this difference between (simple) Theory and Simulation comes from? Adapting the second implementation of the initial strain is not possible as this model is extended to a more complex one with non linear gradients.
Thanks in advance and best Regards.
I am trying to simulate the scrolling of a thin, flat film/beam into a ring by using an uniaxial strain with a linear gradient. This is done in Comsol using a 2D structural mechanics model with large deformation. The resulting scrolling radius does not match the expected value given by the strain gradient.
From basic mechanical consideration the bending strain (epsb) for a freestanding beam scrolling into a ring can be given by:
epsb(y) = eps0 + (2*PI*(R-y) -2*PI*R)/(2*PI*R) = eps0-y/R
where y is the short axis of the beam, eps0 a (theoretically arbitrary) strain at y=0 and R the radius of the scrolled structure. In other words, the scrolling Radius should solely be determined by the strain gradient. This strain is implemented in Comsol using the 'initial stress and strain' node (see attached model)
solid.eil11 = Y/R
The simulation works well, the beam is bent into a perfect ring and seems to be independent of any solver settings.
However the Radius of the formed ring (as extracted from the maximum x value) is always larger by a constant value then the R used in the strain definition. E.g. if R= 60[um] then max(x) = 64.5[um],R= 70[um] then max(x) = 74.5[um].
Astonishingly, if the strain is defined as solid.eil11 = (Y-3/4*tBeam)/R the correct Radius is obtained.
This would imply that the radius is actually dependent on eps0 which doesn't make any sense to me for this very simple model.
Any Ideas as to where this difference between (simple) Theory and Simulation comes from? Adapting the second implementation of the initial strain is not possible as this model is extended to a more complex one with non linear gradients.
Thanks in advance and best Regards.
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5 Replies Last Post 11 apr 2014, 09:57 GMT-4
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Posted:
1 decade ago
Hi Silvan,
I would vary the thickness of your beam and see if the offset depends on it.
If you find any constant ratio in that - this would be suspicious.
Best regards,
Sven
I would vary the thickness of your beam and see if the offset depends on it.
If you find any constant ratio in that - this would be suspicious.
Best regards,
Sven
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Posted:
1 decade ago
Hi Sven,
Many thanks for your reply.
Indeed you're correct: For tBeam 3[um]-10[um] (range of intrest for me) the offset/tBase ratio is almost perfectly constant at 1.49 (1.498...1.494).
In which way exactly do you mean that this is suspicious? Is there an error in my model?
Best Regards,
Silvan
Many thanks for your reply.
Indeed you're correct: For tBeam 3[um]-10[um] (range of intrest for me) the offset/tBase ratio is almost perfectly constant at 1.49 (1.498...1.494).
In which way exactly do you mean that this is suspicious? Is there an error in my model?
Best Regards,
Silvan
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Posted:
1 decade ago
Hi,
The most fundamental problem in your model is that when you are using geometric nonlinearity, the strains are measured as Green-Lagrange Strains. The initial strains must be interpreted as such, so for your simple problem you would need to enter Y/R+0.5*(Y/R)^2.
You also have to be careful about how you define the radius and how you measure it. With your geometry the expected result of the max(x) evaluation is 73 (=R+t) and that is what you will get when you include the nonlinear term above. If you center your geometry around Y=0, then you will measure 71.5 (R+t/2) since you measure on the outside and not at the centerline.
Regards,
Henerik
The most fundamental problem in your model is that when you are using geometric nonlinearity, the strains are measured as Green-Lagrange Strains. The initial strains must be interpreted as such, so for your simple problem you would need to enter Y/R+0.5*(Y/R)^2.
You also have to be careful about how you define the radius and how you measure it. With your geometry the expected result of the max(x) evaluation is 73 (=R+t) and that is what you will get when you include the nonlinear term above. If you center your geometry around Y=0, then you will measure 71.5 (R+t/2) since you measure on the outside and not at the centerline.
Regards,
Henerik
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Posted:
1 decade ago
Hello Henrik,
Thank you, this clarifies the issue a lot. However, I'm confused when it comes to eps0. The final radius still depends on eps0 if I either implement it as:
solid.eil11 = (eps0-Y/R) + 1/2(eps0-Y/R)^2
(which should be the correct form according to Green-Lagrange)
or as
solid.eil11 = eps0 - Y/R + 1/2(-Y/R)^2
This doesn't make sense to me, as a constant strain offset should result in a simple elongation of the structure? Does this mean that the theoretical bending strain is already wrong or can it not so simply be modeled in Comsol?
Best Regards,
Silvan
PS:
The strain in my first post and the model should have been solid.eil11 = -Y/R so R is equal to the Radius on the outside of the structure.
Thank you, this clarifies the issue a lot. However, I'm confused when it comes to eps0. The final radius still depends on eps0 if I either implement it as:
solid.eil11 = (eps0-Y/R) + 1/2(eps0-Y/R)^2
(which should be the correct form according to Green-Lagrange)
or as
solid.eil11 = eps0 - Y/R + 1/2(-Y/R)^2
This doesn't make sense to me, as a constant strain offset should result in a simple elongation of the structure? Does this mean that the theoretical bending strain is already wrong or can it not so simply be modeled in Comsol?
Best Regards,
Silvan
PS:
The strain in my first post and the model should have been solid.eil11 = -Y/R so R is equal to the Radius on the outside of the structure.
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Posted:
1 decade ago
Hi,
Your first expression ( (eps0-Y/R) + 1/2(eps0-Y/R)^2 ) is the correct one.
Think of the physics the following way:
1. Curl up the structure to the intended radius
2. Now add eps0. Your circle will grow radially, and thus the radius will increase.
Regards,
Henrik
Your first expression ( (eps0-Y/R) + 1/2(eps0-Y/R)^2 ) is the correct one.
Think of the physics the following way:
1. Curl up the structure to the intended radius
2. Now add eps0. Your circle will grow radially, and thus the radius will increase.
Regards,
Henrik