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Boundary conditions in terms of integrated forces and moments
Posted 12 juin 2014, 06:06 UTC−4 Modeling Tools & Definitions, Parameters, Variables, & Functions, Results & Visualization, Structural Mechanics 4 Replies
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Hello,
I am interested in simulating with COMSOL (version 4.2a) the following
(I think) simple and rather academic elasticity problem.
Let us consider a cylindrical rod (3D system with axis along Z),
free on its lateral surfaces, but loaded on its bottom and top surfaces.
However, I do not want to specify pointwise loads but to impose only
the values of the integrated forces and torques. More explicitly, I want
to constraint the system by these integrals of the surface tractions and
moments:
integral( stress_xz ) = F_x ( --> shear )
integral( stress_yz ) = F_y ( --> shear )
integral( stress_zz ) = F_z ( --> axial deformation )
integral( stress_zz * y ) = M_x ( --> bending )
integral( stress_zz * x ) = M_y ( --> bending )
integral( stress_yz * x - stress_xz * y ) = M_z ( --> torsion )
where F_x, F_y, F_z, and M_x, M_y, M_z are GIVEN values.
My first doubt is whether this problem is well posed for
simulation with COMSOL. The theoretical literature seems to
indicate that it is indeed mathematically well-posed.
If this is the case, how to implement it within the Structural
Mechanics module? Obviously this kind of boundary conditions are
not immediately available (although something similar is available
in the Terminal and Floating Potential boundary conditions of
the AC/DC module), but would it be possible to implement them with
other more advanced features, such as adding Weak Contributions or
Weak Constraints?
I would be immensely grateful for some help on this topic.
Regards,
Alberto.
I am interested in simulating with COMSOL (version 4.2a) the following
(I think) simple and rather academic elasticity problem.
Let us consider a cylindrical rod (3D system with axis along Z),
free on its lateral surfaces, but loaded on its bottom and top surfaces.
However, I do not want to specify pointwise loads but to impose only
the values of the integrated forces and torques. More explicitly, I want
to constraint the system by these integrals of the surface tractions and
moments:
integral( stress_xz ) = F_x ( --> shear )
integral( stress_yz ) = F_y ( --> shear )
integral( stress_zz ) = F_z ( --> axial deformation )
integral( stress_zz * y ) = M_x ( --> bending )
integral( stress_zz * x ) = M_y ( --> bending )
integral( stress_yz * x - stress_xz * y ) = M_z ( --> torsion )
where F_x, F_y, F_z, and M_x, M_y, M_z are GIVEN values.
My first doubt is whether this problem is well posed for
simulation with COMSOL. The theoretical literature seems to
indicate that it is indeed mathematically well-posed.
If this is the case, how to implement it within the Structural
Mechanics module? Obviously this kind of boundary conditions are
not immediately available (although something similar is available
in the Terminal and Floating Potential boundary conditions of
the AC/DC module), but would it be possible to implement them with
other more advanced features, such as adding Weak Contributions or
Weak Constraints?
I would be immensely grateful for some help on this topic.
Regards,
Alberto.
4 Replies Last Post 15 juin 2014, 03:37 UTC−4
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Posted:
1 decade ago
Hi,
The problem is not well posed. There is an infinite number of stress distributions which will fulfill your six equations. As an example, any shear stress distribution having sxz(X)=-sxz(-X) will not affect any of your resultants.
The simplest solution is to use the Rigid Connector, to which you can apply the resultant forces and moments directly. The assumption there is that the whole loaded boundary behaves like it was rigid. With that assumption the problem is well posed.
If the rigidity assumption is not acceptable, you need to figure out some other criteria which better match your physics and at the same time allows for a unique solution. That may not be trivial, though.
Regards,
Henrik
The problem is not well posed. There is an infinite number of stress distributions which will fulfill your six equations. As an example, any shear stress distribution having sxz(X)=-sxz(-X) will not affect any of your resultants.
The simplest solution is to use the Rigid Connector, to which you can apply the resultant forces and moments directly. The assumption there is that the whole loaded boundary behaves like it was rigid. With that assumption the problem is well posed.
If the rigidity assumption is not acceptable, you need to figure out some other criteria which better match your physics and at the same time allows for a unique solution. That may not be trivial, though.
Regards,
Henrik
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Posted:
1 decade ago
Hi All
Its clear that there will be infinite number of stress distributions which satisfy the required conditions. But I think by Saint-Venant’s principle the strain, stress and displacements caused by all the possible stress distributions far away from the loading points will approximately be the same. So the problem will still be well-posed .
Regards,
H.
Its clear that there will be infinite number of stress distributions which satisfy the required conditions. But I think by Saint-Venant’s principle the strain, stress and displacements caused by all the possible stress distributions far away from the loading points will approximately be the same. So the problem will still be well-posed .
Regards,
H.
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Posted:
1 decade ago
Hi again,
I agree in a sense with both Henrik and Heruy. An elasticity problem in a finite body
requires a pointwise specification of the applied loads to be well posed, but it is also
true that (according to Saint-venant principle) for a slender body the deformation
in the central part is determined only by the total (integrated) values of the forces
and torques applied on the remote ends. There are even analytical solutions for this
case that can be found in the literarure (see, e.g., Handbook of Continuum Mechanics,
by Jean Salençon).
Maybe I should have specified that I am actually interested in the Saint Venant problem.
Therefore, my question remains:
What is the best form to implement the constraining conditions associated to prescribed
integrated forces and torques?
There must be an elegant way to implement that in COMSOL , but after careful scrutiny
of the discussion forum and COMSOL tutorials I have been unable to find it.
I really would appreciate any help on this point.
Alberto.
I agree in a sense with both Henrik and Heruy. An elasticity problem in a finite body
requires a pointwise specification of the applied loads to be well posed, but it is also
true that (according to Saint-venant principle) for a slender body the deformation
in the central part is determined only by the total (integrated) values of the forces
and torques applied on the remote ends. There are even analytical solutions for this
case that can be found in the literarure (see, e.g., Handbook of Continuum Mechanics,
by Jean Salençon).
Maybe I should have specified that I am actually interested in the Saint Venant problem.
Therefore, my question remains:
What is the best form to implement the constraining conditions associated to prescribed
integrated forces and torques?
There must be an elegant way to implement that in COMSOL , but after careful scrutiny
of the discussion forum and COMSOL tutorials I have been unable to find it.
I really would appreciate any help on this point.
Alberto.
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Posted:
1 decade ago
Hi Alberto,
In case that the local solution is not important, I would recommend using the rigid connector.
Another approach could be to enter boundary loads like Fz=A+B*X+C*Y and then determine the unknowns A, B, C, ... using Global Equations and integration operators.
Regards,
Henrik
In case that the local solution is not important, I would recommend using the rigid connector.
Another approach could be to enter boundary loads like Fz=A+B*X+C*Y and then determine the unknowns A, B, C, ... using Global Equations and integration operators.
Regards,
Henrik