Discussion Forum

It seems all derivative of variables along radius direction in solid mechanics modula can not be show appropriately.

Hi,

In axial symmetry, the analytical expression for the strain in the circumferential direction is



When the radius approaches zero (on the z-axis) this expression becomes problematic from the numerical point of view.

So when you take a derivative of the stress, the underlying operation will be to compute the r-derivative of the expression above. This is why you see disturbances close to the z-axis. The result will be very sensitive to the mesh size at small r-coordinates.

In the formulation. we do take measures to avoid zero-divide in the strain itself, but that does not protect against taking derivatives of it.

Regards,
Henrik

Hi Henrik,

Thanks so much for your explanation. Now I understand why disturbances will appear when close to the z-axis.

As you mentioned, Comsol takes measures to avoid zero-divide in the strain itself in that formulation while does not protect against taking derivatives of it. However, if I want to take derivatives of it, are there any methods to avoid this disturbance? Thanks you!

Best,
Rong

Hi

I would propose split your domain around r=0 i.e. at a r=r_min and have at least 3 elements along the radial direction in this small cercle.

Or just omit the central part at r<r_min from your model. remains to define what walie to put on r_min

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Good luck
Ivar

Hi,

I just realized the major issue here: The derivatives should be taken with respect to to 'R', not 'r'.

A quote from the user's guide:

"From a simulation perspective it is desirable to solve the equations of solid mechanics on a fixed domain, rather than on a domain that changes continuously with the deformation. In COMSOL Multiphysics this is achieved by defining a displacement field for every point in the solid, usually with components u, v, and w. At a given coordinate (X, Y, Z) in the reference configuration (on the left of Figure 2-4) the value of u describes the displacement of the point relative to its original position. Taking derivatives of the displacement with respect to X, Y, and Z enables the definition of a strain tensor, known as the Green-Lagrange strain (or material strain). This strain is defined in the reference or Lagrangian frame, with X, Y, and Z representing the coordinates in this frame. The displacement is considered as a function of the material coordinates (X, Y, Z), but it is not explicitly a function of the spatial coordinates (x, y, z). It is thus only possible to compute derivatives with respect to the material coordinates."

Regards,
Henrik

Hi Henrik,

Thanks so much. Now I can get right derivatives of stress with respect to R.

However, my problem is the coupling of diffusion and stress. In diffusion part, there is no material deformation. So I cannot get derivatives of concentration with respect to R. If I still used the derivatives of concentration with respect to r while the derivatives of stress with respect to R, I think there mush be some problems. Do you have some advices?

Best,
Rong

Hi Ivar,

I think your method is a good way to deal with the singularity near the axis. I will try it! Thanks so much.

Best,
Rong

Hi,

As long as the deformations are not so large that there is a significant change in the geometry, there should not be any problem in mixing derivatives with respect to 'r' and 'R' as long as you respect the type of dependency that a certain variable has.

If the deformations are large, then several difficult aspects will arise. Some are:
a) the dependence of the diffusivity on strain
b) spatial and material orientations start to differ

Regards,
Henrik

Hi Henrik,

You are right, thanks so much. Now I can understand it.

Meanwhile, I have another question:

as you said, the derivatives of the deformation should be with respect to the material coordinate X,Y,Z, like d(u,X), d(u,Y). However, in my model, although there is a singularity in calculating d(u,r), I can still get the derivatives of the deformation with respect to the spatial coordinate z.

Moreover, I tried a 3D model and found I can get the derivatives of the deformation with respect to the spatial coordinate x,y, z.

I am a little confused about it. If you said the deformation should be with respect to the material coordinate, What is those derivatives I get? are there any relationship between d(u,x) and d(u,X)? Thanks!

Best,
Rong

Hi Henrik,

You are right, thanks so much. Now I can understand it.

Meanwhile, I have another question:

as you said, the derivatives of the deformation should be with respect to the material coordinate X,Y,Z, like d(u,X), d(u,Y). However, in my model, although there is a singularity in calculating d(u,r), I can still get the derivatives of the deformation with respect to the spatial coordinate z.

Moreover, I tried a 3D model and found I can get the derivatives of the deformation with respect to the spatial coordinate x,y, z.

I am a little confused about it. If you said the deformation should be with respect to the material coordinate, What is those derivatives I get? are there any relationship between d(u,x) and d(u,X)? Thanks!

Best,
Rong

Hi,

As long as the deformations are not so large that there is a significant change in the geometry, there should not be any problem in mixing derivatives with respect to 'r' and 'R' as long as you respect the type of dependency that a certain variable has.

If the deformations are large, then several difficult aspects will arise. Some are:
a) the dependence of the diffusivity on strain
b) spatial and material orientations start to differ

Regards,
Henrik