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# Periodic Boundary Conditions for Non-orthogonal Unit Cell

Hey guys,

I'm currently doing my master thesis on failure analysis of braided composites.

I want to apply Periodic Boundary Conditions on my non orthogonal (say for a braiding angle of 30°, it becomes a parallelogram) unit cell, and then load the whole thing by applying concentrated forces to a set of dummy nodes (which I call ConstraintsDrivers). I use the equation option in Abaqus. PBCs are defined in global XYZ axes (see attached pic), but I define my loading in a rotated CSYS (x axis from bottom left point to upper right).

Now here's the problem:

I started with a state of simple shear, but I get coupling effects, meaning that I produce a mixed state of shear and tension when I apply only shear.

I compensated that by superimposing an axial force, but this isnt really a good solution.

When i tried to apply a state of pure shear, I also get these unwated coupling effects. Additionally, the thing doesn't deform symmetrically (I would assume the upper right point would stay on a line connecting the bottom left point and the upper right point of I apply pure shear)

I'm not sure If that's a problem of the non-orthogonal unit cell, that cannot be undone if PBCS are defined for an orthogonal CSYS.

Does anyone have experience with non-orthogonal unit cells?

Thanks for your input in advance!

Cheers,

Tobias

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## Re: Periodic Boundary Conditions for Non-orthogonal Unit Cell

Have you tried to run the test on an isotropic and homogeneous unit cell first?

-- Biswajit

## Re:Periodic Boundary Conditions for Non-orthogonal Unit Cell

1) Can you mesh a unit cell that is parallel to the coordinate axes? (see http://imechanica.org/node/11145 )

2) "I started with a state of simple shear, but I get coupling effects" - Is this a manifestation of the Poynting effect? "Topics in Finite Elasticity" by Morton Gurtin has a short and great explanation of this.

Nachiket

## why do you need it?

Before going into how to make a non-orthogonal unit cell work, why do you need it? You could simply make a unit cell in a rotated CSYS and have the usual orthogonal unit cell that has braids parallel to x- (or y-) axis. This would eliminate any problems you may be creating with 30 degree unit cell.

## Thanks for your

Thanks for your input.

1) Can you mesh a unit cell that is parallel to the coordinate

axes?

Since I'm modeling a regular braid (2x2) (equivalent to twill) with a braiding angle of 30°, using beam elements embedded into the unit cell, I do have to rely on a non-orthogonal unit cell ( the unit cell would be orthogonalif the braiding angle would be 45°, i.e the yarns would interlace perpendicular to each other)

If i use an orthogonal unit cell for the 30° braid, it wouldnt be a periodic structure anymore.

Have you tried to run the test on an isotropic and homogeneous unit cellfirst?

-- Biswaji

Yes, i left out the yarns and ran the whole thing on a isotropic unit cell. Similar coupling effects as with the yarns (x tension + shear)

2) "I started with a state ofsimple shear, but I get coupling effects" - Is this a manifestation of

the Poynting effect? "Topics in Finite Elasticity" by Morton Gurtin has a

short and great explanation of this.

I'll look into that. I'll see if I can get the book anywhere.

Cheers,

Tobias

## Re: Non-orthogonal unit cell

I'm assuming:

1) You're using linear elasticity

2) You're computing the stress components in the correct coordinte system

Thn, if you get tensions in an isotropic and homogeneous unit cell when you think you are applying only a shear, you're probably not actually applying only a shear traction. There will be some edge effects, but the volume averaged stresses should not have any tensile components in the correct coordinate system.

I think you're actually applying both normal and shear tractions and that can be found by summing tractions at the edges of the unit cell. Of course, that's just a guess and more can't be said without seeing exactly what you've done.

## Are you applying displacement BCs?

If your constraint equations are correct and if you apply shear strain (not traction), then the deformation should not show any tensile strain. (And for isotropic material, it shouldn't show any tensile stresses either). So if you are not getting back what you are specifying in terms of strains, the constraint equations would be first thing to check.

Of course, all these quantities are volume averaged.