Strained shell crystolecule structure

This is about pretty much all crystolecule structures that
closing up on themselves to make cylindrical shapes.
Plus shapes that are single or multi-threadedly helical.

Use cases

Strained shell crystolecule structures will be useful for quite a variety of tings. Including:

• torsional torque and angular speed transmitting axles
• inner and outer and in-between shells of slide bearings
• gears with the additional challenge of the teeth
• excenters (and cam following profiles that may even be radially deformed usin ting large diameter rings)
• screws for positioning, mechanical transmissions, fastening
• and as odd one out: use cases exploiting high intentional residual compressive and or tensile stress

All these structures need the atoms shifted away
from their natural position in an non-strained flat crystal lattice
to some lesser or stronger degree. Aka strain.

The several means of induction for strain

Strain can be induced by several means:

• just stretching alone (linking stress to strain by just the stiffness tensor, this is not the case for the other cases)
• elemental substitutions (which can reduce the local internal stresses by a lot and the global average internal stress almost to zero)
• intentional precisely placed defects (exactodefects) of all the known types
• Quasiamorphicity: Bond topology that locally looks disordered but actually is intentional and large scaly symmetric (usually threefold or higher as two-fold is hardly round and one-fold is no symmetry at all, more like kaehler brackets.

The natural ways to model and characterize strained shell structures

Internal stress fields are always interesting as they can hint on excessively stressed ones
that might become thermally chemically and mechanically unstable. And even a high energy material in bulk.
Fire and explosion risk.

Strain fields are interesting as there are several approaches.
One needs to pick an origin of zero strain.
The cylinder axis makes no sense as this is the one point without tangential direction.

Strain fields from plane crystal

For just stretched ones one will will likely want to pick an atom near the neutral fiber of bending.
This will only be useful for the local environment of the picked origin as the strain shifts go to the size of the entire.
If angular segments as symmetry units are small in angle this might be useful for an entire segment.

Strain field from pre-transformed coordinate system

One can pre-transform the crystal lattice
by wrapping the space to cylindrical using a fittingly picked neutral fiber radius.
Then apply a molecular relaxation.

Now the relative strain field from that is defined
from the yet non-relaxed atom positions
to the relaxed atom positions.

The magnitude of that strain filed will depend on how well one has picked teh neutral fiber and
the effect that compression is stiffer than stretching especially so for larger degrees of strain.

Ho to deal with dislocations and quasi-amorphicity?

If one uses dislocations then there these can be modeled as dislocation fields.
But how does that map to discrete per atom atomistic?
Apply that first before looking at an even smaller strain dislocations from that point?

That relates to space transformations accounting for dislocations in planar space
that is not distorted like here to wrap up on itself to a cylinder (possibly helcially).

More related math challenges

Linking the atomistic per atom quantities to bulk continuum model values.
Hunch: This may even involve exotic math like fractional derivatives.
This is related to multi-scale finite (volumetric) element modelling.

Misc

High internal stress (intentional residual stress) might actually be the main motivation in some cases. See page: Thick walled tube segment squeezing

Manufacturing via mechanosynthesis and jigs

For the case of planar crystal that is just bent high force applying jigs are necessary. This may turn out challenging.

For the cases with
★ elemental substitutions inclusive or
★ deliberate precise defects inclusive or
★ quasi-amorphicity:
There might be no need for a force applying jig.
Stick-n-place post assembly doing seamless covalent welding might suffice.
Heck in some cases (axial mechanosynthesis) there might not even be a need for a jig at all.

Mechanosynthesis buildplate templates might be needed
That in conjunction with seamless covalent cleaving
exploiting designed stress concentrations to get a clean reproducible cleaving break-off of crystolecules.

Related

• Thick walled tube segment squeezing & High pressure
• Design of Crystolecules#avoid too high interface pressure in sleeve bearings
• Atomically precise slide bearing
• Negative pressure bearing
• Atomically precise bearings & Atomically precise roller gearbearings
• Atomically precise gears
• Generalized gears

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• Scale transposed prototyping

External links

Wikipedia

• https://en.wikipedia.org/wiki/Strain_(mechanics)
• https://en.wikipedia.org/wiki/Strain_(mechanics)#Strain_tensor

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• https://en.wikipedia.org/wiki/Alternative_stress_measures
• https://en.wikipedia.org/wiki/Stress_(mechanics)
• https://en.wikipedia.org/wiki/Residual_stress
• https://en.wikipedia.org/wiki/Stress_concentration
• https://en.wikipedia.org/wiki/Thermal_stress

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• https://en.wikipedia.org/wiki/Hooke's_law#Matrix_representation_(stiffness_tensor)
• https://en.wikipedia.org/wiki/Stiffness_matrix

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• https://en.wikipedia.org/wiki/Bending