understanding "work / strain hardening" ?

[Widdekind]

http://en.wikipedia.org/wiki/Work_hardening

 

[math]\sigma = \sigma_y + G \alpha b \rho_{\perp}^{1/2}[/math]

 

[math]\sigma \approx \sigma_y + K \epsilon_{p}^{0.2 - 0.5}[/math]

 

So, simplistically stated, the perpendicular / transverse dislocation density (# / m²) is proportional to the plastic strain ?

 

[math]\rho_{\perp} \approx \epsilon_p[/math]

 

Now, [math]\rho_{\perp}^{1/2} = 1/<d>[/math] where <d> = average distance between dislocations. So, seemingly,

 

[math]\sigma \approx \sigma_y + \frac{\left( \alpha G b \right)}{<d>}[/math]

 

And, a 1/d dependency resembles viscous fluid flow, [math]F = \mu A v / \Delta h[/math].

 

So, the material under strain is "trying" to flow away from the middle of the material body, "left & right" towards the two vice grips grabbing the specimen studied...

but, dislocations impede the plastic fluid-like flow, somewhat similar to viscous drag friction forces ? i.e. the material is forced to flow "up over (down under) and around" the dislocations (hence the presence of the shear modulus G) ?

 

Also, the "necking down" of the strained specimen seemingly concentrates dislocations, increasing the perpendicular dislocation density ? So, would Poisson's ratio, commonly close to a half, have some sort of conceivable connection, to this particular process ? For [math]\nu = 1/2[/math], if the strain doubled, the perpendicular area would have to halve, and so for a constant number of dislocations, their perpendicular density would double, [math]\rho_{\perp} \approx \epsilon[/math]. And, if [math]\nu \ll 1/2[/math], then nearly no "necking down" occurs, and so a constant number of dislocations would be concentrated allot less, seemingly suggesting, that the strain's dependency on stress, would decrease, as seems said in the second equation considered.

 

( producing a plot, i notice no clear correlation, between n <----> v )

[Enthalpy]

Crystallography and the like use to give answers of limited usefulness for the user and even the producer of alloys, so I haven't invested much in it, sorry for that. It's complete books of theories not easy to check experimentally, and when you need a simple answer they don't give it. Worse, there are so many possible processes and explanations that crystallography explains equally well everything and its opposite.

 

Resembles a viscous fluid:

time and speed have only a secondary (but real) importance in the flow of solids, while they're paramount in liquids. As well, flows of solids show thresholds that simple liquids don't have - though many liquids aren't just linear.

 

The behaviour of an alloy under strain depends fundamentally of said alloy and its thermal and deformation history. This contrasts with a liquid. Austenitic stainless steel, commonly used for cold bending and deep drawing, gains much strength through deformation; this makes places already deformed more resistent, so the ongoing deformation affects in priority the other places, and the global deformation is evenly spread. Other alloys behave differently and aren't as good for cold bending. Simple deformation models, independent of the alloy, can't tell that.

 

Also important, deformation can change the metallurgical state, typically from austenitic to martensitic in stainless steel. Some people wrongly believe it's the only cause of hardening.

 

Poisson near 1/2: no! That would leave the solid's volume unchanged under uniaxial stress, which is not the case. Most alloys are near 1/3 (...though reasoning would imply 1/4).

Edited August 5, 2013 by Enthalpy

[Widdekind]

logically, the crystal lattice of some sample specimen straining under tensile stress, is a little like a long "chain of hands", atoms "gripping nearest neighbors"

 

when the lattice is stressed, some atoms wind up having a "firmer hold" of the "hands" to one side or another...

 

some of the lattice atoms on one side of the mid-plane will actually become pulled towards the other side... et vice versa

 

opposing sets of "chains of hands" lie along each other, and slide past each other, towards opposite poles of the stress apparatus

 

as they slide past each other, they would induce shear-like tensions in each other (and their flow would seemingly somewhat resemble viscous fluid-like flow)

 

if you stopped the stressing, and sawed the specimen in half... then you would see what you could conceivably call "dislocations" in each smoothly sawed face...

 

those "dislocations" would actually have been "locations" connected to the crystal lattice in the other half (they would, then, "line up" when the halves were placed face to face)

 

the more dislocations per cross-sectional area, the more viscous-like-fluid-flow-forces of shear would have been opposing the pulling

 

at some point, during "necking down", the "last in the lines" of the "chains of hands" would slip past each other, at failure

 

inter-penetrating portions of the crystal lattice, pulled oppositely past each other (on nano-to-micro-scopic scales), inducing shear-like viscous-resembling force effects, could common-sensically account, for the equations with-on Wikipedia

[Enthalpy]

Crystal deformation works by moving dislocations. Good single-crystals tend to be very hard and brittle. They're a very bad model to understand metals, whose behaviour fully depends on crystal imperfections.

 

In a dislocation, some atoms have a less-than-perfect position in the crystal. When the dislocation moves, these atoms get a better position (for their bonds) but their neighbours get the worse position. This is energeticaly neutral, and the energy hill is small, so it happens at room temperature.

http://en.wikipedia.org/wiki/Dislocation

 

By the way, dislocations are easily seen at the surface of a good single-crystal. I once scratched the rear side of a monocrystalline silicon wafer to mark it, insisted a bit, and the markings were visible at the front side. This was not a plastic deformation, which is impossible with silicon.

[Widdekind]

http://en.wikipedia.org/wiki/Grain_boundary_strengthening

http://en.wikipedia.org/wiki/Stress_intensity_factor

 

 

the yield strength of many materials depends upon the sizes (D) of the grains comprising the sample, [math]\sigma = \sigma_0 + \frac{K}{\sqrt{D}}[/math]

 

that is the same mathematical dependency, as the stress occurring w/ cracks, of length L, [math]\sigma \approx \frac{K}{\sqrt{L}}[/math]

 

logically, grains = "bricks", and the amorphous material "mortaring" them together = "mortar"

 

the characteristic size of the bricks and surrounding mortar could conceivably set the scale of size of cracks inside the sample, somewhat similar to bricks cracking loose from their surrounding mortar, in a brick wall, most cracks become about as big as the bricks...

 

except would be in 3D inside some crystal lattice

 

seemingly, supposing that the size scale of the bricks & mortar, sets / limits the size scale of cracks, could account for both processes... the amorphous mortar, and also the mis-alignments between bricks, contains cracks inside grains...

 

and the grains block the propagation of cracks, down through seems of mortar, between the faces of opposite grains, when the cracks try to propagate out to the sides, for then they "run into" the sides of other grains...

 

the dis-order of the mortar and mis-alignments from grain to grain, block the propagation of dislocations within the ordered lattices inside grains... and the order of the grains, blocks the jagged faults w/in the mortar

 

grain hardening seemingly plausibly represents "crack containment" w/in the crystal lattice

 

 

http://en.wikipedia.org/wiki/Carbon_nanotube

 

if so, then by analogy, crack containment could be crucial, in carbon-fibre cables... stronger carbon cables could conceivably be composed, of multiple types of "zig zag" & "arm chair" and other "m,n" configurations of carbon tubes... such that cracks occurring in one, would be contained, by the atomic mis-alignments, of its carbon atoms, w/ those of adjacent fibers... although the separate-ness of the separate fibers could conceivably already contain cracks into single fibers

[Enthalpy]

I wouldn't follow the analogy with concrete, because

- concrete is fragile while acceptable alloys are ductile, so the failure mode is radically different

- joints aren't very dissimilar to grains in alloys.

 

Neither would I make any parallel with carbon nanotubes, first because their behaviour in significant amounts is widely unknown. Spreading the stress despite bends and dissimilar lengths is what makes an acceptable rope, and nanotues have still to show their fitness. Expect to lose a factor of 5+ in tenacity provided nanotubes behave as well as adequate fibres do, and then carbon ropes won't be very good.

 

Comparisons up to now are made between single nanotubes and complete polymer ropes, which is extremely unfair. Checking against single polyethylene or carbon fibres show that nanotubes aren't a revolution, and they still have to show their capability as a rope. If nanotube ropes get usable, their performance won't probably enable space elevators for instance.

[studiot]

I don't really catch where this thread is leading but a couple of comments.

 

The mortar in brickwork does not stick the bricks together.

It is there to hold the bricks apart.

Masonry in general is a compression based system.

 

Many promising very strong fibres are also very slippery.

It is all very well creating a strong fibre but a problem if you can't transfer the load into or out of it.

Steel reinforcement in concrete copes with this by the use of hooks and other shapes. This option is not available to ropelike fibres.

This problem has beset many existing uses of carbon fibres.

 

Creep is another issue with many materials, particularly glasslike structures, plastic and semiplastic materials.

Edited August 10, 2013 by studiot