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The Plastic Phenomenon in Polycrystals
The second point to be fixed is that so far, we have discussed in terms of a single crystal, but in practice we have polycrystals. How the plastic phenomenon applies to polycrystals. Another scientist evaluated how many grains which have their own crystal planes and directions maybe have under the application of a normal stress. He tried to generalize the Schmid factor. The conclusion was, for slip to occur in a polycrystalline material, at least 5 independent slip systems are required. It means that when we have a single crystal, the most suitable oriented plane will start to flow to make dislocations move. In polycrystalline material, at least 5 independent slip systems must be predisposed to make dislocations move at room temperature. At higher temperature, slip is easier. At high temperature, more slip systems may become active and polycrystalline materials which are brittle in room temperature may become ductile in high temperature.
When we have polycrystalline materials, we have different...
kinds of grain boundaries. Low angle grain boundaries are favorable to make dislocations cross the grain boundaries, however when they have high angle boundary, dislocations can't cross and are restricted to a single grain. We can also rationalize polycrystalline materials in terms of directionality. It means that in a single crystal, the orientation of the lattice w.r.t to the normal stress is important to make dislocations move. This means that single crystals are highly anisotropic, so plastic deformation of single crystals depends on the crystal structure and how it is oriented w.r.t applied load. What if the material is polycrystalline? Because we have such a large number of grains, as a whole this large spectrum will make the material isotropic. Crystal anisotropy of each grain disappears when these grains are populated in a polycrystalline material, therefore we can say it behaves isotropically. Macroscopically - Isotropic Microscopically - Each grain is anisotropic SLIDE 7Will such polycrystalline material deform under the application of a stress? On the top left of the slide, we can see different grains with different lattices and orientation. They have their own theta and phi.
If the polycrystalline material is like the first figure in the right-hand side, and we apply a normal stress, what would be the behavior of these grains? The material will not allow any void or overlap. The polycrystalline material will never deform by allowing voids and overlap. Each grain will follow the plastic deformation induced by stress by making consistent plastic deformation and the grains will be in contact with each other.
Therefore, the figure D shows that if the grains fulfill this condition, always must make their own surface compatible. In order to fulfill this condition, the grain boundaries will create a stress system which makes these grains bonded to each other. It is a special role of grain boundaries to fulfill the consistency condition so that the grains may deform.
without causing voids or overlap. Only when the stress reaches the maximal value, a crack will nucleate in the weakest bonding of the grains.slip first. So, we must think of progressive plastic deformation in all of the grains, and whenever the grain will reach saturation, the grain will fail, because no more dislocations will be allocated there because all the systems will have been saturated.
We conclude that plastic deformation in a polycrystalline material may be homogeneous, in terms of direction. But the yielding of the material will not be homogenous, therefore deformation is not homogeneous, because it varies from grain to grain.
SLIDE 9
Even inside one grain we may have inhomogeneous plastic deformation. Why? Consider 2 grains in contact with each other by a common grain boundary and we apply a normal stress. We know that the normal stress will provide a shear stress to make dislocation motion. Consider that this possibility is only possible when the grains are large enough and contain dislocations. When we increase the stress gradually, what do we expect?
(1) Consider what happens in the cross section along the dashed line.
Specifically consider point number 1 and 2. These 2 points are contained in the boundary of the lower grain. When we apply a normal stress, the region that feels it first is the grain boundary. At point 1 and 2 the stress initially will be equal to the applied stress. Within the grain itself, stress is not homogenously felt by the grain, but the GB will be loaded first. Initially, the resistance of the grain at this point will be equal to the applied stress. Since the resistance of sigma B is larger than sigma AP, it will not fail. The resistance of the material in the grain will be lower.
(2) What happens if sigma AP (applied) increases? Dislocations will appear in the GB. The cross section of this grain will provide that the peak in the GB will not increase anymore. Because dislocations will increase in density, the thickness will increase (dislocations are expanding around the grain boundaries. This can be shown by a plateau in the tensile graph.) The inner grains still feel a lower stress.
(3)
By continuing to increase the applied stress, dislocations will continue to expand. After the inclined boundaries, the horizontal boundary will be involved in the end. Then we must expect that the resistance of the material overall will equalize the applied stress. The overall material (boundary and grain) will respond with the same resistance, and therefore, it will yield. We call this micro-yielding.
SLIDE 10
How will the Schmid law change from a single crystal to a polycrystalline material? Given a normal stress, given a multitude of grains which are statistically oriented in a different manner, which is the condition of orientation of the slip planes? There are a lot more slip systems now. We still may use the Schmid law. Statistically the optimal condition still holds with the Schmid factor equal to 0.5. In this case the product of cosines is given as the maximum in an infinite number of grains. But in order to have such condition applied we must require at least 5 active slip systems. If just
one of the slip systems doesn’t work the material will fail and the Schmid law won’t hold.
SLIDE 11
Now I want to show you an important property of a polycrystalline material, which can be deduced from the tensile test curves.
We know when we evaluate the plastic behavior of materials, we must rely on the tensile test curve, but NEVER use the engineering stress-strain curve.
It is okay to use the engineering curve to identify the position of UTS.
But for plastic behavior we must consider the true stress-strain curve, which is the one which always increases.
In the relation, we say sigma f for sigma maximal. Since it is a maximum in the engineering curve, the derivative of P is 0.
Saying the volume is constant is saying the cross-section area times length will be constant. We differentiate the volume too and express dA.
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