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Chapter 3 ISOLATOR DEVICES AND SYSTEMS
3.2 PLASTICITY OF METALS
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Friction devices behave in a similar way to the extrusion damper, are simple but may require maintenance. Changes may occur in the friction coefficient due to age, environmental attack, temperature or wear during use. A further problem is that of 'stick-slip', where after a long time under a vertical load the device requires a very large force to initiate slipping. A dramatic example of a system isolated by this means is the Buddha at Kamakura; a stainless steel plate was welded to the base of the statue and it was rested on a polished granite base without anchoring.
Figure 3.1: (a) Stress-strain curves for a typical metal which changes from elastic to plastic behaviour at the yield point (B).
(b) Stress-strain curves for a typical mild steel under cyclic loading.
It should be noted that the area ABCE in Figure 3.1(a) represents input work while the area DCE represents elastic energy stored in the metal at point C and released on unloading to point D.
The difference area ABCD represents the hysteretic energy absorbed in the metal.
In the case of lead, the absorbed energy is rapidly converted into heat, while in the case of mild steel it is dominantly converted to heat, but a small fraction is absorbed during the changes of state associated with work hardening and fatigue.
Since metal-hysteresis dampers involve cyclic plastic deformation of the metal components, it is appropriate to consider the stress-strain relationship for a metal cycled plastically in various strain ranges, as shown in Figure 3.1(b) for a metal with the features typical of mild steel. Included in Figure 3.1(b) is the initial stress-strain curve of Figure 3.1(a). Notice the increasing stress levels with increasing strain range, and the lower yield levels during plastic cycling. With lead, the hysteretic loops are almost elastic-plastic, i.e., an elastic portion is followed by yield at a constant stress (zero slope in the plastic region). Typical operating strains are much greater than the yield strain, the loop tops are almost level, and the loop height is not significantly influenced by strain range.
To understand the behaviour of a metal as it is plastically deformed, it is necessary to look at it on an atomic scale. Previous to the 1930's, the plastic deformation of a metal was not understood, and theoretical calculations predicted yield stresses and strains very different from those observed in practice. It was calculated that a perfect crystal, with its atoms in well-defined positions, should have a shearing yield stress y of the order of 1010 Pa, and should break in a brittle fashion, like a piece of chalk, at a shear strain y of the order of 0.1. In practice, metal single crystals start to yield at a stress of 106 to 107 Pa (a strain of 10-4 to 10-3) and continue to yield plastically up to strains of 0.01 to 0.1 or more. The weakness of real metal crystals could in part be attributed to minute cracks within the crystal, but the model failed in that it did not indicate how the crystal could be deformed plastically (Van Vlack (1985); Read (1953); Cottrell (1961)).
The dislocation model was then devised and overcame these difficulties. Since its inception the dislocation model has been extremely successful in explaining the strength, deformability and related properties of metal single crystals and polycrystals.
(a) (b)
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The plastic deformation in a crystalline solid occurs by planes of atoms sliding over one another like cards in a pack. In a dislocation-free solid it would be necessary for this slip to occur uniformly in one movement, with all the bonds between atoms on one slip plane stretching equally and finally breaking at the same instant, where the bond density is of the order of 1016 bonds per square centimetre. In most crystals, however, this slip, or deformation, is not uniform over the whole slip plane but is concentrated at dislocations. Figure 3.2(a) is a schematic drawing of the simplest of many types of dislocation, namely an edge dislocation with the solid spheres representing atoms. The edge dislocation itself is along the line AD and it is in the region of this line that most of the crystal distortion occurs. Under the application of the shear stress this dislocation line will move across the slip plane ADCB, allowing the crystal to deform plastically.
The bonds, which must be broken as the dislocation moves, are of the order of 108 per centimetre, and are concentrated at the dislocation core, thus enabling the dislocation to move under a relatively low shear stress. As the dislocation moves from the left-hand edge of the crystal (Figure 3.2a) it leaves a step in the crystal surface, which is finally transmitted to the right-hand side. Figure 3.2(b) shows the other major type of dislocation, namely a simple screw dislocation, which may also transmitplastic deformation by moving across the crystal.
Figure 3.2: Atomic arrangements corresponding to:
(a) an edge dislocation, (b) a screw dislocation.
Here b is the Bergers vector, a measure of the local distortion and AD is the dislocation line.
The dislocations in crystals may be observed using electron microscopy, while the ends of dislocations are readily seen with the optical microscope after the surface of the crystal has been suitably etched. Typical dislocation densities are 108 dislocations per square centimetre in a deformed metal and about 105 per square centimetre in an annealed metal, namely one which has been heated and cooled slowly to produce softening. Dislocations are held immobile at points where a number of them meet, and also at points where impurity atoms are clustered.
The three main regions of a typical stress-strain curve are interpreted on the dislocation model as follows:
(1) Initial elastic behaviour is due to the motion of atoms in their respective potential wells;
existing dislocations are able to bend a little, causing microplasticity.
(2) A sharp reduction in gradient at the yield stress is due to the movement of dislocations.
(a) (b)
(3) An extended plastic region, whose gradient is the plastic modulus or strain-hardening coefficient, occurs when further dislocations are being generated and proceed to move. As they tangle with one another, and interact with impurity atoms, they cause work hardening.
It is also possible to model a polycrystalline metal as a set of interconnected domains, each with (different) hysteretic features of the type conferred by dislocations, which give the general stress-strain features displayed by the hysteresis loops of Figure 3.1(b).
Since dislocations are not in thermal equilibrium in a metal, but are a result of the metal's history, there is no equation of state which can be used to predict accurately the stress-strain behaviour of the metal. However, the behaviour of a metal may be approximately predicted in particular situations, if the history and deformation are reasonably well characterised.