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Dynamic Plasticity: Dislocations

1.1. Introduction: how to describe plasticity?

The microscopic description of the plastic strain of metals and alloys mainly relies on the knowledge of the mechanisms for the generation and propagation of dislocations. Though the modeling of macroscopic observations using these microscopic mechanisms has made great progress from a qualitative point of view, much remains to be done from a quantitative point of view.

Under these conditions, the use of a descriptive approach to macroscopic observations remains of great interest. These observations are made on the basis of an examination of mechanical tests in which the strain experienced by the metal as a result of the action of a given system of stresses.

It is necessary here to distinguish the mechanical properties where the time and the rates of strain play only a secondary role (plasticity in the usual sense of the term) from those where time and/or strain rates play an important role (creep, fatigue, dynamic plasticity, etc.).

The mechanical properties reflect the microscopic behavior of the material, and the strain observed at the macroscopic level is the result of local strains on a much finer scale. This microscopic aspect is fundamental for the understanding of different phenomena.

That is why the mechanical tests are often supplemented by a local physical study of the strain mechanism (observations using X-rays, optical microscopes, electronic scanning microscopes, transmission electron microscopes, etc.).

In order to better identify the fundamental mechanisms, it is essential to work on well-defined systems. This is why many studies of the plasticity have been carried out on single crystals.

However, the most commonly used materials are polycrystalline, meaning that they are made up of a more or less isotropic group of monocrystals. It should be noted that for polycrystalline materials, it is more correct to speak of quasi-isotropy than of isotropy.

If the technological material deviates significantly from an isotropic state, then it is said to have a texture. More precisely, a material has a texture if the orientation of its monocrystalline grains, which are generally very numerous, is not totally random but instead presents specific directions which are prevalent. Textures are created at the time the material solidifies, or during an anisotropic strain. They can be transformed by annealing or through phase change. They are of interest economically insofar as they make it possible to improve certain properties of the materials.

Despite the interest industry has in textured materials, we will only consider quasi-isotropic materials in this book.

The application of the experimental laws of plasticity from polycrystalline materials to the calculation of structures in plastic strain, or to the study of processes for forming, can be done by associating two tensors with the stresses and strains shown in the tests with two quantities:

• the generalized stress (or equivalent stress) ;
• the generalized strain (or equivalent strain) .

It can then be shown that the curve of the variations of on the basis of those of is independent of the type of mechanical test performed. The value of for a given load can thus be deduced from the results of the tensile test, which gives a very particular interest to this type of simple test.

1.1.1. Equivalence between forming processes and mechanical tests

A forming method is in itself a mechanical test. Therefore, it is equivalent to any mechanical test conducted under the conditions indicated above.

A number of simple modeling methods allow for a practical implementation of the laws of plasticity.

REMARK.- The forming processes are generally carried out in dynamic plasticity, with strain speeds from 10 to 104 s-1 (including cases such as forging, wire drawing, stamping, rolling, metal cutting with machining at high speeds, pulsed magnetic fields, explosive, etc.). These cases involve a dynamic rheology, which will be the subject of a dedicated chapter.

1.1.2. Early stages of strain

The yield strength is usually defined as the stress above which a strain does not return to zero once the material is no longer subjected to a load.

This problem of the elastic limit is at the origin of the theory of dislocations formulated by Orowan, Polanyi and Taylor, who noted that in order to attain a plastic strain on a material, it is not necessary to deform it over the entirety of its volume. Rather, it suffices to propagate a dislocation line along a slip plane. However, a macroscopic plastic strain may only result from the propagation of a large number of dislocations. Before a deformation, each crystal contains within itself an initial network of dislocations, known as the Frank network, though there is only a small number of these dislocations.

Thus, it will only be possible to observe a macroscopic plastic strain if the dislocations of the Frank network multiply. This can occur in ways such as through the Frank-Read mechanism.

1.1.3. Multiplication of dislocations

The initial density of dislocations present in a crystal does not give rise to a plastic deformation of very large amplitude. For example, let us consider a grain of 100 µm on each side, containing 10-8 cm-2 dislocations of Burgers vector 2.5 × 10-8 cm. Provided that they stretch across the entire section of the grain, we would have, at most, a plastic strain given by the formula:

We now need to imagine a mechanism capable of multiplying dislocations as can be observed experimentally. Frank and Read have proposed a reel-like mechanism. Let us suppose that there is a dislocation anchored for various reasons at two points, A and B, distant from L (Figure 1.1).

Under the effect of the split exerted on the slip plane, the dislocation occurs. It adopts a curvature such that an arched element dl is in equilibrium under the effect of the force f = tb, and the line tension t is assumed to be constant regardless of the nature of the dislocation.

Figure 1.1. Source mechanisms. (a) Successive positions of a dislocation anchored at A and B and subjected to an increasing force