October 14, 2011 13:37 World Scientific Book - 9in x 6in master Chapter 1 Introduction 1.1 Plasticity Metals, and, to a lesser extent, most solid materials, can undergo a per- manent change of shape when submitted temporarily to external forces of sufficient magnitude. This mechanical property is called plasticity. It has been used since the beginning of the Bronze Age, in order to manufacture tools or weapons by turning pieces of metals into desired shapes. This was achieved with the help of bending or hammering forces, this action being made more efficient through heating the material to high temperatures. The present book aims at presenting the main ideas which constitute the microscopic physical explanation of the plastic behaviour of solids. The explanation will refer, almost exclusively, to the case of crystalline solids, i.e. solids which are built from a spatially periodic assembly of atoms. Two reasons justify this restriction. The main one is that the atomic-scale periodicity of crystals is an es- sential element of the theory which accounts for the plastic behaviour of solids. Indeed, this theory, developed progressively between 1920 and 1960, gives a central role to specific defects of crystals. These defects, which are called dislocations, are put in motion when suitable external forces are applied to a solid. This motion as well as the interaction of dislocations with different types of internal forces due to other defects, or to the atoms constituting the solid, are the basic ingredients used to analyze the various characteristics of the plastic behaviour. Another reason is that metals and alloys, which are the solids display- ing in the most spectacular way the plastic behaviour, are crystals. More precisely, a piece of a metal or alloy is, generally, an assembly of grains, each grain being a crystal of micronic size, whose individual properties de- 1 October 14, 2011 13:37 World Scientific Book - 9in x 6in master 2 Introduction termine, to a large extent, the mechanical properties of the metal or the alloy. In order to further clarify the object of the book let us recall some characteristics of the mechanical properties of solids. Plastic range σ F Yield stress A σA Hardening σ Y Y Ductility Effect of temperature Elastic range σ = F/S ε = δl/l ε O εY ε2 εF Figure 1.1 Schematic relationship between stress and strain in a solid material. OY is the elastic range and YF the plastic range. 1.1.1 Mechanical properties of solids Figure 1.1 summarizes the schematic mechanical behaviour of a solid ma- terial. It shows the complex relationship between the mechanical stress σ applied to a solid and the strain (also called deformation) ǫ resulting from this application. In a simplified approach, this relationship is deduced, for instance, from the measurement of the relative change of length ǫ = δl/l of a rod of section S submitted to a force F = σ.S pulling on the ends of the rod. Five aspects of this relationship can be emphasized. i) When the magnitude σ of the stress is smaller than a value σY , called the yield strength or alternately, the elastic limit, the mechanical behaviour of the solid is reversible. Thus, the value of the strain ǫ(σ) is the same for increasing values of the stress or for decreasing ones. In particular, if the value of the stress is brought back to zero, the induced strain vanishes. This behaviour corresponds to the elastic range labelled OY of the curve plotted on Fig. 1.1. In this range, one can generally consider, to a good approximation, that the strain is proportional to the stress: σ Cǫ. ≃ The C coefficient of proportionality between the strain and the stress, is a measure of the elastic stiffness of the considered solid material. It is an October 14, 2011 13:37 World Scientific Book - 9in x 6in master 1.1. Plasticity 3 important characteristics of its mechanical properties. For metals, the value of the stiffness lies in the range 1010-1011Pascals (∼1-10 tons per square millimeter). Such a value means that a weight of one kilogram suspended to a wire of section one square millimeter will determine a relative elongation 5 6 δl/l of the wire of 10− 10− . − As for the the yield strength σY , it characterizes the hardness of a solid material. Its value does not only depend of the nature of the material, but also, in a pronounced manner, of the mechanical and thermal processing imposed to it, as well as of other parameters such as, for instance its tem- 6 2 perature. In a “soft” material, σY can be less than 10 Pascals (100g/mm ), whereas in a “hard” one it can be higher than 109 Pascals (100 kg/mm2). A current value of the strain ǫY corresponding to the yield strength is ∼1%. ii) For σ > σY , the material becomes plastic. In this range of σ val- ues (YF on the plot) the deformation becomes non-reversible. Bringing the value of the stress down to zero does not cancel the strain. For in- stance, starting from point A, the strain decreases along the line Aǫ2, and a permanent strain ǫ2 remains at zero applied stress. iii) The application of a moderate stress to the material initially in the state (ǫ = ǫ2,σ = 0), determines, again, a reversible evolution along ǫ2A 1 with a quasi-linear relationship σ C′(ǫ ǫ ) With a further increase ≃ − 2 of the stress, a new elastic limit σA is reached. The schematic situation represented on Fig. 1.1, in which YA has a positive slope, characterizes a material in which the elastic limit σA after deformation is larger than σY . The material is hardened by the plastic deformation. iv) The end point F , corresponds to the fracture (or rupture) of the solid-sample, i.e. the loss of its cohesion. It is reached after a permanent elongation ǫF . In certain metals, one can elongate a rod to one hundred times its initial length before fracture occurs. Such a metal is very ductile. The ductility of a material is associated to the value of the deformation ǫF . The larger ǫF , the larger the ductility. When almost no ductility exists the material is brittle. In this case, the fracture occurs just above the yield strength (ǫY,σY ) and F then almost coincides with Y . Similarly to the yield strength, ductility is a function of the “mechanical history” of the sample considered, as well as of its temperature. 1 On the plot, the rigidity C′ has been represented as equal to C: the line ǫ2A is parallel to OY . This corresponds to the simplest type of behaviour in which the rigidity is approximately the same in all the reversible ranges. This behaviour will be justified in chapter 8 by the fact that the perfection of the crystalline order is almost preserved in the volume of the material. October 14, 2011 13:37 World Scientific Book - 9in x 6in master 4 Introduction v) Temperature modifies significantly all the mechanical properties of a material. In general, heating a solid makes easier its plastic deformation: it decreases the value of the yield strength (ǫY , σY ), and it increases the ductility. It also decreases the hardening (the slope of YA). The table below provides a few examples of values measured in currently used solid materials at room temperature. Material longitudinal stiffness (1010Pa) yield strength (107 Pa) Lead 5 ∼0,01 Aluminum 11 ∼0,1 Copper 17 ∼0,1 (1.1) Gold 19 ∼0,1 Iron 24 1-10 Silicon 17 10-100 Diamond 100 >100 1.1.2 Microscopic mechanisms In the above macroscopic description (cf. Fig. 1.1), the elastic and plas- tic deformations of a solid appear as two successive steps of a continuous process. Actually, these sequences have a very different microscopic back- ground. To grasp their difference of nature, it is worth giving, prior to a description of the plastic behaviour, some indications on the microscopic origin of the elastic behaviour. Elastic behaviour The interpretation of the elastic behaviour of a solid, from the standpoint of its atomic structure, derives from a simple principle. Thus, the assembly of atoms constituting a solid possesses an equilibrium spatial configuration, in which the interatomic distances are well defined.2 In the absence of external forces applied to the solid, any deviation of atomic configuration (Fig. 1.2), with respect to the equilibrium one, will increase the total energy of the solid, and has therefore a tendency to regress. Such a deviation, consisting in small changes in the distances between atoms, will actually 2At very low temperatures this configuration corresponds to the minimum of the energy of the set of particles composing the solid. October 14, 2011 13:37 World Scientific Book - 9in x 6in master 1.1. Plasticity 5 be induced by the macroscopic deformation, imposed to the solid by an external stress. When the stress is suppressed, the atomic configuration will tend to return to equilibrium, thus accounting for the reversibility of the behaviour. Moreover, as the deviations provoked by strains are very small (less than a percent) the effective forces acting on the atoms to bring their distances back to their equilibrium values, will therefore be appproximately proportional to the small deviations, thus justifying the linearity of the elastic behaviour. Figure 1.2 Schematic representation of the atomic displacements induced by a macro- scopic shear. There are changes of the interatomic distances and of the angles between chemical bonds. Restoring forces are generated which tend to bring back the distances and angles to their initial equilibrium values.