DEFORMATION MECHANISMS AND DEFORMATIONMECHANISM MAPS
1.1 Atomic Processes and Deformation Mechanisms
1.2 RateEquations
1.3 DeformationMechanism Maps
1.4 A Warning
CRYSTALLINE Solids deform plastically by a number of alternative, often competing, mechanisms. This book describes the mechanisms, and the construction of maps which show the field of stress, temperature and strainrate over which each is dominant. It contains maps for more than 40 pure metals, alloys and ceramics. They are constructed from experimental data, fitted to modelbased rateequations which describe the mechanisms. Throughout, we have assumed that fracture is suppressed, if necessary, by applying a sufficiently large hydrostatic confining pressure.
The first part of the book (Chapters 13) describes deformation mechanisms and the construction of deformationmechanism maps. The second part (Chapters 416) presents, with extensive documentation, maps for pure metals, ferrous and nonferrous alloys, covalent elements, alkali halides, carbides, and a large number of oxides. The final section (Chapters 1719) describes further developments (including transient behavior, the influence of pressure, behavior at very low and very high strain rates) and the problem of scaling laws; and it illustrates the use of the maps by a number of simple case studies.
The catalogue of maps given here is, inevitably, incomplete. But the division of materials into isomechanical groups (Chapter 18) helps to give information about materials not analyzed here. And the method of constructing maps (Chapter 3) is now a wellestablished one which the reader may wish to apply to new materials for himself
Plastic flow is a kinetic process. Although it is often convenient to think of a polycrystalline solid as having a well defined yield strength, below which it does not flow and above which flow is rapid, this is true only at absolute zero. In general, the strength of the solid depends on both strain and strainrate, and on temperature. It is determined by the kinetics of the processes occurring on the atomic scale: the glidemotion of dislocation lines; their coupled glide and climb; the diffusive flow of individual atoms; the relative displacement of grains by grain boundary sliding (involving diffusion and defectmotion in the boundaries); mechanical twinning (by the motion of twinning dislocations) and so forth. These are the underlying atomistic processes which cause flow. But it is more convenient to describe polycrystal plasticity in terms of the mechanisms to which the atomistic processes contribute. We therefore consider the following deformation mechanisms, divided into five groups.
The mechanisms may superimpose in complicated ways. Certain other mechanisms (such as superplastic flow) appear to be examples of such combinations.
Plastic flow of fullydense solids is caused by the shearing, or deviatoric part of the stress field, σ_{s}. In terms of the principal stresses σ_{1}, σ_{2} and σ_{3}:
_{} 
(1.1) 
or in terms of the stress tensor σ_{ij}:
_{} 
(1.2) 
where 
_{ } 
(Very large hydrostatic pressures influence plastic flow by changing the material properties in the way described in Chapter 17, Section 17.4, but the flow is still driven by the shear stress σ_{s}.)
This shear stress exerts forces on the defects—the dislocations, vacancies, etc.—in the solid, causing them to move. The defects are the carriers of deformation, much as an electron or an ion is a carrier of charge. Just as the electric current depends on the density and velocity of the charge carriers, the shear strainrate, _{ }, reflects the density and velocity of deformation carriers. In terms of the principal strainrates _{ }, _{ } and _{ }, this shear strainrate is:

(l .3) 
or, in terms of the strainrate tensor _{ }:

(1.4) 
For simple tension, σ_{s} and _{ } are related to the tensile stress σ_{1} and strainrate _{ } by:

(1.5) 
The macroscopic variables of plastic deformation are the stress σ_{s}, temperature T, strainrate _{ } and the strain γ_{ }or time t. If stress and temperature are prescribed (the independent variables), then the consequent strainrate and strain, typically, have the forms shown in Fig. l.la. At low temperatures ( ~ 0.1 T_{M}_{ } , where T_{M} is the melting point) the material workhardens until the flow strength just equals the applied stress. In doing so, its structure changes: the dislocation density (a microscopic, or state variable) increases, obstructing further dislocation motion and the strainrate falls to zero, and the strain tends asymptotically to a fixed value. If, instead, T and _{ } are prescribed (Fig. 1.1b), the stress rises as the dislocation density rises. But for a given set of values of this and the other state variables S_{i} (dislocation density and arrangement, cell size, grain size, precipitate size and spacing, and so forth) the strength is determined by T and _{ }, or (alternatively), the strainrate is determined by σ_{s} and T.
At higher temperatures (~ 0.5T_{M}) , polycrystalline solids creep (Fig. 1.1, centre). After a transient during which the state variables change, a steady state may be reached in which the solid continues to deform with no further significant change in S_{i}. Their values depend on the stress, temperature and strainrate, and a relationship then exists between these three macroscopic variables.
Fig. 1.1. The way in which σ_{s} ,T, _{ } and γ are related for materials (a) when σ_{s} and T are prescribed and (b) when _{ } and T are prescribed, for low temperatures (top), high temperatures (middle) and very high temperatures (bottom).
At very high temperatures (~ 0.9T_{M}) the state variables, instead of tending to steady values, may oscillate (because of dynamic recrystallization, for instance: Fig. 1.1, bottom). Often, they oscillate about more or less steady values; then it is possible to define a quasisteady state, and once more, stress, temperature and strainrate are (approximately) related.
Obviously, either stress or strainrate can be treated as the independent variable. In many engineering applications—pressure vessels, for instance—loads (and thus stresses) are prescribed; in others—metalworking operations, for example—it is the strainrate which is given. To simplify the following discussion, we shall choose the strainrate _{ }as the independent variable. Then each mechanism of deformation can be described by a rate equation which relates _{ } to the stress σ_{s}, the temperature T, and to the structure of the material at that instant:

(1.6) 
As already stated, the set of i quantities S_{i} are the state variables which describe the current microstructural state of the materials. The set of j quantities P_{j} are the material properties:: lattice parameter, atomic volume, bond energies, moduli, diffusion constants, etc.; these can be regarded as constant except when the plastic properties of different materials are to be compared (Chapter 18).
The state variables S_{i} generally change as deformation progresses. A second set of equations describes their rate of change, one for each state variable:
_{} 
(1.7) 
where t is time.
The individual components of strainrate are recovered from eqn. (1.6) by using the associated flow rule:

(1.8) 
or, in terms of the stress and strainrate tensors:

(1.9) 
where C is a constant.
The coupled set of equations (1.6) and (1.7) are the constitutive law for a mechanism. They can be integrated over time to give the strain after any loading history. But although we have satisfactory models for the rateequation (eqn. (1.6)) we do not, at present, understand the evolution of structure with strain or time sufficiently well to formulate expressions for the others (those for dS_{i}/dt). To proceed further, we must make simplifying assumptions about the structure.
Two alternative assumptions are used here. The first, and simplest, is the assumption of constant structure:
_{} 
(1.10) 
Then the rateequation for _{ }completely describes plasticity. The alternative assumption is that of steady state:
_{} 
(1.11) 
Then the internal variables (dislocation density and arrangement, grain size, etc.) no longer appear explicitly in the rateequations because they are determined by the external variables of stress and temperature. Using eqn. (1.7) we can solve for S_{1}, S_{2}, etc., in terms of _{ }σ_{s} and T, again obtaining an explicit rateequation for _{ }.
Either simplification reduces the constitutive law to a single equation:

(1.12) 
since, for a given material, the properties P_{j} are constant and the state variables are either constant or determined by σ_{s} and T. In Chapter 2 we assemble constitutive laws, in the form of eqn. (1.12), for each of the mechanisms of deformation. At low temperatures a steady state is rarely achieved, so for the dislocationglide mechanisms we have used a constant structure formulation: the equations describe flow at a given structure and state of workhardening. But at high temperatures, deforming materials quickly approach a steady state, and the equations we have used are appropriate for this steady behavior. Nonsteady or transient behavior is discussed in Chapter 17, Section 17.1; and ways of normalizing the constitutive laws to include change in the material properties P_{j} are discussed in Chapter 18.
It is useful to have a way of summarizing, for a given polycrystalline solid, information about the range of dominance of each of the mechanisms of plasticity, and the rates of flow they produce. One way of doing this (Ashby, 1972; Frost and Ashby, 1973; Frost, 1974) [13] is shown in Fig. 1.2. It is a diagram with axes of normalized stress σ_{s}/μ and temperature, T/T_{M} (where µ is the shear modulus and T_{M} the melting temperature). It is divided into fields which show the regions of stress and temperature over which each of the deformation mechanisms is dominant. Superimposed on the fields are contours of constant strainrate: these show the net strainrate (due to an appropriate superposition of all the mechanisms) that a given combination of stress and temperature will produce. The map displays the relationship between the three macroscopic variables: stress as, temperature T and strainrate _{ }. If any pair of these variables are specified, the map can be used to determine the third.
There are, of course, other ways of presenting the same information. One is shown in Fig. 1.3: the axes are shear strainrate and (normalized) shear stress; the contours are those of temperature. Maps like these are particularly useful in fitting isothermal data to the rateequations, but because they do not extend to 0 K they contain less information than the first kind of map.
A third type of map is obviously possible: one with axes of strain rate and temperature (or reciprocal temperature) with contours of constant stress (Figs. 1.4 and 1.5). We have used such plots as a way of fitting constantstress data to the rateequations of Chapter 2, and for examining behavior at very high strainrates (Chapter 17, Section 17.2).
Fig. 1.2. A stress/temperature map for nominally pure nickel with a grain size of 0.1 mm. The equations and data used to construct it are described in Chapters 2 and 4.
Fig. 1.3. A strainrate/stress map for nominally pure nickel, using the same data as Fig. 1.2.
Fig. 1.4. A strainrate/temperature map for nominally pure nickel, using the same data as Fig. 1.2.
Fig. 1.5. A strainrate/reciprocal temperature map for nominally pure nickel, using the same data as Fig. 1.2.
Finally, it is possible to present maps with a structure parameter (S_{1},_{ }S_{2}, etc.) such as dislocation density or grain size as one of the axes (see, for example, Mohamed and Langdon, 1974) [4]. Occasionally this is useful, but in general it is best to avoid the use of such microscopic structure variables as axes of maps because they cannot be externally controlled or easily or accurately measured. It is usually better to construct maps either for given, fixed values of these parameters, or for values determined by the assumption of a steady state.
Fig. 1.6. A threedimensional map for nominally pure nickel, using the same data as Figs. 1.2, 1.3 and 1.4.
Three of the maps shown above are orthogonal sections through the same threedimensional space, shown in Fig. 1.6. In general, we have not found such figures useful, and throughout the rest of this book we restrict ourselves to twodimensional maps of the kind shown in Figs. 1.2, 1.3 and, occasionally, 1.4.
One must be careful not to attribute too much precision to the diagrams. Although they are the best we can do at present, they are far from perfect or complete. Both the equations in the following sections, and the maps constructed from them, must be regarded as a first approximation only. The maps are no better (and no worse) than the equations and data used to construct them.