U.F. Kocks

Kinetics of flow and strain-hardening☆

Abstract The kinetics of glide at constant structure and the kinetics of structure evolution are correlated on the basis of various experimental observations in pure f.c.c. mono- and polycrystals. Two regimes of behavior are identified. In the initial regime, the Cottrell-Stokes law is satisfied, hardening is athermal, and a single structure parameter is adequate. With increasing importance of dynamic recovery, be it at large strains or at high temperatures, all of these simple assumptions break down. However, the proportionality between the flow stress and the square-root of the dislocation density holds, to a good approximation, over the entire regime; mild deviations arc primarily ascribed to differences between the various experimental techniques used. A phenomenological model is proposed, which incorporates the rate of dynamic recovery into the flow kinetics. It has been successful in matching many experimental data quantitatively.

Laws for Work-Hardening and Low-Temperature Creep

The true stress-strain curves of polycrystalline aluminum, copper, and stainless steel are shown to be adequately represented by an exponential approach to a saturation stress over a significant range. This empirical law, which was first proposed by Voce, is expanded to describe the temperature and strain-rate dependence, and is put on a physical foundation in the framework of dislocation storage and dynamic recovery rates. The formalism can be applied to the steady-state limit of creep in the same range of temperatures and strain rates; the stress exponent of the creep rate must, as a consequence, be strongly temperature dependent, the activation energy weakly stress dependent. Near half the melting temperature, where available work-hardening data and available creep data overlap, they match. Extrapolation of the proposed law to higher temperatures suggests that no new mechanisms may be necessary to describe high-temperature creep. A new differential equation for transient creep also follows from the empirical work-hardening law.

A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable

Abstract The axisymmetric deformation behavior of 0.9999 Cu is investigated at strain rates from 10−4 to 104 s−1. The variations of the flow stress and of the mechanical threshold stress (the flow stress at 0 K), which is used as an internal state variable, with strain rate and strain are measured. The experimental results are analyzed using a model proposed by Kocks and Mecking: results at constant structure are described with thermal activation theory; structure evolution (strain and strain rate evolution of the mechanical threshold stress) is treated by the sum of dislocation generation and dynamic recovery processes. A significant result is that the athermal dislocation accumulation rate, or Stage II hardening rate, becomes a strong function of strain rate at strain rates exceeding 103 s−1. This leads to the apparent increased strain rate sensitivity seen in a plot of flow stress at a given strain vs the logarithm of strain rate. An explanation is proposed for the strain rate dependence of this initial strain hardening rate based on the limiting dislocation velocity and average distance between obstacles.

The relation between polycrystal deformation and single-crystal deformation

A phenomenological description of crystallographic slip and pencil glide in single crystals is outlined, with emphasis on the behavior under prescribed strains. Theoretical relations are established between these single-crystal properties and the behavior of quasi-homogeneous, quasi-isotropic polycrystals deforming uniformly on a macroscopic scale, at subdiffusive temperatures. Experimental comparisons between single crystals and polycrystals are reviewed, considering flow stress, work hardening, temperature and strain rate effects, and various effects of grain size.

A statistical theory of flow stress and work-hardening

Abstract The consequences of a random distribution of point-like obstacles to the movement of dislocations are examined. Firstly, it is proven that a critical applied stress exists; it is determined by a critical value of the fraction of passable obstacles such that the average dislocation segment can keep moving. There is thus no limit to the expandability of a convex dislocation loop; however, the moving dislocation leaves debris behind in the form of concave loops around areas surrounded entirely by impassable obstacles. The rate of this primary dislocation storage with strain, if one considers monopoles only, is very low and again depends on the critical fraction of passable obstacles only, not on any properties of the distribution as long as it is non-singular. The absence of large-scale regularity in the obstacle structure alone (whatever the nature of the obstacles may be) can thus explain “stage II” hardening which is characterized by a value of about 1/300 of the shear modulus, independent of all...

New observations on the mechanisms of dynamic strain aging and of jerky flow

Abstract The rate sensitivity of the flow stress was investigated as a function of pre-strain for a commercial nickel alloy (INCONEL 600) and for polycrystalline Al-1% Mg. over a wide temperature range encompassing the regime of jerky flow. Anomalies were observed which suggest that the dynamic strain aging mechanism operates over a wider temperature range than that in which jerky flow is found. A detailed analysis shows that a negative rate sensitivity is associated with the strain hardening contribution to the flow stress from the beginning of deformation. The total rate sensitivity becomes negative only after this strain hardening contribution dominates. This suggests an interpretation of the ‘critical strain’ for the appearance of jerky flow, which does not rely on the production of vacancies and moreover casts doubt on any interpretation that associates dynamic strain aging with the interaction of mobile solutes with the moving dislocations. Instead, a dislocation arrest model is used in which solute atoms diffuse along dislocations at forest intersections.

A model for texture development dominated by deformation twinning: Application to zirconium alloys

Abstract In many polycrystals of less than cubic crystal symmetry, plastic deformation is dominated by twinning. In particular, we will treat the case of Zr and Zr alloys in detail. We propose a new method for modelling grain reorientation due to twinning, which is based on a Volume Fraction Transfer (VFT) scheme; the scheme is also applied to the slip modes. We find that this method predicts textures that are, when twinning is the dominant mode, considerably different from and in better agreement with experiment than the conventional schemes which reorient an entire grain when some criterion has been met. Various combinations of slip and twinning modes and of the associated critical stresses are systematically investigated for the case of rolling, tension and compression of Zr alloys. A comparison of various predicted and experimental textures leads to the conclusion that twinning must, indeed, be controlling texture development.

The effect of dislocation self-interaction on the Orowan stress

Abstract The effect of dislocation self-interaction on the Orowan stress was determined for collinear, impenetrable circular obstacles. The line tension approximation was dropped and an appropriate integro-differential equation governing static equilibrium was solved by an algorithm based on curvature adjustments. When the interactions are taken into account, the Orowan stress varies as In where is the harmonic mean of the obstacle size and spacing. In addition, the configuration of a bowing dislocation at the Orowan stress is strongly influenced by the interactions: as the interactions increase in strength, i.e. as the ratio of obstacle size to spacing decreases, the bow-out and area swept are suppressed. The essential features of the interaction can be incorporated into a line tension framework by treating the impenetrable obstacles as if they were penetrable. Based on this line tension analogue, an approximate form for the flow stress of a random distribution of impenetrable obstacles was developed.

Operational texture analysis

Abstract A comprehensive overview is presented of the elements that enter quantitative techniques of texture measurement, analysis and representations. There are many potential errors to be corrected and many choices to be made in all these stages, and we present those that we consider most appropriate. They have been implemented in a software package that is available publicly. A number of novel techniques are used including, for example, in the representation of measured textures, both for quantitative visual inspection and for use in the prediction of anisotropic properties. The symmetry of the test sample is allowed to be general, and that of the crystal structure may be as low as trigonal.

Theory of torsion texture development

Abstract A method previously proposed which takes into account the influence of grain shape on the slip geometry of polycrystal deformation, is applied here to the large strain torsion of f.c.c. metals, and in particular to the prediction of texture. The results of the relaxed constraints (RC) theory on the one hand and of the full constraints (FC) classical theory (Taylor/Bishop and Hill) on the other are compared with those of experiments. Both the RC and FC theories describe the texture development well up to shear strains of about 3. At larger strains, however, the RC results correspond far better to the experimental textures, which tend towards a unique end orientation. It is also shown that the average Taylor factor decreases until γ = 3 and then increases towards an ultimate value of √3.