G.J. Dienes

Calculations of the energy and migration characteristics of carbon and nitrogen in α-iron and vanadium☆

Various properties of carbon and nitrogen in iron, and nitrogen in vanadium have been investigated with the aid of computer techniques. The metal-metal interaction, represented by a two-body central force which matches the elastic moduli, is sharply repulsive at close separation, and goes to zero midway between the second and third neighboring atoms. 531 atoms surrounding the defect were treated as individual particles, while the remainder of the crystal was treated as an elastic continuum with atoms imbedded in it. The iron-carbon interaction was represented by a cubic equation with the parameters chosen to yield the experimental value for the carbon migration energy, the activation volume (effectively zero), and the binding energy of a carbon atom to a vacancy. With this model the approximate experimental value of the energy of a carbon atom in solution in iron relative to Fe3C was obtained. The model also gives the correct relaxation strength for internal friction, but does not reproduce closely the formation volume. The migration path for carbon, on the basis of this calculation, is a straight line from octahedral to octahedral position with the tetrahedral position as the saddle point. The behavior of nitrogen in iron is almost identical. The same method of calculation for nitrogen in vanadium reproduced closely the sizable experimental activation volume for this system. Some calculations of di-carbon formation and motion are also reported.

Kinetics of order-disorder transformations

Abstract A simple theory of the kinetics of order-disorder transformations, based on chemical rate theory, is presented. The theory is constructed entirely in terms of the long range order parameter and reduces to the Bragg and Williams theory at equilibrium. There are several unusual features in the theoretical rate curves. The origin of these characteristics is the order parameter dependence of the energy of ordering. It is shown that ordering must start by fluctuations which are expected to be particularly large for an AB 3 alloy. Thus, the shapes of the ordering curves remind one of nucleation and growth processes, although they are derived entirely from simple rate theory. The theory is shown to be in agreement with the experimentally known qualitative behavior of order-disorder systems. For example, the theory predicts a maximum ordering rate slightly below the critical temperature in agreement with experiment. An improvement of the Bragg and Williams equilibrium theory is also suggested, based on a nonlinear dependence of the ordering energy on the long-range order parameter, which leads to excellent agreement with experiment. The kinetic theory is modified accordingly, resulting in quantitative but not qualitative changes in the predicted rate curves. There are insufficient experimental rate data for a critical quantitative evaluation of the kinetic theory.

Molecular dynamic simulations of crack propagation

Abstract Molecular dynamic simulations of crack propagation are reviewed with emphasis on the investigations of the last 5 yr. The major part of the review is based on the work of the authors and their colleagues and covers the following main topics: the role of the interatomic potential, the local stresses at the crack tip, dymamic simulations with emphasis on the crack propagation velocity, and some environmental effects on fracture.

A kinetic theory of nucleation in dilute solid solutions; Application to the precipitation of carbon and nitrogen in α-iron

Abstract A kinetic theory of nucleation and growth of precipitate particles in solid solutions is derived. This theory, which is applicable at low solute concentration and rather low solute-solute binding energy, is obtained by application of equilibrium conditions to the reversible chemical rate equations of cluster formation. The equilibrium approximation permits sufficient cancellation of terms in the equations so that the remaining equations are integrable. Computer solutions for a variety of parameters of the kinetic equations were used to check the range of validity of the approximation necessary to obtain the analytic expression. The analytic expression describes the experimentally observed time dependence of the precipitation of carbon and nitrogen in α-iron. The analytic theory also predicts an average binding energy of precipitate atoms in a cluster which is in reasonable agreement with experimentally derived carbon-carbon and nitrogen-nitrogen binding energies in iron. The size of the fundamental precipitate nucleus, as well as the distribution of sizes of precipitate particles, is also predicted, but no experiments exist to check these predictions.

Effect of atomic crack tip geometry on local stresses

Abstract The results of a molecular dynamic investigation, designed to explore the sensitivity of the crack tip stress distribution to changes in crack length and crack tip geometry (shape) on the atomic scale, are described. The simulation was carried out on a two-dimensional sample of 10704 atoms arranged in a triangular lattice and interacting with a Lennard-Jones (6–12) or a Johnson potential. Surprisingly, no significant difference in the stress distribution was observed between narrow (sharp) and wide (blunt) cracks. However, wide cracks with irregularly shaped tips (termed the wide jagged crack) displayed appreciably lower stresses in the near tip region. The magnitude of the maximum stress concentration (at the tip, i.e. at r = 0) was about a factor of ten lower than that predicted by classical continuum treatments of elliptically shaped cracks. For cracks of length 2a at distances r from the crack tip the results of the simulations in the 0.1 r a range are in good agreement with stresses predicted by the continuum elastic theory of elliptical cracks. The lack of variation in the stress distribution in going from narrow to wide cracks suggests that there is an effective radius of the crack tip controlling the magnitude of the stresses. The simulation results indicate that the discreteness of the lattice limits the degree of geometric blunting—at least within the size scale readily accessible with the present simulations. Conventional continuum analysis was found to account for the functional dependence of stress on crack length but not the magnitude of the stresses, whereas the nonlocal theory of Eringen was found to account for the order of magnitude of the stresses but not the functional dependence.

Lattice parameter and short-range order☆

Abstract A relation has been derived between the nearest-neighbor distance, r, the composition, x, and the shortrange order parameter, σ, for binary alloys based on nearest-neighbor interactions, additivity of bond energies and quadratic expansion of energy as a function of distance. The relation is shown to be applicable to random alloys of copper and gold and silver and gold. The relation describes the dependence of r on short-range order and the theoretical predictions are compared to experiments on Cu3Au. For this alloy the right direction and about the right magnitude of Δ r are predicted upon ordering. The relation between r and σ is, in general, nonlinear, but a linear approximation is quite satisfactory for Cu3Au.

Environmentally induced crack nucleation and brittle fracture

Brittle fracture is known to occur in ductile metals in special gaseous environments. Often, these gases form thin solid films on the metal. Computer simulations were used to study the propagation of cracks coated with thin elastically hard films in a ductile two-dimensional material. It was found that under certain conditions secondary crack nucleation occurred in the simulations and brittle fracture was observed. The dynamics of the crack nucleation and the detailed spatial distribution of local forces were examined. A model has been developed for this new mode of crack nucleation and brittle fracture. The role of the elastically hard film in suppressing dislocation generation and in initiating the secondary crack is discussed. This mode of initiating brittle fracture is entirely different from usual models for gaseous embrittlement.

Atomistics of crack propagation

Abstract The molecular dynamic technique is used to investigate static and dynamic aspects of crack extension. The material chosen for this study was the 2D triangular solid with atoms interacting via the Johnson potential. The 2D Johnson solid was chosen for this study since a sharp crack in this material remains stable against dislocation emission up to the critical Griffith load. This behavior allows for a meaningful comparison between the simulation results and continuum energy theorems for crack extension by appropriately defining an effective modulus which accounts for sample size effects and the non-linear elastic behavior of the Johnson solid. The simulation results for the energetics of quasi-static crack extension are in very good agreement with continuum predictions. During quasi-static crack extension under constant load boundary conditions the ratio of the work done by the external loads to the increase in elastic strain energy is ~ 1.94 which is close to the continuum prediction of 2.00 for a linear elastic solid. Good agreement was also obtained between the simulation results for the critical stress and the predictions of the Griffith criterion. Normalized crack velocity-crack length curves are presented for a variety of sample sizes and a variety of loading conditions. The measured terminal velocity was independent of sample size and loading condition and was 0.25 of the longitudinal sound velocity. The method of loading has some influence on acceleration of the crack to the terminal velocity. The details of the energy balance during dynamic crack extension are presented for various sample sizes. For the largest sample examined the crack reaches terminal velocity well before any elastic wave reflections occur at the sample boundary. Simulation results are presented for the stress fields of moving cracks and these dynamic results are discussed in terms of the dynamic crack propagation theories of Mott, Eshelby, and Freund.

Effect of mobile vacancy-impurity complexes on vacancy annealing kinetics

Abstract The kinetics of annealing of excess vacancies with mobile vacancy-impurity complex formation has been studied both analytically and with the use of a computer. The computer solutions indicate that an approximate equilibrium is maintained between free vacancies and vacancy-impurity complexes past an initial transient. When this equilibrium approximation is valid the defect concentration decreases exponentially with the annealing time. The effective rate constant of this exponential decay is a composite quantity involving the activation energies for migration of vacancies and complexes, the binding energy of the vacancy-impurity complex and the impurity concentration, which can all be determined by suitably controlled experiments. It is shown that the equilibrium approximation is a good one, for reasonable values of the energy parameters, when the impurity concentration is 5–50 times as large as the vacancy concentration. Expressions are also given for the transient and for the amount of impurities deposited at vacancy sinks.

Computer Modeling of Cracks

The theoretical techniques used in modeling cracks in crystalline lattices are reviewed. It is shown that there is generally a trade-off between sample size and realistic interatomic potentials. Infinite discrete one and two-dimensional lattices can be handled by the methods of lattice statics, but only with simple unrealistic potentials. In the hybrid lattice statics models a very small crystalline region, where the calculations are done with realistic potentials, is imbedded in an infinite elastic continuum. In this approach the boundary matching between the two regions is the difficulty. At the other end of the scale, molecular dynamic techniques can be used on an unconstrained system of a “large” number of atoms interacting with a reasonably realistic interatomic potential (this is the only way dynamic simulations have been done so far). Here, of course, the question is how “large” is large enough to simulate the behavior of the corresponding infinite system.