R. N. Stevens

The effect of volume fraction of precipitate on ostwald ripening

Abstract The classical theory of the coarsening of precipitate particles by competitive growth (Ostwald ripening) is extended to include the effect of encounters between growing particles using the method outlined by Lifshitz and Slyosov [J. phys. Chem. Solids19, 35 (1961)]. In the modified theory the rate of change of the mean particle size is still proportional to the cube root of the time but the rate constant is a function of volume fraction and so is the relative particle size distribution. The effect of encounters is to increase the growth rate although the effect is not great; the rate constant varying by a factor of approximately three over the whole volume fraction range. This is in broad agreement with current experimental evidence but contradicts alternative modifications of the classical theory. The relative particle size distributions are greatly altered in the modified theory, becoming flattened, more symmetrical and having a much broader range of particle sizes than the unmodified theory. The broadening of the relative particle size distribution is also in good accord with experiment and is rather greater than predicted by alternative modifications.

Precipitation in Ni-Co-Al alloys

Etude de la precipitation entre 673 et 973 K par microscopie optique et microscopie electronique en transmission

The chemical stress applied to creep and fracture theories—II. Application to the growth of sub-critical Griffith cracks

Abstract The formalism, developed in an accompanying paper, concerning the thermodynainic driving force for diffusion, is applied here to the specific case of a Griffith-type crack growing by diffusion of vacancies from the lattice to the crack tip. The pertinent literature is reviewed and it is shown how an expression can be derived for the boundary value of the chemical driving force at the crack tip. This is used to derive the flux of vacancies to the crack tip and hence the crack tip velocity. The effect of the elastic interaction of the vacancy with the stress gradient present at the crack tip is also included in the calculation. The first-order size effect and second-order inhomogeneity interaction is considered. Although these interactions do not affect the thermodynamic stability of a crack, they do modify the kinetics of crack growth through drift terms. A numerical calculation for alumina indicates that, for short cracks, these drift terms cause the crack velocity to be retarded by about 5 per cent while for large cracks, the velocity is enhanced by about 25 per cent. The magnitude of such effects is considered to be relatively minor.

The effect of stacking fault energy on the plastic deformation of polycrystalline Ni-Co alloys

The deformation characteristics of a series of polycrystalline Ni-Co alloys have been studied by measuring activation, volume flow stress, Cottrell-Stokes ratio (CSR) and work hardening rate in the temperature range 4.2 → 480 K. The alloys contain 0–70 per cent cobalt and have a range of stacking fault energy (SFE) from 240 → 10 mJm2. The relative contributions of short range (τs + τi) and long range stresses (τL) to the total flow stress (τ) have been estimated. In the initial stages of a stress-strain curve (e < 0.06) short range stresses determine the flow stress; whereas at large strains the long range stresses are the dominant part. The CSR is obeyed approximately only at large strains. The magnitude of τL is determined by the ease with which cross-slip takes place, being easier at higher temperatures and for higher SFE alloys. The degree of cross-slip determines the rate of work hardening.

The chemical stress applied to creep and fracture theories—I. A general approach

Abstract The theory of diffusion is usually formulated in terms of the chemical potential of the diffusing species. In crystalline systems fluxes of defects of various kinds will accompany the flux of the diffusing species and the free energy changes of the system will depend on their kind and number. The free energy changes accompanying diffusion by various defect mechanisms is studied in some detail. It is concluded that the potential driving diffusion is not the chemical potential defined by macroscopic thermodynamics, but that it depends on the chemical potentials of defects. However, given reasonable assumptions, the potential is independent of the defect mechanism of diffusion. Expressions for this potential are given for the case of both single and multi-component systems, and a comparison is made with alternative treatments. Consideration is given to the effect of non-hydrostatic stresses on the chemical potentials and it is concluded that the theory holds for this case also. The boundary values of the potential driving diffusion are considered. A simple but completely rigorous method of deriving the boundary values is given, and it is shown that although the potential driving diffusion is not the classical chemical potential, the boundary values of the driving force are equal to the local macroscopic chemical potential for atoms providing that local equilibrium can be maintained. The results of this study have particular relevance to the problems of diffusion creep, sintering, dislocation climb and high temperature fracture. In an accompanying paper, the above formalism is applied to the case of fracture resulting from the diffusional growth of Griffith-type cracks.

Physical interpretation of fracture-toughening mechanisms

The basic idea behind the toughening of materials by the introduction of energy-absorbing or dissipating artefacts is critically re-examined. It is shown that energy dissipation by plastic deformation or other dissipative processes at the tip of a growing crack does not contribute to increasing the effective surface energy or the crack resistance of the material. Erroneous interpretations of toughening by the presence of fibres or by phase transformations occuring at the tip of a growing crack are discussed. It is argued that all processes which dissipate energy at the crack tip produce crack shielding and that this effect must be an important contribution to toughening. It is concluded that most of the features and properties embodied in methods of toughening can be explained by shielding effects and that the increase in toughness is due to a reduction in the local value of the crack extension force, or its equivalent stress intensity factor, and not to an increase in energy dissipated.

Self-consistent forms of the chemical rate theory of Ostwald ripening

The chemical rate theory of Ostwald ripening introduced by A. D. Brailsford and P. Wynblatt (Act. Metall.27 (1979) 498) determines the mean growth rate of particles of a particular size class by solving the diffusion equations for a representative particle (radius r) surrounded by a shell of matrix (the averaging sphere, radius rA) outside which there is a homogeneous effective medium averaging the emission and absorption of solute atoms by the remainder of the particles. Brailsford and Wynblatt set r = rA, in effect removing the matrix shell. It is argued herein that the feature of the theory so omitted is a very important one and we therefore use it to develop and extend the theory to make it self-consistent in the sense that the mean ratio of the particle and averaging sphere volumes is equal to the volume fraction of particles. Three self-consistent versions are developed, two of which have rA relatively constant for small particles and slowly increasing for particles greater than approximately average size. These were motivated by the observation from numerical simulations that small particles are little influenced by their neighbours whereas larger particles are much more strongly affected by the environment. Analytical expressions in terms of experimentally observable variables are given for the probability distributions for particle sizes, and tables of the parameters required to evaluate the distribution functions as a function of volume fraction are provided. It is concluded that the properties of the Brailsford and Wynblatt effective medium are closely reproduced by the alternative analytical theories, but that the idea of a matrix shell round the representative particle is unique to the chemical rate theory. It is argued that this feature makes the theory flexible and adaptable. This adaptability could be used to reproduce the results of sophisticated numerical simulations in a form which would be computationally efficient to include in wider simulations involving, say, the effect of particle growth on long term mechanical properties.

Dislocation configurations and internal stresses in the creep of Nimonic 91

The internal stress, σi, developed during the creep of Nimonic 91 was determined as a function of applied stress, σa, using the strain transient dip technique. Transmission electron microscope observations of thin films of crept specimens showed Orowan dislocation loops to exist aroundγ′ phase particles at low stresses with partial dislocation loops around faultedγ′ particles at high stresses. The numbers of loops per specimen volume were counted and the resulting internal stress calculated. The results indicate that a significant part of the mechanically measured internal stress can be attributed to Orowan loops aroundγ′ particles which are stabilized against climb by the superlattice fault resulting from partial penetration of theγ′ particle by the dislocation. The variation of internal stress with applied stress can be accounted for qualitatively by the variation of loop density with stress at low stresses and the initiation of relaxation processes involving partial or complete shearing ofγ′ particles by loops at high stresses. It is suggested that creep in Nimonic 91 is dependent on the magnitude of the effective (σa-σi) and that the internal stress is determined largely by the density of dislocation loops aroundγ′ particles.

Internal stress and unloading experiments in creep

Internal stresses are developed during deformation and have an important role in determining the mechanical properties and, in particular, the creep properties of crystalline materials. The strain transient dip test is the generally accepted method for the determination of internal stresses developed during creep. The strain transient dip test has been analysed using a number of very general creep models and it is concluded that, for glide-controlled creep, the dip test can only be interpreted if the relation between dislocation velocity and the force on the dislocation is linear. When this is the case it measures not an average internal stress but an average back stress for all the dislocations, mobile and immobile, where the back stress is the resolved component of the internal stress plus the glide component of the line tension force divided by the Burgers vector. The dip test does not allow separation of the back stress into internal stress and line tension components. For recovery models the results of the dip test cannot be simply interpreted because expressions for the creep rate do not define a unique average internal stress or back stress. However, for the recovery model in which strain occurs by athermal or jerky glide there will be a reverse yield stress, i.e. there will be a stress reduction below which there will be “instantaneous” reverse strain followed by reverse creep. By averaging the instability condition for all the dislocations participating in jerky glide it is shown, subject to assumptions, that the sum of the average internal stress experienced by dislocations involved in both forward and reverse creep can be obtained from the reverse yield stress. Separate values for these internal stresses cannot be obtained, however. Determination of the reverse yield stress for recovery creep is the experiment equivalent to the strain transient dip test for glide-controlled creep.

High-temperature creep of the particlehardened commercial Al-Li-Cu-Mg alloy 8090

Creep of the particle-hardened commercial Al-Li 8090 alloy has been studied at temperatures of 425 and 445 K. The measured stress sensitivity of the minimum creep rates changes abruptly at a given applied stress with stress exponents being around 4–6 at low stresses and 30–40 at high stresses. Creep activation enthalpies were determined by both temperature cycling and by comparing creep rates at two temperatures at a given applied stress, the results from both gave the same unrealistically high values. The internal stresses, σi, developed during creep were determined using the strain-transient dip test. These increased linearly with the applied stress, σa, at low stresses and were effectively constant at high stresses. The minimum creep rate was found to be a simple function of the effective stress, σa-σi, with a stress exponent of between 5 and 6, at all applied stresses. The dislocation and precipitate structure of the alloy was examined before and after creep using thin-film electron microscopy. The initial structure consisted of pancake grains with a well-developed {1 1 0}〈1 1 2〉 type texture. The grains contained well-developed sub-cells and δ′ and S precipitates. The structure developed during creep consisted of dislocation pairs, single dislocations and dislocations loops. There was evidence to suggest that slip took place on both {1 0 0} and {1 1 1} planes. The dislocation loops were most likely to have been Orowan in character and around the rodlike S precipitate, with the coherent δ′ precipitate being sheared by pairs of dislocations. The measured internal stresses result from inhomogeneity of plastic deformation. These stresses increase continuously with applied stress up to the observed macroscopic yield stress, and then become constant. The internal stresses are likely to have arisen from the Orowan loops around S and the behaviour of sub-grain boundaries. The increases in internal stress may have resulted from an increased loop density with increasing applied stress. This rate of increase is likely to slow down if S particles are sheared or fractured at high applied stresses.