Volker Mohles

Simulations of dislocation glide in overaged precipitation-hardened crystals

Abstract Precipitation hardening of metals and alloys is investigated by means of computer simulations of dislocations gliding through a matrix with spherical coherent precipitates. The critical resolved shear stress is derived. Most realistic models are applied in these simulations: the radius distribution and the three-dimensional spatial arrangement of the particles are close to those of an actual Ostwald ripened crystal. Unlike the surrounding matrix, the precipitates are long-range ordered. A dislocation cutting through them generates an antiphase boundary and hence senses an obstacle stress inside the precipitates, which impedes dislocation glide. Such a strengthening mechanism is effective for instance in the nickel-based superalloy Nimonic PE16. The simulations are based on the local stress equilibrium along the dislocation line. The linear elastic interaction of the dislocations with themselves and with other dislocations is fully allowed for, similar to the approaches of Brown and of Bacon. Overaged crystals are considered in which dislocation glide is governed by the Orowan process. The average particle radius r and the particle volume fraction ƒ are varied. The strengthening contribution of these particles is derived as a function of the parameters r and f. This function is compared with a recent theoretical function suggested by Nembach; after adding a minor correction term to the latter function, it represents the present simulation results well.

Double strengthening of copper by dissolved gold-atoms and by incoherent SiO2-particles: how do the two strengthening contributions superimpose?

Abstract Copper single crystals have been simultaneously strengthened by solved gold-atoms and by incoherent SiO 2 -particles. The critical resolved shear stress (CRSS) τ s of a binary copper–gold solid solution and the CRSS τ t of the same solid solution additionally strengthened by SiO 2 -particles have been measured in the temperature range 90–283 K. From the experimentally established temperature dependences of τ s and τ t and from the theoretically known one of τ p (=CRSS if the SiO 2 -particles are the only strengtheners present), the function τ t ( τ s , τ p ) has been derived: τ t =( τ k s + τ k p ) 1/ k , with k ≈1.8. This result is at variance with a linear relationship suggested earlier by Ebeling and Ashby [Phil. Mag. 13 (1966) 805]. The bearings of the present findings on the evaluation of experimental data on dispersion strengthening are evident.

Orowan process controlled dislocation glide in materials containing incoherent particles

Dislocation glide in a matrix containing incoherent spherical particles is simulated on a computer. The critical resolved shear stress (CRSS) is derived. The simulations are based on the local stress equilibrium along the dislocation line. The stress terms taken into account are the external applied stress, a viscous drag stress, the obstacle stress and the dislocation self-stress. The latter stress term is similar to the approaches of Brown [Phil. Mag. 10 (1964) 441] and Bacon [Phys. Stat. Sol. 23 (1967) 527]; it replaces the line tension model and fully allows for the elastic interaction of a dislocation with itself and with other dislocations. The particles have a radii distribution and a three-dimensional spatial arrangement, both of which are close to those of an actual crystal. The CRSS is calculated as a function of the mean particle radius and of the particle volume fraction. This function is compared with a recent theoretical one. If a minor correction to this latter function is permitted, it represents the present simulation results well.

On the additivity of solid solution and dispersion strengthening

The superposition of solid solution and dispersion strengthening has been experimentally investigated for a copper-rich copper gold solid solution dispersion strengthened by incoherent SiO 2 -particles. On the basis of the different temperature dependences of r, [total critical resolved shear stress (CRSS) of the simultaneously solid solution and dispersion strengthened single crystals], of τ p [contribution of the SiO 2 -particles to τ l ], and of τ s [CRSS of the binary copper-gold matrix solid solution] the superposition law τ t (τ s . τ p ) has been derived: τ k t = τ k s + τ k p with k 1.8. This result contrasts with Ebeling and Ashbys (Phil. Mag. 13 (1966) 805) former one that k equals unity.

Dislocation dynamics modelling of dislocation–loop interactions in irradiated metals

The resistance dislocation loops provide to the glide of dislocations is an important element of several features of plasticity of irradiated metals. In the present work, a dislocation dynamics model based on the elasticity treatment of dislocations, with self-stress effects included, has been used to investigate the critical applied stress required for a gliding dislocation to overcome the stress field of a row of sessile loops near the glide plane. The critical stress has been determined for wide ranges of loop size, spacing and distance from the glide plane. Various approximations to the loop distribution have been tested, including an edge dipole, a single dislocation and a row of infinitesimal loops. The infinitesimal loop model, in particular, is shown to give good agreement with the accurate simulation data over a wide range of loop size and spacing. The ranges of applicability are discussed and compared with previous treatments in which such approximations have been used to investigate hardening due to dislocation–loop interactions.

Computer simulations of particle strengthening: lattice mismatch strengthening

Abstract Precipitation hardening is investigated by simulating the dislocation glide through obstacle fields and determining the critical resolved shear stress. In these simulations, the elastic interaction of the dislocations with themselves and with other dislocations is fully allowed for, similarly as has been done by Brown [1] and Bacon [2] . The obstacles considered are coherent spherical particles with a lattice mismatch (e.g. Cu-rich Cu Co-alloys). The respective obstacle stress is derived from Eshelbys elastic stress tensor of spherical inclusions. The distribution of the particle radii and their three dimensional spatial arrangement are the same ones as in an Ostwald ripened crystal. The mean particle radius r is varied over a wide range, covering both the underaged and the overaged region. Small precipitates are cut and large ones are circumvented by the Orowan process. The volume fraction f of the particles and the constrained lattice mismatch parameter e are varied as well. Thus, the critical resolved shear stress is obtained as a function of all three parameters r , f and e .

Computer simulations of the glide of dissociated dislocations in lattice mismatch strengthened materials

Abstract Computer simulations of dislocation glide in a matrix containing coherent spherical precipitates are presented. These particles have a lattice mismatch. Their strengthening contribution (critical resolved shear stress, CRSS) is derived. The size distribution and three dimensional spatial arrangement of the particles are close to those of an actual Ostwald-ripened crystal. An example for such a system is a Cu-rich Cu–Co alloy. The simulations are based on the local stress equilibrium along the dislocation line. The elastic dislocation self- interaction is fully allowed for. Perfect dislocations as well as pairs of Shockley partials are considered. The effect of dislocation dissociation on the CRSS is worked out for large and for small particles. Two particle arrangements of different lattice mismatch and volume fraction are considered. For the simulations of the dissociated dislocation two approaches are used. The first one is straightforward: a pair of partials enclosing a stacking fault is simulated. The second approach is a major simplification which significantly reduces the numerical effort. It is shown that the simplified model comprehends the basic features of dislocation dissociation.

Energy conserving orientational force for determining grain boundary mobility

Current experimental methods are not able to determine the mobility of flat grain boundaries across the large misorientation phase space. We find that the synthetic driving force method proposed to achieve this feat by simulation has a deficiency concerning numerical accuracy. We introduce a new synthetic driving force method by defining a new way to differentiate between crystal orientations. In contrast to the former method, this has the advantage that energy is correctly preserved during the simulation and is thus more reliable. This also results in a closer match of the applied energy difference to the thermodynamic free energy. This reduces the necessity of a post-simulation correction of the applied energy per atom to resulting driving pressure. We compare the newly proposed version to the old one for two grain boundaries and investigate the influence of simulation parameters on the resulting mobility values. For future simulations using a synthetic grain boundary driving force, we recommend using this newly proposed version over previous methods.

The Effect of Grain Boundary Junctions on Grain Microstructure Evolution: 3D Vertex Simulation

A 3D Vertex Model has been successfully implemented to investigate the evolution of a special grain assembly during grain growth. The model considers the mean curvature as driving force for the motion of the vertices and allows the consideration of all parameters affecting the motion of the system, i.e., grain boundary energy and line tension of the triple lines, as well as grain boundary (GB), triple line (TL) and quadruple point (QP) mobility as well. The used special configuration makes it possible to study the influence of all structural elements of a grain boundary network on the evolution of the system by allowing the steady-state motion of the boundaries of a shrinking grain. In the present work the different mobilities have been systematically varied and the evolution of the grain size with time has been studied as a function of TL and QP mobility. The results of the simulations are finally linked to the different kinetic regimes reached by the system.

Correlation between Supersaturation of Solid Solution and Mechanical Behaviour of Two Binary Al-Si-Alloys

Cooling of age-hardening Al-alloys after solution annealing is a critical step with respect to distortion and residual stresses. In order to predict their extent by simulation models, the mechanical behaviour must be known in a wide range of conditions and compositions. Therefore, experimental data is needed both for calibration and validation of the mechanical model. It is known for Al-Mg-Si alloys that supersaturation of the solid solution leads to a significant increase of strength during cooling. In order to understand the influence of single alloying elements on the strengthening effect, the mechanical properties of different binary alloys are investigated experimentally. The precipitation behaviour during cooling was investigated by Differential Scanning Calorimetry in a wide cooling rate range. A methodology to determine the degree of supersaturation of the solid solution based on the calorimetric results is presented. This approach is compared to atom probe tomography data. The mechanical behaviour of the alloys after various heat treatments was analysed by mechanical tests performed in a quenching and deformation dilatometer. Flow curves with high resolution at small strains (< 3 %) were measured at different temperatures. The results of the different experimental techniques are discussed in comparison and with respect to their testing limitations.