J.A. Spittle

A review of measurement techniques for the thermal expansion coefficient of metals and alloys at elevated temperatures

Metallurgical operations at elevated temperatures, such as those that involve solidification and/or mechanical deformation, can be critically influenced by the thermal stresses and strains that result from expansion and contraction of the material as a function of temperature. With the increasing use of computer-based process models for these operations, there arises a greater need for quantitative data on the thermal expansion coefficient of the relevant alloy at the temperatures involved. After briefly reviewing some existing sources of data for this property, the various techniques for its measurement at elevated temperatures are then described. These include mechanical dilatometry, optical imaging and interference systems, x-ray diffraction methods and electrical pulse heating techniques. Finally the implications, for process modelling, of the available data and measurement techniques are discussed.

A cellular automaton model of steady-state columnar-dendritic growth in binary alloys

A two-dimensional cellular automaton model has been developed to examine the evolution and coarsening behaviour of solid-solution dendrites during steady-state columnar freezing. Using an empirical rule to account for interface growth, realistic dendrite geometries were obtained for different assumed compositions and process conditions. Coarsening occurred by a coalescence mechanism associated with bridging of adjacent dendrite arms.

Numerical determination of liquid flow permeabilities for equiaxed dendritic structures

Abstract Darcys law has been applied to the 3D finite difference numerical determination of the influence of solid fraction and geometry on the permeability of equiaxed dendritic structures. The micro-model computes the permeability for flow through a domain equivalent to the volume ultimately occupied by a single solid solution dendritic grain in an Al3Cu3Si alloy. Evolution of the dendrite shape during solidification was simulated using a novel cellular automaton-finite difference technique. Numerically determined permeabilities compare well with reported experimental data for aluminium alloys. For solid fractions in excess of ∼20%, there is also reasonable correlation with the Kozeny–Carman (KC) expression for a KC constant of unity. A significant feature of the micro-model is that it is able to account for the isolation of interdendritic liquid pools in calculating the effective values of the solid–liquid interfacial area and of the fraction liquid.

Modelling of non-equilibrium solidification in ternary alloys: comparison of 1D, 2D, and 3D cellular automaton–finite difference simulations

Abstract A cellular automaton–finite difference (CAFD) computer model is presented that describes the solidification of multicomponent multiphase alloys at the microscopic level. The objective of the model is to enable the prediction of microsegregation patterns and the appearance of non-equilibrium constituents during non-equilibrium freezing. To support the development of the model, coupling with a thermodynamic software package, ThermoCalc, has been achieved to obtain accurate thermodynamic data for multicomponent alloys. The CAFD model has been used to generate evolving dendritic structures in two and three dimensions. A simple one-dimensional (1D) CAFD plate model, which assumes that adjacent dendrite arms are plates, has also been developed. Recently, experimental results of a study carried out on a directionally solidified Al–3.95Cu–0.8 Mg alloy (cooling rate of 0.378 K s-1) have been reported. In the present investigation, a comparison between 1D, 2D, and 3D simulations of microsegregation and these experimental results is made, with respect to the amounts of non-equilibrium constituents and solute profiles in the primary -Al phase, for the same alloy and solidification conditions.

A cellular automaton model of the steady-state “free” growth of a non-isothermal dendrite

Abstract A 2D cellular automaton model has been developed to study the steady-state “free” growth of a non-isothermal dendrite. The model incorporates rules to account for heat diffusion, the influence of curvature on the equilibrium freezing temperature and latent heat evolution. The model predicts a V ∝ δTb growth rate-undercooling relationship for the various dendrite tip growth temperatures selected. The prediction of the values of b accords reasonably with analytical models and reported experimental observations.

Numerical modelling of permeability variation with composition in aluminium alloy systems and its relationship to hot tearing susceptibility

Abstract A numerical micromodel has been developed to simulate the evolution of equiaxed primary phase grains during the solidification of alloys in the systems Al–Cu, Al–Si, Al–Mg, and Al–Zn. The microstructures generated have then been used to model liquid permeability as a function of composition for each system, for a given solid fraction and cooling rate. In all systems a marked minimum occurred in the permeability curve at a value < ∼1 wt-% solute. The composition corresponding to the minimum permeability tended to increase with increasing equilibrium partition coefficient. It is argued, for each system, that the composition displaying minimum permeability would correspond to that composition exhibiting maximum susceptibility to hot tearing. Comparison of the permeability data with experimental hot tear test data for the same systems reveals the limitations of most hot tear tests.

A 3D numerical model of the temperature - time characteristics of specimens tested on a Gleeble thermomechanical simulator

A three-dimensional (3D) explicit finite difference model has been developed to simulate the thermal behaviour of specimens subjected to controlled thermal cycling patterns using direct resistance heating in a Gleeble thermomechanical simulator. Model predictions have been compared with temperature measurements performed on 2024 aluminium alloy, 0.2% plain carbon steel and 316 austenitic stainless steel specimens in the simulator. The specimen materials were chosen to provide widely differing thermophysical properties and, in the case of the steels, to allow a comparison of the predictions with and without an to phase transition.

A 3D cellular automaton model of coupled growth in two component systems

Abstract A 3D cellular automaton model has been developed to illustrate the evolution and coupled radial growth for a two phase cell (grain). Coupled growth occurs in various types of phase transformation in materials systems and the resulting microstructures can significantly influence final properties. The model simulates the redistribution of the atomic/molecular species taking place during a transformation and the establishment of coupled growth.

Prediction of the effective thermal conductivity of three-dimensional dendritic regions by the finite element method

Abstract The finite element method has been used to predict the variation in effective thermal conductivity, k eff , in evolving three-dimensional dendritic mushy zones. The model demonstrates the influence of a dendrite-like geometry and volume fraction of solid on k eff assuming that the conductivities of the solid and liquid phases remain constant and uniform as the structure evolves. The three-dimensional dendrite-like shapes have been generated by a cellular automaton-finite difference model. Numerical solutions are presented for both columnar and equiaxed dendritic zones.

Numerical prediction of the effective thermal conductivity of dendritic mushy zones

A numerical method has been proposed for predicting the effective thermal conductivity of dendritic mushy zones as a function of fraction solid. The proposed method estimates the value of the effective thermal conductivity by the finite-difference method using known morphologies of the mushy zone. Numerical solutions are presented for some simple two-dimensional shapes approximating columnar and equiaxed dendrites and some complex dendrites generated by a cellular-automaton-finite-difference model.