Consistent copper - nonlinear modelling

Materials World magazine
1 Jul 2010

Abhay Bulsari, from Nonlinear Solutions Oy, Turku, Finland, outlines the benefits of nonlinear modelling for hardness and thermal conductivity.

Copper alloys are commonly used in applications that require high electrical and thermal conductivity, high hardness and good resistance to softening at elevated temperatures. One alloy with a good combination of these properties is the UNS alloy C18150, having a nominal composition of one per cent chromium, 0.1% zirconium and the rest copper. Hardness of some products of this alloy is achieved partly by the amount of cold work and partly by precipitation hardening time and temperatutre. In this case, the cold work is usually carried out prior to precipitation hardening treatment.

Nonlinear modelling of hardness and conductivity after precipitation hardening of cold worked material, where the amount of cold work was a variable, has previously been described. Here, hot extruded wire rods of the same alloy, which undergo a constant amount of cold work before precipitation hardening, are being described. Cold work is, therefore, not a variable.

Production starts with billet casting. Melting and casting is carried out in highly controlled conditions to get homogeneous material and properties. Solution annealing is first conducted at a high enough temperature to dissolve the alloying elements. Then it goes through hot extrusion where the diameter is determined, so that the required mechanical and electrical properties are achieved after further processing. After hot extrusion, cold drawing is performed with pull blocks and drawing benches. These wire rods then undergo precipitation hardening to form precipitates of chromium and zirconium in proper amounts and sizes, to get the desired mechanical and electrical properties, particularly hardness.

Modelling types

Mathematical modelling can be performed in different ways, and a variety of these are suitable for different situations. It is not possible to use physical modelling in many situations, as they require assumptions and simplifications. Empirical or semi-empirical modelling is often a better alternative.

Traditional empirical modelling is based on linear statistical techniques. Nothing in nature is absolutely linear, so, it helps to take nonlinearities into account. New techniques of nonlinear modelling, based on artificial neural networks, can estimate nonlinearities without specifying in detail. Feed-forward neural networks have several features making them highly efficient tools for nonlinear empirical modelling. Nonlinear modelling has been used successfully in several industrial sectors, including metals, concrete, plastics, glass, semiconductors, power generation, pharmaceuticals, medicine, biotechnology and food materials.

A total of 13 experiments have been carried out in a laboratory oven at five temperatures for periods of up to three hours, resulting in 66 observations. Hardness was measured at half the radius of the wire rods, besides electrical conductivity. The quality of the experimental data was good, and no observations were discarded.
The graphs (left and below, right) show clear trends for hardness as well as conductivity in the experimental data. Hardness increases with time, but at higher temperatures overaging starts to take place and the hardness reduces. Conductivity increases at higher temperatures, and it helps to have a larger initial conductivity. Considering hardness, a lower initial conductivity is better.

Nonlinear models for hardness and conductivity have been developed from the experimental data using NLS 020 software. The models had three input variables – heat treatment time, temperature and initial conductivity. The diagram (see table, below left) shows a schematic of the nonlinear models.

These models were in the form of feed-forward neural networks with sigmoidal activation functions on the hidden layer, in terms of transformed output variables. The nonlinear models show good statistical characteristics. The root mean square (rms) error (the standard deviation of the prediction errors) was 1.39HV for hardness and 2.16% International Annealed Copper Standard (IACS) for conductivity, which amount to correlations of 99.01% and 98.05% respectively. For hardness, the fractional rms error is less than one per cent, which is approximately the measurement accuracy.
The charts (below, right) show comparisons of the measured values with those predicted by the nonlinear models of hardness and conductivity. Most of the points lie close to the ideal line.

The nonlinear models show the effects of the input variables as expected. At higher temperatures, the hardness increases, then overaging starts.

























Multi-purpose models

Nonlinear models serve several purposes. They contain a lot of valuable quantitative knowledge in a highly condensed form. They can be used for ‘what-if’ analysis or as alternatives to carrying out experiments. The most important of all, however, is to determine good conditions for operating the process. To determine suitable operating conditions, one can specify upper and/or lower limits on the variables in the models, or fix their values. In addition, one can maximise or minimise one of the variables. 

With the right mathematical tools such as LUMET systems, it becomes easy to determine suitable solutions, if they exist. This is not always feasible by trial and error experimentation.  It would be tedious determining a suitable annealing condition by trial and error. With an initial conductivity of 50%IACS, there is no feasible region. It would take excessive trial and error experimentation before realising there may not be a solution. 


The initial conductivity of the wire rods, however, varies from one end to the other. The variation might be as high as 10%IACS. The more interesting problem of the production unit is to determine the best annealing time and temperature with which the variation in the hardness is minimised, in addition to keeping the conductivity and lowest hardness above desired levels. With these types of mathematical tools, it is easier to determine the best annealing time and temperature for each wire rod.

Precipitation hardening is an important heat treatment process for several alloys. Determining the best operating conditions can significantly improve the production economics by reducing annealing time, energy consumption and rejects, leading to enhanced competitiveness. Nonlinear modelling is a powerful tool for describing the complicated process in quantitative form. With suitable mathematical software, it becomes easy to determine the best operating conditions.

Further information

Nonlinear Solutions Oy, Kaivokatu 10A 21, 20520 Turku, Finland. Tel: +358 2 2154721. E-mail: Website: Thanks to Ilkka Vuoristo and Ilpo Koppinen, Luvata Oy, Pori, Finland.