The raw material in this study is a copper war bar of 99.86 wt% purity from Usines Chimiques de Shituru/Gécaminesgroupe centre (US/GCM) and manufactured into a rod at LC/GCM. Chemical analysis of the material was carried out by atomic absorption spectrometry in the chemical analysis EMT Gecamines laboratory.

Rods are prepared from a 120 kg war bar, which has been separated into three parts of 40 kg each. The resulting portions are then pre-heated in a diesel furnace at 850˚C for two hours and then pressed into a 150 mm diameter container to give rod round shape. The rod is subsequently processed to remove the oxide layer and reduce its diameter to 5 mm and its mass to 3 kg. The equipment used for this purpose was: a saw, a diesel furnace, a 750-ton press, a receiver, a pointing device, a die, a die holder, studs, a counter-die, a pin, an extrusion disc or pad, a spanner, a pair of pliers, a scale, a calliper, a tape measure.

Extrusion tests of the billets were carried out in the laboratory of LC/Gecamines Groupe Ouest. They have consisted of billet pre-heating in a diesel furnace up to the temperature of 850˚C, loading the billet in the receiver to evaluate the flow speed of the material according to the formula (1).

v = ( 2 p / ρ ) 1 / 2 (1)

where ρ is the copper density and v is the flow rate.

At the end of the test, the quality control of the resulting blank is performed by:

- Sizing the billet (diameter: 145 ± 2 mm, cross-section: 1.65 ± 0.01 m², length: 0.250 ± 0.001 m, volume: 0.41 ± 0.01 m³ and mass of the billet: 37 kg).

- Calculation of the ratio between sections R = A 0 A 1 (2) where A₀ and A₁ are

the sections of the billet and the blank respectively.

- Calculation of the extrusion force F = p × A 0 ln R (3); p is the pressure applied

by the press [22].

The extrusion die and pin will determine the external and internal diameter of the blank. Thus, three factors influence their choice: the billet pre-heating temperature of 850˚C (taken as constant in this study), the pressure supplied by the press of 245 GPa (constant in this study) and the material flow speed.

The optimization method chosen is the non-deterministic method. It is a stochastic method of the simplex algorithm with the objective of optimizing the die section and the material flow rate [2] [23] [24] [25] [26].

At each iteration, a set of points is generated, by eliminating the point corresponding to the highest value of z and determining a new point using the reflection operation. The position of this point is then modified by contraction or expansion depending on the value of z at that point. When the optimal value is achieved and no previous operation is satisfactory, the points are contracted in all directions, reducing the search area [27] [28] [29] (Table 1).

To carry out the mathematical modelling of the extrusion of copper tubes, the approach adopted was as follows:

1) The further statements [2] [30] [31]:

- The material (copper) used is perfectly bonded, any chemical or physical transformation of the material is ignored. The internal energy is dependent on the absolute temperature T (e =e(T)) via the specific heat capacity c of the material.

- The volume rate term of the received heat (r ≅ 0) is negligible. The energy creation is defined by the mechanical work rate.

- The deformation of the billet is uniaxial. Since the material under consideration is isotropic, the deformation tensor is equal to the value of the deformation.

- The material flow is in a permanent flux.

- Copper heated to 850˚C is viscous, it acts like an incompressible fluid.

{ ∇ ⋅ ( 2 η ( γ ˙ ) ε ˙ ( v ) ) − ∇ p = 0                                                                 ( 4 ) ∇ ⋅ v → = 0                                                                                                               ( 5 ) ρ c ^ d T d t = − ∇ ⋅ q + σ : ε ˙ ( v )                                                                 ( 6 )

γ ˙ = 2 ∑ i , j − 1 3 ε ˙ i j 2 (7)

With:

c ^ : expansion factor

η : the medium viscosity

ε ˙ : the tensor of strain rates

γ ˙ : is the generalized shear rate

T: the temperature in celcius

v: the flow of the material

p: the pressure

q: the heat flux

The heat flux can be determined by Fourier’s law:

q = k A ( T 1 − T 2 ) L (8)

With:

k: thermal conductivity. For copper,k = 388 W/mK

A: transferred surface (m²)

L: length.

2) The Law of Material Performance

It is experimentally established that molten metals do not have Newtonian behaviour: their viscosity is not constant but changes with the shear rate. The molten copper is taken as a plastic ideal fluid whose coefficient will be determined by the following formula [31]:

τ = τ 0 + η ( Δ V Δ h ) (9)

With:

τ: shearing stress

∆V: increase in particle velocity

∆h: fluid film thickness

To carry out the numerical modelling of the extrusion of copper tubes, the main considerations are as follows [32] [33] [34]:

1) Assumptions

- The billets used for extrusion have a round shape.

- The modeling is performed on the LS DYNA finite element software.

- The material is viscoplastic, i.e. a thermomechanically coupled law of behaviour.

2) Geometry

The copper billet is round with the dimensions of 145 ± 2 mm in diameter, 1.65 ± 0.01 m² in cross-section, 0.25 ± 0.001 m in length and 0.41 ± 0.01 m³ in volume. It had a mass of 37 kg and was then heated to a temperature of 850˚C.

3) The following material behaviour law:

The Johnson-Cook model was used to define the performance of the material during the extrusion of the billet, it is frequently used in hot forming, particularly in compression and forging, and offers an excellent correlation with the experimental data [12] [13] - [31]:

With:

A: the average flow stress (MPa)

B: the work-hardening modulus (MPa)

n: the work hardening coefficient

C: the coefficient of strain rate sensitivity

ϵ ˙ 0 p : the reference strain rate (s-1)

ε p : the plastic strains

ε ˙ p : the strain rate (s-1)

T: the temperature of the deformed billet (˚C)

T limit : the ambient temperature of the material (˚C)

T f : material melting temperature.

4) Limit conditions:

The limit conditions are such that:

- The die is recessed.

- The die penetrates the billet prior to the extrusion operation for the inner diameter of the blank.

- The extrusion of the billet is carried out at temperature 850˚C and the operation is carried out in an adiabatic environment.

5) Finite element model:

The tube starts its stroke outside the dies. A parallel displacement to the die axis is specified at the nodes at the end of the tube. The displacement is a function of the percentage reduction and the simulation time. However, for all simulations, the time is one second in order to simplify the calculations [31] [32].

The chemical composition of the war bar is shown in Table 2.

The aim of the billet extrusion was to manufacture a tube blank that could be used to produce a tube during cold drawing with a dimension of 17/14 mm [22]. The results of the extrusion tests are given in Table 3.

As the press force is a function of die diameter and billet temperature, the smaller die diameter and billet temperature, the greater the force. After the extrusion process, the blank obtained is of D_ext = 54 ± 2 mm, d_int = 48.4 ± 2 mm and thickness of 2.8 ± 0.1 mm when applying an extrusion force of 493.52 kN.

The extrusion temperature of 850˚C resulted in easier billet flow and deformation. However, the maximum extrusion rate is reduced because of the high

temperature peaks which could initiate melting of the billet.

The extrusion force is dependent on the billet temperature and the section ratio. The latter gave us a value of 493.5145 kN for a ratio of 3700.

Several factors were optimized while keeping the same operating mode [23]:

- The diameter of the die which was obtained as a function of the pressure.

- The flow rate of the material was optimized by considering a permanent flow.

The results of this optimization are reported in Table 4.

The tooling optimization, using the non-deterministic method with the simplex algorithm, was carried out keeping the same material characteristics as at the industrial scale. The press force resulted in a blank with a dimension of 19.2/14 mm external and internal diameter, a thickness of 2.6 mm and a total blank length of 28 m with an extrusion force of 565.6 kN.