CONSTITUVE MODELLING FOR CREEP OF DRAWN COPPER WIRE

The high-temperature creep behaviour of drawn coppe r wire was studied by different constant stresses 98,108 and 118 MPa and under the temperatu res of 250, 290 and 340°C. This study deals with the creep based prediction modelling of an industrial copper wire. The proposed unified creep damage constitutive equations were de t rmined using experimental data achieved for materials at applied stress. The comparison of experimental and predicted effective creep strain curves is carried out for all applied stress es applied on the drawn copper wire. The evaluated stress exponent n = 9.21, 10 and 14 and t he activation energy Q = 22.25, 31.75 and 50.75 indicated that the creep deformation of the d rawn copper wire is controlled by the dislocation creep. The evaluated of the mean relati v error from 5.18 % to 10.11 confirmed the creep strain predicted by the proposed model well a gree with experimental data.


Introduction
Copper in particular has attracted attention due to its good properties such as low resistivity.Due to its high ductility which is the ability to be easily drawn into wires, copper dawning is very attractive manufacturing process.The wire drawing is a process that is used for the manufacturing of metal wires.Copper wire has long been the preferred conductor material.The demands of electrical technology require copper to have higher mechanical properties and to be capable of using at elevated operating temperatures while still retaining the good conductivity for which it is selected in the first place [1].Creep is the process by which plastic flow occurs when a constant stress is applied to a metal for a prolonged period of time.After the initial strain ε0 which follows the application of the load, creep usually exhibits a rapid transient period of flow before is settles down to the linear steadystage, which eventually gives way to tertiary creep and fracture [2].However, creep phenomena development is still not sufficiently recognized, especially under uniaxial stress conditions.However, further complex microscopic creep investigations are required to achieve a better understanding of the nature of the process.In a uniaxial creep curve tertiary creep is observed as the increase of the creep rate.The shape of the final part of the creep curve and the duration of the tertiary creep stage depend on the material composition, the stress and the temperature [3].including the formation, growth and coalescence of voids on grain boundaries, coarsening of precipitates and environmental effects [4][5][6][7][8][9][10].
In order to describe the creep behaviours of metals and alloys, many researchers have tried to establish constitutive models, such as continuum damage equation and Andrade's equation [11][12][13][14][15][16][17].For example Authors [11] performed a review on the development and the use of internal state variable theory based on the Coleman and Gurtin thermodynamics formulations for dislocations, creep, and continuum damage mechanics.Authors [12] proposed a model which integrated the power-law creep, diffusional creep and a simple damage term for simulating the creep-failure behaviour of the inter-critical heat affected zone.Authors [13]

Material and experimental methods
The material used in this investigation is an industrial copper wire of composition 99.9 Cu, 0.001 Bi, 0.002 Sb, 0.002 As, 0.005 Fe, 0.002 Fe, 0.002 Ni, 005 Pb, 0.002 Sn, 0.004S, 0.004 Zn and 0.073 (wt.%), others elements.Samples having gauge length of 100 mm and diameter of 1.8 mm obtained after cold wire-drawing process were annealed at 500°C for 2 hours.The creep specimens were tested at temperature 250, 290 and 340 °C and under stress 98, 108, and 118 MPa.

Approach Modelling
A relationship for describing high-temperature plastic deformation within the structural phenomenological model of the Andrade [18] is based on the plastic deformation ε, arising from a constant stress σ, is expressed in the Eq.1: whereε 0 is the instantaneous plastic deformation, B and m are material constants.The term tm models the primary creep and is the creep rate.We know that the Andrade's equation plot only first and second stage of the creep curve, but not plot third stage.For that reason we have to substitute Eq. 2 (relationship of Kachanov [19] and Rabotnov [20]) into Eq.(1).A variable w (0<w<1) was introduced by [21] to present the state of damage of material.

  
The material constants A, n, and l can be then determined from the stationary creep.Let • be minimum creep rates for constant stresses σ 1 and σ 2 , respectively.Then the material constants n and A can be estimated from: within the temperature ranges considered in this work, increasing temperature caused an increase in the parameters A and B, for that the material constants A and B should be replaced by the functions of temperature.Assuming the Arrhenius type temperature dependence the following relations can be utilized (Eq.( 7) and Eq. ( 8)).Substituting Eq. ( 7) and Eq. ( 8) into Eq.( 3) yields the creep model Eq. ( 11): 3 Results and Discussion Fig. 1 present the curves of creep strain versus time of copper draw wire, obtained at different applied stress (118, 108 and 98 MPa) at temperature 250, 290 and 340°C.Generally, the time dependent elevated temperature creep deformation can be represented by the creep strain time curve which is usually distinguished by the primary, secondary and tertiary stages.Following the initial strain on loading, the creep rate gradually decreases during the primary stage.The creep rate continues to decrease and reach a minimum or secondary value during the secondary  The transient creep in Ref [22], is represented the second part in Eq. ( 3); this means that it can be written the following Eq.( 12).
We can calculate the transient creep time exponent m with the Eq. ( 13) (Fig. 3).The values of the m are changing with temperature and stress, for example in Fig. 4a, m = 0.415, 0.56 and 0.60 under stress 98,108 and 118 MPa, respectively at temperature 250°C.In Ref [3], they calculated the parameter m = 0.55 for copper.These curves are predicted data by using the determined material constants, listed in Table 1.Fig. 4 show that the comparisons of the measured and predicted creep strains by Eq. (11).
Obviously, an agreement between measured and calculated values is satisfactory under the creep temperature of 250, 290 and 340°C, which indicates that the proposed constitutive models can give a good estimate of the first, secondary and tertiary stages creep deformation for drawn copper wire.Where and are the experimental strain and prediction strain, respectively.We have calculated the mean relative error (Error M ) for all creep testes as in Table 2, using Eq.(15).We observed that the values of the Error M are between 5.18 and 10.11 %, which is less than 10 %.N presents the number of data points on each creep curve (between curve experimental and prediction), which we calculated the relative error.
where Q tr and Q are the activation energies of creep and transient creep, respectively.

1 lnDOI 10 .
12776/ams.v20i4.411p-ISSN 1335-1532 e-ISSN 1338-1156 stage.During the tertiary stage, due to the increase of cavitations and cracks in the specimens, the creep rate rapidly increases, which lead to the final fracture.

Fig. 2 aFig. 3
Fig.2 a)Minimum creep rates versus stress for copper wire at temperature of 250, 290 and 340°C, b) Relation between transient creep strain and temperature for copper wire under different stress 98,108 and 118 MPa.c) Relation between minimum creep rate and temperature for copper wire under different stress 98,108 and 118 MPa

Fig. 4
Fig.4 Comparison of experimental and predicted effective strain creep curves for copper wire at temperature of a) 250°C, b) 290°C and 340°C The origins of tertiary creep are progressive damage processes DOI 10.12776/ams.v20i4.411p-ISSN 1335-1532 e-ISSN 1338-1156

Table 2
The value of the mean relative error (Error M ).Constitutive model to describe the high temperature creep behaviour of drawn copper wire based on the modification of the Andrade's equation.The creep curves of drawn copper wire predicted by proposed model well agree with experimental results, which confirm that the developed creep constitutive model can give an accurate and precise estimate of the high temperature creep behaviour for drawn copper wire.