The effect of elastic and plastic stresses on the electrical resistivity of conductive materials

The mobility sector has been responsible for large emissions of harmful greenhouse gases worldwide for years. E-mobility can contribute to a reduction of these emissions. For a faster spread and acceptance of electric vehicles, the achievable range and thus the efficiency of the individual components in the drive and charging train is essential. In addition to the battery and the electric machine, losses in the cables and bus bars are playing an increasingly important role. The cross-sectional shape, material and manufacturing technology of these conductive materials depends on the voltage and power to be transmitted and may still change in the future, especially in the field of fast charging and increased voltage. The aim of this work is to investigate the influence of the mechanical stress condition of copper, brass and aluminium materials on their electrical conductivity. The conductivity is measured by means of 4-wire measurements under defined tensile and compressive loads. The effects of elastic and plastic strains are taken into account. The findings can be used for an optimized manufacturing strategy for conductive components for future applications.


Introduction
In order to achieve sustainability goals in general, but especially in the automotive sector, efficient generation, provision and transportation of energy is necessary.Electromobility is characterized by a high level of efficiency when converting electrical energy stored in batteries into mechanical energy through the electric motor.Compared with efforts to achieve compact and high-capacity batteries, storage with high energy density and efficient energy utilization in the electric motor, the focus has not yet been on the efficiency of the current carrying structures.[1] In addition to the achievable range, fast charging of cars is crucial for the acceptance of the future transport transition.Very high power levels are achieved with fast charging using DC current.This leads to an increase in the required cross-sections, which also reduces the flexibility of the current-carrying cables and conductors.Where flexible cables were often used in the past, it is now necessary to use rigid power conductors manufactured by forming technology.Similar to current collectors and bus bars, there is a change in electrical properties in the formed areas, e.g.sharp radii.[1] For the realization of structural current-conducting components, different forming processes are used, which lead to a different residual stress state in the formed area of the manufactured product.The aim of this work is to investigate typical current conductor materials with respect to the change in electrical resistivity as a function of the degree of forming.The electrical resistivity is measured using a four-wire measurement on round specimens for uniaxial tension.

State of the Art
According to equation (1), for an electrical conductor material with isotropic properties, the electrical resistivity ρ is defined as the quotient of the magnitude of the electric field E by the magnitude of the current density J [2]: Elastic and plastic strains influence the value of ρ.While an increase in electrical resistivity with increasing strain was observed for most of the metallic materials examined, for example, nickel shows a decrease at low strains [3].Grain boundaries [4] and dislocation density [5] were identified as influencing variables responsible for the change in ρ at the microstructure level.ρ can also be affected by the heat treatment history [6].Furthermore, ρ varies with temperature T, whereby the following linearized relationship in equation ( 2) with the temperature coefficient α can be derived in the range around a reference temperature T0: The electrical resistivity ρ can be determined by measuring the electrical resistance R on a known geometry.For a cylindrical conductor of length l, cross-section  and homogeneous current density J, the equation (3) follows: For determining small values of R, the four-wire measurement method is established.For the measurement of electrical steel sheets, this method is specified in more detail in DIN EN IEC 60404-13 [7].During the measurement, the sample is contacted at four points.An electric current is applied to the two outer contact points, causing a homogeneous current density in between.
At the two remaining contact points, located between the outer contact points, the potential difference U due to the electrical resistance is sensed.By using two separate pairs of connecting electrodes, the wiring and contact resistance are eliminated [2].As a result, the electrical resistance R according to Ohm's law is quantified.

Experimental Setup
Strain controlled tensile and compression tests were performed on an Allround Line 150 kN tensile-compression testing machine from ZwickRoell GmbH, Ulm, Germany.The change in the cross-section of the specimens was recorded using a stereo DIC system, GOM Aramis SRX from Carl Zeiss GOM Metrology GmbH, Braunschweig, Germany.For the four-wire measurement a Keithly 2400 SourceMeter was used as the current source and a Keithly 2182A nanovoltmeter from Keithley Instruments SRL, Cleveland, Ohio, USA was used to record the voltage.The temperature of the samples was recorded during the tests using a PT-1000 sensor.The experimental setup is shown in Figure 1.The materials listed in Table 1 are typically used in electrical conductors and cover a broad variety of electrical conductivity:  The electrical resistivity is a material constant that depends on the temperature and the residual stress state.The four-wire measurement is used to determine the electrical resistivity.In the multi-conductor measurement, the resistance is measured using two voltage taps U1 and U2 at a defined distance from the resistor while current flows through the conductor.The current flows between the two external conductors I1 and I2.To maintain the distance between the contacting pins, a rigid mounting device is used.The schematic structure of the four-wire measurement is explained in Figure 2: The specific resistance ρ is calculated according to the following equation ( 4) from the measured voltage U1-2, the flowing current I1-2, the current cross-sectional area A and the linear distance of the voltage measurement: The main test parameters and the holding times for the cyclic tensile and compression tests are summarized in the following table 2: Table 2. Experimental and process parameter.An electrical simulation was carried out to ensure a uniform distribution of the electrical distribution for the voltage measurement area.The simulation was performed using Abaqus 2022, Dassault Systèmes, Johnston, RI, USA.An electrical current was applied to two contact points at a distance of 130 mm, analogue to the test setup.For a distance of 90 mm between the voltage measuring points, a homogeneous distribution of the EPG (electrical potential gradient) is maintained as shown in Figure 3.

Results
Strain-controlled cyclic tensile tests were performed to determine the electrical resistivity as a function of the degree of deformation.In order to achieve a high resolution in the area of elastic deformation, the strain per cycle was chosen to be lower.After the start of the flow, the elongation steps per cycle were increased.The temperature was recorded for all tests, whereby no significant sample heating occurred due to the current flow or material deformation.The temperature change during the tests was within the measurement accuracy range of ± 0.1 °C for all materials.

Results for Steel
For the steel material Figure 4 shows the force curve over the increasing elongation and the resulting relative change in the specific resistance per cycle is shown in Figure 5.The strain-controlled hysteresis had an increase of 0.005 mm/cycle until the point of yielding and 0.05 mm/cycle afterwards:   The specific resistance for steel increases steadily with increasing uniaxial tensile load.With the onset of yielding, there is a slight decrease in the specific resistance, which then increases until failure.As expected, the breakpoints show higher resistance values in the loaded state of hysteresis.The initial electrical resistivity used for the relative change of the steel specimen is 149.58*10 - Ω *  2  .

Results for Copper (Cu-ETP)
Figure 6 shows the force curve for each hysteresis cycle over the increasing elongation of the copper material and the resulting relative change in the specific resistance per cycle is shown in Figure 7.The increase of the strain was chosen to be 0.005 mm/cycle before yielding and 0.05 mm/cycle after.
The initial electrical resistivity used for the rel.change of the Cu-ETP specimen is 16.55*10 -3 Ω *  2  .
Figue 6. Hysteresis of the force-elongation diagram for the tensile testing of the copper specimen.The tested copper specimen shows an increase in the specific electrical resistance with increasing uniaxial tensile load.In the yielding range, there is a slight decrease in resistance for the unloaded and a strong increase during the loaded plateaus of the hysteresis.Due to the high ductility of the copper material, necking only occurs at very high strain values.In the area before necking, a reduction in the specific electrical resistance occurs at an elongation of approx.1.2 mm before it subsequently increases again.

Results for Aluminium (EN AW-6060)
For the brass material Figure 10 shows the force curve with each hysteresis cycle over the increasing elongation and the resulting relative change in the specific resistance per cycle is presented in .The strain-controlled hysteresis increased by 0.005 mm/cycle until the point of yielding was passed and  The aluminium specimen has the smallest change in electrical resistivity of all tested samples.While there is an increase of 1 % in the area before yielding, the value at the point of yielding the initial value of the resistance.After yielding, there is a very fluctuating behaviour and a strong decrease in the unloaded hysteresis plateaus in the area before constriction.In this area, the uniform reduction in diameter of the tensile specimen is superimposed on the decreasing voltage of the four-wire measurement.

Results for Brass (CuZn39Pb3, EN12164)
For the brass material Figure 10 shows the force curve with each hysteresis cycle over the increasing elongation and the resulting relative change in the specific resistance per cycle is shown in Figure 11.The strain-controlled hysteresis had an increase of 0.005 mm/cycle until the point of yielding was passed and 0.05 mm/cycle afterwards.The initial electrical resistivity used for the relative change of the brass specimen is 68.22*10 -3 Ω *  2
The course of the specific resistance of the brass tensile specimen shows a steady increase with increasing tensile load.In the area shortly before necking at an elongation of approx.3 mm, there is a slight decrease in the electrical resistance.After necking began, the four-wire measurement lost contact with the sample, meaning that the measured values after necking could not be utilised.

Results for Brass (CuZn39Pb3, EN12164), compression testing
In addition to the previous tests, a pressure test was performed on a brass specimen.The compression test was carried out with the same test setup as the tensile tests, but the control of the compression was done force with an increase of 500 N per hysteresis cycle.Due to the length of the sample, no significant plastic deformation occurred.The test was stopped when the specimen buckled.
Figure 12 shows the increasing compression force with each hysteresis cycle over time and the resulting relative change in the specific resistance is shown in Figure 13.The initial electrical resistivity used for the relative change of the brass specimen is 68.20*10 -3 Ω *  2  .The Force curve indicates that there is no yielding occurring while performing the test.In contrast to the tensile tests, the compression test shows a decrease in the relative specific resistance shortly before the specimen buckles.

Conclusion and Outlook
Tensile tests were carried out on steel, aluminium, brass and copper materials.The work shows a change in specific electrical conductivity of up to 8 % due to uniaxial loading.This must be taken into account when designing and dimensioning future conductors.An exemplary compression test for the brass material shows a decrease in the specific resistance due to a uniaxial compressive load.
The investigations of the materials with tensile test provide an initial indication of the sensitivity of the specific resistance.Based on the promising results of the compression test and the potential increase in electrical conductivity, tests with pressure samples will be carried out in further work.As electrical components are usually bent, further work will be executed for bent samples.Production processes with different mechanisms and the resulting residual stress states, such as pressure-superimposed free-form bending or rotational tensile bending, can be used to utilise the advantages and increase the performance of electrical conductors.As the results show that the electrical conductivity decreases with plastic elongation due to tensile stresses, it would be interesting to investigate the possibility of stress relief annealing in greater depth for subsequent studies.Stress relief annealing can reduce residual stresses and possibly improve electrical conductivity.

Figure 2 .
Figure 2. Specimen and contacting system for the four-wire measurement.

Figure 3 .
Figure 3. Distribution of the EPG (el.potential gradient) for the four-wire measurement.

Figure 4 .
Figure 4. Hysteresis of the force-elongation diagram for the tensile testing of the steel specimen.

Figure 5 .
Figure 5. Relative change in specific resistance and elongation for tensile testing Cu-ETP for hysteresis cycle.

Figure 7 .
Figure 7. Relative change in specific resistance and elongation for tensile testing of Cu-ETP for hysteresis cycle.The tested copper specimen shows an increase in the specific electrical resistance with increasing uniaxial tensile load.In the yielding range, there is a slight decrease in resistance for the unloaded and a strong increase during the loaded plateaus of the hysteresis.Due to the high ductility of the copper material, necking only occurs at very high strain values.In the area before necking, a reduction in the specific electrical resistance occurs at an elongation of approx.1.2 mm before it subsequently increases again.

Figure 8 .
Figure 8. Hysteresis of the force-elongation diagram for tensile testing of the aluminium specimen.

Figure 9 .
Figure 9. Relative change in specific resistance and elongation for tensile testing of the AlMgSi0.5 material for hysteresis cycle.

Figure 10 .
Figure 10.Hysteresis of the force-elongation diagram for tensile testing of the brass specimen.

Figure 11 .
Figure 11.Relative change in specific resistance and elongation for tensile testing of the brass material for hysteresis cycle.

Figure 12 .
Figure 12.Hysteresis of the force diagram for the compression testing of the brass specimen.

Figure 13 .
Figure 13.Relative change in specific resistance and elongation for compression testing of the brass material for hysteresis cycle.