The electroplastic effect can be predicted and modeled as a 100% bulk heating/softening phenomenon in the quasi-steady-state; however, these same models do not accurately predict flow stress in transient cases. In this work, heterogeneous Joule heating is examined as the possible cause for the transient stress drop during quasi-static pulsed tension of 7075-T6 aluminum. A multiscale finite element model is constructed where heterogeneous thermal softening is explored through the representation of grains, grain boundaries, and precipitates. Electrical resistivity is modeled as a function of temperature and dislocation density. In order to drive the model to predict the observed stress drop, the bulk temperature of the specimen exceeds experiment, while the dislocation density and grain boundary electrical resistivity exceed published values, thereby suggesting that microscale heterogeneous heating theory is not the full explanation for the transient electroplastic effect. A new theory for explaining the electroplastic effect based on dissolution of bonds is proposed called the Electron Stagnation Theory.

The Electroplastic Effect

The effect of electricity on metals during deformation, known as the electroplastic effect, can be utilized to reduce forming loads, springback, and flow stress while increasing ductility during various forming processes [17]. A review of the impact of electricity on various manufacturing processes is reported in Ref. [8].

While the electroplastic effect is well documented and researched, the underlying mechanisms are not fully understood. It has been shown that in steady-state circumstances, the electroplastic effect can be fully modeled and predicted using a conventional energy balance and thermal softening approach that assumes 100% bulk (homogeneous) Joule heating [9,10]. These works use electricity to heat a part but allow the temperature to stabilize before starting deformation.

The steady-state modeling approach implemented into a finite element analysis (FEA) cannot predict the transient stress drop caused by pulsed electric current even though the correct temperature profile is predicted [1113].

Experimental results comparing the flow stress results of specimen heated in a furnace to the same temperatures seen through electrically assisted deformation found that electrically assisted deformation had a lower stress than thermally assisted, suggesting bulk Joule heating cannot fully explain the transient electroplastic effect [7,1418]. Theories have been developed to explain the transient nature of the electroplastic effect; these include (a detailed review of electroplastic theories and models can be found in Ref. [19]):

  1. (1)

    Electron wind. The flow of electrons from the electrical pulse causes collisions between electrons and dislocations resulting in a momentum transfer leading to reduced flow stress [20,21].

  2. (2)

    Magnetoplasticity. The electric current creates a resultant magnetic field, the magnetic field adds energy to dislocation motion allowing the dislocations to overcome obstacles more easily, leading to reduced flow stress [22,23]

  3. (3)

    Heterogeneous Joule heating. Electrical resistivity is influenced by temperature, dislocation density, stacking faults, voids, and interstitials [24]. Deformation of a metal will create dislocations and stacking faults leading to increased resistivity. It is theorized that this effect will be greatest at grain boundaries due to the increased dislocation density resulting from the increased number of obstacles for dislocations to overcome. The increased local electrical resistivity will lead to hot spots within the metal leading to higher magnitude thermal softening at localized regions within the metal.

  4. (4)

    Dissolution of metallic bonds. Metallic bonds require the sharing of electrons in an electron cloud between ion cores. If electron sharing were to be decreased, then the bonds between ion cores could weaken or dissolve allowing for increased stress relief and decreased recrystallization times [25].

The objective of this research is to investigate the heterogeneous Joule heating theory through FEA. Electrical resistivity is modeled as a function of temperature and dislocation density using the specific resistivity of a dislocation density from the literature.

Model Setup and Evaluation

Four different finite element models were evaluated in ABAQUS 6.14 with its commercially available thermal-structural-electric implicit solver. The models were examined in increasing steps of complexity listed below. All four models represented 7075-T6 aluminum specimens with temperature-dependent material properties for the coefficient of thermal expansion, specific heat, thermal conductivity, electrical conductivity, and elastic modulus, taken from the JAHM material database. All properties were entered into ABAQUS as tabular data given at 10 K increments from 290–800 K.

Models examined:

  1. (1)

    Bulk model. Boundary conditions and meshing to match existing FEA models from the literature [12,13].

  2. (2)

    Scaled model. A small section from the central region of the tensile specimen was meshed with grains and grain boundaries and solved only during the electrical pulse. The initial conditions were given to match the bulk model's stress and temperature results prior to the electrical pulse.

  3. (3)

    Precipitate model. Equivalent to the Scaled Model but smaller in size with elevated electrical resistivity MgZn2 precipitates uniformly added.

  4. (4)

    Hexagonal model. Equivalent to the Scaled Model but using a hexagonal representation of grains and grain boundaries. This model was used to evaluate the sensitivity of grain boundary thickness due to the ability to easily scale the hexagon shapes.

All experimental data presented in this paper are from the author's previous work [26]. 7075-T6 aluminum specimens were cut according to ASTM E8 from a single sheet with a thickness of 1 mm. The front of the specimen was painted black outside of the grip area to allow temperature data acquisition through a FLIR A40 thermal camera (maximum temperature 580 °C, resolution 0.1 °C, sampled at 12 Hz). The specimens were deformed with an electrically insulated Instron 1332 servohydraulic tension/compression machine at a constant platen velocity of 2.5 mm/min. Electricity was applied with a nominal current density of 60 A/mm2 (applied current is calculated based on the initial specimen size and not adjusted throughout the duration of the test) in the gage region with a pulse duration of 1 s and pulse period of 60 s using a Darrah 4 kA power supply controlled through LabVIEW.

Model 1: Existing Model From the Literature, Bulk Model.

A half-symmetry model of an ASTM E8 dogbone specimen was created with a thickness of 1 mm in ABAQUS 6.14. The model setup is shown in Fig. 1, and is described below:

  • The bulk model was meshed with Q3D8 thermal-structural-electrical linear hexahedral elements with five elements through the thickness totaling 2200 elements and 3000 nodes.

  • The gripped region was held at constant ambient temperature (300 K), due to the high thermal conductivity of aluminum inside of large steel grips.

  • The exposed region was subject to both convection and radiation with a combined coefficient of 22.5 W/m2 K.

  • The left clamped region was fully fixed.

  • The right clamped region could only move in the axial direction and was given a velocity of 2.5 mm/min.

  • Electricity was applied to the left side and grounded at the right side of the specimen to give a nominal current density of 60 A/mm2 in the gage region.

  • The solution increment was set to 0.1 s.

  • The Johnson–Cook plasticity model was used for flow stress calculation.

  • A symmetry condition was applied at the center of the specimen in the length direction.

This model was used without electricity to evaluate the Johnson–Cook parameters for 7075-T6 found in Ref. [27]. For elevated temperature, the specimen was given a uniform and constant temperature boundary condition.

Johnson–Cook Parameter Evaluation.

The Johnson–Cook model is composed of three terms, and the full equation is shown in Eq. (1). The first term represents the flow stress of a metal in quasi-static uniaxial tension. The second term represents the strain rate effects, which will be absent in this model; all testing is done at the quasi-static rate with a strain rate insensitive metal (C = 0). The final term is temperature sensitivity. The parameters A, B, and n were validated using a room temperature tensile test, with the results shown in Fig. 2. To check parameter m, elevated temperature tensile tests were conducted by an outside lab at 150 °C (highest temperature reached during the electrically assisted experiments), the resulting stress–strain curve is shown in Fig. 2. The elevated temperature test model loses accuracy past 15% strain; most electrically pulsed tensile specimens studied in this work broke near this strain. The parameters used in the Johnson–Cook model are shown in Table 1, the yield stress, A, was modified to 500 MPa from 546 MPa to match experiment 
σ=A+Bεn1+Clnε˙ε0˙1T*m
(1)

Electrically Pulsed Bulk Model.

The bulk model predicted the correct temperature as shown in Fig. 3, but could not predict the correct stress during the electrical pulse. The predicted stress was higher than experiment, which matched the findings of Jones [11] and Hariharan et al. [12].

Electrical resistivity was modeled as a function of temperature and dislocation density; this model roughly represents a single crystal material without grain boundaries as a source for dislocation stacking, meaning the dislocation density in this model was lower than a model with grain boundaries.

It has been shown that electrical resistivity is composed of three main aspects: temperature or thermal resistivity, dislocation density-based resistivity, and stacking fault area-based resistivity [24]. As a modeling simplification, this work focused on dislocation density and thermal-based electrical resistivity (ignoring stacking fault resistivity) such that the total resistivity of each element was found using Eq. (2). Where ρe is the total electrical resistivity, ρt is the thermal resistivity, SRd is the specific resistivity of a dislocation density which is between 1.2–3.3 × 10−19 Ω cm3 for aluminum [2830] (3.3 × 10−19 was selected as the specific resistivity of a dislocation density in this work) and, Dd is the dislocation density 
ρe=ρt+SRdDd
(2)
Temperature-dependent electrical resistivity was modeled using a linear function shown in Eq. (3) [31]. Where ρ0 (=4 × 10−8 Ω m) is the room temperature resistivity of the unworked metal, αt (= 0.0039) is the temperature sensitivity of electrical resistivity, and ΔT is the temperature difference between the elevated temperature and room temperature 
ρt=ρ0(1+αtΔT)
(3)
Dislocation density was calculated using Eq. (4) [32], where σ is the flow stress, α is the thermal activation constant, b is the Berger's vector, μ is the shear modulus, and M is the Taylor factor. It has been shown that for aluminum, the thermal activation constant (α) and Taylor factor (M) are 1/3 and 3, respectively, leading to Eq. (5). This paper approached the electroplastic effect as a Joule heating phenomenon where the entire effect is thermal softening; as such, it was assumed that electricity would not interact with the thermal activation constant. In Eq. (5), an additional parameter, Gbd, was added as the dislocation density multiplier to compensate for dislocations stacking at grain boundaries. This term was used later in the grain boundary model, but was set to unity for the bulk model. Equation (5) resulted in an increase in electrical resistivity at the beginning of the electrical pulse which decreased as temperature increased and dislocations annihilated 
Dd=σαμbM2
(4)
 
Dd=Gbdσμb2
(5)
Combining Eqs. (1), (2), and (5), Eq. (6) was derived which represents the total electrical resistivity from dislocations and temperature. Table 2 shows the values the parameter values used in the below equation: 
ρe=ρ01+αtΔT+SRDGbdσμb2
(6)

Equation (6) was solved numerically at 100 °C and 50 MPa increments of temperatures and stress to develop tabular data of electrical conductivity. Then, a user-defined field subroutine was written to set electrical conductivity as a function of temperature and stress values. If the grain boundary factor was left as unity, then the dislocation density is not large enough to change the electrical conductivity, meaning that the temperature and stress results were as shown Fig. 3. If the grain boundary factor was increased, a larger effect on electrical conductivity was found. For example, a grain boundary factor of 17,000 was used, which resulted in a significant effect, shown in Fig. 4, which correlated to dislocation density of the order of magnitude 1019 (by contrast, with a grain boundary factor of 1, conductivity shows no relation to stress).

Model 2: Grain and Grain Boundary Model (Scaled Model).

To study the heterogeneous Joule heating theory, grains and grain boundaries were modeled during the tensile test. To reduce computational load while studying a microscale phenomenon, a small piece of the gage region of the tensile specimen was meshed with grains and grain boundaries and solved only during the pulse duration. The microscale model was assigned homogeneous properties (no grain boundary or dislocation density based properties) and checked to ensure it produced results similar to the bulk model during the electrical pulse. Then the grain boundary dislocation density multiplier was applied along with the subroutine for dislocation density and temperature-based electrical resistivity.

Grain size was determined to be 8.9±2.5 μm on an electron back scatter diffraction (EBSD) image of as-received 7075-T6 aluminum using the line intercept method with 20 repetitions. The grains and grain boundaries from half of the image were traced and used to create grain boundary partitions in the model, shown in Fig. 5. The specimen size was set to a rectangle with dimensions 1 × 0.09 × 0.01 mm to reduce computational load and to prevent patterning of the copied grain boundaries. The length was much greater than the width such that stress bands resulting from the end boundary conditions would not interfere with the grain boundary region. This part thickness assumed that modeling a single grain thickness was representative of a multigrain layered part. The thickness was to 0.01 mm to prevent aspect ratio errors from the greatly reduced mesh size (153,000 elements, one element through the thickness). The grain size model represented a small piece of the central gage region, shown in Fig. 6; this region was at a uniform axial stress level prior to the electrical pulse.

The model was evaluated for 3 pulses to match experiment before specimen fracture of the pulsed tensile test shown in Fig. 3. The scaled model's setup was the same as the bulk model with the following changes:

  • Current density was set to match the bulk model's current density prior to the electrical pulse.

  • A velocity boundary condition was applied at the right end and adjusted such that the strain rate of the scaled model matches the strain rate of the bulk model within the gage region.

  • The initial temperature and axial stress were set to the gage region temperature/stress of the bulk specimen prior to the electrical pulse.

  • The solution time increment was selected as 0.05 s.

  • An increment of 0.01 s was tested and found not to have an effect.

  • The mesh consisted of 153,000 elements and 302,000 nodes.

Figure 7 shows an example of the validation of the scaled model by comparing it to the bulk model and the experimental data only during the electrical pulses. The scaled model set to unity must match the bulk model to draw further conclusion when heterogeneous properties are applied. The experimental data are shown to demonstrate the difference the heterogeneous Joule heating theory must account for. The entire scaled model specimen reached a uniform temperature, which was expected since a region larger than the microspecimen heated uniformly in the bulk model (Fig. 7). The temperature and stress results at the end of the 3 electrical pulses (pulse time = 1 s) are shown in Table 3. Scaled model represents the validation of the scaled model compared with the bulk model; the material properties of this model are set to homogenous. Scaled model fit represents the results from the scaled model with dislocation density and temperature-dependent electrical resistivity with a dislocation density multiplier on grain boundary elements. The dislocation density multiplier was selected such that the scaled model could predict the same stress at the end of the 1 s pulse as experiment. This resulted in grain boundary multipliers near 17,000 for each of the 3 pulses. This assumed that the entire heterogeneous resistivity difference was caused by dislocations and ignored lattice misalignment at grain boundaries as well as stacking fault density. Figure 8 shows an example of the fitted scaled model compared to experiment and the bulk model. To predict the correct stress drop, the temperature at the end of the pulse (time = 1 s) was higher than the bulk temperature found during the experiment, shown in Table 3.

The fitted scaled model for the first pulse resulted in a temperature of 120 °C, an 8% increase over experiment and 17% increase over the bulk model's prediction. The fitted scaled model for the second pulse resulted in a temperature of 143 °C, a 17% increase over experiment and 20% increase over the bulk model. The fitted model for the third pulse resulted in a temperature of 164 °C, a 16% increase over the experiment and a 20% increase over the bulk model. If the initial stress was directly matched to experiment and all other parameters are assumed to be correct the required temperature for the stress drop increased to 127 °C, 14% higher than experiment for the first pulse, 149 °C, 22% higher than experiment for the second pulse, and 178 °C, 26% higher than experiment for the third pulse. The entire microscale model heated uniformly even though the heterogeneous resistivity field caused a nonuniform current density as shown in Fig. 9, similar results were found in Ref. [33]. The current was concentrated in areas where there are fewer grain boundaries and larger grains due to the lower resistance to electrical flow. Changing the grain boundary dislocation density multiplier does not change the current density distribution but does change the magnitude. However, even at higher heterogeneous current densities fields, above 200 A/mm2 the temperature field remained uniform. The grain boundaries are much smaller than the grains, and as such, the grains act as large heat sinks; if the grain boundaries heat up, the entire part does as well, assuming they produce enough heat to heat the larger grains.

The dislocation density to obtain the required stress drops was on the order of 1019/m2; this exceeds the range for dislocation density of severely worked metals (1018) [34]; at the first pulse, the axial strain was only 0.05, far from severely worked. The ratio of the electrical resistivity of the grain boundaries compared to the grains is 190 before the electrical pulse, much higher than 2.5–10 ratio found in the literature [35]. These results suggest that heterogeneous Joule heating is unable to fully account for the transient electroplastic effect. If the grain boundaries heat up enough to generate the appropriate softening to predict the transient stress drop, the predicted bulk temperature is higher than what is found in the experiment. The greater the strain when the electrical pulse is used, the greater the difference between the grain boundary model's temperature prediction and experiment. In addition, the bulk and scaled model's prediction of flow stress before the electrical pulse was typically around 10 MPa lower than experiment, which would result in an even higher required temperature in order to predict the stress drop if the scaled model started at the experiment's flow stress.

As the grain boundaries are traced from the EBSD image, the thickness was larger than what was realistic. This was explored by modifying the thickness of the grain boundaries in a specimen meshed with hexagon grains; see model 4.

Model 3: Precipitate Model.

7075-T6 is a precipitate-strengthened aluminum, where T6 is the strongest precipitated version of 7075 commercially available, strengthened through uniform precipitation throughout its matrix. The precipitates are intermetallic MgZn2. which has an electrical resistivity of 25.6 × 10−6 Ω cm [36]. As the precipitates are more resistive than the base aluminum, they may influence the current density and resultant temperature results. The precipitates' lower conductivity may force the current density to be near uniform, leading to a greater current density flowing through grain boundaries and a higher grain boundary temperature than the scaled model. To examine this effect, precipitates were added to a section of the scaled model from the Model 2: Grain and Grain Boundary Model (Scaled Model) section. The precipitates were assumed round and evenly dispersed with a diameter of 67 nm measured as the average size of the as-received material using a high-resolution transmission electron microscope (HRTEM) operating at 300 kV, shown in Fig. 10. Similar precipitate sizes were found for 7075-T6 in Ref. [37], the precipitate density was set to 15/μm2. The precipitates and grain boundaries used in the FEA model are shown in Fig. 11. The model is meshed with 235,000 linear tetrahedral elements and 454,000 nodes.

The model was tested with and without precipitates to determine the effect of precipitation on current density and resultant temperature field, the current density results are shown in Fig. 12. The current density field contours are the same for both models when precipitates are added there is a small increase in current density of around 1 A/mm2. There was no significant effect on resultant Joule heating or temperature distribution as the current density field was not more uniform in the presence of precipitates.

Model 4: Sensitivity of Grain Boundary Thickness.

The grain boundaries from the traced EBSD image are larger than realistic. As such, the sensitivity of the scaled model to grain boundary thickness is explored in this section. Hexagons were used in the place of traced grains for their stackability and ease of patterning in computer aided design software. The models were meshed with roughly 76,000 linear hexahedral elements and 152,000 nodes. The grain boundary thickness from the EBSD image was not uniform making a modification of this thickness with a uniform factor difficult. The grains were sized such that the tips of the hexagon touched the diameter of a 9 μm circle. The shape was then offset to create a grain boundary, followed by patterning across the specimen, an example is shown in Fig. 13.

The grain boundary thickness (tgb) was determined using Eq. (7) [38], where k is a material constant and dg is the grain size (9 μm), k was set to a midpoint value equal to 0.125 [17]. Other values of k were created around the midpoint to study the effect of grain boundary thickness, shown in Table 4. Temperature-dependent electrical resistivity for grain boundary elements was multiplied by a constant to simulate increased resistivity at grain boundaries. The grain boundary resistivity multiplier constant was found by fitting the flow stress between the scaled model and experiment; the results are shown in Table 4  
tgb=kdg
(7)

Smaller thickness grain boundaries required a higher electrical resistivity in order to predict the flow stress found during the experiment. This was due to the increasing grain/grain boundary size ratio that led to more heat being taken by the grains from the grain boundaries without significant temperature rise. The grain boundary thicknesses tested are on the order of 100 nm, while it is found experimentally that grain boundaries typically are in the 1–9 nm range [35]. This reduces the likelihood that heterogeneous Joule heating from the microscale model is responsible for causing the transient electroplastic effect. Grain boundaries that are larger than realistic values are unable to predict the transient stress drop with a reasonable electrical resistivity, making them smaller will increase the electrical resistivity at the grain boundaries.

A power curve fit model was found to have the best fit of the grain boundary multiplier data shown in Table 4; the resultant curve is shown in Fig. 14. This model predicted that using a realistic grain boundary thickness of 10 nm, the required grain boundary resistivity factor to predict the correct flow stress during the second electrical pulse of the experiment is 4400. This value is unrealistic but would be even higher if applied to the scaled model with EBSD grains since the current density is much less uniform.

This sensitivity study offered a simplified view of what would happen to the resistivity at the grain boundaries, though the resistivity multipliers seem low compared to the 190 found on the actual grain and grain boundary model shown previously in this paper. The reason for this is caused by the electrical current path. Since the hexagons were a patterned feature, the current density also followed a pattern, shown in Fig. 15. More grain boundary area in the hexagon model led to a lower required resistivity multiplier, yet the multiplier is still too high to be reasonable. Also, note that high current densities arise at the top of the hexagon part where the least number of grain boundaries and resultant electrical resistance exist.

The Mechanisms of the Transient Electroplastic Effect

The section examines existing theories for the transient electroplastic effect and their ability to explain the stress drop in a pulsed tensile test.

The Heterogeneous Joule Heating Theory.

This paper studied the microscale Joule heating theory and found it unable to fully account for the electroplastic effect. One of the primary arguments in favor of this theory was grain boundary melting observed by Fan et al. in brass [7]. However, given the results of this research, the authors postulate that the grain boundary melting was instead due to arcing across a microcrack or void that had formed in the metal during deformation unless the melting point of the metal were lowered (see theory at the end of this paper). If microcracks or voids were added to the model with increased electrical resistivity, similar to the grain boundaries (crack) or precipitates (voids) in this research, the electricity would avoid the cracks or voids due to their elevated electrical resistivity. If electricity was to flow across the cracks or voids, which may happen when a large number of cracks and voids are present, then high magnitude hot spots may form. This would not explain differences between model and experiment during early pulses at lower strain values.

A similar work modeled nanocrystalline titanium as grains and grain boundaries for a simple square geometry, with 20× resistivity on grain boundaries [33]. It was found that a complex current density field existed, similar to this research. However, it was also found that the grain boundary temperature increased only 0.1 °C above the surrounding grains; this work did not examine flow stress.

The Electron Wind Theory.

The electron wind theory is often dismissed as the potential cause for the electroplastic effect as multiple mathematical models have shown that the electron wind force is orders of magnitude too low to account for electroplastic stress reductions [11,22,39,40]. However, the electron wind itself is a well-documented phenomenon arising from the study of electromigration (EM). Electromigration is atomic diffusion driven by an electric field as current passes through a metal and is studied as a potential failure mechanism for thin film connectors in circuits [41]. Over a long period of time, the electron-driven diffusion leads to fracture of thin film electronic connectors. The driving mechanism behind EM is the electron wind, where collisions of electrons with ion cores result in mass diffusion toward the grounded side of the metal [42]. Over a long period of time, this leads to thinning and fracturing of the metal toward the side where electricity was applied. It is known that the electron wind in EM is magnified at grain boundaries due to increased obstacles and electron scattering from collisions. However, due to the long time-scale required for EM, it is unlikely that the electron wind is solely responsible for the transient stress drop from the electroplastic effect.

The Magnetoplasticity Theory.

The magnetoplasticity theory, as proposed to explain the electroplastic effect by Molotskii stems from the field of electromagnetic forming [22,23]. Electromagnetic forming is a forming technology that uses pulsed magnetic fields to apply forming loads to materials with high electrical conductivity [43]. Accomplished by applying a high frequency pulsed current, ranging from 5–100 kHz, to a metal coil placed near the workpiece. Most electrically assisted forming papers do not use coils of wire, and the pulse frequency is much lower than electromagnetic forming. This suggests that magnetoplasticity cannot be solely responsible for the electroplastic effect given the current pulse frequency and lack of coiled wires.

The Dissolution of Bonds Theory.

The dissolution of metallic bond theory can be understood as the electrical flow brought on by an applied current pushes extra electrons into the lattice, saturating the electron cloud allowing for reduced sharing between ion cores and subsequently improved ion core mobility within the electron cloud. In theory, if the current density is high enough, it may be possible to completely dissolve the bonds of the metal. The dissolution of bonds theory remains untested and still has potential to explain the electroplastic effect.

A Note on In Situ Transmission Electron Microscope Studies.

Recent in situ observation of dislocation motion in an electrically assisted tensile test under a transmission electron microscope (TEM) has shown that there is no significant change to dislocation motion in single crystal copper [40] and 5052 aluminum [39]. However, in the case of single crystal copper, the strain was held constant (no deformation) during the electrical pulse while dislocations were studied on the [1 1 0] zone axis [40]. The electron wind and magnetoplasticity theories as applied to deformation mechanics rely on dislocation motion assistance to overcome obstacles through electron collision based momentum transfer, meaning that this theory is not tested in the TEM studies since deformation (dislocation motion) is stopped during the electrical pulse. Instead, this TEM study is showing that electromigration does not exist during short duration electrical pulses.

The thickness of TEM specimens (typically near 100 nm) leads to some uncertainty on results from in situ TEM studies. The resultant surface area to thickness ratio may cause a significant skin effect from the electric current, where most of the current runs along the outer edges of the parts rather than through the thickness, resulting in the observation of a nonexistent electron-dislocation interaction.

Taking a TEM picture requires exposure time for the camera to process the current electron image, longer exposure times typically correlate to higher quality images. However, this presents a problem if true in situ dislocation observation during an electrically assisted deformation process is to be observed. If dislocations are continuously moving while a picture is attempted, the exposure time of the camera becomes a problem for imaging as the image is continually changing throughout the exposure time of the camera due to dislocation motion.

Electron Stagnation Theory.

It is the authors' view that a proper electroplastic effect theory must be able to explain five major phenomena, which show distinct differences between conventional bulk heating deformation and electroplastic methods. First, the general theory will be explained and then each of the five points below will be addressed.

  1. (1)

    Flow stress difference. The transient stress drop caused during pulsed tension, which cannot be predicted using Joule heating and thermal softening [11,12]. Along with the difference between furnace heated and electroplastic deformation (electroplastic has a lower flow stress) [7,14,16,17].

  2. (2)

    Deformation mechanisms and grain boundary melting. A shift in deformation mechanisms, most often observed in magnesium through elimination or reduction of twinning in the presence of electricity [44].

  3. (3)

    Threshold effect. Some metals will not have a reaction to electricity during electrically assisted deformation until a threshold current density is reached, after which, a large benefit is found, often observed as a large reduction in flow stress [45].

  4. (4)

    Cold work and grain size dependency. The influence of strain hardening and grain size on the electroplastic effect, smaller grains, and larger degrees of strain hardening increase the stress reduction from the electroplastic effect [46,47]

  5. (5)

    Time dependency. The absence of an electroplastic effect at high strain rate during the short time period tension testing. There must be a time dependency on the proposed electroplastic mechanisms.

The application of electric current to a metal will result in an increased number of electrons in the electron cloud proportional to the applied current density. The electrons will move at a drift velocity with emphasized directionality toward ground. As the electrons move through the metal, they will encounter precipitates, grain boundaries, dislocations, and other obstacles, all acting as impediments to electron flow.

Electrons flowing toward a precipitate of increased electrical resistivity will be forced to flow around the precipitate, similar to water flowing around a rigid body within its flow. At the central point where the water contacts the rigid body there is a stagnation point, where the velocity of the water reaches 0 and water behind it slows. A similar phenomenon may happen with electrons moving past obstacles. However, electrons do not flow in a straight line; instead, they move about in random patterns until colliding with an obstacle or other electrons, which redirect their path. This will result in electron stagnation points on the precipitate resulting in increased electron density, likely with an emphasis on the left side of the obstacle if ground is on the right side of the specimen/obstacle, shown in Fig. 16.

This increased electron density will reduce sharing of electrons by ion cores near the impeding edges of the precipitate. If there is an electron near the stagnation point, it will be hit with continuous collisions of electrons as they reach the stagnation point and slow, resulting in an increased electron density and dissolution or weakening of bonds. A similar reaction would occur when electrons encounter dislocations or other small obstacles; the increased electrical resistivity will force the electrons to move around the obstacle causing stagnation points on the obstacle and increased electron density.

As dislocations encounter voids, interstitials, and precipitates, dislocation loops may form around the obstacle leading to an effectively larger stagnation zone around the obstacle.

This effect would be magnified at ground boundaries as increased dislocation tangles and lattice misalignment will lead to increased electron stagnation and a greater electron density on the impeding side of the grain boundary. Due to the length of grain boundaries, the stagnation regions will be larger resulting in a greater electron density than a single obstacle such as precipitate. As such, it is expected that dislocation motion resistance will be lowest at grain boundaries.

The electron stagnation theory can be thought of using the following analogy. Think of sports fans (electrons) trying to leave a sporting event at a large stadium, there are a large number of fans that want to move toward the parking lot (electrical ground) from the main seating (grain region) when the game has ended. To get to the parking lot, they must move through a limited number of exits from the seating area (passable areas within the grain boundary). When everyone tries to leave the game, the constriction of the exits leads to a back-log and congestion of people behind the exit area and into a portion of the seating area. This is coupled with a decrease in the movement speed of the people due to the large number and close proximity of the people in the exit area. The same thing may happen at grain boundaries in a metal. Dislocations tangles and lattice misalignment at grain boundaries reduce the number of free paths (exits) for electrons to flow through the metal resulting in excess electrons that are slowed down near these exit points, similar to people slowly leaving a sporting event. This analogy will be used to address the five major phenomena presented earlier.

Flow Stress Differences.

A weakening of metallic bonds through dissolution due to an increased electron density near electron stagnation points on an obstacle would result in easier dislocation motion, leading to a stress drop beyond what is explainable through thermal softening and a lower flow stress in the presence of electricity compared to equivalent temperature furnace heated specimens.

Deformation Mechanisms.

If a large enough electron density due to electron stagnation appears near grain boundaries, the resultant weakening or dissolution of bonds may lead to localized changes in slip mechanisms from crystal slip to grain boundary sliding. This phenomenon may explain why magnesium does not twin as significantly in the presence of electric current [44].

If the dissolution of bonds and increased energy state is high enough at the grain boundaries, then it may be possible to witness grain boundary melting below the normal melting temperature of a given metal, similar to what was observed in Ref. [7]. With more of the bonds near the grain boundary already dissolved or weakened, the remaining bonds that must be broken through thermal application to achieve melting would be decreased. This would result in a lower melting temperature and the potential for grain boundary melting, though restricted to areas of elevated electron density, such as grain boundaries.

Threshold, Grain Size, and Cold Work Effect.

Using the stadium analogy, assume a stadium has 10 exits and 100 people in the stands. There will not be any traffic when the game ends and the people leave. If there are 1 million people in the stands at the end of the game, then suddenly there is going to be a large amount of traffic and a backlog of people near the exits. The traffic at the exits will be a function of the total number and size of the exits. However, local obstacles (precipitates and dislocations) in exit areas or in the seating area that must be passed to get to the exits will require fewer people to cause congestion at the exits and a backlog due to less straight paths to the exits.

In order for the electroplastic effect to be activated, there must be electron stagnation, theorized to be the most significant at the grain boundaries. If there are not enough electrons pushed through the metal to cause significant stagnation, then the electroplastic effect will be weak or absent. While a threshold current density has not been found for all metals, it is likely possible to experience a threshold on all metals, based on the treatment of the metal. Siopis et al. showed that increasing grain size increased the threshold current density in copper [47]. Larger grains have a lower grain boundary-grain area ratio (more exits for a given area), resulting in less electron stagnation across the metal for a given current density. As such, a larger current density is required to reach the required electron stagnation for dissolution of bonds to take significant effect. Size and density of precipitates prior cold work will also affect the threshold current density and electroplastic effect.

A large number of precipitates will result in more localized electron stagnation allowing for localized dissolution of bonds aiding in dislocation motion near these objects. A metal with a large number of precipitates would be expected to be absent of or possess a lower threshold current density compared to a similar metal with the same grain size and fewer precipitates. Cold work would present a similar effect, both through smaller localized regions where dislocations are present, along with grain boundaries where dislocation tangles tend to form. A metal that is heavily cold worked will have a lower threshold current density than the same metal absent cold work.

Time Dependency.

Using the stadium analogy, if a camera is watching the exits of a stadium as a game is ending, it will be able to see the progression and creation of traffic near the exits. If the camera only films for a couple of seconds after the end of the game, then it will not observe the traffic that will build up as more of the crowd of people begins to leave. However, if the camera films for a long period of time, it will observe a buildup of people near the exits, assuming there are enough people in the stadium to cause traffic at the exits. If there is not enough time given to create significant electron stagnation, it is unlikely that the electroplastic effect will be observed. As such, with short test times, such as the high strain rate testing in Ref. [47], there was likely a lack of an effect due to low-temperature rise and an inadequate amount of time for electron stagnation to develop.

Conclusions

The heterogeneous Joule heating theory for explaining the transient electroplastic effect is evaluated in this research through the creation of multiscale models which model the entire bulk tensile specimen along with a small region of the gage length where grains and grain boundaries are partitioned. Electrical resistivity is modeled as a function of dislocation density and temperature. The following conclusions are drawn:

  • With a Joule heat fraction of 1, the correct temperature profile can be predicted using the traditional conservation of energy approach, but this does not allow for the correct prediction of flow stress during an electrical pulse.

  • In order to predict the correct stress, drop in the presence of an electrical pulse based on increased resistivity at grain boundaries the bulk temperature of the part must exceed experiment. The resultant resistivity and dislocation density are outside of acceptable ranges. It is found that thinner grain boundaries require a higher electrical resistivity to produce a required stress drop, meaning the electrical resistivity of grain boundaries are likely higher than what was found in this work. The heterogeneous Joule heating is unlikely to be the only fundamental mechanism of the electroplastic effect.

  • A revision to the dissolution of bonds theory is explored where electron stagnation at obstacles and grain boundaries leads to localized weakening and dissolution of bonds, which is capable of explaining many of the electroplastic effect's nonthermal phenomena.

Future Work

In the future, the authors will examine possible methods to evaluate bond strength during an electrically assisted tensile test in order to investigate the dissolution of bonds electroplastic theory.

The influence of slip systems, lattice mis-aslignment, stacking fault density, grain boundary orientation (EBSD), and grain boundary angle will be added to the FEA models presented within this paper to produce an all-encompassing model for microscale Joule heating. The model will be applied to round dogbone specimens to reduce the potential for the skin effect.

Acknowledgment

The authors would like to thank Mr. Gary Lee Mathis for fabricating the fixtures used in this paper. The authors would also like to thank Dr. Taghi Darroudi and Hitachi High Technologies for their assistance and funding of the electron microscopy images seen in this work.

Funding Data

  • Hitachi High Technologies America.

  • National Science Foundation (DGE-1744593).

Nomenclature

     
  • A =

    yield stress

  •  
  • b =

    Berger's vector

  •  
  • B =

    strength coefficient

  •  
  • C =

    strain rate coefficient

  •  
  • dg =

    grain size

  •  
  • Dd =

    dislocation density

  •  
  • Gbd =

    grain boundary dislocation density multiplier

  •  
  • k =

    grain boundary thickness material constant

  •  
  • m =

    temperature sensitivity exponent

  •  
  • M =

    Taylor factor

  •  
  • n =

    strain hardening exponent

  •  
  • SRD =

    specific electrical resistivity of a dislocation density

  •  
  • T =

    temperature

  •  
  • tgb =

    grain boundary thickness

  •  
  • α =

    thermal activation constant

  •  
  • αt =

    temperature sensitivity of electrical resistivity

  •  
  • ε =

    strain

  •  
  • μ =

    shear modulus

  •  
  • ρ =

    electrical resistivity

  •  
  • σ =

    flow stress

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