Failure of thin metallic conductors under electric current loading: Transition from sharp crack to blow-hole

Engineering Fracture Mechanics 208 (2019) 221–237

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Failure of thin metallic conductors under electric current loading: Transition from sharp crack to blow-hole

T

⁎

Deepak Sharma, Praveen Kumar

Department of Materials Engineering, Indian Institute of Science, Bangalore 560 012, India

A R T IC LE I N F O

ABS TRA CT

Keywords: Blow-hole formation and propagation Electric current loading Heat affected zone Sharp crack propagation Self-induced electromagnetic force

Electric current surges of high densities, e.g., > 109 A/m2, may lead to failure of pre-cracked metallic conductors by propagation of either sharp crack due to self-induced electromagnetic forces or blow-holes (akin to blunt cracks) formed at the crack tip. In this paper, we discuss controlling transition from sharp crack propagation to formation and propagation of blow-holes in an edge cracked conductor by manipulating current density and crack length. Experiments revealed that a transition from sharp crack to blow-hole occurs when either the crack length was large or the current density was very high. Particularly, crack length and the current density have a symbiotic relationship, wherein an increase of one reduces the critical value of other required for blow-hole formation. Herein, the role of the heat affected zone (HAZ) ahead of the crack tip in the transition from sharp crack propagation to blow-hole formation was ascertained using the finite element method. It is proposed that the extent of HAZ, which depends on the crack length and the current density, directly controls the probability of formation of blow-holes and its size. Alike crack propagation, the blow-holes also propagated in predictable fashion under repetitive electric current pulse loading, thereby paving a path for using this phenomenon for cutting a material.

1. Introduction Electric current passes through a myriad of metallic structures containing cracks, such as thin film interconnects, exposed structure during lightning strikes, etc. As the electric current traverses around a crack or discontinuity, it gets concentrated near the crack tip; this is called current crowding. At very high current densities (> 109 A/m2), excessive and concentrated Joule heating generated due to the current crowding at the crack tip may induce full or partial melting or even evaporation of the material in the vicinity of the crack tip [1–5]. Once the material is molten, it may be removed or pushed away (sideways) by the electric current induced electromagnetic forces, resulting in the formation of a “blow-hole”. This effect is known as magnetic saw effect [1–5]. Several studies have been conducted in the last a few decades to analyze the magnetic saw effect, especially in the context of rail-guns and, in general, to understand the magnetic saw effect driven blow-hole formation. Furth et al. [1] first observed and described the phenomenon of magnetic saw effect in 1957, while conducting experiments on single-turn and helical coils placed under high transient magnetic fields. They observed that upon passing the magnetic pulses the temperature of the coil increased due to Joule heating (generated by the induced currents), and the regions around a coil edge or other vulnerable places, where some small faults were present, melted. Once the material melted, it was pushed aside by the magnetic field “pressure”, leaving behind a blow-hole. Such an action would further increase the local resistivity of the material near the blow-

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Corresponding author. E-mail address: [email protected] (P. Kumar).

https://doi.org/10.1016/j.engfracmech.2019.01.009 Received 24 October 2017; Received in revised form 28 December 2018; Accepted 7 January 2019 Available online 22 January 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.

Engineering Fracture Mechanics 208 (2019) 221–237

D. Sharma, P. Kumar

H I Im j KIE,d

Nomenclature of used symbols and acronyms α ε γs μ ρe ρ0, e ρm σ Ω a B Bnm

Bna Btm

Bta Cp E Fv f(a/w) g(l/w) h ht

Proportionality constant in linearized resistivity Strain tensor Surface energy Magnetic permeability Electrical resistivity of material Electrical resistivity of material at room temperature Density of the material Stress tensor Electrical conductivity Crack length Magnetic flux density Normal component of magnetic flux density in material Normal component of magnetic flux density in air Tangential component of magnetic flux density in material Tangential component of magnetic flux density in air Specific heat Electric field Body force Geometric factor which is a dimensionless function of (a/w) Geometric factor which is a dimensionless function of (l/w) Convective heat transfer coefficient Thickness of the sample

KIC k l l’ n q Qj Rb rM T Text Tt=0 Tm t tw Ti u We w Y FEM HAZ LEFM

Magnetic Field Electric current Maximum current in the sample Current density Dynamic stress intensity factor due to electric current Critical stress intensity factor of the material in Mode I Thermal conductivity Length of the sample Length of the sample on each side of crack unit vector Heat flux Heat generated due to Joule heating Size of blow hole Melt zone radius Temperature External temperature Initial temperature Melting temperature Time Width of the pulse Traction vector Displacement Strain energy at the crack tip Width of the sample Young’s Modulus Finite element method Heat affected zone Linear elastic fracture mechanics

hole region, thereby further concentrating the Joule heating, resulting in the formation of even larger blow-hole. Subsequently, additional metal near the blow-hole would melt and be pushed away, resulting in a saw-like mark. Thus, Furth et al. [1] named this phenomenon as the “magnetic saw effect” Magnetic saw effect is often prominent in the rail-guns, due to the passage of very high currents through projectile and the armature rails and presence of sharp corners and sliding components. Rail-guns are subjected to very large currents of the order of 106 A over very short period of time (e.g., a few milliseconds) [6–8]. Watt et al. [6,7] investigated the mechanism of the damage in the railguns due to these very high currents and reported melting and ejection of the material from the top and bottom edges of the throat region at high currents. At later stages, and at significantly higher currents, easily discernable cuts due to the magnetic saw effect formed in the throat region of a railgun armature [6]. The geometry of a cut or notch may further concentrate the electric current near the tip, leading to more pronounced magnetic saw effect upon subsequent loading. The electric current, then, makes deep cuts in the conductor, and if it is left unchecked, then it can slice up the entire component [9]. Furthermore, magnetic saw effect becomes prominent in the highly resistive materials due to excessive Joule heating [10]. A few research groups have investigated formation of blow-holes associated with magnetic saw effect in a pre-cracked conductor under the electric pulse loading. Satapathy et al. [2] investigated crack (or rather flaw akin to blow-hole) propagation behavior using a pre-notched 7075 aluminum alloy under pulsed electromagnetic loading. They observed that if the electric currents were above a threshold value, then a flaw grew ahead of the notch, probably, due to the melting of the material near the crack tip [2]. Blow-hole formed at the crack tip due to the magnetic saw effect once the current density was further increased, clearly indicating severe melting at the crack tip [2]. Interestingly, repeated application of electric current pulses on the same sample resulted in the continuous propagation of the flaw by forming new blow-hole at the tip of the previous one. In another work by the same research group, Gallo et al. [3,4] observed melting and cavity growth in a fatigue cracked 6061-T6 Al alloy when short duration pulses of high current density were passed through the sample. Herein, the authors quantified the size of the blow-hole formed ahead of the crack tip based on a singular current field estimate with an assumption that the size of the molten zone ahead of the crack tip is smaller in comparison to other length scales. The equation proposed by them is as follows [3]:

ρ [f (a/ w )]2 rM (t ) = e 2ρm Cp TM a

t

∫ j2 dt

(1)

0

where rM(t) is the radius of the melt zone after time t, a and w are the initial crack length and the sample width, respectively, j is the nominal or far-field current density applied into the specimen, ρe, ρm, Cp and Tm are the electrical resistivity, mass density, specific 222

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heat and melting temperature of the material, respectively, and f(a/w) is a dimensionless function of a/w given as:

2w π a ⎞ tan ⎛ π a ⎝2 w⎠

a f⎛ ⎞= ⎝w⎠

(2)

Although this set of works brought a major advancement in understanding of blow-hole formation due to the magnetic saw effect and its quantification, it does not provide insights into the direction of propagation of blow-hole upon repeated electric current loading. Also, the possibility of propagation of sharp cracks at a current density below a critical value and the conditions responsible for a transition of sharp crack into blow-holes were not addressed clearly. In a recent study exploring the constructive application of the magnetic saw effect, Sitzman et al. [5] produced fine cuts in prenotched metals (an Al-alloy) sheets by using non-contact magnetic cutting. Experimental setup used by Sitzman et al. [5] had a small electromagnet coil placed above the initial crack in the specimen, maintaining a gap between the coil and the sample with the initial crack [5]. As a pulsed electric current was passed through the coil, induced currents were generated in the sample with a high concentration near the crack tip. Due to the current intensification at the crack tip, the material in its vicinity melted due to excessive Joule heating, which was subsequently removed by the magnetic saw effect [5]. Authors showed that if the pulse duration and energy supplied during the process could be controlled, then the blow-hole could be propagated over long distances, and hence a sample can be cut into two pieces [5]. However, repeatability of straightness of the cut due to the propagation of blow-holes was not achieved in the aforementioned study because the cuts were irregular, and they propagated in a random manner with repeated magnetic pulse loading. Furthermore, the splitting of flaw was also reported at higher current densities [5]. However, the reason for splitting of cuts at random locations is still not very well understood, thereby limiting the application of this technique for machining samples. It is evident from the aforementioned discussion that at very high current densities blow-hole in a pre-cracked conductor can form ahead of the crack tip due to the magnetic saw effect. However, the conditions responsible for the formation of blow-holes, instead of propagation of sharp crack with repeated electric current pulse loading, are still not clearly understood. Furthermore, accurate prediction of the “cut-path” fabricated due to the propagation of the blow-hole has not been achieved so far, which hinders the potential application of electromagnetic cutting. This requires a methodological analysis of failure behavior of pre-cracked conductor under the action of very high current densities which can proffer quantitative analysis of (a) the conditions under which blow-holes can form, instead of propagation of sharp crack, (b) size of blow-hole and its correlation with important experimental parameters (such as crack length and current density) and (c) propagation behavior of blow-holes, including the exact direction of propagation, under repeated electric current loading. The above form the major goals of this work, which is achieved by employing a 2-pronged approach, involving numerical analysis using finite element method (FEM) and experimental investigation. 2. Procedure for numerical analysis and experiments 2.1. Numerical analysis Before initiating experimental investigation, a FEM study was performed using COMSOL Multiphysics® to observe the effects of the experimental parameters on self-induced electromagnetic force and Joule heating in a very thin conducting plate with an edge crack. Since, as it will be explained later, short duration electric current pulses were passed in the experiments to observe failure ahead of the crack tip, the same electric current pulse was used as a boundary condition in the FEM study. The profile of a

(a)

(b)

1.5

1

9

2

j (x10 ) (A/m )

Set pulse-width: 0.5 ms

0.5 0.5 ms 0 -0.2

0

0.2

0.4

0.6

0.8

Time (ms) Fig. 1. (a) Profile of a representative pulsed electric current passed through a very thin Al foil (of 12 µm thickness) with an edge crack. The shown electric current pulse had a pulse-width of 0.5 ms and corresponded to a far-field or nominal current density of 1 × 109 A/m2. (b) Schematic illustration of the geometry of the model used for FEM simulation, along with the applied boundary conditions. The boundary condition of j.n was applied at all boundaries, except for the locations of injection of current. 223

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representative electric current pulse is shown in Fig. 1a. The geometry for 2-D FEM model, along with necessary boundary conditions, is shown in Fig. 1b. As shown in Fig. 1b, an electric current pulse was passed through the sample by placing electrodes at two points across the crack. The aforementioned problem of ascertaining the effect of Joule heating, associated with the passage of electric current and induced currents, on the temperature profile near the crack tip was solved using “Electric Currents” and “Heat Transfer in Solids” Physics Modules of COMSOL Multiphysics®. Electromagnetic heat source was used in the model which couples the “Electric Currents” and “Heat Transfer in Solids” Physics Modules. A linearized electrical resistivity, given by ρe = ρ0, e (1 + α (T − T0)) , where T is the actual temperature, ρ0, e is the electrical resistivity of the material at room temperature (i.e., at T0 = 293.15 K) and α is the proportionality constant, was used to accurately calculate the electrical resistivity (and hence Joule heating) as function of temperature. Convective heat transfer was assumed at all the boundaries of the sample in the “Heat Transfer in Solids” Physics Module, where the value of heat transfer coefficient, h, was assumed to be equal to 5 W/m2K; this value of h is moderate and reasonable for the test condition (i.e., ambient condition). Heat radiation was ignored in the FEM simulations. Free triangular mesh elements were used for discretizing the model, wherein the finest mesh elements were used near the crack tip and the coarsest were used far away from the crack for computational efficiency (see Fig. 2a). As shown in Fig. 2b, mesh sensitivity analysis was performed, wherein any noticeable change in the value of the temperature at the crack tip was not observed upon further reducing the maximum element size by 40%. Below, a brief description of the governing equations, constitutive relations, and boundary conditions employed to estimate temperature field in the sample are given. The following set of equations was solved using the Electric Current Physics Module of COMSOL Multiphysics®:

E = −∇V

(3)

∇·j = 0

(4)

j = −ΩE

(5)

where E and V are electric field and electric potential, respectively and Ω is the electrical conductivity. It should be noted that Eqs. (3) and (4) are governing equations, whereas Eq. (5) is the constitutive equation relating electric field and the current density in a metallic conductor. The following boundary condition was imposed at all edges of the sample except at the locations where the electric current was injected:

j·n = 0

(6)

where n is the unit normal vector. At an instant, once the distribution of the electric current was obtained, it was fed into the Heat Transfer Module of COMSOL Multiphysics® to evaluate the temperature distribution in the sample. The following set of equations was solved in this module:

q = − k ∇T

Qj = Cp ρ

(7)

|j|2 (8)

Ω

∂T = −∇ ·k∇T + Qj ∂t

(9)

where q and k are heat flux and thermal conductivity, respectively, and Qj is the heat generated due to Joule heating. The above three equations were solved while satisfying the following boundary condition:

Fig. 2. (a) Discretization of the FEM model using free triangular mesh elements. Mesh elements were refined near the crack tip. (b) Variation of the temperature at the crack (or notch) tip as a function of the reciprocal of the maximum mesh size, showing saturation in the value of maximum temperature at very small mesh size, as enclosed by the rectangle. The arrow shows the size of the mesh used for analysis in this study. 224

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− n . q = h (Text − T )

(10)

and initial condition: (11)

Tt = 0 = 293.15K

where Text is the ambient temperature and Tt=0 is the initial temperature. As an electric current is passed as per the scheme shown in Fig. 1b, it changes its direction of flow across the crack. Such reversal of the electric current will lead to the generation of repulsive electromagnetic force (or Lorentz force) onto the two opposite faces of the crack, leading to the opening of the crack [11,12]. The generated electromagnetic force, which is a body force, would generate a (singular) stress field near the crack tip, giving rise to a stress intensity factor [11,12]. Now, when a short duration electric current pulse is applied, the rate of change of the electric field, dE/dt, remains non-zero in the beginning of the loading. Such a non-zero dE/dt leads to a rapid change in the magnetic field, which then produces induced current opposing the initially applied electric current. The interplay between the rapidly changing magnetic field and the effective applied electric current will load the pre-cracked conductor under a transient or dynamic electromagnetic force induced stress field. In this scenario, the stress intensity factor will also evolve with time, so that it is minimum at the onset of electric pulse loading and maximum a short while after steady state current is established [11]. Now, the stress field and the dynamic stress intensity factor, KIE,d, were solved numerically using “Electric Currents and Magnetic Fields”, and “Solid Mechanics” Physics Modules with time-dependent solver in COMSOL Multiphysics®. The electric current pulse and the sample geometry shown in Fig. 1a and b, respectively, were used in this segment of FEM study also. In the Electric Current and Magnetic Fields Physics Module of COMSOL Multiphysics®, the following governing equations, in addition to Eqs. (3)–(5), were solved: (12)

∇. B = 0

∇×E=−

dB dt

(13)

∇×H=j

(14)

where B and H are magnetic flux density and magnetic field, respectively, which are related by the following constitutive equation:

B = μH

(15)

where μ is the magnetic permeability. It should be noted that we have neglected dD/dt, where D is the electric flux density, term in the time-dependent general Maxwell equations while writing Eq (14). This assumption is reasonable as electromagnetic wave propagation effects are not important for the time scale involved in this problem. Above equations were solved in conjunction with the following boundary conditions, in addition to that given in Eq. (6):

Bnm = Bna

(16)

Hta

(17)

Htm

=

where the superscripts m and a represent material and air, respectively, and subscripts n and t represent normal and tangential directions, respectively. Once the electric and magnetic fields were obtained, electromagnetic force, FV, induced in the specimen at an instant was estimated by solving the following equation:

FV = j × B

(18)

The calculated induced electromagnetic force was then fed into the Solid Mechanics Physics Module of COMSOL Multiphysics® as an applied body force. The following governing equations were solved using Solid Mechanics Physics Module of COMSOL Multiphysics® assuming linear elasticity:

∇ ·σ + FV = ρm ε=

∂ 2u ∂t 2

(19)

1 [(∇u) + (∇u)T ] 2

(20)

where σ and u are stress tensor and displacement, respectively, and ε is the strain tensor. The above set of equations was solved in conjunction with the displacement boundary conditions shown in Fig. 1b. Since the electromagnetic force acts perpendicular to the crack face, the above stress field opens the crack in Mode 1 [11]. As mentioned previously, the stress field and hence the stress intensity factor evolve with time, i.e., they were transient or dynamic in nature. Herein, linear elastic fracture mechanics1 (LEFM) was applied to calculate the electric current induced dynamic stress intensity factor, K1E,d numerically [11]. In order to calculate K1E,d, J-integrals, as described below, were calculated around different 1 It can be inferred from the results on electric current induced fracture of thin films in references [11] and [12] that plasticity may not play a significant role in the crack propagation due to very short duration electric current pulse (50 μs and 0.5 ms) loadings. In these conditions, crack propagation occurred when the applied stress intensity factor (assuming LEFM) was greater than the fracture toughness of the material.

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contours in the FEM model:

J=

∫ ⎧⎨We ni − Ti ∂∂uxii ⎫⎬ dΓ − ∬A Fv ∂∂uxii dA ⎩

(21)

⎭

where We is the strain energy density and Ti is the i-th component of the traction vector. Once the J-integral was calculated, the following equation was employed to calculate the stress intensity factor:

KIE , d =

JY

(22)

where Y is the elastic modulus of the material. 2.2. Experimental procedure A schematic of the experimental setup used to study the effect of electric current pulse on the failure of pre-cracked thin metallic foils is shown in Fig. 3a. Short duration electric current pulses, with pulse-width of either 0.5 ms or 50 μs, were passed through an edge-cracked 12 μm thick Al foil specimen. These short duration electrical pulses were passed through the sample by using two types of power supplies: (1) a constant voltage pulsed power supply, which could pass an electric current pulse of 5–1000 A with a pulsewidth of 0.5 ms, and (2) a surge combination wave generator which could pass an electric current pulse of up to 2500 A with pulsewidth of 50 μs. One may refer to [11] for the details of these power supplies and their working principles. After passage of every electric current pulse through the sample, flaw propagation ahead of the crack tip was observed in-situ using an optical microscope. The mechanism of insertion of Al foil sample into the sample holder is described in Fig. 3b. Firstly, as shown in Fig. 3b(i), two acrylic sheets were placed close to each other so that, later, a cut using a sharp razor blade can be made through the space between them. Subsequently, as shown in Fig. 3b(ii), these acrylic sheets were joined by attaching a high strength tape to their bottom faces so that they do not move while performing subsequent steps of the sample preparation. A 12 μm thick Al foil of size 25 × 40 mm2 was carefully placed above the joined acrylic sheets, as shown in Fig. 3b(iii). Two high-density polymer (HDPE) blocks were placed at the ends of the Al foil, as shown in Fig. 3b(iv). These HDPE and acrylic sheets, besides supporting the Al foil, also provided electrical insulation for the sample from the rest of the fixture, so that electric current passed through the sample only. Subsequently, as shown in Fig. 3b(v), the entire assembly was inserted into the slot of the sample holder, which was then affixed by tightening the screws. Following affixing sample into the sample holder, the high strength tape used to join bottom faces of acrylic sheets in step 2 (see Fig. 3b(ii)) was carefully removed. Subsequently, small Al pads were carefully placed on the edges of the Al foil and were attached to the power supply using appropriate electrical connectors, as shown in Fig. 3b(vi). Finally, an edge crack was formed in the sample using a 200 μm thick razor blade. The length of the crack formed was accurately measured using an optical microscope before conducting an experiment. 3. Results and discussion 3.1. Propagation of sharp crack upon passing electric current Short duration (pulse-width = 0.5 ms) electric current pulses were passed through an edge-cracked conductor with a normalized crack length, a/w, of 0.5 to observe failure ahead of the crack tip under ambient conditions. Critical current density, ‘jc’, was identified in the experiments as the current density at which some detectable flaw propagation ahead of the initial crack tip was

Fig. 3. (a) Schematic illustration of the experimental setup designed to pass short duration electric current pulses through an edge-cracked thin conductor, and (b) steps involved in placing the thin metallic foil into the sample holder and pre-cracking the sample before passing an electric current pulse. 226

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observed. It should be noted that nominal current density was defined as follows:

j=

Im l′ht

(23)

where Im, l’ and ht are maximum current entering the sample, length of the sample on each side of the crack (see Fig. 1b for its definition), and thickness of the foil, respectively. As shown in Fig. 4, small crack propagation occurred at a critical current density of 1.2 × 109 A/m2 in the edge-cracked 12 μm thick Al sample. It should be noted that the crack propagation increased with the increase in the number of electric current pulses [11]. As explained earlier, reversal of electric current around a crack generates self-induced electromagnetic forces that try to open the crack in Mode I. This results in fracture of the material by propagation of sharp crack, as shown in Fig. 4. K1E,d was numerically calculated for the aforementioned test condition using the procedure described in Section 2.1, and it was equal to 5.35 MPa m−0.5. Since the fracture toughness of the Al foil, K1C, at the room temperature is 8.58 MPa m−0.5 [12], it appears that the fracture occurred in the sample sub-critically. This is contrary to the observations reported in our previous studies [11,12], wherein all experiments showed fracture only when KIE,d > KIC. It should be noted that all tests in previous studies [11,12] showing crack propagation upon application of only electric current were performed by dipping the sample in liquid nitrogen. One possible explanation for this apparent “premature” or subcritical fracture in this study can be based on the fact that the temperature ahead of the crack tip increased significantly when the tests were performed under ambient conditions (i.e., without using liquid nitrogen). As Young’s modulus of Al (or most of the metals) decreases with rise in the temperature [13], the material ahead of the crack tip upon passing electric current becomes softer, thereby decreasing the fracture toughness. It should be noted that during such a small period during which self-induced electromagnetic forces act on the sample for propagating the crack (< 0.5 ms), plasticity, which often requires dislocation movement in Al, may not play a significant role. Therefore, the fracture of the foil may occur at lower electromagnetic force and hence at a current density smaller than required at room temperature, as observed in Fig. 4. To support the hypothesis of decrease of the fracture toughness due to the temperature increase associated with Joule heating, temperature distribution in vicinity of the crack tip at applied critical current density (i.e., j = 1.2 × 109 A/m2) was determined using FEM, following the procedure described in Section 2.1. Fig. 5, which shows temperature contour plot for the above test condition, clearly reveals that the temperatures in the vicinity of the crack tip reached as high as 871 K, which is very close to the melting point of Al (∼930 K). At such high temperatures, the bond strength and hence the elastic modulus of Al can drastically reduce. At 871 K, Young’s modulus of Al becomes only ∼ 40 GPa [13], which corresponds to a decrease of ∼ 45% compared to the value of Young’s modulus at the room temperature (∼70 GPa). The back-of-envelope calculation using Griffith criterion for fracture, i.e., σf πa = 2Yγs , which gives a relationship between stress intensity factor (i.e., left-hand side of equation) and material properties (such as Y and γs, which is surface energy, on the right-hand side of equation), and assuming that the surface energy is proportional to the elastic modulus yields a fracture toughness value ∼4.89 MPam−0.5 at 871 K. If KIE,d for the aforementioned test conditions (i.e., ∼5.35 MPam−0.5) is compared with this updated fracture toughness value, the condition of KIE,d > KIC is satisfied. A few more experiments were conducted under ambient temperature conditions, where critical current densities (jc) required to initiate crack propagation was evaluated as a function of the initial crack lengths (a0). The obtained experimental results are shown in Fig. 6, from which it can be readily inferred that the required jc for propagating the crack decreased rapidly, akin to ∼1−a4 dependence, as the initial crack length of the sample increased. jc required to propagate crack at a/w of 0.9 was only 3.4 × 108 A/m2, whereas it was 1.2 × 109 A/m2 for a/w of 0.5. This decrease in the value of jc with a/w can be attributed to the following: (i) the value of KIE,d is proportional to the product of j2 and a1.5 [11], and hence an increase in one will reduce the other to attain the same stress intensity factor (which is required for crack to propagate) and (ii) an increase in the intensity and the size of the heat affected zone ahead of the crack tip with the crack length, as will be discussed in more detail in Section 3.4, will reduce the effective fracture toughness of the material.

Fig. 4. (a) Initial crack in 12 μm thick Al foil before the passage of the short duration electric current pulses. (b) Propagation of sharp crack after passage of 50 short duration 0.5 ms electric current pulses of current density of 1.2 × 109 A/m2. The images were taken in situ using an optical microscope. 227

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Fig. 5. Contour plot showing temperature distribution in an edge-cracked 12 μm thick Al sample upon passing a short duration electric pulse. The sample was kept under ambient condition. Inset shows a magnified view of the crack tip.

14 Pulse width = 0.5 ms Test condition: Ambient

2

8

C

10

8

j (x 10 ) (A/m )

12

6 4

4

j = k(1-(a/w) ); k = 13.3, R = 0.99 c

2 0.4

0.5

0.6

0.7

0.8

0.9

1

a/w Fig. 6. Experimental results showing effect of initial crack length on the critical current density (jc) required to initiate crack growth in an edgecracked 12 μm thick Al foil under the action of short duration 0.5 ms pulsed electric current. The broken curve shows the best fit curve for a 1−a4 dependence and the legend shows the equation for the best fit curve along with the value of regression parameter, R.

3.2. Transition of sharp crack into blow-hole Herein, we identify the test parameters responsible for transition of the mode of flaw propagation from sharp crack to blow-hole by systematically varying the crack length and the current density. In particular, short duration electric current pulses of large current density, greater than required for sharp crack propagation, were passed through the sample. Details of the experiments are as follows: j = varied from 1.2 × 109 A/m2 to 2.09 × 109 A/m2, pulse-width = 0.5 ms, a/w = 0.7, and the tests were performed under ambient conditions. Fig. 7 shows a few instances where blow-holes were formed upon application of only electric currents. As shown in Fig. 7a, propagation of sharp crack occurred initially when a pulse of current density of 1.2 × 109 A/m2 was passed through the sample. However, as the length of the crack reached a normalized length, a/w, of 0.8 after passage of several electric current pulses of the same current density, a transition from the sharp crack into blow-hole occurred. Moreover, as shown in Fig. 7b, if a current pulse of a larger current density of 2.09 × 109 A/m2 was applied at the beginning itself (i.e., when a/w = 0.7), a blow-hole directly formed at the original crack tip (i.e., propagation of sharp crack was not observed at all). Furthermore, as shown in Fig. 7, if the electric current pulse of the same density was continued to be passed through the sample, the size of the consecutive blow-hole increased with each electric pulse. These experiments confirm that the transition of sharp cracks into blow-holes occurred at higher crack lengths and higher current densities, and the size of the blow-holes became larger at longer crack length for fixed current density and at higher current density for fixed crack length. Now, in order to examine the aforementioned effects of a/w and j, and correlate them with the formation as well as the size of the blow-holes, it is important to estimate the extent of heat affected zone ahead of the crack tip; we discuss it next.

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Fig. 7. (a) Sharp crack propagation and subsequent blow-hole formation in an edge-cracked conductor with an initial normalized crack length, a/w, of 0.7. Several electric current pulses of nominal current density of 1.2 × 109 A/m2 were passed. The transition of crack propagation into blow-hole formation occurred when a/w became equal to ∼0.8. (b) Formation of blow-hole in an edge-cracked conductor with a/w = 0.7 upon passage of an electric current pulse of higher current density of 2.09 × 109 A/m2. The vertical arrow in both figures shows the direction of flow propagation. The original crack tip is shown by a pair of the horizontal arrows. The number written next to blow-hole corresponds to the sequence of the current pulse.

3.3. Determination of heat affected zone (HAZ) In order to understand the role of temperature rise associated with Joule heating in the transition of sharp crack into blow-hole, microstructural analysis of the tested sample shown in Fig. 7 at higher magnifications was performed using scanning electron microscope (SEM). Regions near the edges of the flaw in the samples with both small cracks (e.g., a/w = 0.5–0.8), wherein propagation of sharp crack occurred, and long cracks (e.g., a/w > 0.8), wherein blow-holes formed, were observed. As shown in Fig. 8, the spread and prominence of the heat affected zone (HAZ) in the samples with small a/w was relatively limited, and its extent increased considerably at longer a/w when current pulses of the same current density were passed. This observation reveals that the heat affected zone increased as flaw length (a/w) was increased. To understand the role of HAZ in formation of blow-holes, the spread of HAZ in the sample was calculated using FEM. Herein, HAZ was defined as the region near the crack tip where the temperature was ⩾ 500 K (∼0.5 Tm for Al). This temperature limit for defining HAZ for the test sample was selected as some of the relevant materials properties of Al changes significantly beyond 500 K. Fig. 5 shows temperature distribution in an edge-cracked conductor at the end of the passage of an electric current pulse of current density of 1.2 × 109 A/m2, whereas Fig. 9a shows the size of the HAZ formed ahead of the crack tip, as predicted by FEM simulations. Fig. 9b and c show the variation of the maximum and the average temperature in HAZ, as calculated using FEM, as a function of the applied current density and the initial crack length, respectively. As shown in Fig. 9b, an increase in the current density resulted in an increase in both the maximum temperature and the average temperature in the HAZ, thereby enhancing the intensity of HAZ.

Fig. 8. Representative micrographs of an edge-cracked 12 μm thick Al foil sample (shown in Fig. 7a) at various locations of the propagated flaw. The flaw propagated when electric current pulses of pulse-width of 0.5 ms and current density of 1.2 × 109 A/m2 were passed. The initial normalized crack length was 0.7. The insets show higher magnification SEM micrographs of the regions near the edges of the propagated flaw. A clear demarcation between the heat affected zone (HAZ), which appears brighter near the crack edges, and the far away unaffected material can be observed. The boundaries of HAZ in insets are shown by broken lines. 229

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Fig. 9. (a) Identification of the heat affected zone (HAZ), defined as the region in the sample where the temperature was ≥500 K, formed in the vicinity of the crack tip at the end of an electric current pulse. For this simulation, an electric current pulse of current density of 1.2 × 109 A/m2 and pulse-width of 0.5 ms was passed. Variation of maximum and average temperatures in the heat affected zone with (b) current density at the crack tip, jt, and (c) a/w.

Furthermore, the rate of increase of maximum temperature was slightly higher than the rate of increase of the average temperature in the HAZ. Also, as shown in Fig. 9c, the maximum temperature and the average temperature in the HAZ steadily increased with the crack length also; however, a steeper increase, especially in maximum temperature, was observed when the normalized crack lengths, a/w, became larger than 0.7.

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Fig. 10. (a) Variation of the size of heat affected zone, RHAZ, with normalized crack length (a/w) as predicted by FEM. (b) Variation of current density at the crack tip with a/w. (c) Variation of the size of heat affected zone with current density at the crack tip. (d) Effect of crack length on the experimentally observed velocity of flaw propagation. The crack velocity in (d) was calculated by dividing the distance travelled between 2 crack lengths (e.g., from a/w = 0.5 to a/w = 0.6) by the time taken and the datum point is shown against the average of the 2 crack length (e.g., a/ w = 0.55). 230

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As shown in Fig. 10a, FEM simulations showed that the size of HAZ (denoted as RHAZ, which is defined as the distance from the crack tip to the boundary of HAZ along y = 0) increased with the normalized crack length (a/w). HAZ increased gradually, albeit in a linear fashion, till a/w = 0.7; however, at the normalized crack lengths greater than 0.7, a rapid increase in the extent of HAZ with a/ w was observed. In order to analyze this trend of HAZ, variation of current density (jt) at the crack tip was observed as a function of a/ w. As shown in Fig. 10b, jt increased with a/w in a similar way as of the HAZ with a/w. This increase in jt with a/w is a geometric effect, as electric field lines become more concentrated at the crack tip as the distance between crack tip and the edge of the sample along y = 0 (i.e, unbroken ligament) becomes smaller with the increase in the crack length. The variation of HAZ with current density at the crack tip is shown in Fig. 10c. The linear variation of size of HAZ with crack tip current density, as shown in Fig. 10c, confirms that sudden sharp increase in the size of HAZ beyond a/w = 0.7 was mainly due to the sharp increase in the current density at the crack tip. Therefore, Figs. 9 and 10 reveal that both the extent and the intensity of the HAZ increase with crack length as well as current density. Fig. 10d shows the velocity of the flaw or crack propagation with a/w. Fig. 10d readily reveals that the velocity of the stable (and sharp) crack propagation, which occurred at low a/w, was rather slow and it rapidly increased when the blow-hole started forming at higher a/w. The experimental observation of the transition of sharp crack into blow-hole, which propagated rapidly with a/w, seems to be consistent with variation of size of HAZ with a/w, as shown in Fig. 10a, and its intensity, as shown in Fig. 9c. The sharp increase in the extent and intensity of the HAZ enhances the probability of blow-hole formation, as observed in Fig. 7a. This explains the observation of occurrence of the transition from the sharp crack into blow-hole at larger a/w. Therefore, the formation of blow-hole is primarily attributed to the extent and intensity of HAZ. Moreover, this also explains the rapid decrease in critical current density required to initiate crack propagation at longer initial crack length, and an increase in the size of the blow-hole with increase in the crack (or rather flaw) length upon passage of the same current density, as observed in Fig. 7. 3.4. Blow-hole: effects of current density and crack length Herein, the effect of nominal (or, far-field) current density, j, and normalized crack length on the formation of blow-hole will be discussed. In this context, only those blow-holes will be discussed, which directly formed at the initial crack tip, i.e., without any sharp crack propagation. Details of the experiments are as follows: j = varied from 3.75 × 109 A/m2 to 7.5 × 109 A/m2, pulsewidth = 50 µs, a/w = 0.5, and the tests were performed under ambient conditions. As shown in Fig. 11a, when the pulse of current density of 3.75 × 109 A/m2 was passed through a sample, sharp crack with partial melting was observed at the crack tip. It is remarkable to note the shape of the apparent HAZ as shown in Fig. 11b is similar to that obtained in the FEM simulations (see Fig. 9a). As the current density was increased to 5 × 109 A/m2, the region in the vicinity of the crack tip melted and was pushed towards the edges of the crack by the self-induced electromagnetic forces, resulting in the formation of a blow-hole with material pile up at the periphery (see Fig. 11b). This micrograph clearly shows the push of the molten material resulting in a pile up due to the electromagnetic forces, as expected due to the magnetic saw effect. As shown in Fig. 11c, the whole region in the vicinity of the crack tip melted (and perhaps partially evaporated) and was blown away by the self-induced electromagnetic forces upon further increase in the current density to 7.5 × 109 A/m2. This resulted in formation of a very clean and sharp blow-hole. Details of the experiments conducted to evaluate the effect of initial crack length on the blow-hole formation at the crack tip were as follows: j = 3.8 × 109 A/m2, pulse-width = 50 µs, a/w = varied from 0.4 to 0.8, and tests were performed under ambient conditions. As shown in Fig. 12a, at a/w = 0.4, a very small blow-hole with partial melting at the crack tip was observed upon passing a current pulse. As the a/w ratio was increased to 0.6, the region in the vicinity of the crack tip melted upon passing the same electric current pulse and, as shown in Fig. 12b, was pushed at the edges of the crack by electromagnetic forces. Upon further increase in the a/w ratio to 0.8, the whole region in the vicinity of the crack tip melted and was moved away by the self-induced electromagnetic

Fig. 11. Micrographs showing crack propagation and/or blow-hole formation at the original crack tip upon passage of one electric current pulse of 50 μs pulse-width and current density of: (a) 3.75 × 109 A/m2 (sharp crack propagation with well-defined HAZ), (b) 5 × 109 A/m2, and (c) 7.5 × 109 A/m2. Normalized crack length (a/w) in all the cases was 0.5. Vertical arrows show the crack tip in as-fabricated sample. 231

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Fig. 12. Micrographs showing a blow-hole formed at the original crack tip upon passing one electric current pulse of 50 μs pulse-width and current density of 3.8 × 109A/m2 through edge-cracked Al foils with normalized crack length, a/w of (a) 0.4 (very small blow-hole of size ≈10 μm), (b) 0.6, and (c) 0.8. Vertical arrows show the initial location of the crack tip in the sample before the passage of an electric current.

forces, resulting in the formation of a clean and sharp blow-hole (see Fig. 12c). A comparison of Figs. 11 and 12 clearly reveals that the effects of current density and crack length on the formation of blow-hole are qualitatively identical. For estimating the spread of the heat affected zone (RHAZ) formed ahead of the crack tip, an expression for the RHAZ was deduced using FEM analysis, as follows (see Supplemental Material 1 for various graphs used to deduce Eq. (24)): 2 2 j 2 ρe tw RHAZ ⎡f ⎛ a ⎞ ⎤ ⎡g ⎛ l ⎞ ⎤ = ⎢ ⎥ a 2ρm Cp Tm ⎣ ⎝ w ⎠ ⎦ ⎣ ⎝ w ⎠ ⎦

(24)

where tw is the width of the pulse, and f(a/w) and g(l/w) are dimensionless geometric factors. These dimensionless geometric parameters can be given as follows:

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π a −1 π a ⎛k1 ⎞ tan ⎛k1 ⎞ ⎝ 2 w⎠ ⎝ 2 w⎠

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current loading used in this study). Assuming that the size of the blow-hole is proportional to the extent of the HAZ, the following can be written to estimate the size of blow-holes, Rb: (27)

Rb = CRHAZ

where C is a constant and the value of RHAZ is given by Eq. (24). It should be noted that Rb has been defined as the distance from the original crack tip to the boundary of blow-hole along y = 0. Interestingly, besides the fact that Eq. (27) also accounts for the specimen length to width ratio (i.e., g(l/w)) and represents a current loading configuration wherein the current is passed across an edge crack, it is consistent with Eq. (1). Eq. (27) can be used to qualitatively ascertain the variation of the size of the blow-hole formed under experimental conditions as function of important experimental parameters, such as j and a/w. Variation of the experimentally measured size of the blow-hole with current density is shown in Fig. 13a, which is in reasonable accordance with the predictions of both Eqs. (1) and (27). It should be noted that if a value of C equal to 0.52 is used, then Eq. (27) accurately predicts the variation of blow-hole size as function of current density. A value of C smaller than 1 is expected as the size of the blow-hole should be smaller than HAZ. Similarly, as shown in Fig. 13b, variation of Rb with a/w qualitatively agrees with both Eqs. (1) and (27) (using C = 0.52 as obtained from curve fitting exercise using datum points shown in Fig. 13a). Here also, Eq. (27) performs slightly better in terms of quantitative prediction as compared to Eq. (1). It should be noted that prediction of Eq. (1) is quantitatively accurate for small a/w; however, the prediction becomes worse at higher a/w values. This can be attributed to the fact that Eq. (1) assumes small melt zone ahead of the crack tip relative to other length scales. However, as previously discussed, this assumption is not valid for samples with longer crack lengths (a/w > 0.7), because excessive current crowding at the tip of longer crack lengths would result in very large melt zone. In summary, the excellent match between Eq. (27), which is based on Eq. (24), and the experimental results obtained in this study indicates that the extent of the HAZ can be considered to directly influence the size of the blow-hole formed upon passage of an electric current pulse. 3.5. Propagation of blow-holes with repeated electrical loading Effect of repeated short duration electric pulses on the edge-cracked conductor when the current density or crack length was high enough to form a blow-hole was studied experimentally and numerically. Details of the experiments are as follows: j = varied from 3.9 × 109 A/m2 to 5.5 × 109 A/m2, pulse-width = 50 µs, a/w = 0.5, and the tests were performed under ambient conditions. Alike crack propagation (see Fig. 4) [11,12], blow-holes also grew with repeated pulse loading, as shown in Fig. 14. Therefore, the sample can be sliced into two pieces by propagating blow-holes, if a sufficient number of electric pulses (of optimized current density) are applied. This may proffer propagation of fine-tuned sized blow-holes as a viable tool for machining material. Fig. 14b shows that edges of the blow-holes were non-uniform, leading to the formation of myriad features, e.g., splitting, curvature, etc. Initially, the blow-holes propagated in a curved fashion and later at higher a/w ratio blow-hole propagation became straighter, as shown in Fig. 14b. In order to further understand the propagation behavior of blow-holes, experiments and accompanying FEM simulations were conducted at low and high current densities, as described next. 3.5.1. At low current density Passage of electric current pulses of low current density (however, large enough to form blow-holes instead of sharp crack propagation) resulted in the formation of smaller blow-holes with non-uniform edges at the crack tip. This was because the molten material could be only gently pushed sideways to the edges of the blow-hole by the self-induced electromagnetic forces, as described in Section 3.4 (see Fig. 11). Experimental results conducted at low current densities (e.g., j < 4 × 109 A/m2) are shown in Fig. 15, which suggests that (n + 1)th blow-hole was formed at the edge of nth blow-hole. Microstructurally aware FEM simulation using COMSOL Multiphysics® was conducted to quantitatively understand the

Fig. 14. (a) Initial crack before the passage of short duration electric current pulses. (b) Blow-hole propagation after passage of 40 electric current pulses. Formation of myriad features, e.g., splitting of blow-holes, curvature in the cut produced by the propagation of blow-holes, etc., can be easily observed in (b). 233

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Fig. 15. Propagation of blow-holes after (a) n and (b) n + 1 electric current pulses. Non-uniform blow-hole formed at the end of nth as well as (n + 1)th electric current pulse.

propagation of blow-holes at lower current densities. SEM micrographs, as shown in Fig. 16a, shows a blow-hole at the crack tip with non-uniform edges. These non-uniform edges were formed due to the deposition or pushing sideways of the molten material at the edges of blow-hole by the self-induced electromagnetic forces (see Fig. 11b and 12b for high magnification micrographs). Current density and temperature distribution in the vicinity of blow-hole were obtained using the FEM simulation, wherein the micrograph showing the region near and including blow-hole was imported for building the FEM model (see Fig. 16b). A comparison of Fig. 16a and b shows that SEM micrograph was reasonably imported in the FEM software; however, it lost information pertaining to the height profile (i.e., out-of-plane features) while creating the 2-D flat geometry. FEM simulations shown in Fig. 16d and e show that the current density, as well as temperature, was maximum at the two ends of the farthest edge of the blow-hole from the crack tip. This results in melting of the material at any one of these regions; however, not at both locations as formation of the blow-hole at the first location would reduce the driving force at another location under low current density loading, and hence the formation of only one blow-hole could be expected upon passage of the next electric current pulse. This causes blow-hole to propagate in a zig-zag manner, as the blow-hole can be (statistically) formed at any one of the ends of the farthest edge of the blow-hole.

Fig. 16. Steps taken for performing microstructurally aware FEM: (a) SEM micrograph showing the blow-hole formed ahead of the original crack, (b) importing SEM micrograph into COMSOL Multiphysics® as 2-D flat geometry to be used in FEM, (c) discretized FEM model (using triangular mesh), and FEM results showing distribution of (d) current density and (e) temperature in the vicinity of the blow-hole. 234

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Fig. 17 shows the experimental results and FEM simulation results, side by side, validating the hypothesis proposed above in the context of Fig. 16. As the nth electric current pulse was passed, the frontier blow-hole formed having non-uniform edges (see Fig. 17a). The non-uniform edges of the nth blow-hole uniquely concentrated the current density and hence very high temperature was produced in the sharpest region at the edge of blow-hole, as shown in Fig. 17b and c, respectively. As shown in Fig. 17d, as the (n + 1)th electric current pulse was passed through the sample, the new blow-hole formed at the exact location where the current density and hence the temperature were the highest. Therefore, by estimating the temperature profile at the boundary of the existing blow-hole the growth of the next blow-hole can be predicted with high certainty, especially if the high concentration region/spot is unique. 3.5.2. At high current density Experiments conducted at high current densities showed formation of larger blow-holes with very sharp edges, as molten material ahead of the crack tip was completely blown away by the electromagnetic forces (see Section 3.4 for detailed discussion). A representative micrograph showing the propagation of blow-holes at high current density is shown in Fig. 18. At low current density, as shown in Fig. 15, propagation of blow-holes occurred along a zig-zag path, with limited instances of splitting. On the other hand, as shown in Fig. 18, relatively straighter path with several splitting was observed at higher current densities, even though the initial crack length was same in this case as that in the low current density case. In order to understand the aforementioned current density based significant differences in the propagation behavior of blowholes, microstructurally aware FEM simulation was performed by importing micrographs of the sample showing the region in vicinity of the blow-hole as geometry in FEM simulation (see Fig. 16 for procedure). Some of the FEM simulation results at high current density are shown in Fig. 19. Fig. 19 shows that the blow-hole formed upon passage of very high current density had very sharp regions at the edges due to complete removal of material from the blow-hole region, resulting in excessive concentration of electric currents at such locations; these current density values were significantly higher than those attained upon passing current pulses of low current density. Next electric pulse would simultaneously form blow-holes at all such locations, as energy provided in this case (i.e., pulse with high current density) was large enough to simultaneously form blow-holes at two (see Fig. 19a) and three (see Fig. 19b) locations. Therefore, at very high current densities energy provided was sufficient to cause multiple splitting in the blowhole propagation after a single electric pulse. Based on discussion presented in this section, it can be concluded that by identifying locations of high current density (and hence temperature concentration), the direction of blow-hole propagation can be predicted. This can be used to further develop the idea of electromagnetic machining [5,14] based on the formation and propagation of blowholes also. 4. Summary o A thin metal foil with an edge crack may fracture via propagation of sharp crack upon passage of pulsed electric current. The critical current density required to initiate crack propagation decreased with increase in the initial crack length. Reversal of electric current around the crack tip leads to generation of repulsive electromagnetic force on the two faces of the crack, leading to propagation of crack in mode I. For sharp crack propagation, the current density should be above a minimum value and less than the critical value required for blow-hole formation. o Sharp crack propagates incrementally during successive electric current pulse loading till the crack length reaches a critical value,

Fig. 17. (a) Optical micrograph showing the propagation of blow-hole after passage of n electric current pulses through an Al foil sample having an edge crack. Microstructurally aware FEM simulation results showing the expected distribution of (b) electric current density and (b) temperature upon passage of (n + 1)th pulse. (d) Optical micrograph showing the propagation of blow-hole, including the new blow-hole formed after passage of (n + 1)th electric current pulse. 235

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Fig. 18. Propagation of blow-holes upon passage of electric current pulse of high density of 5.5 × 109 A/m2. The horizontal arrow in the micrograph shows the location of the original crack tip. Initial a/w was 0.5.

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Fig. 19. Two examples of propagation of blow-holes, showing splitting, at high current density of 5.6 × 109 A/m2. Set of (i) optical micrograph showing propagation of blow-hole after passage of n electric current pulses through a 12 μm thick Al foil sample having an edge crack, (ii) expected distribution of current density as obtained from microstructurally aware FEM simulations upon passage of (n + 1)th pulse and (iii) optical micrograph showing the propagation of blow-hole, including the new blow-hole formed after the passage of (n + 1)th electric current pulse, at (a) 2 and (b) 3 probable locations.

beyond which it transitions into blow-hole. o Formation of blow-holes was enhanced if the current density was increased substantially more than that required for sharp crack propagation. Similarly, formation of blow-hole became more probable at a current density if the crack length was increased. With increase in the current density as well as the crack length, the current density at the crack tip increased. Due to this increase in the 236

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crack tip current density, the temperature in the vicinity of the crack increased, thereby aiding the formation of blow-holes. o Size of blow-holes monotonically increased with the extent and intensity of the heat affected zone ahead of crack tip and hence it increased with the current density and the crack length. The size of the blow-hole was proportional to the square of the current density and the crack length. o At low current densities, blow-holes propagated in zig-zag path with fewer instances of splitting, whereas at high current densities they propagated in straighter path with splitting at multiple locations. The direction of blow-hole propagation can be ascertained by observing the current crowding locations, thereby making blow-hole based electromagnetic machining a viable technique for machining. Acknowledgements This work was financially supported by Council of Scientific & Industrial Research (CSIR), India through a project funded to PK (Grant number CSIR 0366). Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.engfracmech.2019.01.009. References [1] Furth H, Levine M, Waniek R. Production and use of high transient magnetic fields. II. Rev Sci Instrum 1957;28(11):949–58. [2] Satapathy S, Stefani F, Saenz A. Crack tip behavior under pulsed electromagnetic loading. In: Electromagnetic launch technology. 2004 12th symposium on, 2005, IEEE; 2004. p. 106–10. [3] Gallo F, Satapathy S, Ravi-Chandar K. Melting and crack growth in electrical conductors subjected to short-duration current pulses. Int J Fract 2011;167(2):183–93. [4] Gallo F, Satapathy S, Ravi-Chandar K. Melting and cavity growth in the vicinity of crack tips subjected to short-duration current pulses. IEEE Trans Magn 2009;45(1):584–6. [5] Sitzman AJ, Stefani F, Bourell DL, Trevino E. Use of the magnetic saw effect for manufacturing. IEEE Trans Plasma Sci 2014;42(5):1173–8. [6] Watt T, Stefani F. Experimental and computational investigation of root-radius melting in C-shaped solid armatures. IEEE Trans Magn 2005;41(1):442–7. [7] Watt T, Bryant M. Microstructures in the throat region of recovered aluminum-alloy armatures. IEEE Trans Magn 2007;43(1):422–5. [8] James TE. Current wave and magnetic saw-effect phenomena in solid armatures. IEEE Trans Magn 1995;31(1):622–7. [9] Chen T, Long X, Dutta I, Persad C. Effect of current crowding on microstructural evolution at rail-armature contacts in railguns. IEEE Trans Magn 2007;43(7):3278–86. [10] Melton D, Watt T, Crawford M. A study of magnetic sawing in an aluminum bar. IEEE Trans Magn 2007;43(1):170–2. [11] Sharma D, Reddy SB, Kumar P. Electromagnetic force induced fracture of pre-cracked thin metallic conductors. Int J Fract 2018, 2018,;212(2):182–204. [12] Sharma D, Reddy SB, Kumar P. Fracture of pre-cracked metallic conductors under combined electric current and mechanical loading. Int J Fract 2018;212(2):167–82. [13] McLellan RB, Ishikawa T. The elastic properties of aluminum at high temperatures. J Phys Chem Solids 1987;48(7):603–6. [14] Kumar P, Mishra A, Watt T, Dutta I, Bourell DL, Sahaym U. Electromagnetic jigsaw: Metal-cutting by combining electromagnetic and mechanical forces. Procedia CIRP 2013;6:600–4.

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