Experimental Observation and Modelling of the Electroplastic Effect in Nonferromagnetic Ductile Metals

Abstract

In this study, an innovative methodology to experimentally distinguish the ElectroPlastic (EP) effect from the thermal expansion and thermal softening was developed and applied to validate the proposed theoretical electro-mechanical model. Two series of quasi-static uniaxial tension experiments with titanium and copper were conducted and a quadratic dependence was observed of the EP strain on the electric current density, corrected for increased dislocation density levels. This study experimentally confirmed the existence of EP phenomenon in ductile metals. The significance of the non-EP effects in the electro-mechanical coupling problems was estimated and the experimental procedure limitations were thoroughly discussed as well as the sensitivity and uncertainty, associated with each of the model parameters. Conclusions and suggestions for future work supplement the study. This piloting work on electro-mechanical coupling provides a solid foundation for further modelling and experimental observation of the electroplastic effect.

Appendices

Appendix

Lorenz force (pinch effect) in the electromechanical coupling of ductile metals

Electromagnetic forces in a current carrying conductor

The current induced magnetic field exerts an additional magnetic stress on the current carrying conductor. This effect is referred to as the Lorenz force, named after the Lorenz force law, or as the pinch effect, a term from the plasma physics which describes the self-constriction of a beam of accelerated charged particles. Additionally, an electrostatic force may occur from the unbalanced nonequilibrium surface charge distributions.

For nonferromagnetic metallic conductors with relative permittivity ϵ_r = 1 and relative permeability μ_r ≈ 1, the above-mentioned electromagnetic phenomena can be described with the Maxwell equations in a free space (vacuum) as follows:

$$ \nabla \cdotp \underset{\_}{\mathbbm{E}}={\rho}_q/{\epsilon}_0 $$

(A.1)

$$ \nabla \times \underset{\_}{\mathbbm{B}}-{\epsilon}_0{\mu}_0\underset{\_}{\dot{\mathbbm{E}}}={\mu}_0\underset{\_}{j} $$

(A.2)

$$ \nabla \times \underset{\_}{\mathbbm{E}}+\underset{\_}{\dot{\mathbbm{B}}}=0 $$

(A.3)

$$ \nabla \cdotp \underset{\_}{\mathbbm{B}}=0 $$

(A.4)

where $\underset{\_}{\mathbbm{E}}$ is the electric field vector, $\underset{\_}{\mathbbm{B}}$ is the magnetic induction field vector, ρ_q is the charge density per unite volume, $\underset{\_}{j}$ is the current density vector, ϵ₀ = 8.854 × 10⁻¹² N · V⁻² is the vacuum permittivity, and μ₀ = 1.257 × 10⁻⁸N · A⁻² is the (magnetic) permeability of free space.

The Maxwell equations are used to determine the magnetic and electric fields induced by the electric current that passes through a metallic material. The force density per unit volume (${\underset{\_}{f}}_{EM}$), exerted by those fields on the current-carrying conductor is calculated by the Lorenz force law:

$$ {\underset{\_}{f}}_{EM}={\rho}_q\underset{\_}{\mathbbm{E}}+\underset{\_}{j}\times \underset{\_}{\mathbbm{B}} $$

(A.5)

Another approach for calculating the electromagnetic force is through the Maxwell stress tensor ($\underset{\_}{\mathbbm{T}}$), defined as follows:

$$ {\mathbbm{T}}_{ik}={\epsilon}_0\left({\mathbbm{E}}_i{\mathbbm{E}}_k-{\delta}_{ik}\frac{1}{2}{\mathbbm{E}}^2\right)+\frac{1}{\mu_0}\left({\mathbbm{B}}_i{\mathbbm{B}}_{\mathrm{k}}-{\delta}_{ik}\frac{1}{2}{\mathbbm{B}}^2\right) $$

(A.6)

where δ_ik is the Kronecker delta.

In the next two sections both approaches, the Lorentz force law and the Maxwell stress tensor, are used to estimate the mechanical stresses, induced by the electromagnetic forces in a long titanium wire. It is expected that the results obtained from both approaches should be the same since the Maxwell stress tensor is derived from the Lorentz force law.

Problem statement: A steady electric current with density j = 500 A/mm² flows through a long titanium wire with radius R = 1mm.

Maxwell stress tensor

We choose the direction of the main axis (x₁) of the Cartesian coordinate system (${\hat{x}}_1,{\hat{x}}_2,{\hat{x}}_3$) along the wire axis (and the current direction). The magnetic induction field ($\underset{\_}{\mathbbm{B}}$) inside the conductor is calculated from the following expression:

$$ \underset{\_}{\mathbbm{B}}=\left\{\begin{array}{c}{\mathbbm{B}}_1=0\\ {}{\mathbbm{B}}_2=-\frac{\mu_0j}{2}{x}_3\\ {}{\mathbbm{B}}_3=\frac{\mu_0j}{2}{x}_2\end{array}\right\} for\ \left({x}_1^2+{x}_2^2\right)\le {R}^2 $$

(A.7)

The electric field inside the conductor is uniform:

$$ \underset{\_}{\mathbbm{E}}=\mathbbm{E}{\hat{x}}_1={\rho}_{el}\underset{\_}{j} $$

(A.8)

where ρ_el is the resistivity of the material (ρ_{el, Ti} = 4.2 × 10⁻⁷Ω · m).

The Maxwell stress tensor equation (equation A.6) is then simplified to the following:

$$ \underset{\_}{\mathbbm{T}}=\frac{1}{2}\left\{\begin{array}{ccc}{\epsilon}_0{\mathbbm{E}}^2-{\mu}_0^{-1}{\mathbbm{B}}^2& 0& 0\\ {}0& -{\epsilon}_0{\mathbbm{E}}^2+{\mu}_0^{-1}\left({\mathbbm{B}}_2^2-{\mathbbm{B}}_3^2\right)& 2{\mu}_0^{-1}{\mathbbm{B}}_2{\mathbbm{B}}_3\\ {}0& 2{\mu}_0^{-1}{\mathbbm{B}}_2{\mathbbm{B}}_3& -{\epsilon}_0{\mathbbm{E}}^2+{\mu}_0^{-1}\left({\mathbbm{B}}_3^2-{\mathbbm{B}}_2^2\right)\end{array}\right\} $$

(A.9)

The Maxwell stress tensor is a physical quantity in the electromagnetic theory that resembles the stress tensor in mechanics. Yet, the Maxwell stress tensor does not directly represent the mechanical stress on the material caused by electro-magnetic forces. Computation of the electromagnetic contribution to the mechanical stress state is a boundary value problem, and the Maxwell stress tensor could be used to determine the electromagnetic force density (${\underset{\_}{f}}_{EM}$) as follows:

$$ {\underset{\_}{f}}_{EM}=\nabla \cdotp \underset{\_}{\mathbbm{T}} $$

(A.10)

The electromagnetic force density (${\underset{\_}{f}}_{EM}$) inside the conductor is then calculated by substitution of equation (A.9) into equation (A.10).

$$ {\underset{\_}{f}}_{EM}=-\frac{\mu_0{j}^2}{2}\left\{\begin{array}{c}0\\ {}{x}_2\\ {}{x}_3\end{array}\right\} $$

(A.11)

As evident from equation (A.11) the contribution of the electric field (i.e. the applied voltage) to the calculated electromagnetic forces is cancelled out. Furthermore, the current-carrying wire is not subjected to an additional axial force, such force may occur as an elastic response to the straining in radial direction from magnetic forces.

Equation (A.11) is verified with the Lorentz force law in the following section.

Lorenz force law

We define a cylindrical coordinate system ($\hat{r},\hat{\theta},\hat{z}$) with z-axis along the applied current direction (the wire axis). The magnetic force density is calculated from equation (A.5) as follows:

$$ {\underset{\_}{f}}_{EM}=\underset{\_}{j}\times \underset{\_}{\mathbbm{B}}=-\frac{\mu_0{j}^2}{2}\left\{\begin{array}{c}r\\ {}0\\ {}0\end{array}\right\} $$

(A.12)

Equation (A.12), when converted to a Cartesian coordinate system (${\hat{x}}_1,{\hat{x}}_2,{\hat{x}}_3$), gives the same results as equation (A.11).

The compressive stress (σ_r) acting on the current-carrying wire (the pinch effect) is estimated by integration of equation (A.12) over the wire cross-section as follows:

$$ {\sigma}_r=-\frac{\mu_0{j}^2}{4}\left({R}^2-{r}^2\right) $$

(A.13)

Converted to Cartesian coordinates (${\hat{x}}_1,{\hat{x}}_2,{\hat{x}}_3$) equation (A.13) results in:

$$ \sigma =-\frac{\mu_0{j}^2}{4}\left\{\begin{array}{ccc}0& 0& 0\\ {}0& \frac{\left({R}^2-{r}^2\right)}{r}{x}_2& 0\\ {}0& 0& \frac{\left({R}^2-{r}^2\right)}{r}{x}_3\end{array}\right\},r=\sqrt{x_2^2+{x}_3^2}\le R $$

(A.14)

Substituting j = 500 A/mm², R = 1mm, and x₂ = x₂ = 0 into equation (A.13) gives the maximum compressive electromagnetic stress exerted on the current-carrying conductor:

$$ \min {\sigma}_r\left(r=0\right)=-\frac{\mu_0{j}^2}{4}{R}^2=-7.85\times {10}^{-4}\ MPa $$

(A.15)

When compared to the yield strength of the material, the electromagnetic stress is six magnitudes smaller. Furthermore, the electromagnetic stress is in the radial direction, only a small portion of it is expected to propagate in the axial direction. For precise computation of the axial stress and comparison with the electroplastic stress drop, thorough analysis of the entire mechanical system is required. For a constrained elastic system with the same geometry, the averaged axial contribution of the electromagnetic forces (∆σ_z) was estimated as follows:

$$ \Delta {\sigma}_z\approx -\frac{{\nu \mu}_0{j}^2}{2}\frac{1}{R^2}\underset{0}{\overset{R}{\int }}r\left({R}^2-{r}^2\right) dr=-6.2\times {10}^{-6}\ MPa $$

(A.16)

where ν is the Poisson ration (ν_Ti = 0.32).

Conclusion

The Maxwell stress tensor cannot be simply added to the mechanical stress tensor. Solutions of boundary-value problems and more calculations are necessary to convert the information expressed with the Maxwell stress tensor to mechanical stresses.

In our estimations, the electromagnetic force acting on a current-carrying conductor was found to be magnetic in essence, the electric (electrostatic) part in the Maxwell stress tensor calculations was redundant.

The pinch effect was found to be negligible. The contribution of the electromagnetic forces (the pinch effect) to the mechanical stress state was estimated to be six magnitudes smaller than the yield strength of the material. However, the pinch effect may have to be considered if the applied current density is significantly stronger and the cross-sectional area of the specimen is larger.