Superconducting MgB₂ Wire Drawing Considering Anisotropic Hardening Behavior and Hydrostatic Effect

Abstract

Numerical modeling was conducted to investigate the deformation behavior of powder mixture during multi-pass drawing processes for multi-filamentary MgB₂ wire. A modified Drucker-Prager Cap (DPC) model with an elliptical cap surface using the new material characterization method was developed to capture the anisotropic hardening behavior and hydrostatic effect of the powder mixture. A number of uniaxial die compaction, cold isostatic pressing, diametrical compression, and uniaxial compression tests were conducted using different powder densities to characterize the modified DPC model. A commercial finite element software ABAQUS with a user subroutine was used to simulate the drawing of the MgB₂ wire. The density and area fraction of the powder mixture during the wire-drawing process were verified with experimental results. The difference in packing density at the inner and outer filaments of the MgB₂ wire was successfully captured by simulation. In addition, the effect of the initial packing density on the superconducting properties of MgB₂ wire was numerically studied. It is shown that the increase in the superconducting area, which results from a high initial packing density, should be more effective compared to the increase in the grain connectivity in enhancing the critical current properties for the MgB₂ wire when the final packing density is saturated after a number of drawing processes.

Graphic abstract

Appendix

Based on the incremental elastoplasticity theory, the plastic strain increment $\Delta {\varepsilon }_{ij}^{pl}$ is written as

$$\Delta {\varepsilon }_{ij}^{pl}=\Delta \lambda \frac{\partial Q}{\partial {\sigma }_{{\varvec{i}}{\varvec{j}}}}$$

(21)

where $\Delta \lambda$ is a plastic multiplier indicating the magnitude of plastic deformation and $Q$ is a plastic potential function that is continuously differentiable. In Chapter 2, the plastic potential functions of the modified DPC model are expressed as a function of the first and second stress invariants. The Eq. (21) can be expressed by the following equation by hydrostatic-deviatoric decomposition.

$$\Delta {\varepsilon }_{ij}^{pl}=\Delta \lambda \left(\frac{\partial p}{\partial {\sigma }_{ij}}\frac{\partial Q}{\partial p}+\frac{\partial q}{\partial {\sigma }_{ij}}\frac{\partial Q}{\partial q}\right)=\frac{\Delta \lambda }{3}\frac{\partial Q}{\partial p}{\delta }_{ij}+\frac{3\Delta \lambda }{2q}\frac{\partial Q}{\partial q}{S}_{ij}$$

(22)

where $p$, $q$, ${\delta }_{ij}$ and ${S}_{ij}$ are the hydrostatic pressure, the von Mises equivalent stress, the Kronecker delta, and the deviatoric stress tensor, respectively.

The plastic multiplier, $\Delta \lambda$ which is an unknown parameter, can be determined by the following consistency conditions for the return-mapping method.

$$\Phi =F\left({{\varvec{\sigma}}}_{n}+\Delta{\varvec{\sigma}}\right)-\rho \left({\overline{\varepsilon }}_{n}+\Delta {\overline{\varepsilon }}^{pl}\right)=F\left({{\varvec{\sigma}}}_{n+1}^{trial}-\Delta {\lambda }_{n+1}{{\varvec{C}}}^{{\varvec{e}}}\frac{\partial Q\left({{\varvec{\sigma}}}_{n+1}\right)}{\partial{\varvec{\sigma}}}\right)-\rho \left(\Delta {\overline{\varepsilon }}_{n}^{pl}+\Delta {\lambda }_{n+1}\right)=0$$

(23)

where $F$ is the yield function for the modified DPC model, $\rho$ is the hardening function, ${{\varvec{\sigma}}}_{n+1}^{trial}={{\varvec{\sigma}}}_{n}+{{\varvec{C}}}^{{\varvec{e}}}\Delta{\varvec{\varepsilon}}$ is the trial stress, and ${{\varvec{C}}}^{{\varvec{e}}}$ is the elastic tangent modulus.

In the iterative process, the potential residual $R$ is checked by following violated consistency conditions at each iteration $(k)$. When the potential residual is smaller than the tolerance, the stress integration procedure ends.

$$R=F\left({{\varvec{\sigma}}}_{n+1}^{trial}-\Delta {\lambda }_{n+1}^{(k)}{{\varvec{C}}}^{{\varvec{e}}}\frac{\partial Q\left({{\varvec{\sigma}}}_{n+1}^{(k)}\right)}{\partial{\varvec{\sigma}}}\right)-\rho \left(\Delta {\overline{\varepsilon }}_{n}^{pl}+\Delta {\lambda }_{n+1}^{\left(k\right)}\right)-\Phi$$

(24)

More information about the stress update algorithm can be found in [51,52,53,54,55].