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Two types of unstable eigenmodes of resistive tearing instabilities, namely, symmetric and anti-symmetric modes, in a double current sheet configuration are analyzed by means of both an eigenvalue solver and initial value simulation. It has been clearly identified that these two types of eigenmodes are different from the two independent single tearing modes even though the symmetric eigenmode in a double current sheet configuration shares the same properties of the single tearing mode near each current sheet. In the case with finite separation Δx between two current sheets, an arbitrary phase disturbance between two current sheets can lead to “phase instability,” i.e., the transition from the symmetric mode to the anti-symmetric mode. For a large Δx limit, both anti-symmetric and symmetric modes share the same properties of the single tearing mode. Thus, the superposition of two independent single tearing modes with arbitrary phases and arbitrary amplitudes at two current sheets can become the linear combination of symmetric and anti-symmetric eigenmodes. The same growth rate/eigenvalue of symmetric and anti-symmetric eigenmodes infers that an arbitrary superposition of two independent single tearing modes is still the eigenmode of the double current sheet configuration.

Solid state amorphization is induced by shock, irradiation and deformation, while deformation induced complete amorphization remains a challenge in a bulk solid. Brittle-to-ductile transition (BDT) mechanism is elusive at loading speeds of m/s at nanoscale depth of cut. Existing formula has no effects of shape and radius of cutting edges on the critical depth of cut at BDT. In this study, a new route of deformation induced complete amorphization at nanoscale is proposed in a bulk solid confirmed by transmission electron microscopy (TEM). This is performed by a novel approach of ultraprecision grinding, conducted on a specially designed setup. The grinding is carried out by a developed single diamond grain with a cutting edge radius of 2.5 μm, at depth of cut of 24 nm under a loading speed of 40 m/s. BDT takes place at depth of cut of 419 and 172 nm for Si (100) respectively, ground by single diamond grains with tip radii of 5 and 2.5 μm correspondingly. A new model is suggested for BDT, considering the effects of radius and shape of cutting edges. The findings provide new insights for design and fabrication of high performance devices used in flexible electronics, nanodevices, microelectronics and optoelectronics.