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Shape transition of unstrained flattest single-walled carbon nanotubes under pressure

Single walled carbon nanotube's cross section can be flattened under hydrostatic pressure. One example is the cross section of a single walled carbon nanotube successively deforms from the original round shape to oval shape, then to peanut-like shape. At the transition point of reversible deformation between convex shape and concave shape, the side wall of nanotube is flattest. This flattest tube has many attractive properties. In the present work, an approximate approach is developed to determine the equilibrium shape of this unstrained flattest tube and the curvature distribution of this tube. Our results are in good agreement with recent numerical results, and can be applied to the study of pressure controlled electric properties of single walled carbon nanotubes. The present method can also be used to study other deformed inorganic and organic tube-like structures.