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Minimal energy shapes of closed, elastic shells with twelve pentagonaldisclinations introduced in otherwise hexagonally coordinated crystallinelattice are studied. The geometry and the total energy of shells are studied asa function of the elastic properties of the material they are made of.Particular emphasis is put on the buckling transition of the shells, that is astrong preference of the shell shapes to 'buckle out' in spatial regions closeto the pentagonal disclinations for certain range of the elastic parameters ofthe problem. The transition effectively increases the mean square aspherity ofshapes, making them look more like an icosahedron, rather than a sphere whichis a preferred shape prior to the onset of the transition. The properties ofthe buckling transition are studied in cases when (i) the total volume enclosedby the elastic shell has to be fixed and when (ii) there is an internalpressure acting on the shell. This may be related to maturation process innon-enveloped dsDNA viruses, where the insertion of the genetic material in apre-formed protein shell (viral coating) may effectively impose the fixedvolume and/or pressure constraint. Several scenarios that may explain theexperimentally observed feature of mature viruses being more aspherical(facetted) from their immature precursors are discussed and new predictions forthe elastic properties of viral coatings are obtained on the basis of thepresented studies.Comment: Version that is published in Phys. Rev. E 73, 061915 (2006). Elastic  parameters of viral coatings (HK97) reestimated (Y_3D=1.1 GPa

Buckling transition in icosahedral shells subjected to volume conservation constraint and pressure: Relations to virus maturation