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. 2013 Jul 22;110(32):12893–12898. doi: 10.1073/pnas.1301695110

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Fig. 1. — Our model system: a circular, crystalline sheet of radius W adhered to a spherical substrate of radius . Colors of contours indicate the level of hoop stress: tension (red), vanishing stress (white), and compression (blue). (A) The rest (stress-free) state of the sheet is shown above a schematic of the confinement geometry. (B) The axisymmetric state for confinement : The hoop stress is compressive in a confined zone of width near the perimeter. (C) The wrinkle pattern, in which the hoop stress vanishes in the confined zone as the bendability . Wrinkles require small deformation of the spherical substrate with the attached sheet. (D) The scar pattern that maintains the imposed spherical shape, in which the stress approaches a profile identical to that of C as the defectability . The shaded area indicates the range beyond which dislocation interactions are effectively screened by the boundary . (E and F) The stress fields of the axisymmetric, unstable state (B) and the compression-free state (C and D). (G and H) Profiles of the axisymmetric and compression-free stresses for a negative-curvature (saddle) substrate, in which both radial and hoop stresses collapse in the center of the sheet.

Inline graphic — Our model system: a circular, crystalline sheet of radius W adhered to a spherical substrate of radius . Colors of contours indicate the level of hoop stress: tension (red), vanishing stress (white), and compression (blue). (A) The rest (stress-free) state of the sheet is shown above a schematic of the confinement geometry. (B) The axisymmetric state for confinement : The hoop stress is compressive in a confined zone of width near the perimeter. (C) The wrinkle pattern, in which the hoop stress vanishes in the confined zone as the bendability . Wrinkles require small deformation of the spherical substrate with the attached sheet. (D) The scar pattern that maintains the imposed spherical shape, in which the stress approaches a profile identical to that of C as the defectability . The shaded area indicates the range beyond which dislocation interactions are effectively screened by the boundary . (E and F) The stress fields of the axisymmetric, unstable state (B) and the compression-free state (C and D). (G and H) Profiles of the axisymmetric and compression-free stresses for a negative-curvature (saddle) substrate, in which both radial and hoop stresses collapse in the center of the sheet.