The transition zone near wrinkles in pulled spherical membranes

Abstract

The unwrinkled transition zone between the rigid boundaries and the wrinkled portion of a spherical membrane subjected to a pulling force is analyzed. The strains are first assumed to be small, and two cases are considered: (a) a spherical membrane barrel pulled into a wrinkled cylinder and (b) a hemispherical membrane pulled into a wrinkled cone. It is shown that in both cases the angular size of the transition zone is O(ε), where ε is the prevailing strain, rather then o(√ε) as in the usual nonlinear membrane edge effect problem. Large rotations and substantial variations in the circumferential stresses and strains take place in this narrow zone. The extension to the case of “moderate” strains is then made for the spherical barrel, using a power series approach. It follows from the results that the size of the transition zone becomes O(√ε) as the wrinkled region shrinks to zero. Finally, a complete large strain analysis of a partly wrinkled spherical barrel membrane is made. Included are both the transition zone and the interior wrinkled region. The analysis establishes the ranges of validity of the previous solutions and demonstrates the deterioration of the “edge effect”. A formula for the “wrinkle strain” in the interior is included. Interestingly, the deviation of the slope ᾱ of the deformed membrane from the winkled direction is shown to be always small, so that the problem can be assumed to be geometrically linear in cot ᾱ, but materially nonlinear.