#format jsmath
= Deflection of Shallow Spherical Shells =

Many papers and books, such as [[ www.amazon.com/dp/0071742476 | Roark's Formulas for Stress and Strain ]] cite Reissner, E.: "Stresses and Small Displacements of Shallow Spherical Shells", Journal of Mathematics and Physics, vol. 25, No. 4, 1947.  Part 1, pp. 80-85, Part 2 pp pp 279-300.  This paper is not online, and the 66 y.o. journal is not on most shelves.  Dr. Reissner's article is somewhat difficult to understand, but I will attempt to do so here.  Corrections welcomed!

Reissner considers three cases:
 . (1) Point load at center, edges fixed in angle position and angle
 . (2) Point load at center, edges constrained vertically but free to turn and splay outwards
 * (3) Distributed uniform load in disk area around center, edges constrained vertically but free to turn and splay outwards

Curved-surface deployment resembles case (3).  Not exactly - thinsats are not circular and a detailed finite element model will be needed someday - but an estimate helps.

|| $ v $ || Poisson's ratio - typically 0.33 for aluminum ||
|| $ R $ || radius of curved surface in meters            ||

MoreLater about Reissner paper.


== Guesstimates based on geometry ==

There will be two kinds of thinsat:
 . (1) '''overcurved''' - curved more than the stack average
 . (2) '''undercurved''' - curved less than the stack average

When an overcurved thinsat is flattened into a stack, the outer edges are stretched and the inner disk is compressed.  The opposite happens when an undercurved thinsat is curved extra onto the stack.  Assume the thinsats are round, radius $ r_t $, and that the boundary between compression and stretch is somewhere between $ 0.5 \times r_t $ (equal radial distance) and $ 0.7071 \times r_t $ (equal areas).