A treatment of the formation of a basic core fibril (shish) of the type that is ge nerated by fl ow-indu ced crystallization of a polymer from solution is given that features the concept of cu mulative strain. Multiple nucleation acts by flow-elongated molecules produce an e mbryonic fibril that is a connec ted set of bundlelike nucle i. Surface s tress resulting from repu ls ion of the quasi-random coil c hains in the amorphous zone between the nucle i or c rystallit es builds up at the bundl e e nds as the nucle i mature, lead ing ultimately to a high e nd surface free e nergy, and to volume strain in th e crysta llit es compri sing the core fibril. The theory leads to a s table (or meta sta ble) fibril diameter as and mea n c haracteri sti c length i s with a fix ed ax ial ratio, a nd pred icts why the diameter does not grow further even in a med ium that is supersaturated with polymer. Th e pred ic ted dependence of a s, ls, and th e axia l ratio, on und ercoo ling is ill approximate agreement with expe riment. The latti ce expa nsion in the crystal resulting from volume s tra in is a lso in fair accord with ex perim e nt. The effect of a nn ea ling, including the commonl y e ncou ntered case whe re the volume strain relaxes to give normal la tti ce dimensions, but with a high end surface energy still remaining, is not ed. The effect of vo lume strain and the distribution of core fibril lengths about is on the melting be havior is calculated. The theory ca n reprod uce c rys tallinity versus te mperature data on polye thylene fibrils. Thi s procedure yields an indepe nde nt value of i8 • The ove rall treatment implies that the core fibril is a set of concatenated and substantially exte nded-c hain crystallites with bundlelike ends and a somewhat expanded lattice when una nnealed and under te nsion, the molec ular co nnections between the crystallites consisting of short amorphous ci liary bridges. It is s uggested that pro longed annealing at high temperatures ca n remove a substantial numbe r of the amorphous zones .Key words: Core fibril ; c umulative stress; fl ow-indu ced c rys talli zalion; nucl eation Iheory; polye thyle ne; polymer fiber ; shi sh; vo]um e strain.