  Geometrically frustrated assembly has emerged as an attractive paradigm for
understanding and engineering assemblies with self-limiting, finite equilibrium
dimensions. We propose and study a novel 2D particle based on a so-called
"warped jigsaw" (WJ) shape design: directional bonds in a tapered particle
favor curvature along multi-particle rows that frustrate 2D lattice order. We
investigate how large-scale intra-assembly stress gradients emerge from the
microscopic properties of the particles using a combination of numerical
simulation and continuum elasticity. WJ particles can favor anisotropic ribbon
assemblies, whose lateral width may be self-limiting depending on the relative
strength of cohesive to elastic forces in the assembly, which we show to be
controlled by the range of interactions and degree of shape misfit. The upper
limits of self-limited size are controlled by the crossover between two elastic
modes in assembly: the accumulation of shear with increasing width at small
widths giving way to unbending of preferred row curvature, permitting assembly
to grow to unlimited sizes. We show that the stiffness controlling distinct
elastic modes is governed by combination and placement of repulsive and
attractive binding regions, providing a means to extend the range of
accumulating stress to sizes that are far in excess of the single particle
size, which we corroborate via numerical studies of discrete particles of
variable interactions. Lastly, we relate the ground-state energetics of the
model to lower and upper limits on equilibrium assembly size control set by the
fluctuations of width along the ribbon boundary.
