Geometric mechanics of periodic pleated origami

Origami structures are mechanical metamaterials with properties that arise almost exclusively from the
geometry of the constituent folds and the constraint of piecewise isometric deformations. Here we
characterize the geometry and planar and nonplanar effective elastic response of a simple periodically
folded Miura-ori structure, which is composed of identical unit cells of mountain and valley folds with
four-coordinated ridges, defined completely by two angles and two lengths. We show that the in-plane and
out-of-plane Poisson’s ratios are equal in magnitude, but opposite in sign, independent of material
properties. Furthermore, we show that effective bending stiffness of the unit cell is singular, allowing us to
characterize the two-dimensional deformation of a plate in terms of a one-dimensional theory. Finally, we
solve the inverse design problem of determining the geometric parameters for the optimal geometric and
mechanical response of these extreme structures

Gyrification from constrained cortical expansion

The exterior of the mammalian brain—the cerebral cortex—has
a conserved layered structure whose thickness varies little across
species. However, selection pressures over evolutionary time scales
have led to cortices that have a large surface area to volume ratio in
some organisms, with the result that the brain is strongly convoluted into sulci and gyri. Here we show that the gyrification can
arise as a nonlinear consequence of a simple mechanical instability
driven by tangential expansion of the gray matter constrained by
the white matter. A physical mimic of the process using a layered
swelling gel captures the essence of the mechanism, and numerical
simulations of the brain treated as a soft solid lead to the formation
of cusped sulci and smooth gyri similar to those in the brain. The
resulting gyrification patterns are a function of relative cortical expansion and relative thickness (compared with brain size), and are
consistent with observations of a wide range of brains, ranging
from smooth to highly convoluted. Furthermore, this dependence
on two simple geometric parameters that characterize the brain
also allows us to qualitatively explain how variations in these
parameters lead to anatomical anomalies in such situations as polymicrogyria, pachygyria, and lissencephalia

Aging in complex interdependency networks

Although species longevity is subject to a diverse range of evolutionary forces, the mortality curves of a wide
variety of organisms are rather similar. Here we argue that qualitative and quantitative features of aging can be
reproduced by a simple model based on the interdependence of fault-prone agents on one other. In addition to
fitting our theory to the empiric mortality curves of six very different organisms, we establish the dependence
of lifetime and aging rate on initial conditions, damage and repair rate, and system size. We compare the size
distributions of disease and death and see that they have qualitatively different properties. We show that aging
patterns are independent of the details of interdependence network structure, which suggests that aging is a
many-body effect, and that the qualitative and quantitative features of aging are not sensitively dependent on the
details of dependency structure or its formation.

Optimal control of plates using incompatible strains

A flat plate will bend into a curved shell if it experiences an inhomogeneous
growth field or if constrained appropriately at a boundary. While the forward
problem associated with this process is well studied, the inverse problem of
designing the boundary conditions or growth fields to achieve a particular
shape is much less understood. We use ideas from variational optimization
theory to formulate a well posed version of this inverse problem to determine
the optimal growth field or boundary condition that will give rise to an
arbitrary target shape, optimizing for both closeness to the target shape
and for smoothness of the growth field. We solve the resulting system of
PDE numerically using finite element methods with examples for both the
fully non-symmetric case as well as for simplified one-dimensional and
axisymmetric geometries. We also show that the system can also be solved
semi-analytically by positing an ansatz for the deformation and growth
fields in a circular disk with given thickness profile, leading to paraboloidal,
cylindrical and saddle-shaped target shapes, and show how a soft mode can
arise from a non-axisymmetric deformation of a structure with axisymmetric
material properties.

Programming curvature using origami tessellations

Origami describes rules for creating folded structures from patterns on a flat sheet, but does not prescribe how patterns can
be designed to fit target shapes. Here, starting from the simplest periodic origami pattern that yields one-degree-of-freedom
collapsible structures—we show that scale-independent elementary geometric constructions and constrained optimization
algorithms can be used to determine spatially modulated patterns that yield approximations to given surfaces of constant or
varying curvature. Paper models confirm the feasibility of our calculations. We also assess the diculty of realizing these
geometric structures by quantifying the energetic barrier that separates the metastable flat and folded states. Moreover, we
characterize the trade-o between the accuracy to which the pattern conforms to the target surface, and the eort associated
with creating finer folds. Our approach enables the tailoring of origami patterns to drape complex surfaces independent of
absolute scale, as well as the quantification of the energetic and material cost of doing so.