Universality and hysteresis in slow sweeping of bifurcations.

Bifurcations in dynamical systems are often studied experimentally and numerically using a slow parameter sweep. Focusing on the cases of period-doubling and pitchfork bifurcations in maps, we show that the adiabatic approximation always breaks down sufficiently close to the bifurcation, so the upsweep and downsweep dynamics diverge from one another, disobeying standard bifurcation theory. Nevertheless, we demonstrate universal upsweep and downsweep trajectories for sufficiently slow sweep rates, revealing that the slow trajectories depend essentially on a structural asymmetry parameter, whose effect is negligible for the stationary dynamics. We obtain explicit asymptotic expressions for the universal trajectories and use them to calculate the area of the hysteresis loop enclosed between the upsweep and downsweep trajectories as a function of the asymmetry parameter and the sweep rate.

Wood Warping Composite by 3D Printing.

Wood warping is a phenomenon known as a deformation in wood that occurs when changes in moisture content cause an unevenly volumetric change due to fiber orientation. Here we present an investigation of wood warped objects that were fabricated by 3D printing. Similar to natural wood warping, water evaporation causes volume decrease of the printed object, but in contrast, the printing pathway pattern and flow rate dictate the direction of the alignment and its intensity, all of which can be predesigned and affect the resulting structure after drying. The fabrication of the objects was performed by an extrusion-based 3D printing technique that enables the deposition of water-based inks into 3D objects. The printing ink was composed of 100% wood-based materials, wood flour, and plant-extracted natural binders cellulose nanocrystals, and xyloglucan, without the need for any additional synthetic resins. Two archetypal structures were printed: cylindrical structure and helices. In the former, we identified a new length scale that gauges the effect of gravity on the shape. In the latter, the structure exhibited a shape transition analogous to the opening of a seedpod, quantitatively reproducing theoretical predictions. Together, by carefully tuning the flow rate and printing pathway, the morphology of the fully dried wooden objects can be controlled. Hence, it is possible to design the printing of wet objects that will form different final 3D structures.

Self-Oscillating Membranes: Chemomechanical Sheets Show Autonomous Periodic Shape Transformation.

While living organisms have mastered the dynamic control of residual stresses within sheets to induce shape transformation and locomotion, man-made implementations are rudimentary. We present the first autonomously shape-shifting sheets made of a gel that shrinks and swells in response to the phase of an oscillatory chemical (Belousov-Zhabotinsky) reaction. Propagating reaction-diffusion fronts induce localized deformation of the gel. We show that these localized deformations prescribe a spatiotemporal pattern of Gaussian curvature, leading to time-periodic global shape changes. We present the computational tools and experimental protocols needed to control this system, principally the relationship between the Gaussian curvature and the reaction phase, and optical imprinting of the wave pattern. Together, our results demonstrate a route for developing fully autonomous soft machines mimicking some of the locomotive capabilities of living organisms.

Anomalously Soft Non-Euclidean Springs.

In this work we study the mechanical properties of a frustrated elastic ribbon spring-the non-Euclidean minimal spring. This spring belongs to the family of non-Euclidean plates: it has no spontaneous curvature, but its lateral intrinsic geometry is described by a non-Euclidean reference metric. The reference metric of the minimal spring is hyperbolic, and can be embedded as a minimal surface. We argue that the existence of a continuous set of such isometric minimal surfaces with different extensions leads to a complete degeneracy of the bulk elastic energy of the minimal spring under elongation. This degeneracy is removed only by boundary layer effects. As a result, the mechanical properties of the minimal spring are unusual: the spring is ultrasoft with a rigidity that depends on the thickness t as t^{7/2} and does not explicitly depend on the ribbon's width. Moreover, we show that as the ribbon is widened, the rigidity may even decrease. These predictions are confirmed by a numerical study of a constrained spring. This work is the first to address the unusual mechanical properties of constrained non-Euclidean elastic objects.

Geometry and mechanics of two-dimensional defects in amorphous materials.

We study the geometry of defects in amorphous materials and their elastic interactions. Defects are defined and characterized by deviations of the material's intrinsic metric from a Euclidian metric. This characterization makes possible the identification of localized defects in amorphous materials, the formulation of a corresponding elastic problem, and its solution in various cases of physical interest. We present a multipole expansion that covers a large family of localized 2D defects. The dipole term, which represents a dislocation, is studied analytically and experimentally. Quadrupoles and higher multipoles correspond to fundamental strain-carrying entities. The interactions between those entities, as well as their interaction with external stress fields, are fundamental to the inelastic behavior of solids. We develop analytical tools to study those interactions. The model, methods, and results presented in this work are all relevant to the study of systems that involve a distribution of localized sources of strain. Examples are plasticity in amorphous materials and mechanical interactions between cells on a flexible substrate.