We study the non-Euclidean (incompatible) elastic energy functionals in the description of prestressed thin films, at their singular limits ( @math -limits) as @math in the film's thickness @math . Firstly, we extend the prior results [Lewicka-Pakzad, Bhattacharya-Lewicka-Schaffner, Lewicka-Raoult-Ricciotti] to arbitrary incompatibility metrics that depend on both the midplate and the transversal variables (the "non-oscillatory" case). Secondly, we analyze a more general class of incompatibilities, where the transversal dependence of the lower order terms is not necessarily linear (the "oscillatory" case), extending the results of [Agostiniani-Lucic-Lucantonio, Schmidt] to arbitrary metrics and higher order scalings. We exhibit connections between the two cases via projections of appropriate curvature forms on the polynomial tensor spaces. We also show the effective energy quantisation in terms of scalings as a power of @math and discuss the scaling regimes @math (Kirchhoff), @math (von Karman) in the general case, as well as all possible (even powers) regimes for conformal metrics, thus paving the way to the subsequent complete analysis of the non-oscillatory setting in [Lewicka]. Thirdly, we prove the coercivity inequalities for the singular limits at @math - and @math - scaling orders, while disproving the full coercivity of the classical von Karman energy functional at scaling @math .
On the frontiers of experimental modeling of shape formation, we mention the halftone gel lithography method for polymeric materials that can swell by imbibing fluids @cite_13 @cite_14 @cite_22 @cite_39 . By blocking the ability of portions of plate to swell or causing them to swell inhomogeneously, it is possible to have the plate assume a variety of deformed shapes. Even more sophisticated techniques of biomimetic 4d printing allow for engineering of the 3d shape-morphing systems that mimic nastic plant motions where organs such as tendrils, leaves and flowers respond to the environmental stimuli @cite_43 . Optimal control in such systems has been studied in @cite_25 , see also @cite_20 .
@cite_14 @cite_22 @cite_39 @cite_43 @cite_13 @cite_25 @cite_20
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Nature offers a plethora of astonishing examples of shapes and functions from the aspects of both simplicity and complexity. The creation of synthetic systems that can morph in a controlled manner as seen in nature is of paramount importance in many fields of fundamental and applied sciences. The tremendous interest in self-shaping materials stems from a wide range of applications for these materials, ranging from biomedical devices to aircraft design. This review article highlights recent advances in understanding and designing thin, sheet-like soft materials that can transform into complex three-dimensional structures in a controlled manner by modulating the internal stresses. We review the general principles underlying shape transformation phenomena in natural and synthetic systems, and the significant achievements in fabricating self-shaping of soft materials via representative examples. We conclude with a discussion on the challenges facing the field, and future directions from the perspective of theoretical and experimental methodology and interdisciplinary applications. This communication describes a novel strategy to achieve programmable shape transformation of hybrid hydrogel sheets by modulating both the in-plane and out-of-plane mismatches in mechanical properties. Both our experimental and computational results demonstrate that the shape transformation of hybrid hydrogel sheets shows rich features (e.g., rolling direction, axis, chirality, etc.) and versatile tunability (e.g., via various external stimuli, material properties, pattern geometry, etc.). This work can provide guidance for designing soft materials that are able to undergo more precise and complex shape transformation. Printed hydrogel composites with plant-inspired architectures dynamically change shape on immersion in water to yield prescribed complex morphologies. Self-actuating materials capable of transforming between three-dimensional shapes have applications in areas as diverse as biomedicine, robotics, and tunable micro-optics. We introduce a method of photopatterning polymer films that yields temperature-responsive gel sheets that can transform between a flat state and a prescribed three-dimensional shape. Our approach is based on poly(N-isopropylacrylamide) copolymers containing pendent benzophenone units that allow cross-linking to be tuned by irradiation dose. We describe a simple method of halftone gel lithography using only two photomasks, wherein highly cross-linked dots embedded in a lightly cross-linked matrix provide access to nearly continuous, and fully two-dimensional, patterns of swelling. This method is used to fabricate surfaces with constant Gaussian curvature (spherical caps, saddles, and cones) or zero mean curvature (Enneper’s surfaces), as well as more complex and nearly closed shapes. 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.