Contrasting bending energies from bulk elastic theories

Pure measures of bending for soft plates

This paper, originally motivated by a question raised by Wood and Hanna [Soft Matter, 2019, 15, 2411], shows that pure measures of bending for soft plates can be defined by introducing the class of bending-neutral deformations, which represent finite incremental changes in the plate's shape that do not induce any additional bending. This class of deformations is subject to a geometric compatibility condition, which is fully characterized. A tensorial pure measure of bending, which is invariant under bending-neutral deformations, is described in detail. As shown by an illustrative class of examples, the general notion of a pure measure of bending could be useful in formulating direct theories for soft plates, where stretching and bending energies are treated separately.

Anisotropic swelling of anisotropic elastic panels

While isotropic in-plane swelling problems for thin elastic sheets have been studied extensively in recent years, many shape-programmable materials, including nematic solids and 3D-printed structures, are anisotropic, as are most industrial sheet materials. In this theoretical work, we consider central swelling and shrinkage of plates of aspect ratio and material properties relevant to the manufacture of engineered wood composite panels in which both in-plane swelling and material stiffness are highly orthotropic, leading to multiple separations in energy scales. With transverse swelling in the soft direction, and gradients in the stiff direction, the warped plates adopt two distinct types of configurations, axisymmetric and twisted, which we illustrate with toy models. We employ a two-parameter family of isometries to embed the metric programmed by the swelling, thus reducing the problem to one of minimizing bending energy alone. A simple argument is seen to closely predict averaged axisymmetric curvatures. While purely cylindrical shapes are unobtainable by pure in-plane swelling, they can be closely approximated in a highly anisotropic system. However, anisotropy can favor twisting, and breaks a degenerate soft deformation mode associated with minimal surfaces in isotropic materials. Bifurcations from axisymmetric to twisted shapes can be induced by anisotropy or by certain attributes of a central shrinkage profile. Finally, we note how our findings indicate practical limitations on the diagnosis of moisture inhomogeneities in manufactured panels by observation of warped conformations, due to the sensitivity of the qualitative response to specifics of the profile.

Modeling the behavior of inclusions in circular plates undergoing shape changes from two to three dimensions

Growth of biological tissues and shape changes of thin synthetic sheets are commonly induced by stimulation of isolated regions (inclusions) in the system. These inclusions apply internal forces on their surroundings that, in turn, promote 2D layers to acquire complex 3D configurations. We focus on a fundamental building block of these systems, and consider a circular plate that contains an inclusion with dilative strains. Based on the Föppl-von Kármán (FvK) theory, we derive an analytical model that predicts the 2D-to-3D shape transitions in the system. Our findings are summarized in a phase diagram that reveals two distinct configurations in the post-buckling region. One is an extensive profile that holds close to the threshold of the instability, and the second is a localized profile, which preempts the extensive solution beyond the buckling threshold. While the former solution is derived as a perturbation around the flat configuration, assuming infinitesimal amplitudes, the latter solution is derived around a buckled state that is highly localized. We show that up to vanishingly small corrections that scale with the thickness, this localized configuration is equivalent to that expected for ultra-thin sheets, which completely relax compressive stresses. Our findings agree quantitatively with direct numerical minimization of the FvK energy. Furthermore, we extend the theory to describe shape transitions in polymeric gels, and compare the results with numerical simulations that account for the complete elastodynamic behavior of the gels. The agreement between the theory and these simulations indicates that our results are observable experimentally. Notably, our findings can provide guidelines to the analysis of more complicated systems that encompass interaction between several buckled inclusions.

Modeling the formation of double rolls from heterogeneously patterned gels

Both stimuli-responsive gels and growing biological tissue can undergo pronounced morphological transitions from two-dimensional (2D) layers into 3D geometries. We derive an analytical model that allows us to quantitatively predict the features of 2D-to-3D shape changes in polymer gels that encompasses different degrees of swelling within the sample. We analyze a particular configuration that emerges from a flat rectangular gel that is divided into two strips (bistrips), where each strip is swollen to a different extent in solution. The final configuration yields double rolls that display a narrow transition layer between two cylinders of constant radii. To characterize the rolls' shapes, we modify the theory of thin incompatible elastic sheets to account for the Flory-Huggins interaction between the gel and the solvent. This modification allows us to derive analytical expressions for the radii, the amplitudes, and the length of the transition layer within a given roll. Our predictions agree quantitatively with available experimental data. In addition, we carry out numerical simulations that account for the complete nonlinear behavior of the gel and show good agreement between the analytical predictions and the numerical results. Our solution sheds light on a stress focusing pattern that forms at the border between two dissimilar soft materials. Moreover, models that provide quantitative predictions on the final morphology in such heterogeneously swelling hydrogels are useful for understanding growth patterns in biology as well as accurately tailoring the structure of gels for various technological applications.