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Motivated by pine cones and plant seeds that open or close in response to changes in humidity thanks to the shrinking or swelling of hygroscopic bilayers, and given the recent interest in artificial systems that mimic the hygroscopic motility of seeds, we consider bilayers consisting of two adhering elongated thin layers made of hygroscopic gels, modeled as active, transversally isotropic elastic materials. The direction of transverse isotropy is set as one of the fibers that are present in both biological and synthetic materials. As a special, biologically relevant case, we consider one of the two layers passive. This work develops a three-dimensional model for the coupled evolution of elastic deformations and moisture content (solvent concentration). Based on this model, a numerical scheme for solving the evolution equations coupling mechanical equilibrium and diffusion has been constructed based on the Finite Element Method. The equations are formulated in weak form and solved using Comsol Multiphysics. We have applied our model to compute transient configurations and their steady-state equilibrium limits in some relevant test cases. These configurations are then analyzed to understand the influence of the various geometric and physical parameters.

In this work, an innovative model is proposed as a design tool to predict both the inner and outer radii in rolled structures based on polydimethylsiloxane bilayers. The model represents an improvement of Timoshenko’s formula taking into account the friction arising from contacts between layers arising from rolling by more than one turn, hence broadening its application field towards materials based on elastomeric bilayers capable of large deformations. The fabricated structures were also provided with surface topographical features that would make them potentially usable in different application scenarios, including cell/tissue engineering ones. The bilayer design parameters were varied, such as the initial strain (from 20 to 60%) and the bilayer thickness (from 373 to 93 µm). The model matched experimental data on the inner and outer radii nicely, especially when a high friction condition was implemented in the model, particularly reducing the error below 2% for the outer diameter while varying the strain. The model outperformed the current literature, where self-penetration is not excluded, and a single value of the radius of spontaneous rolling is used to describe multiple rolls. A complex 3D bioinspired hierarchical elastomeric microstructure made of seven spirals arranged like a hexagon inscribed in a circumference, similar to typical biological architectures (e.g., myofibrils within a sarcolemma), was also developed. In this case also, the model effectively predicted the spirals’ features (error smaller than 18%), opening interesting application scenarios in the modeling and fabrication of bioinspired materials.

The control of shape in active structures is a key problem for the realization of smart sensors and actuators, which often draw inspiration from natural systems. In this context, slender structures, such as thin plates, have been studied as a relevant example of shape morphing systems where curvature is generated by in-plane incompatibilities. In particular, in hydrogel plates these incompatibilities can be programmed at fabrication time, such that a target configuration is attained at equilibrium upon swelling or shrinking. While these aspects have been examined in detail, understanding the transient morphing of such active structures deserves further investigation. In this study, we develop a geometrical model for the transient shaping of thin hydrogel plates by extending the theory of non-Euclidean plates. We validate the proposed model using experiments on gel samples that are programmed to reach axisymmetric equilibrium shapes. Interestingly, our experiments show the emergence of non-axisymmetric shapes for early times, as a consequence of boundary layer effects induced by solvent transport. We rationalize these observations using numerical simulations based on a detailed poroelastic model. Overall, this work highlights the limitations of purely geometrical models and the importance of transient, reduced theories for morphing plates that account for the coupled physics driving the evolution of shape. Computational approaches employing these theories will allow to achieve accurate control on the morphing dynamics and ultimately advance 4D printing technologies.

Abstract Nature and technology often adopt structures that can be described as tubular helical assemblies. However, the role and mechanisms of these structures remain elusive. In this paper, we study the mechanical response under compression and extension of a tubular assembly composed of 8 helical Kirchhoff rods, arranged in pairs with opposite chirality and connected by pin joints, both analytically and numerically. We first focus on compression and find that, whereas a single helical rod would buckle, the rods of the assembly deform coherently as stable helical shapes wound around a common axis. Moreover, we investigate the response of the assembly under different boundary conditions, highlighting the emergence of a central region where rods remain circular helices. Secondly, we study the effects of different hypotheses on the elastic properties of rods, i.e., stress-free rods when straight versus when circular helices, Kirchhoff’s rod model versus Sadowsky’s ribbon model. Summing up, our findings highlight the key role of mutual interactions in generating a stable ensemble response that preserves the helical shape of the individual rods, as well as some interesting features, and they shed some light on the reasons why helical shapes in tubular assemblies are so common and persistent in nature and technology. Graphic Abstract We study the mechanical response under compression/extension of an assembly composed of 8 helical rods, pin-jointed and arranged in pairs with opposite chirality. In compression we find that, whereas a single rod buckles (a), the rods of the assembly deform as stable helical shapes (b). We investigate the effect of different boundary conditions and elastic properties on the mechanical response, and find that the deformed geometries exhibit a common central region where rods remain circular helices. Our findings highlight the key role of mutual interactions in the ensemble response and shed some light on the reasons why tubular helical assemblies are so common and persistent.

Based on their shape-shifting capabilities, soft active materials have enabled new possibilities for the engineering of sensing and actuation devices. While the relation between active strains and emergent equilibrium shapes has been fully characterized, the transient morphing of thin structures is a rather unexplored topic. Here, we focus on polymer gel plates and derive a reduced linear model to study their time-dependent response to changes in the fluid environment. We show that independent control of stretching and bending deformations in stress-free conditions allows to realize spherical shapes with prescribed geometry of the mid-plane. Furthermore, we demonstrate that tensile (compressive) membrane stresses delay (accelerate) swelling-induced shape transitions compared to the stress-free evolution. We believe that these effects should be considered for the accurate design of smart systems and may contribute to explain the complexity of natural shapes.

We analyze temperature dynamics in anatomic pathology samples to identify the most efficient refrigeration method and to predict the time available for optimal sectioning before sample heating, thus getting appropriate information for a correct diagnosis by anatomopathologists. A thermal finite element (FE) analysis was carried out with comsolmultiphysics to evaluate temperature variations in paraffin-embedded tissues, i.e., muscle, bone and fat, and the corresponding thermal stresses. Experiments with different tissues and thermocouple-based measurements allowed validating the FE simulations. Simulations allowed to estimate the time needed to bring the sample at the optimal temperature for sectioning (−8 to −4 °C) in different conditions: refrigeration on a cold plate, refrigeration in a cooled environment, and refrigeration in an environment with forced convection. Among the three cooling methods tested, the forced convection at −20 °C and with an air-flow speed of 5 m/s resulted in the shortest cooling time. As compared to the other methods, thermal stresses can be modulated by varying the air-flow speed. For the different conditions, the time needed for the surface of the tissue block to exit from a temperature corresponding to an optimal cutting, when leaving the sample exposed to room temperature after refrigeration, ranged from 12 to 310 s. We quantify the time needed to adequately refrigerate paraffin-embedded tissue samples and the time available before they leave the optimal temperature window for sectioning. We also evaluate the maximum stress attained in the paraffin block during the cooling and the heating transients. This information will help optimize anatomic pathology processes.

We discuss self-folding of a thin sheet by using patterned hydrogel bilayers, which act as hinges connecting flat faces. Folding is actuated by heterogeneous swelling due to different cross-linking densities of the polymer network in the two layers. Our analysis is based on a dimensionally reduced plate model, obtained by applying a recently developed theory [1], which provides us with an explicit connection between (three-dimensional) material properties and the curvatures induced at the hinges. This connection offers a recipe for the fabrication and design of the bilayers, by providing the values of the cross-linking density of each layer that need to be imprinted during polymerization in order to produce a desired folded shape upon swelling.

Polymer gel plates may be programmed to morph into three-dimensional configurations upon swelling. An effective strategy to control such shape transformations consists in patterning the in-plane cross-linking density of the polymer network to realize non-homogeneous swelling. In general, one needs to solve an inverse problem to determine the shear modulus field that produces a given target shape. Here, we propose a computational framework for the solution of such an inverse problem, which we validate against two benchmark problems, i.e. making cones and saddles from gel disks.

The effect of elastic deformations induced by gravity loading on the active circumnutation movements of growing plant shoots is investigated. We consider first a discrete model (a gravitropic spring-pendulum system) and then a continuous rod model which is analyzed both analytically (under the assumption of small deformations) and numerically (in the large deformation regime). We find that, for a choice of material parameters consistent with values reported in the available literature on plant shoots, rods of sufficient length may exhibit lateral oscillations of increasing amplitude, which eventually converge to limit cycles. This behavior strongly suggests the occurrence of a Hopf bifurcation, just as for the gravitropic spring-pendulum system, for which this result is rigorously established. At least in this restricted set of material parameters, our analysis supports a view of Darwin’s circumnutations as a biological analogue to structural systems exhibiting flutter instabilities, i.e., spontaneous oscillations away from equilibrium configurations driven by non-conservative loads. Here, in the context of nutation movements of growing plant shoots, the energy needed to sustain oscillations is continuously supplied to the system by the internal biochemical machinery presiding the capability of plants to maintain a vertical pose.

An elastic bilayer, consisting of an equibiaxially pre-stretched sheet bonded to a stress-free one, spontaneously morphs into curved shapes in the absence of external loads or constraints. Using experiments and numerical simulations, we explore the role of geometry for square and rectangular samples in determining the equilibrium shape of the system, for a fixed pre-stretch. We classify the observed shapes over a wide range of aspect ratios according to their curvatures and compare measured and computed values, which show good agreement. In particular, as the bilayer becomes thinner, a bifurcation of the principal curvatures occurs, which separates two scaling regimes for the energy of the system. We characterize the transition between these two regimes and show the peculiar features that distinguish square from rectangular samples. The results for our model bilayer system may help explaining morphing in more complex systems made of active materials.

Basement membranes (BMs) are thin layers of condensed extracellular matrix proteins serving as permeability filters, cellular anchoring sites, and barriers against cancer cell invasion. It is believed that their biomechanical properties play a crucial role in determining cellular behavior and response, especially in mechanically active tissues like breast glands. Despite this, so far, relatively little attention has been dedicated to their analysis because of the difficulty of isolating and handling such thin layers of material. Here, we isolated BMs derived from MCF10A spheroids—three-dimensional breast gland model systems mimicking in vitro the most relevant phenotypic characteristics of human breast lobules—and characterized them by atomic force microscopy, enhanced resolution confocal microscopy, and scanning electron microscopy. By performing atomic force microscopy height-clamp experiments, we obtained force-relaxation curves that offered the first biomechanical data on isolated breast gland BMs to our knowledge. Based on enhanced resolution confocal microscopy and scanning electron microscopy imaging data, we modeled the system as a polymer network immersed in liquid and described it as a poroelastic material. Finite-element simulations matching the experimental force-relaxation curves allowed for the first quantification, to our knowledge, of the bulk and shear moduli of the membrane as well as its water permeability. These results represent a first step toward a deeper understanding of the mechanism of tensional homeostasis regulating mammary gland activity as well as its disruption during processes of membrane breaching and metastatic invasion.

Visualization and tracking of the facial whiskers is critical to many studies of rodent behavior. High-speed videography is the most robust methodology for characterizing whisker kinematics, but whisker visualization is challenging due to the low contrast of the whisker against its background. Recently, we showed that fluorescent dye(s) can be applied to enhance visualization and tracking of whisker(s) ( Rigosa et al., 2017 ), and this protocol provides additional details on the technique.

The J-integral is formulated in a direct manner for a gel consisting of a cross-linked polymer network and a mobile solvent. The form of the J-integral is given for a formulation that exploits the Helmholtz energy density of the gel and expressions are provided for it in both the unswollen reference configuration of the polymer network and in the current swollen configuration of the gel when small strains are superimposed on the swollen state. Similarly, the form of the J-integral is developed for an approach that exploits the Landau energy density of the gel and its reference and current configuration expressions are also developed. The Flory-Rehner model of the gel is used to obtain expressions for both the densities of Helmholtz energy and the Landau energy, with the chemical potential of the solvent derived from the Helmholtz energy used in the Legendre transformation that generates the Landau energy. Both the Helmholtz and Landau energies are expanded asymptotically for small strains superimposed on the swollen state of the gel. The results for the various forms of the energies are then used to obtain the elasticity law and the incompressibility constraint for the gel, each derived from both the Helmholtz and the Landau energies. The results are then inserted into the J-integral and fracture mechanics insights obtained for the rapid and slow loading of a gel body with a stationary crack and for a gel body with a crack that is experiencing slow, steady propagation. It is found that the Landau energy form of the J-integral is particularly useful for the slow loading of stationary cracks and for the slow steady propagation of the crack. It is noted that solvent flux during crack growth can cause an increase in the effective fracture toughness of the gel. However, it is found that there is an absence of such diffusional toughening in the rapidly loaded stationary crack case, the very slowly loaded stationary crack case and for the crack experiencing extremely slow but steady propagation. It is further found that, for cracks propagating very slowly, diffusional toughening rises linearly with crack propagation rate up to a critical crack growth rate, above which the diffusional toughening becomes insensitive to the crack propagation rate. The critical crack propagation rate for this transition is found to be dependent on the linear dimension of the gel body and on constitutive parameters for the gel elasticity and solvent diffusion.

Visualization and tracking of the facial whiskers is required in an increasing number of rodent studies. Although many approaches have been employed, only high-speed videography has proven adequate for measuring whisker motion and deformation during interaction with an object. However, whisker visualization and tracking is challenging for multiple reasons, primary among them the low contrast of the whisker against its background. Here, we demonstrate a fluorescent dye method suitable for visualization of one or more rat whiskers. The process makes the dyed whisker(s) easily visible against a dark background. The coloring does not influence the behavioral performance of rats trained on a vibrissal vibrotactile discrimination task, nor does it affect the whiskers’ mechanical properties. DOI: http://dx.doi.org/10.7554/eLife.25290.001

We derive a multiphysics model that accounts for network elasticity with spontaneous strains, swelling and nematic interactions in liquid crystal gels (LCGs). We discuss the coupling among the various physical mechanisms, with particular reference to the effects of nematic interactions on chemical equilibrium and that of swelling on the nematic-isotropic transition. Building upon this discussion and using numerical simulations, we explore the transient phenomena involving concurrent swelling and phase transition in LCGs subject to a temperature change. Specifically, we demonstrate separation in time scales between solvent uptake and phase change, in agreement with experiments, which determines a kinetic decoupling between shape and volume changes. Finally, we discuss possible applications in the context of microswimmers, where such a kinetic decoupling is exploited to achieve non-reciprocal actuation and net motion in Stokes flow.

Brittle materials fail catastrophically. In consequence of their limited flaw-tolerance, failure occurs by localized fracture and is typically a dynamic process. Recently, experiments on epithelial cell monolayers have revealed that this scenario can be significantly modified when the material susceptible to cracking is adhered to a hydrogel substrate. Thanks to the hydraulic coupling between the brittle layer and the poroelastic substrate, such a composite can develop a toughening mechanism that relies on the simultaneous growth of multiple cracks. Here, we study this remarkable behaviour by means of a detailed model, and explore how the material and loading parameters concur in determining the macroscopic toughness of the system. By extending a previous study, our results show that rapid loading conveys material toughness by promoting distributed cracking. Moreover, our theoretical findings may suggest innovative architectures of flaw-insensitive materials with higher toughness.

We derive a large-strain plate model that allows to describe transient, coupled processes involving elasticity and solvent migration, by performing a dimensional reduction of a three-dimensional poroelastic theory. We apply the model to polymer gel plates, for which a specific kinematic constraint and constitutive relations hold. Finally, we assess the accuracy of the plate model with respect to the parent three-dimensional model through two numerical benchmarks, solved by means of the finite element method. Our results show that the theory offers an efficient computational framework for the study of swelling-induced morphing of composite gel plates.

We rigorously derive a Kirchhoff plate theory, via Γ-convergence, from a three-dimensional model that describes the finite elasticity of an elastically heterogeneous, thin sheet. The heterogeneity in the elastic properties of the material results in a spontaneous strain that depends on both the thickness and the plane variables x′. At the same time, the spontaneous strain is h-close to the identity, where h is the small parameter quantifying the thickness. The 2D Kirchhoff limiting model is constrained to the set of isometric immersions of the mid-plane of the plate into ℝ3, with a corresponding energy that penalizes deviations of the curvature tensor associated with a deformation from an x′-dependent target curvature tensor. A discussion on the 2D minimizers is provided in the case where the target curvature tensor is piecewise constant. Finally, we apply the derived plate theory to the modeling of swelling-induced shape changes in heterogeneous thin gel sheets.

Soft membranes are commonly employed in shape-morphing applications, where the material is programmed to achieve a target shape upon activation by an external trigger, and as coating layers that alter the surface characteristics of bulk materials, such as the properties of spreading and absorption of liquids. In particular, polymer gel membranes experience swelling or shrinking when their solvent content change, and the non-homogeneous swelling field may be exploited to control their shape. Here, we develop a theory of swelling material surfaces to model polymer gel membranes and demonstrate its features by numerically studying applications in the contexts of biomedicine, micro-motility, and coating technology. We also specialize the theory to thermo-responsive gels, which are made of polymers that change their affinity with a solvent when temperature varies.

Brittle materials propagate opening cracks under tension. When stress increases beyond a critical magnitude, then quasistatic crack propagation becomes unstable. In the presence of several precracks, a brittle material always propagates only the weakest crack, leading to catastrophic failure. Here, we show that all these features of brittle fracture are fundamentally modified when the material susceptible to cracking is bonded to a hydrogel, a common situation in biological tissues. In the presence of the hydrogel, the brittle material can fracture in compression and can hydraulically resist cracking in tension. Furthermore, the poroelastic coupling regularizes the crack dynamics and enhances material toughness by promoting multiple cracking.

We describe swelling-driven curving in originally straight and non-homogeneous beams. We present and verify a structural model of swollen beams, based on a new point of view adopted to describe swelling-induced deformation processes in bilayered gel beams, that is based on the split of the swelling-induced deformation of the beam at equilibrium into two components, both depending on the elastic properties of the gel. The method allows us to: (i) determine beam stretching and curving, once assigned the characteristics of the solvent bath and of the non-homogeneous beam, and (ii) estimate the characteristics of non-homogeneous flat gel beams in such a way as to obtain, under free-swelling conditions, three-dimensional shapes. The study was pursued by means of analytical, semi-analytical and numerical tools; excellent agreement of the outcomes of the different techniques was found, thus confirming the strength of the method.

It has been demonstrated experimentally that pre-stretch affects the swelling of an elastomeric membrane when it is exposed to a solvent. We study theoretically the one-dimensional swelling of a pre-stretched thin elastomeric sheet, bonded to an impermeable rigid substrate, to quantify the influence of pre-stretch. We show that the solvent uptake increases when pre-stretch increases, both at equilibrium and during the swelling transient, where it exhibits two different scaling regimes. The coupling between the solvent uptake and pre-stretch may be practically exploited to design soft actuators where the swelling-induced deformations can be controlled by varying the pre-stretch.

A thermodynamically consistent modeling of the non-linear multiphysics behavior of ionic polymers is presented and discussed. A key ingredient of the model is the treatment of the electrically induced deformations, resulting in a large bending motion when ionic polymers are employed in IPMCs (Ionic Polymer-Metal Composites). With specific reference to the mechanics, the position of the linear models for ionic polymers is discussed and set within a formalized linearization procedure. A set of numerical experiments is developed, showing the capability of the model to capture the fundamental features of the actuation mechanism.

The purpose of the research is to describe the swelling-induced large deformations in polymer gels: a theoretical model is developed, and then implemented and solved using the finite element method. The model is firstly assessed with two well-known bench- mark problems; moreover, the proposed approach is benchmarked against a recent experiment involving localized exposure of the gel boundary to a solvent, where large bending deformations appear during solvent absorption. In both cases, our results are quite satisfying.

When stretched uniaxially, a thin elastic sheet may exhibit buckling. The occurrence of buckling depends on the geometrical properties of the sheet and the magnitude of the applied strain. Here we show that an elastomeric sheet initially stable under uniaxial stretching can destabilize when exposed to a solvent that swells the elastomer. We demonstrate experimentally and computationally that the features of the buckling pattern depend on the magnitude of stretching, and this observation offers a new way for controlling the shape of a swollen homogeneous thin sheet., 7 pages, 5 figures, submitted