Bifurcations in dynamical systems are often studied experimentally and numerically using a slow p... more 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 that 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 is a phenomenon known as a deformation in wood that occurs when changes in moisture ... more 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 for...
Living organisms have mastered the dynamic control of internal stresses to perform an array of fu... more Living organisms have mastered the dynamic control of internal stresses to perform an array of functions, such as change shape and locomote. State-of-the-art attempts to replicate this ability in synthetic materials are rudimentary in comparison. Here we present the first experimental realization of a self-oscillating gel in a thin sheet configuration. We show that internal signaling produces stresses that drive lifelike shape changes, that the materials response is accurately modelled with the theory of non-Euclidean elasticity and that the internal signaling can be programmed with light. Together, our results demonstrate a complete route for developing fully autonomous soft machines.
We present the first example of an autonomous, extended, shape-transforming sheet made from a gel... more We present the first example of an autonomous, extended, shape-transforming sheet made from a gel that shrinks and swells in response to the Belousov-Zhabotinsky (BZ) reaction. BZ waves traveling within the gel induce periodic flapping of the sheets. We compute the relationship between the reference Gaussian curvature and the reaction phase using the theory of non-Euclidean elasticity and show that this quantitatively predicts the evolving three-dimensional shape. We account for deviations from the purely geometric theory due to finite-thickness effects. This system demonstrates behavior reminiscent of biological organisms: autonomous shape changes of a soft slender structure powered by chemical free energy extracted from the environment. Our results constitute an important step towards the realization of autonomous soft machines.
Geometrically frustrated elastic ribbons exhibit, in many cases, significant changes in configura... more Geometrically frustrated elastic ribbons exhibit, in many cases, significant changes in configuration depending on the relation between their width and thickness. We show that the existence of such a transition, and the scaling at which it occurs, strongly depend on the system considered. Using an asymptotic approach, treating the width as a small parameter, we find the leading energy terms resulting from the frustration and predict the existence and scaling of the shape transition. We study in detail 5 different types of frustrated ribbons with a different morphological dependence on ribbon’s width: a sharp shape-transition at a critical width, a moderate transition with an intermediate regime, and no transition at all. We show that the predictions of our approach match experimental results from two different experimental systems: prestressed rubber bilayers and 4D printed thermoplastics, in a wide variety of geometric settings.
We present a new definition of defects which is based on a Riemannian formulation of incompatible... more We present a new definition of defects which is based on a Riemannian formulation of incompatible elasticity. Defects are viewed as local deviations of the material's reference metric field, $\bar{\mathfrak{g}}$, from a Euclidian metric. This definition allows the description of defects in amorphous materials and the formulation of the elastic problem, using a single field, $\bar{\mathfrak{g}}$. We provide a multipole expansion of reference metrics that represent a large family of two-dimensional (2D) localized defects. The case of a dipole, which corresponds to an edge dislocation is studied analytically, experimentally and numerically. The quadrupole term, which is studied analytically, as well as higher multipoles of curvature carry local deformations. These multipoles are good candidates for fundamental strain carrying entities in plasticity theories of amorphous materials and for a continuous modeling of recently developed meta-materials.
While living organisms have mastered the dynamic control of residual stresses within sheets to in... more 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.
In this work we study the mechanical properties of a frustrated elastic ribbon spring-the non-Euc... more 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.
We present a new definition of defects which is based on a Riemannian formulation of incompatible... more We present a new definition of defects which is based on a Riemannian formulation of incompatible elasticity. Defects are viewed as local deviations of the material's reference metric field, $\bar{\mathfrak{g}}$, from a Euclidian metric. This definition allows the description of defects in amorphous materials and the formulation of the elastic problem, using a single field, $\bar{\mathfrak{g}}$. We provide a multipole expansion of reference metrics that represent a large family of two-dimensional (2D) localized defects. The case of a dipole, which corresponds to an edge dislocation is studied analytically, experimentally and numerically. The quadrupole term, which is studied analytically, as well as higher multipoles of curvature carry local deformations. These multipoles are good candidates for fundamental strain carrying entities in plasticity theories of amorphous materials and for a continuous modeling of recently developed meta-materials.
Proceedings of the National Academy of Sciences, 2015
We study the geometry of defects in amorphous materials and their elastic interactions. Defects a... more 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 dist...
Bifurcations in dynamical systems are often studied experimentally and numerically using a slow p... more 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 that 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 is a phenomenon known as a deformation in wood that occurs when changes in moisture ... more 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 for...
Living organisms have mastered the dynamic control of internal stresses to perform an array of fu... more Living organisms have mastered the dynamic control of internal stresses to perform an array of functions, such as change shape and locomote. State-of-the-art attempts to replicate this ability in synthetic materials are rudimentary in comparison. Here we present the first experimental realization of a self-oscillating gel in a thin sheet configuration. We show that internal signaling produces stresses that drive lifelike shape changes, that the materials response is accurately modelled with the theory of non-Euclidean elasticity and that the internal signaling can be programmed with light. Together, our results demonstrate a complete route for developing fully autonomous soft machines.
We present the first example of an autonomous, extended, shape-transforming sheet made from a gel... more We present the first example of an autonomous, extended, shape-transforming sheet made from a gel that shrinks and swells in response to the Belousov-Zhabotinsky (BZ) reaction. BZ waves traveling within the gel induce periodic flapping of the sheets. We compute the relationship between the reference Gaussian curvature and the reaction phase using the theory of non-Euclidean elasticity and show that this quantitatively predicts the evolving three-dimensional shape. We account for deviations from the purely geometric theory due to finite-thickness effects. This system demonstrates behavior reminiscent of biological organisms: autonomous shape changes of a soft slender structure powered by chemical free energy extracted from the environment. Our results constitute an important step towards the realization of autonomous soft machines.
Geometrically frustrated elastic ribbons exhibit, in many cases, significant changes in configura... more Geometrically frustrated elastic ribbons exhibit, in many cases, significant changes in configuration depending on the relation between their width and thickness. We show that the existence of such a transition, and the scaling at which it occurs, strongly depend on the system considered. Using an asymptotic approach, treating the width as a small parameter, we find the leading energy terms resulting from the frustration and predict the existence and scaling of the shape transition. We study in detail 5 different types of frustrated ribbons with a different morphological dependence on ribbon’s width: a sharp shape-transition at a critical width, a moderate transition with an intermediate regime, and no transition at all. We show that the predictions of our approach match experimental results from two different experimental systems: prestressed rubber bilayers and 4D printed thermoplastics, in a wide variety of geometric settings.
We present a new definition of defects which is based on a Riemannian formulation of incompatible... more We present a new definition of defects which is based on a Riemannian formulation of incompatible elasticity. Defects are viewed as local deviations of the material's reference metric field, $\bar{\mathfrak{g}}$, from a Euclidian metric. This definition allows the description of defects in amorphous materials and the formulation of the elastic problem, using a single field, $\bar{\mathfrak{g}}$. We provide a multipole expansion of reference metrics that represent a large family of two-dimensional (2D) localized defects. The case of a dipole, which corresponds to an edge dislocation is studied analytically, experimentally and numerically. The quadrupole term, which is studied analytically, as well as higher multipoles of curvature carry local deformations. These multipoles are good candidates for fundamental strain carrying entities in plasticity theories of amorphous materials and for a continuous modeling of recently developed meta-materials.
While living organisms have mastered the dynamic control of residual stresses within sheets to in... more 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.
In this work we study the mechanical properties of a frustrated elastic ribbon spring-the non-Euc... more 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.
We present a new definition of defects which is based on a Riemannian formulation of incompatible... more We present a new definition of defects which is based on a Riemannian formulation of incompatible elasticity. Defects are viewed as local deviations of the material's reference metric field, $\bar{\mathfrak{g}}$, from a Euclidian metric. This definition allows the description of defects in amorphous materials and the formulation of the elastic problem, using a single field, $\bar{\mathfrak{g}}$. We provide a multipole expansion of reference metrics that represent a large family of two-dimensional (2D) localized defects. The case of a dipole, which corresponds to an edge dislocation is studied analytically, experimentally and numerically. The quadrupole term, which is studied analytically, as well as higher multipoles of curvature carry local deformations. These multipoles are good candidates for fundamental strain carrying entities in plasticity theories of amorphous materials and for a continuous modeling of recently developed meta-materials.
Proceedings of the National Academy of Sciences, 2015
We study the geometry of defects in amorphous materials and their elastic interactions. Defects a... more 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 dist...
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