Geometric incompatibility, the inability of a material's rest state to be realized in Euclidean space, underlies shape formation in natural and synthetic thin sheets. Classical Gauss and Mainardi-Codazzi-Peterson incompatibilities explain many patterns in nature, but they do not exhaust the mechanisms that frustrate thin elastic sheets. We identify a new incompatibility that forbids smooth stretching-free configurations, even when the rest state of the elastic sheet locally satisfies the Gauss and Mainardi-Codazzi-Peterson compatibility conditions. We demonstrate this principle in a model of surface growth with positive Gaussian curvature, where a geometric horizon forms, leading to the onset of frustration. Experiments, simulations, and theory show that the sheet responds by nucleating periodic d-cone-like dimples. We show that this obstruction to stretching-free configurations is topological, and we point to open questions concerning the origin of frustration.

Disordered solids, straddling the solid-fluid boundary, lack a comprehensive continuum mechanical description. They exhibit a complex microstructure wherein multiple meta-stable states exist. Deforming disordered solids induces particles rearrangements enabling the system to transition between meta-stable states. A dramatic consequence of these transitions is that quasistatic deformation cycles modify the reference state, facilitating the storage and release of mechanical energy. Here we develop a continuum mechanical theory of disordered solids, which accounts for the absence of a reference state and the lack of conserved potential energy. Our theory, which introduces a new modulus describing non-conservative mechanical screening, reduces to classical elasticity in the absence of screening. We analytically derive predictions for the deformation field for various perturbations and geometries. While our theory applies to general disordered solids, we focus on a two-dimensional disordered granular system and predict accurately the non-affine displacement fields observed in experiments for both small and large deformations, along with the observable vanishing shear modulus. The new proposed moduli satisfy universal relations that are independent of the specific experimental realization. Our work thus forms the basis of an entirely new family of continuum descriptions of the mechanics of disordered solids.

Disordered granular matter exhibits mechanical responses that occupy the boundary between fluids and solids, lacking a complete description within a continuum theoretical framework. Recent studies have shown that, in the quasi-static limit, the mechanical response of disordered solids to external perturbations is anomalous and can be accurately predicted by the theory of "odd dipole screening." In this work, we investigate responsive granular matter, where grains change size in response to stimuli such as humidity, temperature, or other factors. We develop a geometric theory of odd dipole-screening, incorporating the growth field into the equilibrium equation. Our theory predicts an anomalous displacement field in response to non-uniform growth fields, confirmed by molecular dynamics simulations of granular matter. Although the screening parameters in our theory are phenomenological and not derived from microscopic physics, we identify a surprising relationship between the odd parameter and Poisson's ratio. This theory has implications for various experimental protocols, including non-uniform heating or wetting, which lead to spatially varying expansion fields.

Growth and form are deeply interconnected, in a manner often mediated by mechanical instabilities arising from geometric incompatibilities. Although Gauss incompatibility has long been recognized as the source of morphing in naturally growing slender organs, here we show that the growth profile of rose petals remains Gauss compatible. Their distinctive shape emerges from a different type of geometric incompatibility, the Mainardi-Codazzi-Peterson (MCP) incompatibility, which leads to the formation of localized cusps along the petal margins. We validated this mechanism in model disc petals theoretically, computationally, and experimentally. Our study reveals distinct morphological regimes, ranging from smooth edges to cusp-forming configurations, and demonstrates how stress focusing at cusps influences subsequent petal growth. These findings position MCP incompatibility as a generic mechanism for cusp formation in both natural and manmade self-morphing sheets.

Miura-Ori, a celebrated origami pattern that facilitates functionality in matter, has found multiple applications in the field of mechanical metamaterials. Modifications of Miura-Ori pattern can produce curved configurations during folding, thereby enhancing its potential functionalities. Thus, a key challenge in designing generalized Miura-Ori structures is to tailor their folding patterns to achieve desired geometries. In this work, we address this inverse-design problem by developing a new continuum framework for the differential geometry of generalized Miura-Ori. By assuming that the perturbation to the classical Miura-Ori is slowly varying in space, we derive analytical relations between geometrical properties and the perturbation field. These relationships are shown to be invertible, allowing us to design complex curved geometries. Our framework enables porting knowledge, methods and tools from continuum theories of matter and differential geometry to the field of origami metamaterials.

Predicting crack trajectories in brittle solids remains an open challenge due to their non-local nature and the way they modify the surrounding medium. Here, we develop a framework for analytically predicting crack paths, analogous to predicting particle motion in Newtonian mechanics. We show that a crack can be described as a distribution of elastic charges, with its interaction with background stress dominated by a singular geometric charge at the crack tip. The crack’s motion is then governed by the propagation of this singular charge. We apply our analytical approach to study crack trajectories near defects and validate it through experiments on flat elastomer sheets containing an edge dislocation. The experimental results show excellent agreement with theoretical predictions, including the convergence of curved crack trajectories toward a focal point. We conclude by discussing the analytical power of our method within Linear Elastic Fracture Mechanics and its extension to multiple interacting cracks. Predicting crack trajectories in materials is a complex problem. Here it is shown that cracks can be described as a distribution of elastic charges, and that crack motion can be precited based on a single charge within the unperturbed stress field.

Thin elastic sheets appear in systems ranging from graphene to biological membranes, where phenomena such as wrinkling, folding, and thermal fluctuations originate from geometric nonlinearities. These effects are treated within weakly nonlinear theories, such as the Foppl-von Karman equations, which require small slopes and fail when deflections become large even if strains remain small. We introduce a methodological progress via a geometric reformulation of thin-sheet elasticity based on a stress potential and a curvature potential. This formulation preserves the structure of the classical equations while extending their validity to nonlinear, multivalued configurations, and geometrically frustrated states. The framework provides a unified description of thin-sheet mechanics in regimes inaccessible to existing theories and opens new possibilities for the study of elastic membranes and two-dimensional materials.

Finding evidence of non-trivial pairing states is one of the greatest experimental challenges in the field of unconventional superconductivity. Such evidence requires phase-sensitive probes susceptible to the internal structure of the order parameter. We report the measurement of the Little-Parks effect in the unconventional superconductor candidate 4Hb-TaS2. In half of our rings, which are fabricated from single-crystals, we find a π-shift in the transition-temperature oscillations. According to theory, such a π-shift is only possible if the order parameter is non-s-wave. In the absence of crystallographic defects, the shift provides evidence of a multi-component order parameter. Thus, this observation increases the likelihood of the two-component order parameter scenario in 4Hb-TaS2. Furthermore, we show that Tc is enhanced as a function of the out-of-plane field when a constant in-plane field is applied, which we explain using a two-component order-parameter.

A geometric theory for manipulating energy landscapes and mechanical properties of thin elastic sheets.

Using a mean field approach and simulations, we study the non-linear mechanical response of the vertex model (VM) of biological tissue to compression and dilation. The VM is known to exhibit a transition between solid and fluid-like, or floppy, states driven by geometric incompatibility. Target perimeter and area set a target shape which may not be geometrically achievable, thereby engendering frustration. Previously, an asymmetry in the linear elastic response was identified at the rigidity transition between compression and dilation. Here we show that the asymmetry extends away from the transition point for finite strains. Under finite compression, an initially solid VM can completely relax perimeter tension, resulting in a drop discontinuity in the mechanical response. Conversely, an initially floppy VM under dilation can rigidify and have a higher response. These observations imply that re-scaling of cell area shifts the transition between rigid and floppy states. Based on this insight, we calculate the re-scaling of cell area engendered by intrinsic curvature and write a prediction for the rigidity transition in the presence of curvature. The shift of the rigidity transition in the presence of curvature for the VM provides a new metric for predicting tissue rigidity from image data of curved tissues in a manner analogous to the flat case.

When amorphous solids are subjected to simple or pure strain, they exhibit elastic increase in stress, punctuated by plastic events that become denser (in strain) upon increasing the system size. It is customary to assume in theoretical models that the stress released in each plastic event is redistributed according to the linear Eshelby kernel, causing avalanches of additional stress release. Here we demonstrate that contrary to the uniform affine strain resulting from simple or pure strain, each plastic event is associated with a non-uniform strain that gives rise to a displacement field that contains quadrupolar and dipolar charges that typically screen the linear elastic phenomenology and introduce anomalous length-scales and influence the form of the stress redistribution. An important question that opens up is how to take this into account in elasto-plastic models of shear induced phenomena like shear-banding.

The vertex model of epithelia describes the apical surface of a tissue as a tiling of polygonal cells, with a mechanical energy governed by deviations in cell shape from preferred, or target, area, A0, and perimeter, P0. The model exhibits a rigidity transition driven by geometric incompatibility as tuned by the target shape index, . For with p*(6) the perimeter of a regular hexagon of unit area, a cell can simultaneously attain both the preferred area and preferred perimeter. As a result, the tissue is in a mechanically soft compatible state, with zero shear and Young's moduli. For p0 < p*(6), it is geometrically impossible for any cell to realize the preferred area and perimeter simultaneously, and the tissue is in an incompatible rigid solid state. Using a mean-field approach, we present a complete analytical calculation of the linear elastic moduli of an ordered vertex model. We analyze a relaxation step that includes non-affine deformations, leading to a softer response than previously reported. The origin of the vanishing shear and Young's moduli in the compatible state is the presence of zero-energy deformations of cell shape. The bulk modulus exhibits a jump discontinuity at the transition and can be lower in the rigid state than in the fluid-like state. The Poisson's ratio can become negative which lowers the bulk and Young's moduli. Our work provides a unified treatment of linear elasticity for the vertex model and demonstrates that this linear response is protocol-dependent.

Holes in mechanical metamaterials, quasilocalized plastic events in amorphous solids, and bound dislocations in a hexatic matter are different mechanisms of generic stress relaxation in solids. Regardless of the specific mechanism, these and other local stress relaxation modes are quadrupolar in nature, forming the foundation for stress screening in solids, similar to polarization fields in electrostatic media. We propose a geometric theory for stress screening in generalized solids based on this observation. The theory includes a hierarchy of screening modes, each characterized by internal length scales, and is partially analogous to theories of electrostatic screening such as dielectrics and Debye-Hückel theory. Additionally, our formalism suggests that the hexatic phase, traditionally defined by structural properties, can also be defined by mechanical properties and may exist in amorphous materials.

The method of stress function in elasticity theory is a powerful analytical tool with applications to a wide range of physical systems, including defective crystals, fluctuating membranes, and more. A complex coordinates formulation of stress function, known as the Kolosov-Muskhelishvili formalism, enabled the analysis of elastic problems with singular domains, particularly cracks, forming the basis for fracture mechanics. A shortcoming of this method is its limitation to linear elasticity, which assumes Hookean energy and linear strain measure. Under finite loads, the linearized strain fails to describe the deformation field adequately, reflecting the onset of geometric nonlinearity. The latter is common in materials experiencing large rotations, such as regions close to the crack tip or elastic metamaterials. While a nonlinear stress function formalism exists, the Kolosov-Muskhelishvili complex representation had not been generalized and remained limited to linear elasticity. This paper develops a Kolosov-Muskhelishvili formalism for the nonlinear stress function. Our formalism allows us to port methods from complex analysis to nonlinear elasticity and to solve nonlinear problems in singular domains. Upon implementing the method to the crack problem, we discover that nonlinear solutions strongly depend on the applied remote loads, excluding a universal form of the solution close to the crack tip and questioning the validity of previous studies of nonlinear crack analysis.

Finding unambiguous evidence of non-trivial pairing states is one of the greatest experimental challenges in the field of unconventional superconductivity. Such evidence requires phase-sensitive probes susceptible to the internal structure of the order parameter. We measure the Little-Parks effect to provide clear evidence of an unconventional superconducting order parameter in 4Hb-TaS$_2$. Namely, we find a $π$-shift in the transition-temperature oscillations of rings made of a single crystal. We argue that such an effect can only occur if the underlying order parameter belongs to a two-dimensional representation, in other words there are two degenerate order parameters right at the transition. Additionally, we show that $T_c$ is enhanced as a function of the out-of-plane field when a constant in-plane field is applied. Such an increase is consistent with a chiral state, which again, in general only emerges from a two-component order parameter. In combination with previous experiments, our results strongly indicate that 4Hb-TaS$_2$ indeed realizes a chiral superconductor.

The concept of mechanical screening is widely applied in solid-state systems. Examples include nucleation of defects in crystalline materials, scars and pleats in curved crystals, wrinkles in strongly confined thin sheets, and cell-rearrangements in biological tissue. Available theories of such screening usually contain a crucial ingredient, which is the existence of an ordered reference state, with respect to which screening elements nucleate to release stresses. In contradistinction, amorphous materials in which a unique reference state does not exist, nevertheless exhibits plastic events that act as screening geometric charges with significant implications on the mechanical response. In a recent paper [Phys. Rev. E 104, 024904] it was proposed that mechanical strains in amorphous solids can be either weakly or strongly screened by the formation of low or high density of plastic events. At low densities the screening effect is reminiscent of the role of dipoles in dielectrics, in only renormalizing the elastic moduli. The effect of high density screening has no immediate electrostatic analog and is expected to change qualitatively the mechanical response, as seen for example in the displacement field. On the basis of experiments and simulations, we show that in granular matter, strong screening results in significant deviation from elasticity theory. The theoretical analysis, which accounts for an emergent inherent length scale, the experimental measurements and the numerical simulations of frictional granular amorphous assemblies are in agreement with each other, and provide a strong support for the novel continuum theory.

Amorphous solids under mechanical strains are prone to plastic responses. Recent work showed that in amorphous granular systems these plastic events, that are typically quadrupolar in nature, can screen the elastic response. When the density of the quadrupoles is high, the gradients of the quadrupole field act as emergent dipole sources, leading to qualitative changes in the mechanical response, as seen for example in the displacement field. In this paper we examine the effect of screening in classical glass formers. These are made of point particles that interact via binary forces. Both inverse power law forces and Lennard-Jones interactions are examined, and it is shown that in both cases the elastic response can be strongly screened, in agreement with the novel theory. The degree of deviation from classical elasticity theory is parametrized by a proposed measure that is shown to have a functional dependence on the amount of energy lost to plastic responses.

Recent progress in studying the physics of amorphous solids has revealed that mechanical strains can be strongly screened by the formation of plastic events that are typically quadrupolar in nature. The theory stipulates that gradients in the density of the quadrupoles act as emergent dipole sources, leading to strong screening and to qualitative changes in the mechanical response, as seen, for example, in the displacement field. In this Letter we first offer direct measurements of the dipole field, independently of any theoretical assumptions, and second we demonstrate detailed agreement with the recently proposed theory. These two goals are achieved by using data from both simulations and experiments. Finally, we show how measurements of the dipole fields pinpoint the theory parameters that determine the profile of the displacement field.

Vertex models, such as those used to describe cellular tissue, have an energy controlled by deviations of each cell area and perimeter from target values. The constrained nonlinear relation between area and perimeter leads to new mechanical response. Here we provide a mean-field treatment of a highly simplified model: a uniform network of regular polygons with no topological rearrangements. Since all polygons deform in the same way, we only need to analyze the ground states and the response to deformations of a single polygon (cell). The model exhibits the known transition between a fluid/compatible state, where the cell can accommodate both target area and perimeter, and a rigid/incompatible state. We calculate and measure the mechanical resistance to various deformation protocols and discover that at the onset of rigidity, where a single zero-energy ground state exists, linear elasticity fails to describe the mechanical response to even infinitesimal deformations. In particular, we identify a breakdown of reciprocity expressed via different moduli for compressive and tensile loads, implying nonanalyticity of the energy functional. We give a pictorial representation in configuration space that reveals that the complex elastic response of the vertex model arises from the presence of two distinct sets of reference states (associated with target area and target perimeter). Our results on the critically compatible tissue provide a new route for the design of mechanical metamaterials that violate or extend classical elasticity.