Dipole screening in pure shear strain protocols of amorphous solids.

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 nonuniform 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 elastoplastic models of shear induced phenomena like shear banding.

Finite elasticity of the vertex model and its role in rigidity of curved cellular tissues.

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.

Anomalous elasticity and emergent dipole screening in three-dimensional amorphous solids.

In recent work, we developed a screening theory for describing the effect of plastic events in amorphous solids on its emergent mechanics. The suggested theory uncovered an anomalous mechanical response of amorphous solids where plastic events collectively induce distributed dipoles that are analogous to dislocations in crystalline solids. The theory was tested against various models of amorphous solids in two dimensions, including frictional and frictionless granular media and numerical models of amorphous glass. Here we extend our theory to screening in three-dimensional amorphous solids and predict the existence of anomalous mechanics similar to the one observed in two-dimensional systems. We conclude by interpreting the mechanical response as the formation of nontopological distributed dipoles that have no analog in the crystalline defects literature. Having in mind that the onset of dipole screening is reminiscent of Kosterlitz-Thouless and hexatic transitions, the finding of dipole screening in three dimensions is surprising.

Geometric theory of mechanical screening in two-dimensional solids.

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.

Nonlinear extension of the Kolosov-Muskhelishvili stress function formalism.

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.

The role of non-affine deformations in the elastic behavior of the cellular vertex model.

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.

Anomalous elasticity in classical glass formers.

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.

Anomalous elasticity of a cellular tissue vertex model.

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.

Direct measurement of dipoles in anomalous elasticity of amorphous solids.

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.

Anomalous elasticity and plastic screening in amorphous solids.

Amorphous solids appear to react elastically to small external strains, but in contrast to ideal elastic media, plastic responses abound immediately at any value of the strain. Such plastic responses are quasilocalized in nature, with the "cheapest" one being a quadrupolar source. The existence of such plastic responses results in screened elasticity in which strains and stresses can either quantitatively or qualitatively differ from the unscreened theory, depending on the specific screening mechanism. Here we offer a theory of such screening effects by plastic quadrupoles, dipoles, and monopoles, explain their natural appearance, and point out the analogy to electrostatic screening by electric charges and dipoles. For low density of quadrupoles the effect is to normalize the elastic moduli without a qualitative change compared to pure elasticity theory; for higher density of quadrupoles the screening effects result in qualitative changes. Predictions for the spatial dependence of displacement fields caused by local sources of strains are provided and compared to numerical simulations. We find that anomalous elasticity is richer than electrostatics in having a screening mode that does not appear in the electrostatic analog.

Geometric charges and nonlinear elasticity of two-dimensional elastic metamaterials.

Problems of flexible mechanical metamaterials, and highly deformable porous solids in general, are rich and complex due to their nonlinear mechanics and the presence of nontrivial geometrical effects. While numeric approaches are successful, analytic tools and conceptual frameworks are largely lacking. Using an analogy with electrostatics, and building on recent developments in a nonlinear geometric formulation of elasticity, we develop a formalism that maps the two-dimensional (2D) elastic problem into that of nonlinear interaction of elastic charges. This approach offers an intuitive conceptual framework, qualitatively explaining the linear response, the onset of mechanical instability, and aspects of the postinstability state. Apart from intuition, the formalism also quantitatively reproduces full numeric simulations of several prototypical 2D structures. Possible applications of the tools developed in this work for the study of ordered and disordered 2D porous elastic metamaterials are discussed.

Erratum: Geometric Frustration and Solid-Solid Transitions in Model 2D Tissue [Phys. Rev. Lett. 120, 268105 (2018)].

This corrects the article DOI: 10.1103/PhysRevLett.120.268105.

Nonlinear mechanics of thin frames.

The dramatic effect kirigami, such as hole cutting, has on the elastic properties of thin sheets invites a study of the mechanics of thin elastic frames under an external load. Such frames can be thought of as modular elements needed to build any kirigami pattern. Here we develop the technique of elastic charges to address a variety of elastic problems involving thin sheets with perforations, focusing on frames with sharp corners. We find that holes generate elastic defects (partial disclinations), which act as sources of geometric incompatibility. Numerical and analytic studies are made of three different aspects of loaded frames-the deformed configuration itself, the effective mechanical properties in the form of force-extension curves, and the buckling transition triggered by defects. This allows us to understand generic kirigami mechanics in terms of a set of force-dependent elastic charges with long-range interactions.

Kirigami Mechanics as Stress Relief by Elastic Charges.

We develop a geometric approach to understand the mechanics of perforated thin elastic sheets, using the method of strain-dependent image elastic charges. This technique recognizes the buckling response of a hole under an external load as a geometrically tuned mechanism of stress relief. We use a diagonally pulled square paper frame as a model system to quantitatively test and validate our approach. Specifically, we compare nonlinear force-extension curves and global displacement fields in theory and experiment. We find a strong softening of the force response accompanied by curvature localization at the inner corners of the buckled frame. Counterintuitively, though in complete agreement with our theory, for a range of intermediate hole sizes, wider frames are found to buckle more easily than narrower ones. Upon extending these ideas to many holes, we demonstrate that interacting elastic image charges can provide a useful kirigami design principle to selectively relax stresses in elastic materials.

Cutaneous presentations of omphalomesenteric duct remnant: A systematic review of the literature.

BACKGROUND: Disorders of the umbilicus are commonly seen in infancy, including hernias, infections, anomalies, granulomas, and malignancies. Meticulous inspection of the umbilicus at birth might reveal a persisting embryonic remnant, such as an omphalomesenteric duct (OMD), manifested by a variety of cutaneous signs, such as an umbilical mass, granulation tissue, or discharge. OBJECTIVE: To systematically review the available data regarding the presence and management of OMD remnant with cutaneous involvement to suggest a practical approach for diagnosis and treatment. METHODS: A systematic review of the literature evaluating OMD anomalies presenting with cutaneous symptoms was performed. In addition, an index case of an 11-month-old patient is presented. RESULTS: We included 59 publications reporting 536 cases; 97% of the patients whose age was noted were infants (mean age 11 months). In 7.5% of the cases, diagnosis was established only after treatment failure. In 6.4% of patients, nonlethal complications were reported, and in 10.3%, the outcome was death, partly due to delayed diagnosis or mismanagement. LIMITATIONS: Limited quality of the collected data, reporting bias. CONCLUSION: OMD is relatively rare; however, the clinician must consider this remnant while examining patients with umbilical abnormalities because mismanagement could cause severe morbidity and mortality.

Geometric Frustration and Solid-Solid Transitions in Model 2D Tissue.

We study the mechanical behavior of two-dimensional cellular tissues by formulating the continuum limit of discrete vertex models based on an energy that penalizes departures from a target area A_{0} and a target perimeter P_{0} for the component cells of the tissue. As the dimensionless target shape index s_{0}=(P_{0}/sqrt[A_{0}]) is varied, we find a transition from a soft elastic regime for a compatible target perimeter and area to a stiffer nonlinear elastic regime frustrated by geometric incompatibility. We show that the ground state in the soft regime has a family of degenerate solutions associated with zero modes for the target area and perimeter. The onset of geometric incompatibility at a critical s_{0}^{c} lifts this degeneracy. The resultant energy gap leads to a nonlinear elastic response distinct from that obtained in classical elasticity models. We draw an analogy between cellular tissues and anelastic deformations in solids.

Malignant melanoma clinically mimicking pyogenic granuloma: comparison of clinical evaluation and histopathology.

Amelanotic melanomas (AMMs) account for a small proportion of all melanomas. They pose a risk of delayed diagnosis and, consequently, poor prognosis. AMMs may atypically present as a pyogenic granuloma-like lesion. This study sought to investigate the prevalence and clinical and histological features of AMM masquerading as pyogenic granuloma. The database of a tertiary medical center was screened for all patients pathologically diagnosed with melanoma in 2005-2016. Those with a suspected primary (i.e. pre-excision) clinical diagnosis of pyogenic granuloma were identified, and their demographic, clinical, histologic, and outcome data were collected from the medical files. Of 2038 patients diagnosed with melanoma, 10 (∼0.5%) had a pyogenic granuloma-like AMM. The mean±SD age at lesion presentation was 56±18.9 years and the mean time from lesion appearance to diagnosis was 91.5±117.1 months. Nine tumors were located on the skin surface, and one on the oral mucosa. The mean lesion size was 19.6±14.1 mm and the mean Breslow's depth was 6.47±3.1 mm; all tumors presented in the vertical growth phase. Seven (70%) patients had lymph node involvement or metastasis at diagnosis. Two patients died of the disease within 1 year of diagnosis. Given the potential lethality of AMM and the benign nature of pyogenic granuloma, clinician recognition of pyogenic granuloma-like AMMs is crucial. In the presence of a pyogenic granuloma-like lesion, findings of older patient age and large tumor size should raise the index of suspicion and prompt a biopsy study, thereby ensuring early and accurate treatment.

Internal Stresses Lead to Net Forces and Torques on Extended Elastic Bodies.

A geometrically frustrated elastic body will develop residual stresses arising from the mismatch between the intrinsic geometry of the body and the geometry of the ambient space. We analyze these stresses for an ambient space with gradients in its intrinsic curvature, and show that residual stresses generate effective forces and torques on the center of mass of the body. We analytically calculate these forces in two dimensions, and experimentally demonstrate their action by the migration of a non-Euclidean gel disc in a curved Hele-Shaw cell. An extension of our analysis to higher dimensions shows that these forces are also generated in three dimensions, but are negligible compared to gravity.

Geometry and mechanics of two-dimensional defects in amorphous materials.

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 distribution of localized sources of strain. Examples are plasticity in amorphous materials and mechanical interactions between cells on a flexible substrate.

Shape transformations of soft matter governed by bi-axial stresses.

Rational design of the programmable soft matter requires understanding of the effect of a complex metric on shape transformations of thin non-Euclidean sheets. In the present work, we explored experimentally and using simulations how simultaneous or consecutive application of two orthogonal perturbations to thin patterned stimuli-responsive hydrogel sheets affects their three-dimensional shape transformations. The final shape of the sheet is governed by the metric, but not the order, in which the perturbations are applied to the system, and is determined by the competition of small-scale bidirectional stresses. In addition, a new, unexpected transition from a planar state to an equilibrium helical shape of the hydrogel sheet is observed via a mechanism that is yet to be explained.