arXiv subjects

Ryuichi Tarumi

Publications and source records attributed to Ryuichi Tarumi.

10 recordsLinked to original sources

arXiv2026

Numerical and experimental framework for bending elasticity of highly flexible slender structures

Slender structures are highly flexible, spanning several orders of magnitude in length scale. Their deformation depends on the slenderness of their cross sections, highlighting that the elasticity and geometry of structures are intrinsically coupled. The deformation of the cross-section becomes significant, particularly when tubes and pipes are subjected to bending, known as the Brazier instability. Although the bending performance of slender structures is quantified experimentally using a canonical three-point bending test, their numerical counterparts remain under-explored because complex contact mechanics must be implemented in simulations. In this study, we develop a computational framework to simulate experimental three-point bending tests using a hybrid material point method (hybrid-MPM) approach, which integrates Lagrangian finite element and Eulerian finite difference frameworks. We adapt our framework to elastic tubes and tape springs as canonical examples that exhibit characteristic bending deformation in which the cross-sectional and lengthwise bending are coupled. The predictions of numerical simulations are validated against desktop experiments and classical theory. The excellent agreement between the simulation and the experiments implies that the hybrid-MPM framework provides a robust computational framework for predicting the large deformation of structures involving complex contact, such as soft robots and deployable structures.

cond-mat.soft↗

arXiv2026

Where Humpty Dumpty Breaks: Geometry-Driven Fracture in Ellipsoidal Shells

Fracture networks are ubiquitous in nature, spanning scales from millimeter-sized cracks in botanical peels to hundred-kilometer-long lineae on planetary satellites. The propagation of a crack is a complex, nonlinear phenomenon governed by the interplay of mechanical properties, rheological behavior, and system geometry. While fracture mechanics has long addressed structural failure, the relationship among fracture, elasticity, and nonlinear geometry has recently revived as a focal point in condensed matter and biophysics. However, a unified framework that systematically explains how surface geometry prescribes the transition between disparate fracture morphologies remains elusive. Here we show that shell curvature provides a geometric blueprint for fracture, governing the evolution of complex crack networks through induced stress anisotropy. By internally pressurizing thin, bilayer spheroidal shells, we demonstrate that a rich diversity of crack morphologies across lateral, longitudinal, and random orientations depends on the curvature ratio between the pole and the equator. We find that these patterns arise from the nonlinear mechanics of the shell, which can be leveraged to effectively control crack growth. Our results establish a direct link between structural curvature and fractures, providing a predictive framework that integrates nonlinear geometry with the classical Griffith and von Mises criteria. Beyond our model system, we find that the disparate fracture patterns observed in ripening muskmelons and in the icy crust of Europa follow the same geometric principles. We expect that this unified understanding of crack morphogenesis will inform the design principles of novel functional materials that are resilient to fracture and provide insights into the mechanical performance of curved biological and geophysical architectures.

cond-mat.soft↗

arXiv2025

Analytical Expression for Fracture Profile in Viscoelastic Crack Propagation

We derive an analytical expression for the strain field during steady-state crack propagation in viscoelastic solids described by the standard linear solid (Zener) model. This expression reveals three regions in the fracture profile and in the strain field ahead of the crack tip, each distinguished by power-law exponents that evolve with distance from the crack tip. These features explain the experimentally observed crack-tip sharpening in rubbers and gels as the crack-propagation velocity increases, often associated with catastrophic failure triggered by a velocity jump. Furthermore, we establish de Gennes' viscoelastic trumpet on a continuum-mechanical foundation, previously based only on a scaling argument.

cond-mat.soft↗

arXiv2024

Revisiting Volterra defects: Geometrical relation between edge dislocations and wedge disclinations

This study presents a comprehensive mathematical model for Volterra defects and explores their relations using differential geometry on Riemann--Cartan manifolds. Following the standard Volterra process, we derived the Cartan moving frame, a geometric representation of plastic fields, and the associated Riemannian metric using exterior algebra. Although the analysis naturally defines the geometry of three types of dislocations and the wedge disclination, it fails to classify twist disclinations owing to the persistent torsion component, suggesting the need for modifications to the Volterra process. By leveraging the interchangeability of the Weitzenböck and Levi-Civita connections and applying an analytical solution for plasticity derived from the Biot--Savart law, we provide a rigorous mathematical proof of the long-standing phenomenological relationship between edge dislocations and wedge disclinations. Additionally, we showcase the effectiveness of novel mathematical tools, including Riemannian holonomy for analysing the Frank vector and complex potentials that encapsulate the topological properties of wedge disclinations as jump discontinuities. Furthermore, we derive analytical expressions for the linearized stress fields of wedge disclinations and confirm their consistency with existing results. These findings demonstrate that the present geometrical framework extends and generalizes the classical theory of Volterra defects.

cond-mat.mtrl-sci↗

arXiv2024

Snap and Jump: How Elastic Shells Pop Out

Grip, walk, crawl, and jump. Soft robots are integrated functional structures composed of compliant mechanisms, whose activity spans various industrial applications such as surgery, healthcare, surveillance, and even planetary exploration. One of their promising mobility mechanism is snap-buckling; the instability mode of flexible structures passing from one equilibrium state to another can instantaneously generate large power for its motion. Predicting their performance with even simple geometry requires disentangling material, geometric nonlinearity, and contact, thereby still being a challenging problem to date. Here, we study the jumping dynamics of hemispherical elastic shells driven by snap-buckling, as a model system of soft jumping mechanisms, combining experiments, simulations, and analytical theory. We find that the contact transition dynamics trigger the jumping phenomenon upon snap-buckling by constructing the analytical predictions with shell elasticity in excellent agreement with both experiments and simulations. Despite the simple geometry of the shell, its dynamical performance primarily relies on a complex interplay between elasticity, geometry, and contact friction. By elucidating the dynamics of the building blocks of soft robots that undergo large deformations, we can build their predictive experimental and numerical framework. Our research paves the way for designing soft robots suitable for the required loading conditions or structural requirements without empirical methods.

cond-mat.soft↗

arXiv2024

Curling morphology of knitted fabrics: Structure and Mechanics

Knitted fabrics are two-dimensional-like structures formed by stitching one-dimensional yarn into three-dimensional curves. Plain stitch or stockinette stitch, one of the most fundamental knitting stitches, consists of periodic lattices of bent yarns, where three-dimensional (3D) curling behavior naturally emerges at the edges. The elasticity and geometry of knitted fabrics have been studied in previous studies, primarily based on 2D modeling. Still, the relation between 3D geometry and the mechanics of knitted fabrics has not been clarified so far. The curling behavior of knits is intricately related to the forces and moments acting on the yarns, geometry of the unit knitted loops, mechanical properties, and contacts, hence requiring a 3D analysis. Here, we show that the curling of plain knits emerges through the elasticity and geometry of the knitted loops, combining desktop-scale experiments and reduced elasticity-based simulations. We find that by changing the horizontal and vertical knitting numbers, three types of curl shapes emerge: side curl and top/bottom curl shapes, which are curled only horizontally and vertically, and double curl shape, in which both curl shapes appear together. The fundamental mechanism of intricate shape deformation is clarified through the force and moment balance along yarn whose centerline shape is discretized through the B-spline curves where elastic stretching, bending, and contact mechanics are taken into account. We reveal that the 3D structure of the single-knitted loop plays a critical role in the curling behavior. Our results imply that the change in shape per a single knitted loop has the potential to control the 3D natural overall shape of knitted fabrics, and could be applied in predicting or designing more complex 3D shapes made of knitted fabrics.

cond-mat.soft↗

arXiv2024

Biot-Savart law in the geometrical theory of dislocations

Universal mechanical principles may exist behind seemingly unrelated physical phenomena, providing novel insights into these phenomena. This study sheds light on the geometrical theory of dislocations through an analogy with electromagnetics. In this theory, solving Cartan's first structure equation is essential for connecting the dislocation density to the plastic deformation field of the dislocations. The additional constraint of a divergence-free condition, derived from the Helmholtz decomposition, forms the governing equations that mirror Ampère's and Gauss' law in electromagnetics. This allows for the analytical integration of the equations using the Biot-Savart law. The plastic deformation fields of screw and edge dislocations obtained through this process form both a vortex and an orthogonal coordinate system on the cross-section perpendicular to the dislocation line. This orthogonality is rooted in the conformal property of the corresponding complex function that satisfies the Cauchy-Riemann equations, leading to the complex potential of plastic deformation. We validate the results through a comparison with the classical dislocation theory. The incompatibility tensor is crucial in the generation of the mechanical field. These findings reveal a profound unification of dislocation theories, electromagnetics, and complex functions through their underlying mathematical parallels.

cond-mat.mtrl-sci↗

arXiv2024

Geometrical frustration in nonlinear mechanics of screw dislocation

The existence of stress singularities and reliance on linear approximations pose significant challenges in comprehending the stress field generation mechanism around dislocations. This study employs differential geometry and calculus of variations to mathematically model and numerically analyse screw dislocations. The kinematics of the dislocation are expressed by the diffeomorphism of the Riemann--Cartan manifold, which includes both the Riemannian metric and affine connection. The modelling begins with a continuous distribution of dislocation density, which is transformed into torsion $τ$ through the Hodge duality. The plasticity functional is constructed by applying the Helmholtz decomposition to bundle isomorphism, which is equivalent to the Cartan first structure equation for the intermediate configuration $\mathcal{B}$. The current configuration is derived by the elastic embedding of $\mathcal{B}$ into the standard Euclidean space $\mathbb{R}^3$. The numerical analysis reveals the elastic stress fields effectively eliminate the singularity along the dislocation line and exhibit excellent conformity with Volterra's theory beyond the dislocation core. Geometrical frustration is the direct source of dislocation stress fields, as demonstrated through the multiplicative decomposition of deformation gradients. By leveraging the mathematical properties of the Riemann--Cartan manifold, we demonstrate that the Ricci curvature determines the symmetry of stress fields. These results substantiate a long-standing mathematical hypothesis: the duality between stress and curvature.

cond-mat.mtrl-sci↗

arXiv2022

Weaving paper strips for designing of general curved surface with geometrical elasticity

This study proposes 'amigami' as a new method of creating a general curved surface. It conducts the shape optimization of weaving paper strips based on the theory of nonlinear elasticity on Riemannian manifolds. The target surface is split into small curved strips by cutting the medium along with its coordinates, and each strip is embedded into a flat paper sheet to minimize a strain energy functional due to the in-plane deformation. The weak form equilibrium equation is derived from a Lie derivative with the virtual displacement vector field, and the equation is solved numerically using the Galerkin method with a non-uniform B-spline manifold. As a demonstration, we made catenoid and helicoid surfaces which are made by waving 54 paper strips. The papercraft reminds us of the isometric transformation from the catenoid to the helicoid and vice versa. We also provide strain estimates for paper strips with rigorous mathematical proof. This estimating process is a generalization of the classical beam theory of Euler-Bernoulli to a modern geometrical elasticity.

math-ph↗

arXiv2022

Geometrical Modelling and Numerical Analysis of Dislocaion Mechanics

This study undertakes the mathematical modelling and numerical analysis of dislocations within the framework of differential geometry. The fundamental configurations, i.e. reference, intermediate and current configurations, are expressed as the Riemann-Cartan manifold, which equips the Riemannian metric and Weitzenböck connection. The torsion 2-form on the intermediate configuration is obtained through the Hodge duality of the dislocation density and the corresponding bundle isomorphism is subjected to the Helmholtz decomposition. This analysis introduces the boundary condition for plastic deformation. Cartan first structure equation and stress equilibrium equation are solved numerically using weak form variational expressions and isogeometric analysis. The numerical analysis carried out for this study reveals the distribution of plastic deformation fields around screw and edge dislocations for the first time. It also demonstrates stress fields around dislocations of which the distant fields show full agreement with the classical Volterra theory, while at the same time eliminating the singularity otherwise introduced at the dislocation by classical methods. The stress fields include several characteristic features due to the geometrical nonlinearity included therein. We also demonstrate that free surfaces affect both plastic and elastic deformation, but in different ways. The mathematical framework of this study is applicable to an arbitrary configuration of dislocations.

math.NA↗