Kazuhito Yamasaki

A deformed medium including a defect field and differential forms

We consider basic equations for a deformed medium including a defect field on the basis of differential forms. To make our analysis, we extend three basic equations: (I) an incompatibility equation; (II) the Peach-Kohler equation; (III) the Navier equation based on the Hodge duality of the deformed medium. By combining two exterior differential operators, we derive (I) an incompatibility equation that extends the compatibility equation to include a defect field. The Hodge dual of the incompatibility equation becomes a generalized stress function, which includes previously derived stress functions such as Beltramis, Moreras, Maxwells and Airys stress functions. By applying homotopy operators, we extend (II) the Peach-Kohler equation to include disclinations. In this case, we can define the basic quantities of stress space by analogy with the monopole theory. By combining exterior differential operators and star operators, we extend (III) the Navier equation to include a defect field. In this analysis, we define a Navier operator that is related to the Laplace operator through Hodge duality. We consider gauge conditions for a defect field based on the differential geometry of a deformed medium. This suggests a duality between yielding and fatigue fractures. The gauge condition in strain space-time is interpreted as basic relations in polycrystalline plastic deformation.

HODGE DUALITY AND CONTINUUM THEORY OF DEFECTS

Dual material space-time with defect field is presented in the language of differential forms: one is the strain space-time whose basic equation is the continuity equation for the dislocation 2-form; the other is the stress space-time whose basic equation is the continuity equation for the couple-stress and angular momentum 2-form. Continuity and kinematic equations in each space can be derived by the transformation from p-form to (p + 1)-form. Moreover, several constitutive equations can be recognized as the transformation between the p-form of the strain space-time and the (4-p)-form of the stress space-time. These kinematic, continuity and constitutive equations can be interpreted geometrically as Cartan structure equations, Bianchi identities and Hodge duality transformations, respectively.

Differential geometric structures of stream functions: incompressible two-dimensional flow and curvatures

The Okubo–Weiss field, frequently used for partitioning incompressible two-dimensional (2D) fluids into coherent and incoherent regions, corresponds to the Gaussian curvature of the stream function. Therefore, we consider the differential geometric structures of stream functions and calculate the Gaussian curvatures of some basic flows. We find the following. (I) The vorticity corresponds to the mean curvature of the stream function. Thus, the stream-function surface for an irrotational flow and that for a parallel shear flow correspond to the minimal surface and a developable surface, respectively. (II) The relationship between the coherency and the magnitude of the vorticity is interpreted by the curvatures. (III) Using the Gaussian curvature, stability of single and double point vortex streets is analyzed. The results of this analysis are compared with the well-known linear stability analysis. (IV) Conformal mapping in fluid mechanics is the physical expression of the geometric fact that the sign of the Gaussian curvature does not change in conformal mapping. These findings suggest that the curvatures of stream functions are useful for understanding the geometric structure of an incompressible 2D flow.

Species-area curve for land snails on Kikai Island in geological time

Abstract Historical changes in the coastline of Kikai Island of the Ryukyu Islands in the southeast part of Japan were estimated by using a numerical simulation based on a glacio-hydro-isostasy model. Temporal changes in the area of the island during the last 40 Kyr were compared with temporal changes in species diversity in fossil land snails of the island. The species number in the past was theoretically estimated by the area of Kikai Island in the past and a species-area relationship among the modern land snail fauna of the Ryukyu Islands. The theoretical species numbers are very close to the actual ones. This suggests that the change in island area is the main cause of the change in species diversity in Kikai Island. In addition, we discuss causes other than the area, such as island elevation, distance to the nearest large island, climate change, human activity, and imperfection of fossil data. We also discuss the change in Fishers alpha and body size against the change in the area.

Jacobi stability for dynamical systems of two-dimensional second-order differential equations and application to overhead crane system

Geometric structures of dynamical systems are investigated based on a differential geometric method (Jacobi stability of KCC-theory). This study focuses on differences of Jacobi stability of two-dimensional second-order differential equation from that of one-dimensional second-order differential equation. One of different properties from a one-dimensional case is the Jacobi unstable condition given by eigenvalues of deviation curvature with different signs. Then, this geometric theory is applied to an overhead crane system as a two-dimensional dynamical system. It is shown a relationship between the Hopf bifurcation of linearized overhead crane and the Jacobi stability. Especially, the Jacobi stable trajectory is found for stable and unstable spirals of the two-dimensional linearized system. In case of the linearized overhead crane system, the Jacobi stable spiral approaches to the equilibrium point faster than the Jacobi unstable spiral. This means that the Jacobi stability is related to the resilience of deviated trajectory in the transient state. Moreover, for the nonlinear overhead crane system, the Jacobi stability for limit cycle changes stable and unstable over time.

Geometry of surfaces with Caputo fractional derivatives and applications to incompressible two-dimensional flows

Geometric structures of surfaces are formulated based on Caputo fractional derivatives. The Gauss frame of a surface with fractional order is introduced. Then, the non-locality of the fractional derivative characterizes the asymmetric second fundamental form. The mean and Gaussian curvatures of the surface are defined in the case of fractional order. Based on the fractional curvatures, incompressible two-dimensional flows are discussed. The stream functions are obtained from a fractional continuity equation. The asymmetric second fundamental form of stream-function surface is related to the path dependence of flux. Moreover, the fractional curvatures are calculated for the stream-function surfaces of uniform and corner flows. The uniform flow with fractional order is characterized by the non-vanishing mean curvature. The non-locality of corner flow is expressed by the mean and Gaussian curvatures with fractional order. In particular, the fractional order within the stream-function of corner flow determines the change of sign of Gaussian curvature. Therefore, the non-local property of incompressible flows can be investigated by the fractional curvatures.

Differential geometric approach to the stress aspect of a fault: Airy stress function surface and curvatures

We considered the two-dimensional stress aspect of a fault from the viewpoint of differential geometry. For this analysis, we concentrated on the curvatures of the Airy stress function surface. We found the following: (i) Because the principal stresses are the principal curvatures of the stress function surface, the first and the second invariant quantities in the elasticity correspond to invariant quantities in differential geometry; specifically, the mean and Gaussian curvatures, respectively; (ii) Coulomb’s failure criterion shows that the coefficient of friction is the physical expression of the geometric energy of the stress function surface; (iii) The differential geometric expression of the Goursat formula shows that the fault (dislocation) type (strike-slip or dip-slip) corresponds to the stress function surface type (elliptic or hyperbolic). Finally, we discuss the need to use non-biharmonic stress tensor theory to describe the stress aspect of multi-faults or an earthquake source zone.

Differential geometric structure of non-equilibrium dynamics in competition and predation: Finsler geometry and KCC theory

Abstract We considered the differential geometric structure of non-equilibrium dynamics in non-linear interactions, such as competition and predation, based on Kosambi-Cartan-Chern (KCC) theory. The stability of a geodesic flow on a Finslerian manifold is characterized by the deviation curvature (the second invariant in the dynamical system). According to KCC theory, the value of the deviation curvature is constant around the equilibrium point. However, in the non-equilibrium region, not only the value but also the sign of the deviation curvature depend on time. Next, we reapplied KCC theory to the dynamics of the deviation curvature and determined the hierarchical structure of the geometric stability. The dynamics of the deviation curvature in the nonequilibrium region is accompanied by a complex periodic (node) pattern in the predation (competition) system.

Geometry of stress function surfaces for an asymmetric continuum

A two-dimensional stress field of dislocation or fault is geometrically studied for an asymmetric continuum. For geometric surfaces of the stress and couple-stress functions, the mean and Gaussian curvatures are derived. The mean curvature of couple-stress function surface is connected with the asymmetric of stress tensor. Moreover, the Gaussian curvature of stress function surface is characterized by both the stress and couple-stress. On the other hand, the mean curvature of stress function surface is not affected by the asymmetry of stress. Based on these geometric expressions, the Coulomb’s failure criterion and the friction coefficient are expressed by the curvatures of couple-stress function surface. Moreover, geometric structures of stress and couple stress function surfaces are shown for edge and wedge dislocations as faults. The curvatures of these surfaces show that the effect of couple-stress is constrained around the dislocations only.

KCC Analysis of the Normal Form of Typical Bifurcations in One-Dimensional Dynamical Systems: Geometrical Invariants of Saddle-Node, Transcritical, and Pitchfork Bifurcations

The Jacobi stability of the normal form of typical bifurcations in one-dimensional dynamical systems is analyzed by introducing the concept of the production process (time-like potential) to KCC th...