Kinematics of Hypersurfaces in Riemannian Manifolds

Kadianakis, N.; Travlopanos, Fotios

doi:10.1007/s10659-012-9399-9

Cited by 5 publications

(17 citation statements)

References 26 publications

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“…In this section we recall the main concepts, notation and facts concerning the geometry and the kinematics of an m -dimensional, oriented, differentiable hypersurface M in an ( m + 1 )-dimensional Riemannian manifold N . A more detailed description can be found in sections 2 and 3 of [7]. The canonical embedding of M in N is denoted by j : M ↪ N and for each X ∈ M we write J X = d j X : T X M → T j ( X ) N for the differential of j at X . The sets of vector fields on M and N are denoted by X ( M ) and X ( N ) , respectively, and X ¯ ( M ) is the set of vector fields defined on M with values on the tangent bundle of the ambient manifold N (also called vector fields along M ).…”

Section: Geometry and Kinematics Of A Hypersurfacementioning

confidence: 99%

“…Using a form of the polar decomposition theorem generalized to manifolds for the relative deformation gradient F t ( τ ) we get (see [7])…”

Section: Geometry and Kinematics Of A Hypersurfacementioning

confidence: 99%

“…Splitting the velocity field v into tangential and normal parts as v = v | | + v n n = J v | | + v n n , where v | | ∈ X ( M t ) , we get the following relations for the stretching (see section 3 in [7]).…”

Section: Geometry and Kinematics Of A Hypersurfacementioning

confidence: 99%

“…First we recall the concept of variation of the Levi-Civita connection. A τ -dependent geometry on the instantaneous hypersurface M t has been defined in [7] by pulling back on M t the geometry of M τ , using the motion. Following this procedure the τ -dependent metric tensor and the τ -dependent shape operator S t ( τ ) are defined by…”

Section: Variation Of the Levi-civita Connection Of A Moving Hypermentioning

confidence: 99%

“…This work contributes to the general study of the variation of the intrinsic and extrinsic geometry of a hypersurface [6,7], and it is an attempt towards a rational and coordinate-free description of the kinematics of continua (see also [8][9][10][11][12][13]). For this purpose we use an adapted version of the polar decomposition theorem introduced in [14].…”

Section: Introductionmentioning

confidence: 99%

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Infinitesimally affine deformations of a hypersurface

Kadianakis

Travlopanos²

2016

Mathematics and Mechanics of Solids

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Affine deformations serve as basic examples in the continuum mechanics of deformable three-dimensional bodies (usually referred to as homogeneous deformations). They preserve parallelism of straight lines, and are often used as an approximation to general deformations. However, when the deformable body is a membrane, a shell or an interface modeled by a surface, parallelism is defined by the affine connection of this surface. In this work we study the infinitesimally affine time-dependent deformations (motions) of such a continuum, but in a more general context, by considering that it is modeled by a Riemannian hypersurface. First we prove certain equivalent formulas for the variation of the connection of the hypersurface. Some of these formulas are expressed in terms of geometrical quantities, and others in terms of kinematical quantities of the deforming continuum. The latter is achieved by using an adapted version of the polar decomposition theorem, frequently used in continuum mechanics to analyze motion. We also apply our results to special motions like tangential and normal motions. Further, we find necessary and sufficient conditions for this variation to be zero (infinitesimal affine motions), providing insight on the form of these motions and the kind of hypersurfaces that allow such motions. Finally, we give some specific examples of mechanical interest which demonstrate motions that are infinitesimally affine but not infinitesimally isometric.

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Section: Geometry and Kinematics Of A Hypersurfacementioning

confidence: 99%

“…Using a form of the polar decomposition theorem generalized to manifolds for the relative deformation gradient F t ( τ ) we get (see [7])…”

Section: Geometry and Kinematics Of A Hypersurfacementioning

confidence: 99%

Section: Geometry and Kinematics Of A Hypersurfacementioning

confidence: 99%

Section: Variation Of the Levi-civita Connection Of A Moving Hypermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Infinitesimally affine deformations of a hypersurface

Kadianakis

Travlopanos²

2016

Mathematics and Mechanics of Solids

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A Geometric Theory of Nonlinear Morphoelastic Shells

et al. 2016

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Many thin three-dimensional elastic bodies can be reduced to elastic shells: two-dimensional elastic bodies whose reference shape is not necessarily flat. More generally, morphoelastic shells are elastic shells that can remodel and grow in time. These idealized objects are suitable models for many physical, engineering, and biological systems. Here, we formulate a general geometric theory of nonlinear morphoelastic shells that describes both the evolution of the body shape, viewed as an orientable surface, as well as its intrinsic material properties such as its reference curvatures. In this geometric theory, bulk growth is modeled using an evolving referential configuration for the shell, the so-called material manifold. Geometric quantities attached to the surface, such as the first and second fundamental forms are obtained from the metric of the three-dimensional body and its evolution. The governing dynamical equations for the the body are obtained from variational consideration by assuming that both fundamental forms on the material manifold are dynamical variables in a Lagrangian field theory. In the case where growth can be modeled by a Rayleigh potential, we also obtain the governing equations for growth in the form of kinetic equations coupling the evolution of the first and the second fundamental forms with the state of stress of the shell. We apply these ideas to obtain stress-free growth fields of a planar sheet, the time-evolution of a morphoelastic circular cylindrical shell subject to time-dependent internal pressure, and the residual stress of a morphoelastic planar circular shell.

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Transformation Cloaking in Elastic Plates

Golgoon

Yavari

2021

J Nonlinear Sci

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Kinematics of Hypersurfaces in Riemannian Manifolds

Cited by 5 publications

References 26 publications

Infinitesimally affine deformations of a hypersurface

Infinitesimally affine deformations of a hypersurface

A Geometric Theory of Nonlinear Morphoelastic Shells

Transformation Cloaking in Elastic Plates

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