### Citations

193 |
Theoretical Elasticity,
- Green, Zerna
- 1968
(Show Context)
Citation Context ...d products of these quantities in comparison with their first powers. With respect to such an approximation in the classical (infinitesimal) theory of elasticity, the covariant strain tensor becomes (=-=[8]-=-, p.149) γij = 1 2 (vi |j + v j |i) (13) But, generally, it is introduced by the relation γij = 1 2 (˜gij − gij), (14) with the metric tensors defined by gij =< ei;ej > at the point P and by ˜gij =< ˜... |

113 |
Total Mean Curvature and Submanifolds of Finite Type, World Scientific,
- Chen
- 1984
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Citation Context ... where [δ] denotes the column-matrix of the displacement at each point of M. To establish the relation (30) we start with the energy E(φ) associated to the diffeomorphism φ defined by the expression (=-=[7]-=-, p.202) E(φ) = 1 2 ∫ M ‖φ∗‖ 2 dω, (31) where φ∗ is the differential of φ and dω is the volume element of M. The Euclidean norm ||.|| here present acts on the tangent space T ˜ P ( ˜M) at each image ˜... |

59 |
A Course in Differential Geometry
- Klingenberg
- 1978
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Citation Context ...f vectors ( ˜ei,˜ei+1) of a moving frame in M associated to the orbit . This can be explained by the similitude of the scalar components of the tensor ω with the coefficients (that define curvatures, =-=[12]-=-) in the Cartan’s equations associated to the Frenet’s moving frame {C(t);f1,f2,f3} along a curve C in E3 : ˙C(t) = ∑ i αi (t)fi(t) fj(t) ˙ = ∑ i ωi j(t)fi(t), where the coefficients of the second equ... |

4 | Aspects of global Riemannian geometry - Petersen - 1999 |

2 | Unsheared triads and extended polar decompositions of the deformation gradient - Boulanger, Hayes - 2001 |

2 |
Differential Geometry with Applications to General Relativity (in Romanian
- Ianus
- 1983
(Show Context)
Citation Context ...(see [1]) are compact manifolds. However, if B is a non-compact differentiable manifold there exists an open neighborhood U of every point of B such that the restriction p | U is a regular embedding (=-=[9]-=-). In order to describe a body-manifold deformation we consider the placement p to be ”of reference” and choose another placement ˜p of B in E 3 as a new element of C k (B). Let us denote by M and ˜M ... |

1 |
Geometric Approach to the Continuum Deformation Analysis
- Boja, Brailoiu
- 1997
(Show Context)
Citation Context ...ed with a differentiable (even Riemannian) manifold structure. Endowed with a real 3-dimensional differentiable manifold structure, an image M by the elements of C k (B) is called a body-manifold (see=-=[1]-=- for supplementary conditions). The considered above embedding is regular, that is B and its image M(⊂ E 3 ) are homeomorphic, because the body-manifolds in our acceptance (see [1]) are compact manifo... |

1 |
Body-manifold deformation from the differential geometry point of view
- Boja, Brailoiu
- 2001
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Citation Context ...d(P, P ′ ) = d(δt(P),δt(P ′ )), where d is the distance function in E 3 . Otherwise, φ is a motion of M in E 3 . So, with the help of the family ∆I one can describes any deformation φ : M → ˜ M (see =-=[2]-=-). Here φ must be understood not as much as a description of the body state after deformation but as a mapping describing it continuously during the deformation process. Thus, the image by φ of any po... |

1 |
On the deformation energy of a body-manifold
- Boja
- 1997
(Show Context)
Citation Context ...,α =M. Then, the union of these atlases for all t ∈ I is a differentiable atlas on M, defining the differentiable manifold structure on it. We say that M is a time deformed trace-manifold. In [2] and =-=[3]-=- were studied deformations when M and ˜M are looked as submanifolds of M. Namely, M = M0 (at the moment t = 0) contains all the reference positions p0(λ) . = P of the particles λ of B in E 3 . Now we ... |

1 |
On the mean rotation in finite deformation
- Jaric, Cowin
- 1998
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Citation Context ...tue of the enumerated above properties, it is a transformation of the domain of E 3 occupied by M into another domain occupied by ˜M by composing some of the following elementary transformations (see =-=[10]-=-): translations, rotations, and stretches; each of them defines an elementary state of local deformation of the continuum. However, by superposition can also be considered rigid-body motions, compress... |

1 |
delasticit en grande deformations; Sem. dAnalyse Convexe
- Moreau, Lois
- 1979
(Show Context)
Citation Context ... by admissible placements are called admissible positions. Further, for the geometry of continuous material media, we have in view only placements of P(B) in accordance with the following hypothesis (=-=[13]-=-): H1. All placements of a material medium in an Euclidean affine space are admissible. So, for p ∈ C k (B), if λ ∈ B is a particle, p(λ) . = P will denote its admissible position, and their union M =... |

1 | Some open problems of finite elastic equilibrium - Sburlan - 1998 |