On exact expressions of the bending tensor in the nonlinear theory of thin shells

Some equivalent exact expressions of the bending tensor in the nonlinear theory of thin shells are reviewed. It is noted that the bending tensor, proposed by Shen et al. (2010) [X.Q. Shen, K.T. Li, Y. Ming “The modified model of Koiter’s type for the nonlinearly elastic shells”, Appl. Math. Model. 34 (2010) 3527-3535] as a third-degree polynomial of displacements, is an approximate expression, not the exact one. Then integrability of the fourth kinematic boundary condition, associated with two different but equivalent exact expressions of the bending tensor, is briefly discussed. Finally, a few modified definitions of the bending tensor proposed in the literature are recalled. Within the first-approximation theory they all lead to energetically equivalent models of elastic shells.     

The ODEs forms of linearly elastic shell model and its numerical computing

In this paper, we successfully use 1D model to approximate the 3D problems. Firstly the PDEs (Partial Differential Equations) forms of Koiter’s Model for 2D linear elastic shell is proposed on special curvature coordinate system, i.e., spherical-coordinate system. Then the ODEs (Ordinary Differential Equations) forms of Koiter’s Model for 2D linear elastic shell is proposed under the assumption that the shell is axis-symmetric. Finally, we do numerical experiments to verify validity and accuracy of 1D models.     

Un théorème d'existence pour une coque non linéairement élastique ≪ en flexion ≫

We establish an existence theorem for the two-dimensional equations of a nonlinearly elastic “flexural” shell, recently justified by V. Lods and B. Miara by the method of formal asymptotic expansions applied to the corresponding three-dimensional equations of nonlinear elasticity. To this end, we show that the associated energy has at least one minimizer over the corresponding set of admissible deformations. The strain energy is a quadratic expression in terms of the “exact” change of curvature tensor, between the deformed and undeformed middle surfaces; the set of admissible deformations is formed by the deformations of the undeformed middle surface that preserve its metric and satisfy boundary conditions of clamping or simple support.     

Asymptotic justification of the intrinsic equations of Koiter's model of a linearly elastic shell

We show that the intrinsic equations of Koiter's model of a linearly elastic shell can be derived from the intrinsic formulation of the three-dimensional equations of a linearly elastic shell, by using an appropriate a priori assumption regarding the three-dimensional strain tensor fields appearing in these equations. To this end, we recast in particular the Dirichlet boundary conditions satisfied by any admissible displacement field as boundary conditions satisfied by the covariant components of the corresponding strain tensor field expressed in the natural curvilinear coordinates of the shell. Then we show that, when restricted to strain tensor fields satisfying a specific a priori assumption, these new boundary conditions reduce to those of the intrinsic equations of Koiter's model of a linearly elastic shell.     

A nonlinear shell model of Koiter's type

We define a new two-dimensional nonlinear shell model “of Koiter's type” that can be used for the modeling of any type of shell and boundary conditions and for which we establish an existence theorem. The model uses a specific three-dimensional stored energy function of Ogden's type that satisfies all the assumptions of John Ball's fundamental existence theorem in three-dimensional nonlinear elasticity and that is adapted here to the modeling of thin nonlinearly elastic shells by means of specific deformations that are quadratic with respect to the transverse variable.     

A confinement problem for a linearly elastic Koiter's shell

In this Note, we propose a natural two-dimensional model of “Koiter's type” for a general linearly elastic shell confined in a half space. This model is governed by a set of variational inequalities posed over a non-empty closed and convex subset of the function space used for modeling the corresponding “unconstrained” Koiter's model. To study the limit behavior of the proposed model as the thickness of the shell, regarded as a small parameter, approaches zero, we perform a rigorous asymptotic analysis, distinguishing the cases where the shell is either an elliptic membrane shell, a generalized membrane shell of the first kind, or a flexural shell. Moreover, in the case where the shell is an elliptic membrane shell, we show that the limit model obtained via the asymptotic analysis of our proposed two-dimensional Koiter's model coincides with the limit model obtained via a rigorous asymptotic analysis of the corresponding three-dimensional “constrained” model.     

Une approche intrinsèque d'un modèle non linéaire de la théorie des coques

We consider the model of a nonlinearly elastic “shallow” shell proposed by L.H. Donnell, V.Z. Vlasov, K.M. Mushtari & K.Z. Galimov, and W.T. Koiter. We show that the linearized change of curvature and nonlinear strain tensor fields appearing in the energy of this model can be taken as the sole unknowns of the problem, instead of the displacement field as is customary. In order to justify this “intrinsic approach” to this nonlinear model, we identify nonlinear compatibility conditions that these new unknowns must satisfy. These conditions are of Donati type, in the sense that they take the form of integral orthogonality relations against divergence-free tensor fields.     

Intrinsic formulation of the displacement-traction problem in linear shell theory

We recast the Dirichlet boundary conditions satisfied by the displacement field of the middle surface of a linearly elastic shell as boundary conditions satisfied by the corresponding linearized change of metric and of curvature tensor fields. This in turn allows us to give an intrinsic formulation of the linear shell model of W.T. Koiter with these two tensor fields as the sole unknowns.     

Another approach to linear shell theory and a new proof of Korn's inequality on a surface

We propose a new approach to the quadratic minimization problems arising in Koiter's linear shell theory. The novelty consists in considering the linearized change of metric and change of curvature tensors as the new unknowns, instead of the displacement vector field as is customary. This approach also provides a new proof of Korn's inequality on a surface. To cite this article: P.G. Ciarlet, L. Gratie, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Asymptotic expansions of stress tensor for linearly elastic shell

In this paper, the asymptotic expansions of stress tensor for linearly elastic shell have been proposed by new asymptotic analysis method, which is different from the classical asymptotic analysis. The new asymptotic analysis method has two distinguishing features: one is that the displacement is expanded with respect to the thickness variable of the middle surface not to the thickness; another is that the first order term and the second order term of the displacement variable can be algebraically expressed by the leading term. To decompose stress tensor totally into 2-D variable and thickness variable, we have three steps: operator splitting, variables separation and dimension splitting. In the end, a numerical experiment of special hemispherical shell by FEM (finite element method) is provided. We derive the distribution of displacements and stress fields in the middle surface.     

Prescribing Curvature Problems on the Bakry-Emery Ricci Tensor of a Compact Manifold with Boundary

t The authors consider the problem of conformally deforming a metric such that the k-curvature defined by an elementary symmetric function of the eigenvalues of the Bakry-Emery Ricci tensor on a compact manifold with boundary to a prescribed function. A consequence of our main result is that there exists a complete metric such that the Monge-Amp~re type equation with respect to its Bakry-Emery Ricci tensor is solvable, provided that the initial Bakry-Emery Ricci tensor belongs to a negative convex cone.     

Some Invariant Properties of Semi-symmetric Metric Recurrent Connections and Curvature Tensor Expressions

The projective transformation of the special semi-symmetric metric recurrent connection is studied in this paper. First of all, an invariant under this transformation is granted; Secondly, by inducing of the invariant and making use of the properties that the corresponding covariant derivative keeps being fixed under the distinctness connection, the curvature tensor expression of the Riemannian manifold is posed at the same time.     

M-Eigenvalues of the Riemann Curvature Tensor of Conformally Flat Manifolds

We investigate the M-eigenvalues of the Riemann curvature tensor in the higher dimensional conformally flat manifold. The expressions of Meigenvalues and M-eigenvectors are presented in this paper. As a special case,M-eigenvalues of conformal flat Einstein manifold have also been discussed,and the conformal the invariance of M-eigentriple has been found. We also reveal the relationship between M-eigenvalue and sectional curvature of a Riemannian manifold. We prove that the M-eigenvalue can determine the Riemann curvature tensor uniquely. We also give an example to compute the Meigentriple of de Sitter spacetime which is well-known in general relativity.     

The stress-energy tensor for biharmonic maps

Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy functional. We show that it derives from a variational problem on metrics and exhibit the peculiarity of dimension four. First, we use this tensor to construct new examples of biharmonic maps, then classify maps with vanishing or parallel stress-energy tensor and Riemannian immersions whose stress-energy tensor is proportional to the metric, thus obtaining a weaker but high-dimensional version of the Hopf Theorem on compact constant mean curvature immersions. We also relate the stress-energy tensor of the inclusion of a submanifold in Euclidean space with the harmonic stress-energy tensor of its Gauss map. S. Montaldo was supported by PRIN-2005 (Italy): Riemannian Metrics and Differentiable Manifolds. C. Oniciuc was supported by a CNR-NATO (Italy) fellowship and by the Grant CEEX, ET, 5871/2006 (Romania).     

Convergence of the Solution to General Viscoelastic Koiter Shell Equations

By applying the inequality of Korn＇s type without boundary conditions on a general surface, we prove that the scaled displacement of the two-dimensional linearly viscoelastic Koiter＇s shell converges to the solution of two-dimensional model system of linearly viscoelastic ＂membrane＂ shell.     

Some Results on Contact Metric Manifolds

If the sectional curvatures of plane sections containing the characteristic vector field of a contact metric manifold M are non-vanishing, then we prove that a second order parallel tensor on M is a constant multiple of the associated metric tensor. Next, we prove for a contact metric manifold of dimension greater than 3 and whose Ricci operator commutes with the fundamental collineation that, if its Weyl conformal tensor is harmonic, then it is Einstein. We also prove that, if the Lie derivative of the fundamental collineation along the characteristic vector field on a contact metric 3-manifold M satisfies a cyclic condition, then M is either Sasakian or locally isometric to certain canonical Lie-groups with a left invariant metric. Next, we prove that if a three-dimensional Sasakian manifold admits a non-Killing projective vector field, it is of constant curvature 1. Finally, we prove that a conformally recurrent Sasakian manifold is locally isometric to a unit sphere.     

Cheeger-gromoll type metrics on the (1,1)-tensor bundles

Using a Riemannian metric on a differentiable manifold, a Cheeger-Gromoll type metric is introduced on the (1,1)-tensor bundle of the manifold. Then the Levi-Civita connection, Riemannian curvature tensor, Ricci tensor, scalar curvature and sectional curvature of this metric are calculated. Also, a para-Nordenian structure on the the (1,1)-tensor bundle with this metric is constructed and the geometric properties of this structure are studied.     

Four-dimensional manifolds with degenerate self-dual Weyl curvature operator

It is shown that any four-dimensional Walker metric of nowhere zero scalar curvature has a natural almost para-Hermitian structure. In contrast to the Goldberg–Sachs theorem, if this structure is self-dual and *-Einstein, it is symplectic but not necessarily integrable. This is due to the non-diagonalizability of the self-dual Weyl conformal curvature tensor.      

The Contact Metric Connection on the Heisenberg Group

We prove that there is only one contact metric connection with skew-torsion on the Heisenberg group endowed with a left-invariant Sasakian structure. We obtain the expression of this connection via the contact form and the metric tensor, and show that the torsion tensor and the curvature tensor are constant and the sectional curvature varies between ?1 and 0.     

INVARIANT VARIATIONAL PRINCIPLES INVOLVING VECTOR AND METRIC FIELDS IN 4-DIMENSIONS

Abstract

Variational principles in which the Lagrangian is a scalar density and a function of a metric tensor and a vector field, together with their first derivatives, are investigated in a 4-dimensional space. Associated with such Lagrangians are two expressions, the metric Euler-Lagrange expression and the vector Euler-Lagrange expression. The most general Lagrangians (of this kind) for which either of these Euler-Lagrange expressions vanishes identically, are obtained.

The most general Lagrangian (of this kind) for which the vector Euler-Lagrange equations are precisely Maxwell's equations is obtained. Although this Lagrangian is more general than the one commonly used, it still has essentially the same energy-momentum tensor.

The most general Lagrangian (of this kind) for which the metric Euler-Lagrange expression is precisely the electromagnetic energy-momentum tensor is derived. Although this Lagrangian is also more general than the one commonly used, the associated vector Euler-Lagrange equations are still Maxwell's equations.

Finally it is shown that, in contrast to the situation which obtains in the case of scalar densities which are functions of up to second derivatives of the metric and first derivatives of the vector field, there does not exist a Lagrangian, of the kind under investigation, for which the metric Euler-Lagrange expression is precisely the Einstein tensor and the vector Euler-Lagrange expression vanishes identically.