A New Approach to Linear Shell Theory

The use of differential equations methods in the approach, treatment, and solution of problems in diverse areas of geometry, particularly in affine differential geometry is well known and prolific, where they have proven to be quite fruitful when it comes to the obtainment of definite results. It is perhaps lesser known that the same kind of those very same methods has been and is currently being used to treat developments in some specific areas of applied sciences, such as the theory of shells where, similarly, they can be proven to be quite effective as well. In this paper we precisely show that such is the case in two particular, related instances: the historic approach of the classical, Euclidean part of the theory pursued by Fritz John, in the past century, and the more recent expositions that we ourselves have dedicated to the affine counterpart of the theory.

Discrete & Continuous Dynamical Systems - S, 2018

We prove the existence of a minimizer for a nonlinearly elastic shell model which coincides to within the first order with respect to small thickness and change of metric and curvature energies with the Koiter nonlinear shell model.

Mathematics and Mechanics of Solids, 2014

The paper is concerned with the geometrically non-linear theory of 6-parametric elastic shells with drilling degrees of freedom. This theory establishes a general model for shells, which is characterized by two independent kinematic fields: the translation vector and the rotation tensor. Thus, the kinematical structure of 6parameter shells is identical to that of Cosserat shells. We show the existence of global minimizers for the geometrically non-linear 2D equations of elastic shells. The proof of the existence theorem is based on the direct methods of the calculus of variations using essentially the convexity of the energy in the strain and curvature measures. Since our result is valid for general anisotropic shells, we analyze separately the particular cases of isotropic shells, orthotropic shells, and composite shells.

Journal of Mechanics of Materials and Structures, 2007

The nonlinear kinematics of thin shells is developed in full generality according to a duality approach in which kinematics plays the basic role in the definition of the model. The Kirchhoff-Love shell model is the central issue and is discussed in detail but shear deformable and polar models are also considered and critically reviewed. The analysis is developed with a coordinate-free approach which provides a direct geometrical picture of the shell model. The finite and tangent Green strains of the foliated continuum are explicitly expressed in terms of middle surface kinematics. The new expressions contributed here do not require the splitting of the velocity into parallel and normal components to the middle surface, and provide a computationally convenient context. Finite strain measures for the shell and their tangent and secant rates are analyzed and consistency and nonredundancy properties are discussed. The relations between the finite Green strain, its tangent and secant rates and the corresponding shell strains, are provided. The differential and boundary equilibrium equations of the shell are given in variational terms, both in unsplit and split form. A new expression of the boundary equilibrium equations is contributed and its mechanical soundness with respect to the classical one is emphasized. Equilibrium in a reference placement for the shell model is briefly discussed.

The use of differential equations methods in the approach, treatment, and solution of problems in diverse areas of geometry, particularly in affine differential geometry is well known and prolific, where they have proven to be quite fruitful when it comes to the obtainment of definite results. It is perhaps lesser known that the same kind of those very same methods has been and is currently being used to treat developments in some specific areas of applied sciences, such as the theory of shells where, similarly, they can be proven to be quite effective as well. In this paper we precisely show that such is the case in two particular, related instances: the historic approach of the classical, Euclidean part of the theory pursued by Fritz John, in the past century, and the more recent expositions that we ourselves have dedicated to the affine counterpart of the theory.

The oriented boundary (signed, algebraic) distance function of a set remarkably describes its geometric properties and those of its boundary. When combined with tangential differential operators defined through an extension in a neighborhood of a submanifold, they yield a very powerful and simple intrinsic tangential differential calculus on C 1,1 submanifolds of ℝ N . To illustrate this point of view, a new intrinsic approach has been developed to model and analyze thin shells. We give relevant basic intrinsic differential geometry and provide the correspondence between intrinsic tangential derivatives and classical covariant derivatives using local bases and Christoffel symbols. We extend previous results on the P(1,2) model to the simpler P(1,1) model. We show how the asymptotic membrane equation model and the classical models of Naghdi and Koiter can be expressed in terms of intrinsic operators, and make comparison with the P(1,0) model and an approximations of the P(1,1) model.

2014

The paper is concerned with the geometrically non-linear theory of 6-parametric elastic shells with drilling degrees of freedom. This theory establishes a general model for shells, which is characterized by two independent kinematic fields: the translation vector and the rotation tensor. Thus, the kinematical structure of 6parameter shells is identical to that of Cosserat shells. We show the existence of global minimizers for the geometrically non-linear 2D equations of elastic shells. The proof of the existence theorem is based on the direct methods of the calculus of variations using essentially the convexity of the energy in the strain and curvature measures. Since our result is valid for general anisotropic shells, we analyze separately the particular cases of isotropic shells, orthotropic shells, and composite shells.

We introduce a simplified model for the minimization of the elastic energy in thin shells. This model is not obtained by an asymptotic analysis. The nonlinear simplified model admits always minimizers by contrast with the original one. We show the relevance of our approach by proving that the rescaled minimum of the simplified model and the rescaled infimum of the full model have the same limit as the thickness tends to 0. The simplified energy can be expressed as a functional acting over fields defined on the mid-surface of the shell and where the thickness remains as a parameter. ). The elastic shell is defined by Q δ = Φ(ω×] − δ, δ[) and we consider that it is clamped on a part of its lateral boundary Γ 0,δ = Φ(γ 0 ×] − δ, δ[), where γ 0 ⊂ ∂ω. The energy density is denoted W and we assume that Q δ is submitted to applied body forces f κ,δ whose order with respect to δ depends upon a parameter κ (see the order of f κ,δ below). The total energy is given by

Journal of Elasticity, 2008

Computers & Mathematics with Applications, 2011

In this paper we apply tensor calculus and differential geometry to consider shell structure. Using tensor calculus we examine the stationarity of the Willmore energy under infinitesimal deformations of the surfaces in ℜ 3. We obtain the class of the surfaces which does not change its Willmore energy under infinitesimal deformations. In particular, a special kind of deformation is considered-infinitesimal bending which preserves the arc length. The change of the Willmore energy under such deformations is determined. Also, we give a new proof of a well-known theorem (that reads that the total mean curvature of a surface is stationary under an infinitesimal bending), applying tensor calculus.