**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Lecture# Shells I

Description

This lecture covers linear pressure vessels and provides a brief primer on the differential geometry of surfaces, including covariant and contravariant base vectors, first and second fundamental forms, and the 3D metric tensor. The lecture delves into thin pressure vessels, thin shells, and critical buckling pressure, emphasizing the dimensional reduction from 3D to 2D. It also explores topics such as point indentation (Reissner), nonlinear theory, and obtaining equilibrium equations using linear theory. Additionally, the lecture discusses the geometric description of curved surfaces, the differential geometry of surfaces for plates, and the constitutive nonlinearities and geometric nonlinearities of shells.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (312)

Related lectures (1,000)

Curvilinear coordinates

In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved.

Tensor

In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product.

Covariance and contravariance of vectors

In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In modern mathematical notation, the role is sometimes swapped. A simple illustrative case is that of a vector. For a vector, once a set of basis vectors has been defined, then the components of that vector will always vary opposite to that of the basis vectors. A vector is therefore a contravariant tensor.

Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space.

Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a directed line segment, or graphically as an arrow connecting an initial point A with a terminal point B, and denoted by . A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier".

Shells I: Mechanics of Slender StructureME-411: Mechanics of slender structures

Covers thin pressure vessels, differential geometry of surfaces, and plate buckling theories.

Differential Geometry of SurfacesME-411: Mechanics of slender structures

Covers linear pressure vessels and the basics of differential geometry of surfaces, including covariant and contravariant base vectors.

Shells I: Mechanics of Slender StructuresME-411: Mechanics of slender structures

Covers linear and membrane theories of pressure vessels, differential geometry of surfaces, and the reduction of dimensionality from 3D to 2D.

Linear Shell Theory: Equilibrium and Energy

Covers the expression of the Kirchhoff-Saint Venant energy in a covariant setting and explores equilibrium equations for spherical shells and linear shell theory.

Differential Geometry of SurfacesME-411: Mechanics of slender structures

Covers the fundamentals of differential geometry of surfaces, including the equilibrium of shells, pressure vessels, and the curvature of surfaces.