Lecture

Mechanics of Continuous Media: Stress and Strain

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In course

CIVIL-225: Continuum mechanics (for GC)

Ce cours fondamental couvre les notions de déformations et de contraintes, les grands principes de conservation, les équations d'équilibres, et les lois constitutives. Des applications en mécanique des structures sont abordées en deuxième partie. Le cours ...

Description

This lecture covers the concept of stress and strain in continuous media, focusing on the Cauchy stress tensor and its components. The instructor explains how to calculate stress on a surface, the relationship between stress and normal vectors, and the importance of boundary conditions in determining stress distribution. The lecture also delves into the equilibrium equations and the conditions for mechanical equilibrium. Through examples, the audience learns how to apply the Cauchy relation to analyze stress under different loading conditions.

Instructor

Jean-François Molinari

Professor J.F. Molinari is the director of the Computational Solid Mechanics Laboratory (http://lsms.epfl.ch) at EPFL, Switzerland. He holds an appointment in the Civil Engineering institute, which he directed from 2013 to 2017, and a joint appointment in the Materials Science institute. He started his tenure at EPFL in 2007, and was promoted to Full Professor in 2012. He is currently an elected member of the Research Council of the Swiss National Science Foundation in Division 2 (Mathematics, Natural and Engineering Sciences), and co editor in chief of the journal Mechanics of Materials. J.F. Molinari graduated from Caltech, USA, in 2001, with a M.S. and Ph.D. in Aeronautics. He held professorships in several countries besides Switzerland, including the United States with a position in Mechanical Engineering at the Johns Hopkins University (2000-2006), and France at Ecole Normale Supérieure Cachan in Mechanics (2005-2007), as well as a Teaching Associate position at the Ecole Polytechnique de Paris (2006-2009). The work conducted by Prof. Molinari and his collaborators takes place at the frontier between traditional disciplines and covers several length scales from atomistic to macroscopic scales. Over the years, Professor Molinari and his group have been developing novel multiscale approaches for a seamless coupling across scales. The activities of the laboratory span the domains of damage mechanics of materials and structures, nano- and microstructural mechanical properties, and tribology.

Official source

Related concepts (26)

Cauchy stress tensor

In continuum mechanics, the Cauchy stress tensor , true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length direction vector e to the traction vector T(e) across an imaginary surface perpendicular to e: or, The SI units of both stress tensor and traction vector are N/m2, corresponding to the stress scalar.

Stress (mechanics)

In continuum mechanics, stress is a physical quantity that describes forces present during deformation. An object being pulled apart, such as a stretched elastic band, is subject to tensile stress and may undergo elongation. An object being pushed together, such as a crumpled sponge, is subject to compressive stress and may undergo shortening. The greater the force and the smaller the cross-sectional area of the body on which it acts, the greater the stress. Stress has units of force per area, such as newtons per square meter (N/m2) or pascal (Pa).

Finite strain theory

In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue.

Dirichlet boundary condition

In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take along the boundary of the domain. In finite element method (FEM) analysis, essential or Dirichlet boundary condition is defined by weighted-integral form of a differential equation.

Robin boundary condition

In mathematics, the Robin boundary condition (ˈrɒbɪn; properly ʁɔbɛ̃), or third type boundary condition, is a type of boundary condition, named after Victor Gustave Robin (1855–1897). When imposed on an ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function and the values of its derivative on the boundary of the domain. Other equivalent names in use are Fourier-type condition and radiation condition.

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