Exegesis of the Introduction and Sect. I from

“Fundamentals of the Mechanics of Continua” 1)by E. Hellinger

Simon R. Eugster

October 6, 2016

inm Institute forNonlinear Mechanics

Tullio Levi Civita Lectures 2016, Roma

1) S.R.Eugster, F.dell’Isola. Exegesis of the “Fundamentals of the Mechanics of Continua”, ZAMM, in press.

Encyklopädie der Math. Wissensch.

WITH INCLUSION OF THEIR APPLICATIONS

ENCYCLOPEDIA OF MATHEMATICAL SCIENCES

1898 - 1935 published by B.G. Teubner

supported by the Academy of sciences of Munich, Leipzic,

Göttingen and Vienna

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Vol. I: Arithmetic and Algebra

II: Analysis

III: Geometry

IV: Mechanics

IV-4 Fundamentals of the Mechanics of Continua

V: Physics

VI,1: Geodesy and Geophysics

VI,2: Astronomy

by E. Hellinger

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1935 removed from the faculty as a jewish professor

1938 deportation to concentration camp in Dachau

1939 - 1949 lecturer at Northwestern University, Evanston

1949 visiting professor at Illinois Institute of Technology

28/03/1950 died of cancer in Chicago, USA

30/09/1883 born in Striegau, Germany (now Strzegom, Poland)

1902 - 1907 studies in mathematics in Heidelberg, Breslau, Göttingen

doctoral thesis: “Die Orthogonalinvarianten quadratischer

Formen von unendlich vielen Variablen”, advisor: D.Hilbert

1907 - 1909 assistant of D.Hilbert at University of Göttingen

1909 - 1914 privatdozent at University of Marburg

1914 - 1935 professor in mathematics at University of Frankfurt

Ernst Hellinger

Fundamentals of the Mechanics of Continua

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Contents

1. Introduction

2. The notion of a continuum

I. The foundations of statics

3. The principle of virtual displacements

4. Enhancement of the principle of virtual

displacements

II. The foundations of kinetics

III. The forms of constitutive laws

Introduction

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extremum principles are not adequate, but

explanation of variational principles in continuum mechanics

HAVING THE FORM of the necessary criterion for being an extremum

The Notion of a Continuum

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infinite dimensional configuration manifold

Virtual Displacements

variational family of deformation functions

infinitesimal virtual displacement

variation of deformation functions

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tangent vector to configuration mfld.

Forces and Stresses

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forces per unit volume

mechanics in the spirit of D‘Alembert based on the notion of work

definition of virtual work as primitive quantity

virtual work of continuum - not the most general expression

forces per unit area

interpretation not

clear yet

Interpretation of Stresses as Surface Forces

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forces per unit area

forces per unit volume

“pressure theorem” of Cauchy

Principle of Virtual Displacements

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Truesdell’s Allegation against Hellinger

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C.Truesdell, R.Toupin. The classical field theories, 1960.

Non-Variational Continuum Mechanics

Integral balance laws

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Stress principle of Euler and Cauchy

Cauchy’s stress theorem

G. KirchhoffA. Cauchy

symmetry of stresslocal equilibrium equations

in

on

Localization

L. Euler

Relation to Rigid Body Mechanics

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“center-of-mass theorem” - equilibrium of forces

Symmetry of Stress

2)Axiom of power of internal forces

2) P. Germain. Sur l’application de la méthode des puissances virtuelles en mécanique des milieux, 1972. 14

“law of equal area” - equilibrium of moments

Boltzmann axiom (Hamel, 1908)

P. Germain

Enhancement of the PVD

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Second gradient materials

Media with oriented particles

Variational Continuum Mechanics

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Principle of virtual displacements

local equilibrium equation

interaction between subsystems

Forces as linear forms

G.Piola E.HellingerJ.-L.Lagrange

Axiom of power of internal forces (not formulated by Hellinger) P. Germain

P. Germain

Variational Continuum Mechanics

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Principle of virtual displacements

local equilibrium equation

interaction between subsystems

Forces as linear forms

G.Piola E.HellingerJ.-L.Lagrange

Axiom of power of internal forces (not formulated by Hellinger)