Geometrical Foundations of Continuum Mechanics

An Application to First- and Second-Order Elasticity and Elasto-Plasticity

  • Paul Steinmann

Part of the Lecture Notes in Applied Mathematics and Mechanics book series (LAMM, volume 2)

Table of contents

  1. Front Matter
    Pages 1-22
  2. Prologue

    1. Front Matter
      Pages 1-2
    2. Paul Steinmann
      Pages 3-29
  3. Differential Geometry

    1. Front Matter
      Pages 31-32
    2. Paul Steinmann
      Pages 33-44
    3. Paul Steinmann
      Pages 45-118
    4. Paul Steinmann
      Pages 119-167
    5. Paul Steinmann
      Pages 169-198
  4. Nonlinear Continuum Mechanics

    1. Front Matter
      Pages 199-200
    2. Paul Steinmann
      Pages 201-281
    3. Paul Steinmann
      Pages 283-359
    4. Paul Steinmann
      Pages 361-489
  5. Epilogue

    1. Front Matter
      Pages 491-492
  6. Back Matter
    Pages 501-517

About this book


This book illustrates the deep roots of the geometrically nonlinear kinematics of

generalized continuum mechanics in differential geometry. Besides applications to first-

order elasticity and elasto-plasticity an appreciation thereof is particularly illuminating

for generalized models of continuum mechanics such as second-order (gradient-type)

elasticity and elasto-plasticity.


After a motivation that arises from considering geometrically linear first- and second-

order crystal plasticity in Part I several concepts from differential geometry, relevant

for what follows, such as connection, parallel transport, torsion, curvature, and metric

for holonomic and anholonomic coordinate transformations are reiterated in Part II.

Then, in Part III, the kinematics of geometrically nonlinear continuum mechanics

are considered. There various concepts of differential geometry, in particular aspects

related to compatibility, are generically applied to the kinematics of first- and second-

order geometrically nonlinear continuum mechanics. Together with the discussion on

the integrability conditions for the distortions and double-distortions, the concepts

of dislocation, disclination and point-defect density tensors are introduced. For

concreteness, after touching on nonlinear fir

st- and second-order elasticity, a detailed

discussion of the kinematics of (multiplicative) first- and second-order elasto-plasticity

is given. The discussion naturally culminates in a comprehensive set of different types

of dislocation, disclination and point-defect density tensors. It is argued, that these

can potentially be used to model densities of geometrically necessary defects and the

accompanying hardening in crystalline materials. Eventually Part IV summarizes the

above findings on integrability whereby distinction is made between the straightforward

conditions for the distortion and the double-distortion being integrable and the more

involved conditions for the strain (metric) and the double-strain (connection) being



The book addresses readers with an interest in continuum modelling of solids from

engineering and the sciences alike, whereby a sound knowledge of tensor calculus and

continuum mechanics is required as a prerequisite.




Applied Mathematics Applied Mechanics Differential Geometry Geometrical Foundations of Continuum Mechanics Nonlinear Continuum Mechanics

Authors and affiliations

  • Paul Steinmann
    • 1
  1. 1.University of Erlangen-NurembergErlangenGermany

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag Berlin Heidelberg 2015
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Engineering Engineering (R0)
  • Print ISBN 978-3-662-46459-5
  • Online ISBN 978-3-662-46460-1
  • Series Print ISSN 2197-6724
  • Series Online ISSN 2197-6732
  • Buy this book on publisher's site