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
integrable.
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.
Paul Steinmann is Professor for Mechanics at the Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Germany, where he heads the Institute of Applied Mechanics in the Department of Mechanical Engineering within FAU's Faculty of Engineering. He is also Director of the Glasgow Computational Engineering Centre (GCEC) in the School of Engineering at the University of Glasgow, UK. His research interests include nonlinear continuum mechanics, material modelling, coupled and multiscale problems and corresponding computational methods both for engineering as well as for biomechanical applications. Kenneth Runesson is Professor Emeritus of Material and Structural Mechanics at Chalmers University of Technology, Gothenburg, Sweden. His research interests include the computational modelling of coupled problems for porous media including inelasticity, damage and fracture. A trademark is control of discretization and modelling errors via space-time adaptivity.Computational homogenization and scale-bridging strategies are another major research field.
