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 first- 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.
