Paul Steinmann – författare

This book gives a comprehensive account of the formulation and computational treatment of basic geometrically linear models in 1D. To set the stage, it assembles some preliminaries regarding necessary modelling, computational and mathematical tools. Thereafter, the remaining parts are concerned with the actual catalogue of computational material models. To this end, after starting out with elasticity as a reference, further 15 different basic variants of material models (5 x each of {visco-elasticity, plasticity, visco-plasticity}, respectively) are systematically explored. The presentation for each of these basic material models is a stand-alone account and follows in each case the same structure. On the one hand, this allows, in the true sense of a catalogue, to consult each of the basic material models separately without the need to refer to other basic material models. On the other hand, even though this somewhat repetitious concept may seem tedious, it allows to compare the formulation and resulting algorithmic setting of the various basic material models and thereby to uncover, in detail, similarities and differences. In particular, the response of each basic material model is analysed for the identical histories (Zig-Zag, Sine, Ramp) of prescribed strain and stress so as to clearly showcase and to contrast to each other the characteristics of the various modelling options.

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

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.