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Books > Science & Mathematics > Mathematics > Geometry > Differential & Riemannian geometry
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Geometrical Foundations of Continuum Mechanics - An Application to First- and Second-Order Elasticity and Elasto-Plasticity (Paperback, 2015 ed.)
Loot Price: R6,938
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Geometrical Foundations of Continuum Mechanics - An Application to First- and Second-Order Elasticity and Elasto-Plasticity (Paperback, 2015 ed.)
Series: Lecture Notes in Applied Mathematics and Mechanics, 2
Expected to ship within 10 - 15 working days
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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.
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