Geometrical Foundations of Continuum Mechanics

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<p><b>This book illustrates the deep roots of the geometrically nonlinear kinematics of</b></p><p><b>generalized continuum mechanics in differential geometry. Besides applications to first-</b></p><p><b>order elasticity and elasto-plasticity an appreciation thereof is particularly illuminating</b></p><p><b>for generalized models of continuum mechanics such as second-order (gradient-type)</b></p><p><b>elasticity and elasto-plasticity.</b></p><p><b> </b></p><p><b>After a motivation that arises from considering geometrically linear first- and second-</b></p><p><b>order crystal plasticity in Part I several concepts from differential geometry, relevant</b></p><p><b>for what follows, such as connection, parallel transport, torsion, curvature, and metric</b></p><p><b>for holonomic and anholonomic coordinate transformations are reiterated in Part II.</b></p><p><b>Then, in Part III, the kinematics of geometrically nonlinear continuum mechanics</b></p><p><b>are considered. There various concepts of differential geometry, in particular aspects</b></p><p><b>related to compatibility, are generically applied to the kinematics of first- and second-</b></p><p><b>order geometrically nonlinear continuum mechanics. Together with the discussion on</b></p><p><b>the integrability conditions for the distortions and double-distortions, the concepts</b></p><p><b>of dislocation, disclination and point-defect density tensors are introduced. For</b></p><p><b>concreteness, after touching on nonlinear fir</b></p>st- and second-order elasticity, a detailed<p></p><p><b>discussion of the kinematics of (multiplicative) first- and second-order elasto-plasticity</b></p><p><b>is given. The discussion naturally culminates in a comprehensive set of different types</b></p><p><b>of dislocation, disclination and point-defect density tensors. It is argued, that these</b></p><p><b>can potentially be used to model densities of geometrically necessary defects and the</b></p><p><b>accompanying hardening in crystalline materials. Eventually Part IV summarizes the</b></p><p><b>above findings on integrability whereby distinction is made between the straightforward</b></p><p><b>conditions for the distortion and the double-distortion being integrable and the more</b></p><p><b>involved conditions for the strain (metric) and the double-strain (connection) being</b></p><p><b>integrable.</b></p><p><b> </b></p><p><b>The book addresses readers with an interest in continuum modelling of solids from</b></p><p><b>engineering and the sciences alike, whereby a sound knowledge of tensor calculus and</b></p><p><b>continuum mechanics is required as a prerequisite.</b></p><p> </p><p> </p>

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