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This monograph presents the geometric foundations of continuum
mechanics. An emphasis is placed on increasing the generality and
elegance of the theory by scrutinizing the relationship between the
physical aspects and the mathematical notions used in its
formulation. The theory of uniform fluxes in affine spaces is
covered first, followed by the smooth theory on differentiable
manifolds, and ends with the non-smooth global theory. Because
continuum mechanics provides the theoretical foundations for
disciplines like fluid dynamics and stress analysis, the author’s
extension of the theory will enable researchers to better describe
the mechanics of modern materials and biological tissues. The
global approach to continuum mechanics also enables the formulation
and solutions of practical optimization problems.
Foundations of Geometric Continuum Mechanics will be an invaluable
resource for researchers in the area, particularly mathematicians,
physicists, and engineers interested in the foundational notions of
continuum mechanics.
This contributed volume explores the applications of various topics
in modern differential geometry to the foundations of continuum
mechanics. In particular, the contributors use notions from areas
such as global analysis, algebraic topology, and geometric measure
theory. Chapter authors are experts in their respective areas, and
provide important insights from the most recent research. Organized
into two parts, the book first covers kinematics, forces, and
stress theory, and then addresses defects, uniformity, and
homogeneity. Specific topics covered include: Global stress and
hyper-stress theories Applications of de Rham currents to singular
dislocations Manifolds of mappings for continuum mechanics
Kinematics of defects in solid crystals Geometric Continuum
Mechanics will appeal to graduate students and researchers in the
fields of mechanics, physics, and engineering who seek a more
rigorous mathematical understanding of the area. Mathematicians
interested in applications of analysis and geometry will also find
the topics covered here of interest.
This contributed volume explores the applications of various topics
in modern differential geometry to the foundations of continuum
mechanics. In particular, the contributors use notions from areas
such as global analysis, algebraic topology, and geometric measure
theory. Chapter authors are experts in their respective areas, and
provide important insights from the most recent research. Organized
into two parts, the book first covers kinematics, forces, and
stress theory, and then addresses defects, uniformity, and
homogeneity. Specific topics covered include: Global stress and
hyper-stress theories Applications of de Rham currents to singular
dislocations Manifolds of mappings for continuum mechanics
Kinematics of defects in solid crystals Geometric Continuum
Mechanics will appeal to graduate students and researchers in the
fields of mechanics, physics, and engineering who seek a more
rigorous mathematical understanding of the area. Mathematicians
interested in applications of analysis and geometry will also find
the topics covered here of interest.
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