This monograph is the first in which the theory of groupoids and algebroids is applied to the study of the properties of uniformity and homogeneity of continuous media. It is a further step in the application of differential geometry to the mechanics of continua, initiated years ago with the introduction of the theory of G-structures, in which the group G denotes the group of material symmetries, to study smoothly uniform materials.The new approach presented in this book goes much further by being much more general. It is not a generalization per se, but rather a natural way of considering the algebraic-geometric structure induced by the so-called material isomorphisms. This approach has allowed us to encompass non-uniform materials and discover new properties of uniformity and homogeneity that certain material bodies can possess, thus opening a new area in the discipline.
Contents:
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Preface
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About the Authors
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Introduction
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Fundamentals:
- Continuum Mechanics: Elastic Simple Bodies
- Groupoids
- Algebroids
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Material Groupoid:
- Material Algebroid
- Characteristic Distributions and Material Bodies
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Appendices:
- Foliations and Distributions
- Covariant Derivatives
- Principal Bundles and Connections
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Bibliography
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Index
Readership: Graduate and postgraduate students interested in Continuum Mechanics, Mathematical Physics and Differential Geometry. Researchers in Elasticity, Applied Mathematics and Differential Geometry. And those taking a master or doctorate course that seeks the interaction between mathematics and mechanical engineering.
Key Features:
- The use of groupoids in continuum mechanics is very new and has served to open new lines of research that will afford a new vision for topics such as growth and morphogenesis
- This monograph is also devoted to more recent applications to Continuum Mechanics
- This monograph may be a necessary reference for researchers in elasticity. On the other hand, in the opposite direction, it is a good example of how differential geometry can be applied to mechanics, which will make this monograph attractive also to pure mathematicians
