MATERIAL GEOMETRY: GROUPOIDS IN CONTINUUM MECHANICS

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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. Sample Chapter(s) Preface Chapter 2: Continuum Mechanics: Elastic Simple Bodies Contents: Preface About the Authors Introduction Fundamentals: Continuum Mechanics: Elastic Simple Bodies Groupoids Algebroids Material Groupoid: Material Algebroid Characteristic Distributions and Material Bodies Appendices: Foliations and Distributions Covariant Derivatives Principal Bundles and Connections Bibliography 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.