MATERIAL GEOMETRY: GROUPOIDS IN CONTINUUM MECHANICS

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.