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Material Geometry

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Abstract

Walter Noll’s trailblazing constitutive theory of material defects in smoothly uniform bodies is recast in the language of Lie groupoids and their associated Lie algebroids. From this vantage point the theory is extended to non-uniform bodies by introducing the notion of singular material distributions and the physically cognate idea of graded uniformity and homogeneity.

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Notes

  1. The assumption of smoothness of the inversion is not necessary, since it is implied by the other properties. Mackenzie [15] distinguishes between Lie groupoids and differentiable groupoids, the latter not being necessarily locally trivial.

  2. In the case of a regular one-dimensional distribution, every point is contained in an integral manifold. A regular distribution of higher dimension need not have any integral manifolds.

  3. This terminology is slightly misleading, since \({\mathcal{S}}({\mathcal{P}})\) is technically not a foliation of \({\mathcal{P}}\), which may not even be a manifold.

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Authors and Affiliations

  1. University of Calgary, Calgary, Canada

    Marcelo Epstein

  2. Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Madrid, Spain

    Víctor Manuel Jiménez & Manuel de León

Authors
  1. Marcelo Epstein
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  2. Víctor Manuel Jiménez
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  3. Manuel de León
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Correspondence to Marcelo Epstein.

Additional information

Walter Noll’s scientific legacy stand as a paradigm of Apollonian clarity, but it is also able to inspire the ecstasy of aesthetic inebriation. The first acquaintance with his work has been a decisive turning point in the intellectual lives of many, who are thereby forever grateful. This work is dedicated to his memory.

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Epstein, M., Jiménez, V.M. & de León, M. Material Geometry. J Elast 135, 237–260 (2019). https://doi.org/10.1007/s10659-018-9693-2

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