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Modeling of Solids

Paolo Vannucci

Abstract

This text is the support for the course of Modeling of Solids, of the Master of Mechanics of the University Paris-Saclay - Curriculum MMM: Mathematical Methods for Mechanics, held at Versailles. The course is the continuation of the course Continuum Mechanics - Solids, and as such it is an introduction, for graduate students, to some typical topics of the theory of solid bodies. The different arguments are dealt with in a simple, succinct way, the objective being to give to students the fundamentals of each argument. Only static problems are considered, being the dynamic of structures dealt with in other courses.

Modeling of Solids

Abstract

This text is the support for the course of Modeling of Solids, of the Master of Mechanics of the University Paris-Saclay - Curriculum MMM: Mathematical Methods for Mechanics, held at Versailles. The course is the continuation of the course Continuum Mechanics - Solids, and as such it is an introduction, for graduate students, to some typical topics of the theory of solid bodies. The different arguments are dealt with in a simple, succinct way, the objective being to give to students the fundamentals of each argument. Only static problems are considered, being the dynamic of structures dealt with in other courses.
Paper Structure (169 sections, 9 theorems, 1104 equations, 119 figures)

This paper contains 169 sections, 9 theorems, 1104 equations, 119 figures.

Table of Contents

  1. Preface
  2. Some elements of differential geometry
  3. Curves of points, vectors and tensors
  4. Differentiation of a curve
  5. Integral of a curve of vectors, length of a curve
  6. The Frenet-Serret basis
  7. Curvature of a curve
  8. The Frenet-Serret formulae
  9. The torsion of a curve
  10. Osculating sphere and circle
  11. Surfaces in $\mathcal{E}$, coordinate lines, tangent planes
  12. Surfaces of revolution
  13. Ruled surfaces
  14. First fundamental form of a surface
  15. Second fundamental form of a surface
  16. ...and 154 more sections

Key Result

Theorem 1

Be $\mathbf{r}(t):[a,b]\Rightarrow\mathcal{E}$ a regular curve; then

Figures (119)

  • Figure 1: Mapping of a curve of points.
  • Figure 2: Derivative of a curve.
  • Figure 3: Integral of a vector curve.
  • Figure 4: The Frenet-Serret basis.
  • Figure 5: Curvature of a curve.
  • ...and 114 more figures

Theorems & Definitions (18)

  • Theorem
  • proof
  • Theorem
  • proof
  • Theorem
  • proof
  • Theorem
  • proof
  • Theorem
  • proof
  • ...and 8 more