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Continuum Mechanics

Continuum Mechanics

An interactive book on tensors and the kinematics of deformation

Primary goal

One question runs through this book: what does a deformation do to the matter near a point?

Zoom in far enough and every smooth deformation looks linear: a tensor, the deformation gradient . So the book asks three questions, first of an arbitrary tensor and then of :

  1. Where does it send a vector? A tensor as a map; in kinematics, where a small material fibre goes.
  2. What does it do to lengths, angles, areas and volumes? Scalar products, determinants, orthogonal tensors; in kinematics, , Nanson’s formula and .
  3. What is it made of? Symmetric and skew parts, eigenvectors, the spectral theorem; in kinematics, principal stretches and : a stretch, then a turn.

Part 1 is the vector algebra the course assumes. Part 2 is Favata’s appendix on tensors, and part 3 his first chapter on kinematics, each followed in his order and his notation.

How to read it

  • Drag things. Every section has figures you can move. Arrows, points and frames with a handle can be dragged with a pointer or a finger; with the keyboard, focus a figure, press Enter to pick a handle and the arrow keys to move it (Shift for larger steps). Things snap where something interesting happens, and the text under the figure says what.
  • Colours mean something, the same in the text and in the figures: , , for and their images, for the vector you drag, for eigenvectors, for before and for after (Notation and colours).
  • Dotted links open in place. A reference like Definition 2.1.1 shows what it names right here; click again to close it.
  • Proofs and solutions are folded. Try first, then open.
  • Only a few definitions are essential, marked as such. Everything else is built from them (Index of terms).
  • “Beyond the notes” marks what this book adds to the lecture notes: skipped steps, examples, pictures, and syllabus topics the notes have not reached yet.

Contents

Part 1. Vectors

Points and vectors, bases and components, the scalar and vector products, and index notation: the language the rest is written in.

  1. 1.1 Points, vectors, bases
  2. 1.2 Scalar product and components
  3. 1.3 Vector product and volume

Part 2. Tensors

Second-order tensors as maps of vectors: their matrices, products and transposes, how they change areas and volumes, which ones are rigid, and the spectral theorem that splits a symmetric tensor into three stretches.

  1. 2.1 Tensors as linear maps
  2. 2.2 Transpose and composition
  3. 2.3 Symmetric and skew parts
  4. 2.4 Trace and the scalar product of tensors
  5. 2.5 Inverse and determinant
  6. 2.6 Orthogonal tensors
  7. 2.7 Eigenvalues and the spectral theorem
  8. 2.8 Change of basis

Part 3. Kinematics

A body, a deformation and its gradient : what happens to small fibres, angles, areas and volumes, which deformations are rigid, and how splits into a stretch and a turn.

  1. 3.1 Bodies, placements and deformations
  2. 3.2 The deformation gradient
  3. 3.3 Change in length and angle
  4. 3.4 Change in area and volume
  5. 3.5 Homogeneous and rigid deformations
  6. 3.6 Strain measures

Back matter

  1. Notation and colours
  2. The Invertible Tensor Theorem
  3. Index of terms