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Continuum Mechanics

Continuum Mechanics

An interactive book

An interactive companion to B. Carboni’s Meccanica del continuo lectures, written from my study notes and meant to be enough to study from on its own. It covers the course as far as the lectures have gone: the elementary oscillator (lesson 1), the theory of deformation (lessons 2–6) and the theory of stress (lessons 6–8), with the principal stresses and the linear elastic law that the practice sheets for the first partial already ask for. The constitutive law and the beam will follow when the lectures reach them.

How to read it

  • Drag things. Figures you can move have handles: drag them 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 beside the figure says what.
  • Colours mean something, the same in the text and in the figures (Notation and colours): grey is the reference configuration, violet what you move and what the deformation makes of it, orange principal directions, teal forces, pink normals.
  • Dotted links open in place. A reference 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 lectures” marks what this book adds to Carboni’s lectures: skipped steps, examples, pictures, and the topics the exercise sheets need before the lectures reach them. Mistakes in the handwritten notes are corrected where they occur, in a box that says what the notes have.
  • Every section ends with exercises, from Carboni’s sheets where there are some, with worked solutions. The two “Worked problems” sections run the exam’s practice problems start to finish.

Contents

Part 1. The elementary oscillator

One mass on a spring, then on a spring and a damper: the constitutive law, the equation of motion, the natural frequency, energy dissipation and how damping is measured. Lesson 1.

  1. 1.1 The spring and the equation of motion
  2. 1.2 The free undamped oscillator
  3. 1.3 The damped oscillator
  4. 1.4 Measuring damping

Part 2. Theory of deformation

How a body deforms near one of its points, measured without any reference to forces: the deformation gradient, stretch and shear, the Cauchy–Green tensor and its principal directions, the polar decomposition, volume and area, finite and infinitesimal strain. Lessons 2–6.

  1. 2.1 Configurations and the deformation gradient
  2. 2.2 Stretch, shear and the Cauchy–Green tensor
  3. 2.3 Principal stretches and principal directions
  4. 2.4 Polar decomposition
  5. 2.5 Change of volume and area
  6. 2.6 The Green–Lagrange tensor
  7. 2.7 Infinitesimal strain
  8. 2.8 Worked problems: from a displacement to strain

Part 3. Theory of stress

How forces act inside a body and the equations of motion they obey: Cauchy’s postulates, lemma and theorem, the stress tensor, the balance of momentum, and the same problem written on the reference configuration with the Piola–Kirchhoff tensors. Lessons 6–8, then the principal stresses and the linear elastic law that the exercise sheets use.

  1. 3.1 Forces and mass in two configurations
  2. 3.2 Cauchy's postulates and lemma
  3. 3.3 The Cauchy–Poisson theorem and the stress tensor
  4. 3.4 The balance equations
  5. 3.5 The problem in two configurations, and the first Piola–Kirchhoff tensor
  6. 3.6 The second Piola–Kirchhoff tensor
  7. 3.7 Principal stresses and the linear elastic law
  8. 3.8 Worked problems: stress

Back matter

  1. Notation and colours
  2. When nothing deforms
  3. Index of terms