Micromechanics
Lecture notes — Winter Semester 2026/27
Course information
| Instructors | Jun.-Prof. Dr. Edoardo Milana, Dr. Mohsen Jafarpour (Laboratory for Soft Machines, IMTEK) |
| Semester | Winter Semester 2026/27 |
| Sessions | Monday and Friday, 10:15–12:00, SR 00-010/14 |
| Exam | Friday 26.02.2027, 14:00–17:00, HS 00 036 — written, 180 minutes |
| Communication | Matrix/Element (link in ILIAS) |
Format. Each session starts with a short lecture on the theory of the week, followed by work on the weekly problem set. The schedule lists the topic, chapter and problem set of every session.
Attendance is not mandatory but strongly recommended. There are no prerequisites for the exam; the problem sets prepare you for it, so make sure you have worked through all of them and understand the solutions.
How the course is built
The course looks at the same question — when is a body in mechanical balance? — for three increasingly rich models: the rigid body, the beam and the continuum. For each model we take the same three steps:
- Equilibrium of forces on the body, on an infinitesimal beam element, on an infinitesimal volume element;
- Kinematics: how the model moves and deforms;
- Virtual work: combining forces and kinematics, we derive the balance equations a second time. This reveals which force is conjugate to which deformation, and it is the basis of the energy methods.
| Rigid body | Beam | Continuum | |
|---|---|---|---|
| Equilibrium | \(\sum\vec{F}=0\), \(\sum\vec{M}=0\) (1) | \(Q' = -q\), \(M' = Q\) (4) | \(\nabla\cdot\underline{\sigma} + \vec{f} = 0\) (9) |
| Kinematics | \(\vec{v}_P = \vec{v}_O + \vec{\omega}\times\vec{r}_{P/O}\) (2) | \(\varepsilon = u'\), \(\gamma = w'+\varphi\), \(\chi = \varphi'\) (5) | \(\underline{\varepsilon} = \tfrac{1}{2}(\nabla\vec{u} + \nabla\vec{u}^T)\) (11) |
| Virtual work | \(\vec{F}\cdot\delta\vec{u}_O + \vec{M}\cdot\delta\vec{\theta} = 0\) (3) | \(\int (N\delta\varepsilon + Q\delta\gamma + M\delta\chi)\,\mathrm{d}x\) (6) | \(\int_V \underline{\sigma}:\delta\underline{\varepsilon}\,\mathrm{d}V\) (12) |
The material law (Hooke’s law) then closes the equations, and we apply them to deflections, stresses, failure and stability.
Lecture notes
| Chapter | Topic |
|---|---|
| Part I – Rigid bodies | |
| 1. Forces, moments and equilibrium | Forces, moments, force couples, rigid-body statics |
| 2. Rigid-body kinematics | Rotation matrix, coordinate transformation, rigid-body velocity |
| 3. Constraints and virtual work | Principle of virtual work, supports, static determinacy, instantaneous centers of rotation |
| Part II – Beams | |
| 4. Internal forces in beams | Free body diagrams, \(N\), \(Q\), \(M\), infinitesimal beam element, Dirac and Heaviside |
| 5. Beam kinematics | Displacements \(u\), \(w\), \(\varphi\) and strains \(\varepsilon\), \(\gamma\), \(\chi\) |
| 6. Virtual work for beams | Conjugate pairs, equilibrium and boundary conditions from virtual work |
| 7. Euler–Bernoulli beams | Constitutive relations, Bernoulli hypothesis, \(EIw'''' = q\) |
| 8. Hyperstatic beams | Elastic energy, Castigliano, virtual forces, force method, Müller-Breslau |
| Part III – Continuum | |
| 9. Stress and equilibrium | Stress tensor, equilibrium of a volume element, traction |
| 10. Stresses in beams | \(\sigma_{xx} = Mz/I\), parabolic shear stress |
| 11. Strain and compatibility | Displacement gradient, strain and rotation, compatibility |
| 12. Virtual work in the continuum | \(\underline{\sigma}\)–\(\underline{\varepsilon}\) as conjugate pair, equilibrium from virtual work |
| 13. Hooke’s law and plane problems | Origin of elasticity, elastic constants, plane stress and plane strain |
| 14. Stress transformation and Mohr’s circle | Rotation of the stress tensor, principal stresses, invariants, Mohr’s circle |
| 15. Yield criteria | Tresca, von Mises, Coulomb–Mohr, Drucker–Prager |
| 16. Buckling | Euler buckling, effective length |
The notes collect the notation, the vector and tensor calculus and the main equations; the interactive pages let you play with internal forces and beam deflection.
Please tell us if something in this material is unclear or wrong — we do make mistakes.
Acknowledgements
These notes build on the Micromechanics course material developed by the Simulation group (Prof. Lars Pastewka) and the Laboratory for Soft Machines at the Department of Microsystems Engineering (IMTEK) of the University of Freiburg.