Course detail

Mechanics of Composites

FSI-9MEKAcad. year: 2020/2021

Representative volume element (RVE)concept. Average stress and strain in RVE. Relation between macrofield and microfield parameters. Localization and homogenization. Eigenstrains and eigenstresses. Energy-based approach. Simple estimates on bounds of bulk and shear moduli. Eshelby solution for inclusion. Eshelby's tensor. Application to materials containing microcracks and microvoids. Self-consistent, differential and related averaging metods. Hashin-Shtrikman variational principles. Rate formulation of micromechanical models suitable for material plasticity description. Method of unit cell for solids with periodic microstructure.

Learning outcomes of the course unit

Students will elaborate their knowledge concerning the mechanics of composites. Fundamental concepts togehter with their interpretation will be formulated. Student will be Capability of individual study of professional literature concerning the mechanics of composite materials.


In the field of mechanics: Knowledge of basic concepts of the theory of elasticity (stress, principal stress, deformation, strain, general Hooke law, potential energy). Principle of virtual displacements, principle of virtual work. Elements from the mechanics of materials. In the field of mathematics: Partial differential equations of 2nd order. Elements of variational calculus. Integral and differential calculus.


Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

P. Procházka: Základy mechaniky složených materiálů. Academia
D. Gros, T. Seelig, Fracture mechanics with an introduction to micromechanics , 2nd Edition, Springer Heidelberg Dordrecht London New York, ISBN 978-3-642-19239-5 (EN)
S.Nemat-Nasser, M.Hori: Micromechanics. North-Holland
J.N. Reddy: Mechanics of Laminated Composite Plates and Shells. CRC Press
A.Kelly, C. Zweben: Comprehensive composite materials. Elsevier

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline.

Assesment methods and criteria linked to learning outcomes

Final evaluation is based upon the individual preparation and presentation of a semestral
project completed with discussion over the project.

Language of instruction


Work placements

Not applicable.


The goal of the subject is to make students familiar with basic homogenization techniques and methods of constitutive equations derivation used in problems of the mechanics of composite materials.

Specification of controlled education, way of implementation and compensation for absences

Active participation in the course is controlled individually according to the progression of the semestral project.

Classification of course in study plans

  • Programme D-IME-P Doctoral, 1. year of study, summer semester, 0 credits, recommended
  • Programme D-MAT-P Doctoral, 1. year of study, summer semester, 0 credits, recommended

Type of course unit



20 hours, optionally

Teacher / Lecturer


1. Representative volume element, average stress and stress rate, average strain and strain rate, average rate of stress-work. Interfaces and discontinuities. Potential functions for macro-elements.
2. Statistical homogeneity, average quantities and overall properties. Reciprocal theorem, superposition, Greens function.
3. Overall elastic modulus and compliance tensors. Eigenstrain and eigenstress tensors. Consistency conditions. Eshelbys tensor for special cases. Transformation strains.
4. Estimates of overall modulus and compliance tensors- dilute distribution.
5. Estimates of overall modulus and compliance tensors- self-consistent method.
6. Energy consideration and symmetry of overall elasticity and compliance tensors.
7. Upper and lower bounds for overall elastic moduli. Hashin-Shtrikman variational principle. Part 1.+2.
8. Self consistent, differential and related averaging metods.
9. Solids with periodic microstructure. General properties and field equations. Periodic microstructure and RVE. Periodicity and unit cell.
10. Periodic eigenstrain and eigenstress fields.
11. Mathematical theory of periodic homogenization. Method of asymptotic expansions.
12. Micromechanics of inelastic composite materials.