Course detail

Mechanics of Composites

FSI-9MEKAcad. year: 2023/2024

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.

Language of instruction


Number of ECTS credits


Mode of study

Not applicable.

Entry knowledge

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.

Rules for evaluation and completion of the course

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

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


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.
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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

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

Recommended reading

P. Procházka: Základy mechaniky složených materiálů. Academia

Classification of course in study plans

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