Course detail

Constitutive Relations of Material

FSI-RK0Acad. year: 2020/2021

The coarse provides an comprehensive overview od constitutive dependencies of matters, not only solid (i.e. materials in the sens of mechanical engineering) but liquid and gaseous as well, it defines the term of constitutive models. It deals in detail with materials showing large strains, non-linear elastic as well as non-elastic, isotropic as well as anisotropic. For each of the presented models the basic constitutive equations are formulated on the basis of which the mechanical response of the material is derived by both analytical and numerical (FEM) methods, including applications of the models in ANSYS software.

Learning outcomes of the course unit

Students get an overview of mechanical properties and behaviour of matters and of possibilities of their modelling, especially under large strains. They will have a clear idea of their sophisticated application in design of machines and structures. Within the framework of abilities of the used FE programme systems, they will be made familiar with the practical use of some of the more complex constitutive models (hyperelastic and non-elastic, isotropic and anisotropic) in stress-strain analyses.


Students are expected to have knowledge of basic terms of theory of elasticity (stress, strain, general Hooke's law), as well as some basic terms of hydrodynamics (ideal, Newtonian and non-Newtonian liquids) and thermodynamics (state equation of ideal gas, thermodynamic equilibrium). Fundamentals of FEM and basic skills in ANSYS program system are required as well.


Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Lemaitre J., Chaboche J.-L.: Mechanics of Solid Materials
Janíček P.: Systémové pojetí vybraných oborů pro techniky
Holzapfel G.A.: Nonlinear Solid Mechanics
Články v odborných časopisech

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical application of topics presented in lectures using ANSYS software.

Assesment methods and criteria linked to learning outcomes

The course-unit credit is awarded on condition of having actively participated in seminars and submitted an individual semester project. The exam is based on a written test of basic knowledge and defense of the individual semester project.

Language of instruction


Work placements

Not applicable.


The objective of the course is to provide students a comprehensive and systematic overview of constitutive dependencies of various types of matters, to interconnect their knowledge acquainted in various courses and fields (solid mechanics, hydromechanics, thermomechanics) and, in the same time, to make students familiar with practical applications of some of the constitutive models (in FEA program system ANSYS) useful in modelling of up-to-date materials (e.g. elastomers, plastics, composites with elastomer matrix).

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

Attendance at practical training is obligatory. An apologized absence can be compensed by individual works controlled by the tutor.

Classification of course in study plans

  • Programme M2A-P Master's

    branch M-IMB , 2. year of study, winter semester, 5 credits, compulsory

Type of course unit



26 hours, optionally

Teacher / Lecturer


1. Definition of the term constitutive model. Overview of constitutive models in mechanics, basic constitutive models for individual states of matter.
2. Simple constitutive models - overview. Linear and non-linear models of elasticity.
3. Introduction to tensor calculus, notation and properties of tensors, basic tensor operations. 4. Stress and deformation tensors under large strain conditions, their invariants and decomposition into spherical and deviatoric parts.
5. Hyperelastic models for isotropic hardly compressible elastomers on the polynomial basis.
6. Other hyperelastic models, models for very compressible elastomers (foams), poroelastic models.
7. Anisotropic hyperelastic models of elastomers with reinforcing fibers. Pseudoinvariants of deformation tensor.
8. Models describing inelastic effects of elastomers.
9. Constitutive models of Newtonian and non-Newtoniad liquids.
10. Combined models. Introduction in the theory of viscoelasticity.
11. Models of linear viscoelasticity - response under static and dynamic load.
12. Complex modulus of elasticity, relaxation and creep functions, non-linear viscoelasticity.
13. Other combined models - basic constitutive characteristics. Mikropolar continuum models. Cosserat continuum.

Computer-assisted exercise

13 hours, compulsory

Teacher / Lecturer


1. Revision of applications of linear elastic constitutive model.
2. Matrix and tensor forms of Hooke’s law. Multilinear elastic model.
3. Basic tensor operations – tensor product, double-dot product.
4. Invariants of deformation tensor, modified invariants.
5. Hyperelastic models in ANSYS - testing of elastomers and their input into the constitutive model.
6. Choice of a suitable constitutive model of a hardly compressible elastomer, predictive ability of the model.
7. Adaptation of the constitutive model for the required strain range.
8. Anisotropic hyperelastic models, use of constitutive models of foams.
9. Newtonean fluid. Linear viskoelasticity - behaviour of Maxwell and Voigt models.
10. Linear viskoelasticity - behaviour of Kelvin and generalized Maxwell models.
11. Introducing experimental data into models of linear and non-linear viscosleasticity.
12. Temperature dependence of viscoelastic parameters and application in FE analyses.
13. Semester project, course-unit credit.