Course detail

Nonlinear Problems of Mechanics in FEM

FSI-RNUAcad. year: 2017/2018

The course is a follow-up to basic lectures in solid mechanics, which are traditionally limited to linear problems, and introduces the basic nonlinearities. Material nonlinearity is represented by several models of plastic behaviour.
Next, contact problems, large displacement and large strain problems are presented. Although some classical solutions to selected nonlinear problems are mentioned (Hertz contact, deformation theory of plasticity), attention is given to numerical solution. Above all, the relation between stability and convergence of numerical solution and physical interpretation of the analysed problem is thoroughly inspected in seminars.

Learning outcomes of the course unit

Students learn how to classify basic types of nonlinear behaviour in solid mechanics, they will learn their characteristics and classical solutions for some
types of problems. They can prepare numerical computational model, solve it using some of the commercial FE systems and make a rational analysis of
typical problems with divergence of the iterative process of solution.


Mathematics: linear algebra, matrix notation, functions of one and more variables, calculus, ordinary and partial differential equations. Others: basic theory of elasticity, theory and practical knowledge of the FEM.


Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

G.A.Holzapfel: Nonlinear Solid Mechanics, Wiley, 2000
D.R.J.Owen, E.Hinton: Finite Elements in Plasticity, Pineridge Press, 1980
K.-J.Bathe: Finite Element Procedures, Prentice Hall, 1996
M.A.Crisfield: Non-linear Finite Element Analysis of Solids and Structures 1-2, Wiley, 1991-97
M.Okrouhlík, editor: Mechanika poddajných teles, numerická matematika a superpocítace, Ústav termomechaniky AV CR, Praha, 1997

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 topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

The graded course-unit credit requirements :
- active participation in seminars,
- good results in the written test of basic knowledge,
- individual preparation and presentation of seminar assignments.

Language of instruction


Work placements

Not applicable.


The aim of the course is to provide students with theoretical knowledge and elementary experience with the solution of most frequent types of nonlinear
problems of solid mechanics.

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

Attendance at practical training is obligatory. The absence (in justified cases) is compensated by additional assignments according to the instructions of the tutor.

Classification of course in study plans

  • Programme M2A-P Master's

    branch M-IMB , 1. year of study, summer semester, 5 credits, compulsory

Type of course unit



26 hours, optionally

Teacher / Lecturer


1. Introduction to nonlinear problems of solid mechanics
2. Incremental theory of plasticity and its implementation in FEM systems, Deformation theory of plasticity
3. Elasto-plastic bending of beams, plastic hinge and plastic collaps
4. Elasto-plastic response to cyclic loading
5. Residual stress
6. Contact problems - classical solution
7. Strategy of contact solution in FEM, characteristics of contact elements
8. Large displacement and strain - alternative formulations of strain tensors
9. Large displacement and strain - continued
10. Engineering vs. natural stress and strain, evaluation of materiál flow curve in natural coordinates
11. Stability of thin-walled structures as a nonlinear problem of mechanics
12. Explicit formulation of FEM in nonlinear problems of mechanics
13. Convergence of numerically solved nonlinear problem

Computer-assisted exercise

26 hours, compulsory

Teacher / Lecturer


1. Convergence of iterative solution of nonlinear problem - numerical demonstrations
2. Plasticity in FEM - solution of selected tasks
3. Plasticity in FEM - solution of selected tasks
4. Start of seminar project
5. Plastic collaps
6. Residual stress
7. Tutorial of seminar project
8. Solution of contact problem by FEM
9. Tutorial of seminar project
10. Solution of large displacement problem by FEM
11. Solution of stbility of shell
12. Example of an explicit FEM solver
13. Presentation of seminar projects