Course detail

Nonlinear Mechanics

FAST-NDA028Acad. year: 2022/2023

Index, tensor and matrix notations, vectors and tensors in mechanics, properties of tensors. Types and sources of nonlinear behavior of structures. More general definitions of stress and strain measures that are necessary for geometrical nonlinear analysis of structures. Fundamentals of material nonlinearity. Methods of numerical solution of nonlinear algebraic equations (Picard, Newton-Raphson, modified Newton-Rapshon, Riks). Post critical analysis of structures. Linear and nonlinear buckling. Application of the presented theory for the solution of particular nonlinear problems by a FEM program.

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

doc. Ing. Ivan Němec, CSc.

Department

Institute of Structural Mechanics (STM)

Learning outcomes of the course unit

Students will learn various types of nonlinearities. They will understand the basic differences in the attitude to linear and nonlinear solution of structures. They will learn new definition of stress and strain measures and the principles that are necessary for nonlinear solution of structures by the Newton-Raphson method.

Prerequisites

Linear mechanics, Finite element method, Matrix algebra, Fundamentals of numerical mathematics, Infinitesimal calculus.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Course curriculum

1. Index, tensor and matrix notation, vectors and tensors, properties of tensors, transformation of physical quantities.
2. Fundamental laws i mechanics, kinds of ninlinearities by their sources, Eulerian and Lagrangian meshes, material and space coordinates, fundamentals in geometrical nonlinearity.
3. Srain measures (Green-Lagrange, Euler-Almansi, logarithmick, infinitesimál), their behaviour in large rotation and large deformation.
4. Stress measures (Cauchy, 1st Piola-Kirchhoff, 2nd Piola-Kirchhoff, corotational, Kirchoff) and transformatio between them.
5. Energeticaly konjugate stress and strain measures, two basic formulations in geometyric nonlinearity.
6. Influence of stress on stiffness, geometrical stiffness matrix.
7. Updated Lagrangian formulation, basic laws and tangential stiffness matrix.
8. Total Lagrangian formulation, basic laws and tangential stiffness matrix.
9. Objective stress rates, constitutive matrices, fundamentals of material nonlinearity.
10. Numerical methods of solution of the nonlinear algebraic equations, Picard method, Newton-Rapson method.
11. Modified Newton-Raphsonmethod, Riks method.
12. Linear and nonlinear stability.
13. Postcritical analysis.

Work placements

Not applicable.

Aims

Students will learn various types of nonlinearities that occur in the design of structures. They will understand the basic differences in the attitude to linear and nonlinear solution of structures. They will learn more general definitions of stress and strain measures, the two main formulation of geometrical nonlinearity the same as the fundamentals of material nonlinearity. The main numerical methods of solution of nonlinear algebraic equation will be also explained.

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

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme NPC-SIK Master's 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. Ing. Ivan Němec, CSc.
Ing. Rostislav Lang, Ph.D.

Syllabus

1. Index, tensor and matrix notation, vectors and tensors, properties of tensors, transformation of physical quantities. 2. Fundamental laws i mechanics, kinds of ninlinearities by their sources, Eulerian and Lagrangian meshes, material and space coordinates, fundamentals in geometrical nonlinearity. 3. Srain measures (Green-Lagrange, Euler-Almansi, logarithmick, infinitesimál), their behaviour in large rotation and large deformation. 4. Stress measures (Cauchy, 1st Piola-Kirchhoff, 2nd Piola-Kirchhoff, corotational, Kirchoff) and transformatio between them. 5. Energeticaly konjugate stress and strain measures, two basic formulations in geometyric nonlinearity. 6. Influence of stress on stiffness, geometrical stiffness matrix. 7. Updated Lagrangian formulation, basic laws and tangential stiffness matrix. 8. Total Lagrangian formulation, basic laws and tangential stiffness matrix. 9. Objective stress rates, constitutive matrices, fundamentals of material nonlinearity. 10. Numerical methods of solution of the nonlinear algebraic equations, Picard method, Newton-Rapson method. 11. Modified Newton-Raphsonmethod, Riks method. 12. Linear and nonlinear stability. 13. Postcritical analysis.

Exercise

13 hod., compulsory

Teacher / Lecturer

doc. Ing. Ivan Němec, CSc.
Ing. Rostislav Lang, Ph.D.

Syllabus

1. Demonstration of the differences between linear and nonlinear calculations. 2. Demonstration of the problems with a big rotations. 3. Demonstration of the differences between the 2nd order theory and the large deformations theory. 4. Exdamples on bending of beams with a big rotations of the order of radians. 5. Examples on calculations of cables. 6. Examples on calculations of membranes. 7. Examples on calculations of mechanismes. 8. Examples on calculations of stabilioty of beams. 9. Examples on calculations of stability of shells. 10. Comparison of the Newton-Raphson, modified Newton-Raphson and Picard methods. 11. Examples on postcritical analysis of beams. 12. Examples on postcritical analysis of shells. 13. Demostration of the explicit method in nonlinear dynamics.