Course detail

# Nonlinear Mechanics

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.

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.

Recommended optional programme components

Not applicable.

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Language of instruction

Czech

Work placements

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.

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.

Classification of course in study plans

• Programme N-P-E-SI (N) Master's

branch K , 1. year of study, winter semester, 4 credits, compulsory

• Programme N-K-C-SI (N) Master's

branch K , 1. year of study, winter semester, 4 credits, compulsory

• Programme N-P-C-SI (N) Master's

branch K , 1. year of study, winter semester, 4 credits, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

seminars

13 hours, compulsory

Teacher / Lecturer