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 fundamentals of nonlinear mechanics. They will understand quality and quantity of differences between linear and nonlinear analysis of structures. They will learn different formulations and methods of nonlinear analysis. Use of nonlinear analysis of structures in design praxis is increasing, therefore the knowledge gained in this subject will be very appreciated in praxis.

Prerequisites

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

Co-requisites

In the subject Nonlinear mechanics the konwledge of fundamentals of the tensor calculus are needed. These should be learned in the subject of mathematics.

Recommended optional programme components

Deeper knowledge of nonlinear mechanics including its applications in analysis of structures should be gained on a nonmandatoru specialised seminar.

Recommended or required reading

Not applicable.

Planned learning activities and teaching methods

Lectures are theoretical. They start from the fundamentals of the tensor calculus and then there are lectured the strain and stress measures, basic formulations of the geometrical nonlinearity, fundamentals of the material nonlinearity, methods of solution of nonlinear algebraic equations and postcritical analysis. In the seminaries students work on computers uner the leadership of the teacher. They practically analyze what was theoretically lectured.

Assesment methods and criteria linked to learning outcomes

The attendance of lectures is not mandatory, but serves as one of the bases for evaluation of students. Attendance of exercises is mandatory. The theoretical knowledge which is presented on lectures is needed to pass the written and oral exam.

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-C-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-E-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

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 hours, compulsory

Teacher / Lecturer

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.