Course detail

# Nonlinear Mechanics

Types and sources of nonlinear behavior of structures. New definition of stress and strain measures that is necessary for geometrical nonlinear analysis of structures. Principles of numerical solution of nonlinear problems (Newton-Raphson, modified Newton-Rapshon, arc length). 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

5

Mode of study

Not applicable.

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.Introduction to nonlinear mechanics. Physical and geometrical nonlinearities. Eulerian and Lagrangian nesnes.
2.Strain measures (Green-Lagrange, Euler-Almansi, engineering, logarithmic), their behavior in large strain and large rotation. Stress measures (Cauchy, 1. Piola-Kirchhoff, 2. Piola-Kirchhoff, Biot). Energeticaly conjugate stress and strain measures.
3.Tangent stiffness matrix, Material stiffness, Geometrical stiffness. Influence of nonlinear members of the strain tensor. Newton-Raphson method. Calculation of unbalanced forces.
4.Modified Newton-Raphson method. Postcritical analysis. Deformation control. Arc length method
5.Linear and nonlinear buckling. Von Mises truss, snap through. Physical nonlinearity (supports, beams, concrete, subsoil).
6.Types of materials, introduction into constitutive material models. Linear and nonlinear fracture mechanics. Fracture mechanical material parameters.
7.Problem of strain localization, false sensitivity on the mesh. Restriction of localization. Crack band model. Nonlocal continuum mechanics.
8.Constitutive equations for concrete and other quasi-fragile materials. Fracture-plastic model. Mircroplane model.
9.Influence of size to bearing capacity (size effect). Energetical and statistical causes. Analysis of the influence of size on strength in tension in bending.
10.Presentation of modeling by a software on nonlinear fracture mechanics. Examples of applications. Mechanics of damane.

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 new definition of stress and strain measures and the principles that are necessary for nonlinear solution of structures by the Newton-Raphson method.

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.

Not applicable.

Classification of course in study plans

• Programme N-P-E-SI (N) Master's

branch S , 2. year of study, winter semester, compulsory-optional

• Programme N-K-C-SI (N) Master's

branch S , 2. year of study, winter semester, compulsory-optional

• Programme N-P-C-SI (N) Master's

branch S , 2. year of study, winter semester, compulsory-optional

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

1.Introduction to nonlinear mechanics. Physical and geometrical nonlinearities. Eulerian and Lagrangian nesnes. 2.Strain measures (Green-Lagrange, Euler-Almansi, engineering, logarithmic), their behavior in large strain and large rotation. Stress measures (Cauchy, 1. Piola-Kirchhoff, 2. Piola-Kirchhoff, Biot). Energeticaly conjugate stress and strain measures. 3.Tangent stiffness matrix, Material stiffness, Geometrical stiffness. Influence of nonlinear members of the strain tensor. Newton-Raphson method. Calculation of unbalanced forces. 4.Modified Newton-Raphson method. Postcritical analysis. Deformation control. Arc length method 5.Linear and nonlinear buckling. Von Mises truss, snap through. Physical nonlinearity (supports, beams, concrete, subsoil). 6.Types of materials, introduction into constitutive material models. Linear and nonlinear fracture mechanics. Fracture mechanical material parameters. 7.Problem of strain localization, false sensitivity on the mesh. Restriction of localization. Crack band model. Nonlocal continuum mechanics. 8.Constitutive equations for concrete and other quasi-fragile materials. Fracture-plastic model. Mircroplane model. 9.Influence of size to bearing capacity (size effect). Energetical and statistical causes. Analysis of the influence of size on strength in tension in bending. 10.Presentation of modeling by a software on nonlinear fracture mechanics. Examples of applications. Mechanics of damane.

Exercise

26 hours, compulsory

Teacher / Lecturer

Syllabus

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