Course Details

Nonlinear Mechanics

Academic Year 2025/26

NDB023 course is part of 1 study plan

​ NPC-SIS Winter Semester 2nd year

Course Guarantor

doc. Ing. Ivan Němec, CSc.

Institute

Institute of Structural Mechanics

Language of instruction

Czech

Credits

4 credits

Semester

winter

Forms and criteria of assessment

course-unit credit and examination

Offered to foreign students

Not to offer

Course on BUT site

https://www.vut.cz/en/students/courses/detail/295805

Lecture

13 weeks, 2 hours/week, elective

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

13 weeks, 2 hours/week, compulsory

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.