Course Details
Nonlinear Mechanics
Academic Year 2023/24
NDA028 course is part of 1 study plan
NPC-SIK Winter Semester 1st year
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.
Course Guarantor
Institute
Objective
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.
Knowledge
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.
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.
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.
Prerequisites
Linear mechanics, Finite element method, Matrix algebra, Fundamentals of numerical mathematics, Infinitesimal calculus.
Language of instruction
Czech
Credits
4 credits
Semester
winter
Forms and criteria of assessment
graded course-unit credit
Specification of controlled instruction, the form of instruction, and the form of compensation of the absences
Extent and forms are specified by guarantor’s regulation updated for every academic year.
Offered to foreign students
Not to offer
Course on BUT site
Lecture
13 weeks, 2 hours/week, elective
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 weeks, 1 hours/week, compulsory
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.