Constitutive Equations for IME (FSI-RKI-A)

Academic year 2023/2024
Supervisor: prof. Ing. Jiří Burša, Ph.D.  
Supervising institute: ÚMTMB all courses guaranted by this institute
Teaching language: English
Aims of the course unit:
The objective of the course is to provide students a comprehensive overview of constitutive dependencies of various types of matters, to interconnect their knowledge acquainted in various courses and fields (solid mechanics, hydromechanics, thermomechanics) and to make students familiar with practical applications of some of the constitutive models (in finite element program system ANSYS) useful in modelling of up-to-date materials (e.g. elastomers, plastics, composites with elastomer matrix, metals above the yield limit).
Learning outcomes and competences:
Students get an overview of mechanical properties and behaviour of matters and of possibilities of their mathematical description and modelling, especially of their time dependent as well as large strain behaviour. They will have a clear idea of their sophisticated application in design of machines and structures. Within the framework of capabilities of the used FE programme systems, they will be made familiar with the practical use of some of the more complex constitutive models (hyperelastic and non-elastic, isotropic and anisotropic) in stress-strain analyses.
Prerequisites:
Students are expected to have knowledge of basic terms of theory of elasticity (stress, strain, general Hooke's law), as well as some basic terms of hydrodynamics (ideal, Newtonian and non-Newtonian liquids) and thermodynamics (state equation of ideal gas, thermodynamic equilibrium). Fundamentals of FEM and basic skills in ANSYS program system are required as well.
Course contents:
The course provides a comprehensive overview od constitutive dependencies and constitutive models of matters, not only of solids (i.e. structural materials) but also of liquids and gases. It deals also with time dependence of stress-strain response of materials and describes it using different viscoelastic models. It applies the theory of finite strains of solids in description of non-linear elastic as well as non-elastic behaviour of elastomers and composites with elastomer matrix and of plastic behaviour of metals including their ductile fracture. It presents specific ways of material testing needed for identification of their models. For each of the presented models basic constitutive equations are formulated on the basis of which the response of the material under load is derived using both analytical and numerical (FEM) methods, including applications of the models in ANSYS software.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical application of topics presented in lectures using ANSYS software.
Assesment methods and criteria linked to learning outcomes:
The course-unit credit is awarded on condition of having actively participated in seminars and submitted an individual semester project. The exam is based on a written test of basic knowledge and defense of the individual semester project.
Controlled participation in lessons:
Attendance at practical training is obligatory. An apologized absence can be compensed by individual works controlled by the tutor.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Computer-assisted exercise  13 × 1 hrs. compulsory                  
Course curriculum:
    Lecture

  1. Definition and overview of constitutive models in mechanics, constitutive models for individual states of matter, definition of deformation tensors.

  2. Stress and strain tensors under large strains, hyperelasticity model neo-Hooke.

  3. Mechanical tests of elastomers, polynomial hyperelastic models, predictive capability.

  4. Models Ogden, Arruda Boyce - entropic elasticity.

  5. Incremental modulus. Models of foams. Anisotropic hyperelasticity, pseudoinvariants.

  6. Non-elastic effects (Mullins). Plasticity criteria.

  7. Models of plastic flow, triaxiality factor, Lode parameter.

  8. Models of ductile fracture.

  9. Shape memory alloys

  10. Linear viscoelasticity – introduction

  11. Linear viscoelasticity – behaviour of models under static loading

  12. Linear viscoelasticity - dynamic behaviour, complex modulus

  13. Visco-hyperelasticity – model Bergstrom-Boyce, polar decomposition
    Computer-assisted exercise

  1. Experiment – elastomer testing


2.-3. FEM simulations of tests of elstomers


4.-5. Identification of constitutive models of elastomers


6.-7. Models of plasticity


8.-9. Models of anisotropic behaviour of elastomers


10. Model of Mullinsova efektu


11.-12. Simulation of viscoelastic behaviour


13. Project formulation, course-unit credit.
Literature - fundamental:
1. Lemaitre J., Chaboche J.-L.: Mechanics of Solid Materials. Cambridge University Press, 1994.
2. Holzapfel G.A.: Nonlinear Solid Mechanics. Wiley, 2001.
3. Články v odborných časopisech
Literature - recommended:
1. Němec I. a kol. Nelineární mechanika. VUTIUM, Brno, 2018
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
N-IMB-P full-time study IME Engineering Mechanics -- Cr,Ex 6 Compulsory 2 2 W
N-ENG-Z visiting student --- no specialisation -- Cr,Ex 6 Recommended course 2 1 W