Course detail

# Continuum Mechanics

FSI-S1KAcad. year: 2023/2024

The course deals with the following topics: Introduction, basic terminology, bodies, motions, configurations. Foundation of the theory of finite strains. General equation of balance. Cauchy's I. and II. law of continuum mechanics. Geometrical equations, compatibility conditions, boundary conditions. Thermodynamic background of the theory of constitutive relations. Models of elastic behaviour. Hyperelastic materials. Isotropic elasticity and thermoelasticity. Anisotropic elasticity. Classical formulation of an elastic problem using differential approach. Deformation theory and incremental theory of plasticity. Variational principles in the infinitesimal strain theory. Weak solution. Axisymmetric problems. Plane strain/plane stress. Solution of two-dimensional elasticity problems. Airy's stress function. Foundation of the theory of plates and shells. Fundamentals of linear fracture mechanics. Remarks on Ritz method and FEM in continuum mechanics problems.

Language of instruction

Number of ECTS credits

Mode of study

Guarantor

Entry knowledge

Rules for evaluation and completion of the course

Attendance is required. One absence can be compensated by attending a seminar with another group in the same week, or by elaboration of substitute tasks. Longer absence is compensated by special tasks according to instructions of the tutor. Course-unit credit is awarded on the following conditions: - active participation in the seminars, - good results in seminar tests of basic knowledge, - solution of additional tasks in case of longer excusable absence. Seminar tutor will specify the form of these conditions in the first week of semester.

Aims

Students will be made familiar with basic methods applied for the determination of stress-strain fields in general bodies stemming from differential and variational approach. Knowledge of physical origin of variational formulation of problems of continuum mechanics together with knowledge gained in the course “Numerical Methods III” allows to choose a suitable approach to the preparation of numerical computation. Mastering the basics of the constitutive equation theory offers a good orientation among various material models. Students are also provided with knowledge of a negative influence of cracks upon the durability of cracked bodies.

Study aids

Prerequisites and corequisites

Basic literature

Nečas, J., Hlaváček, I.: Úvod do teorie pružných a pružně plastických těles (CS)

Novacki W.: Teorija uprugosti (CS)

Recommended reading

Janíček, P., Petruška, J.: Úlohy z pružnosti a pevnosti II (CS)

Němec,J. Dvořák,J., Hoschl. C.: Pružnost a pevnost ve strojírenství (CS)

Classification of course in study plans

- Programme N-MAI-P Master's, 2. year of study, winter semester, elective

#### Type of course unit

Lecture

Teacher / Lecturer

Syllabus

Mechanical quantities in the theory of finite deformations. Transport theorem. Euler-Cauchy laws in finite deformations. Piola-Kirchhoff and Cauchy stress tensors.

Introduction to the theory of constitutive equations, axioms and thermodynamical restrictions for constitutive equations.

Models of elastic materials. Hyperelastic material. Isotropic and anisotropic materials. Thermoelastic materials.

Basic equations of the mathematical theory of linear elasticity. Differential equations of equilibrium, geometrical equations, compatibility equations, Hooke law, boundary conditions, classical formulation of basic boundary-value problems of elasticity.

Variational principles of the theory of infinitesimal deformations. Variational formulation and solution of basic boundary-value problems of the theory of elasticity. Weak solution.

Basic elasticity problems in curvilinear coordinates.

2D problems in the theory of elasticity. Airy stress function. Solution to 2D problems in terms of stresses.

Introdution into the theory of plate bending.

Introduction into the theory of shells.

Deformation and incremental theory of plasticity. Mises yield condition. Associated theory of plastic flow. The rule of normality.

Deformation variant of the finite element method for a 2D problem.

Brief resume of the course, time reserve.

Exercise

Teacher / Lecturer

Syllabus

Stress tensors. Principal stresses, invariants. Equations of balance.

Constitutive equations in the continuum mechanics. Thermodynamic laws.

Hyperelastic material. Neo-Hooke law, Mooney-Rivlin law. Hooke law for isotropic and anisotropic bodies.

Selected 3D problems of the linear theory of elasticity.

Variational methods in the theory of infinitesimal deformations.

Basic quantities of the continuum mechanics in curvilinear coordinates.

Axial-symmetric problems of the linear elasticity.

Solution of plane problems using Airy stress function.

Circular and circular plate with concentric hole.

Cylindrical shell.

Axisymmetric membrane shell.

Selected simple problem form the theory of plasticity.

Numerical methods in the elasticity problems. Awarding course-unit credits.