Course detail

# Continuum Mechanics

FSI-S1KAcad. year: 2020/2021

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.

Supervisor

Learning outcomes of the course unit

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.

Prerequisites

In the field of mechanics: Knowledge of basic concepts of the theory of elasticity (stress, principal stress, deformation, strain, Hooke law). Principle of virtual displacements, principle of virtual work. In the field of mathematics: Partial differential equations of 2nd order. Elements of variational calculus. Elements of functional analysis (functional spaces, Hilbert space L2).

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Timoshenko, S.P. Goodier,J.N.: Theory of Elasticity (EN)

Ondráček, E., Vrbka,J., Janíček, P.: Mechanika těles- pružnost a pevnost II (CS)

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

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

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

Novacki W.: Teorija uprugosti (CS)

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

Written test examining the knowledge of basic concepts - examination paper containing 3 examples to be solved - oral discussion over examination papers with an optional additional question.

Language of instruction

Czech

Work placements

Not applicable.

Aims

The goal of the course is to familiarise students with basic concepts and relations of the continuum mechanics of solid state and the ways of formulation and solution of boundary-value problems for elastic and elastic/plastic bodies. Apart from classical formulation, the emphasis is placed on the variational formulation of problems with consideration of consecutive numerical solution. Another goal is for students to learn the basics of finite deformation theory and constitutive equations, which enables better understanding of the application of advanced FEM systems simulating complex processes under consideration of large strains and nonlinear properties of materials.

Specification of controlled education, way of implementation and compensation for absences

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.

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

Syllabus

Theory of finite deformations. Lagrangean and Eulerean formulation of motion. Deformation gradient. Equation of continuity. Polar decomposition of deformation gradient. Deformation and strain measures. Time derivations in finite deformations.

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

39 hours, compulsory

Teacher / Lecturer

Syllabus

Kinematical quantities of the continuum mechanics.

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.