Course detail

# Applied Mechanics

Introduction, basic terminology. Stress and strain tensors, principal stresses. Mathematical theory of elasticity, differential approach (equilibrium equations, Hooke´s law, geometrical equations, boundary conditions). Variational approach, principle of virtual work. Finite element method (FEM), displacement version. Fundamentals of linear fracture mechanics. Associated theory of plasticity. Kinematic and isotropic hardening rule, mixed hardening. Mechanics of composite materials, homogenization and elements of micromechanics. Stiffness and strength of the unidirectional fibre composite (lamina) in longitudinal and transversal direction. Stiffness and strength of the short fibre composites. Hooke's law of anisotropic, orthotropic and transversal orthotropic material in the principal material directions, strength conditions. Mechanisms of toughening of brittle matrix composites.

Learning outcomes of the course unit

Students learn basic methods of determination of stress and strain states at general bodies, based on differential and variational approach. They get practical experience in using of finite element method (Program system ANSYS) in solving stress and strain states of simple structures. The knowledge of the negative influence of cracks on the lifetime and basic knowledge about the mechanical behaviour of composite materials is important as well.

Prerequisites

Knowledge of basic terms of theory of elasticity (stress, strain, general Hooke's law), fundamentals of linear elasticity theory of beams and shells. Fundamentals of theory of limit states (criteria of plasticity and brittle strength).

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Hill,R.: The mathematical theory of plasticity. Oxford U. P., Oxford, 1950
Ondráček,E.,Vrbka,J.,Janíček,P.,Burša,J.: Mechanika těles - pružnost a pevnost II. Akademické nakladatelství CERM, Brno, 2006
Agarwal,B.D., Broutman,L.J.: Vláknové kompozity, SNTL, Praha,1987
Chawla, K.K.: Composite materials. Science and engineering. Springer-Verlag, New York, Berlin, Heidelberg, 1998
Gross, D., Seeling T.: Fracture mechanics. Springer-Verlag, Berlin, Heidelberg, 2006

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

The course-unit credit conferment is based on the successful disputation of the final project, having character of a practical computation of stress and strain states of a simple construction or composite material structure utilizing classic approaches as well as the Finite Element Method (FEM) program system ANSYS and followed by critical judging of results. The exam is combined; it consists of a written review test and of an oral interview.

Language of instruction

Czech

Work placements

Not applicable.

Aims

The aim is to make students familiar with methods and approaches in determination of stress and strain states at general bodies of linear elastic and elasto-plastic materials. Students will learn of the influence of cracks on the stress and strain states and with possibilities of residual lifetime evaluation. In the chapter dealing with composite materials, the students get acquainted with the methods of determination of mechanical properties of a composite material on the basis of knowledge of geometrical structure and properties of individual components. Moreover, students will understand the anisotropic and orthotropic behaviour of composites at the level of continuum as a consequence of the directional structure of the material.

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

Attendance at practical training is obligatory. An apologized absence can be compensed by working out individual projects controlled by the tutor.

Classification of course in study plans

• Programme M2A-P Master's

branch M-MTI , 1. year of study, winter semester, 5 credits, compulsory

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

Syllabus

1.Basic equations of mathematical theory of elasticity. Differential equations of equilibrium, geometrical equations, general Hooke’s law. Boundary conditions.
2.Differential formulation of problem of elasticity in displacements. Possibilities of solution. Variational formulation, virtual work principle, Lagrangean variational principle.
3. Mises condition of plasticity. Kinematic and isotropic stiffening. Prager and Ziegler condition for plasticity surface displacement.
4. Associated theory of plastic creep with combined stiffening. Basic assumptions. Normality rule, strain superposition principle.
5.Deformational variant of finite element method (FEM) for a two-dimensional problem. Triangulation, approximate functions for displacements, problem discretization.
6.FEM equilibrium equation for an element and the whole body. Local and global stiffness matrix.
7. Fundamentals of linear fracture mechanics. Stress intensity factor (SIF) K, J-integral, crack tip opening CTOD. Stress and strain states for the three basic modes I, II and III.
8.Paris-Erdogan’s law. Residual lifetime of the body with a defined crack. Possibilities of SIF evaluation for a generally located crack using FEM.
9.Mechanics of composite materials. Definition and basic terms, classification of composites. Mechanical properties of fibres and of matrix materials
10. Introduction to micromechanics and homogenization of composite materials. Hooke’s law for isotropic, orthotropic and transversally isotropic materials in principal material directions and in general directions. Directional stiffness matrix. Strength conditions.
11. Unidirectional long-fibre composite loaded in longitudinal direction. Elasticity modulus and strength. Critical and minimal volume of fibres.
12.Short-fibre unidirectional composite. Theory of load bearing. Transmission and critical length. Elasticity modulus in tension and strength.
13. Mechanisms of toughening of brittle matrix composites.

Computer-assisted exercise

26 hours, compulsory

Teacher / Lecturer

Syllabus

1.Basic equations of mathematical theory of elasticity. Equilibrium equations. Geometrical equations General Hooke’s law
2. Stress state in a point of body.
3.Differential formulation of problem of elasticity in displacements. Lame’s equations. Yield criterion.
4. Virtual work principle. Lagrange’s principle. Ritz method.
5.Deformational variant of finite element method (FEM). Basic FEM equations. Introduction into FEM program system ANSYS. Basic types of elements.
6. FEM model creation in the FEM program system ANSYS. Solution of a simple two-dimensional beam structure
7.Three-dimensional beam structure in the system ANSYS
8.Plane problems in linear elasticity theory. Calculation of fracture mechanical parameters - stress intensity factor (SIF) K, J-integral, crack tip opening CTOD
9.FEM determination of plastic zone ahead of a crack tip using various yield criteria.
10.Homogenization of fibre composites using FEM -Material characteristics in longitudinal direction.
11. Homogenization of fibre composites using FEM -Material characteristics in transversal direction. Effective temperature expansion of a composite in various directions.
12.Final project.
13.Credit.

E-learning texts

Vrbka, J.: Aplikovaná mechanika. Ústav mechaniky těles, mechatroniky a biomechaniky. FSI VUT v Brně, Brno, 2012 (cs)