# Continuum Mechanics (250952) – Course 2024/25 PDF

# Contents

In this chapter, the basic notation used in the course and a comprehensive review of the main concepts of tensor algebra is introduced. Contents: - Introduction - Vector Algebra - Tensor Algebra - Higher-order tensors - Differential Operators - Integral Theorems Problems

#### Specific Objectives

The objectives are to introduce the tensor notation to be used in the course and do a review of tensor algebra.

#### Dedication

1h 30m Large group + 1h Medium group + 1h 30m Laboratory classes + 5h 36m Self StudyThis topic describes the main assumptions of the Mechanics of Continuous Media and the main concepts of non-linear kinematics of particles are introduced: equation of motion, description of materials and spatial properties of the continuum derived materials, space and convective, displacement, velocity, acceleration, trajectories, streamlines, materials and spatial surfaces and volumes material espacial. Contents: - Definition of Continuous Medium - Motion equation - Material and spatial descriptions - Material and spatial time derivatives - Displacement - Velcoity - Accelerations - Trajectories - Streamlines - Material and spatial surface - Material and spatial volume Problems

#### Specific Objectives

The aim is to introduce the main concepts of nonlinear kinematics of particles.

#### Dedication

1h 30m Large group + 1h 30m Medium group + 1h 30m Laboratory classes + 6h 18m Self StudyIn this chapter the main aspects of nonlinear kinematics of deformation of a continuum medium are introduced. Contents: - Deformation gradient tensor - Material and spatial gradient of displacements tensors - Green-Lagrange and Almansi strain tensors - Volumetric Strain - Variation of the area - Polar decomposition of the deformation gradient tensor - Stretching - Variation of angles - Spatial gradient of velocity tensor - Deformation rate and rotation rate tensors - Material time derivative of different tensors Problems

#### Specific Objectives

The aim is to introduce the main concepts and tension associated with the deformation of a continuous medium.

#### Dedication

2h Large group + 2h Medium group + 3h Laboratory classes + 9h 48m Self StudyThis topic hypothesis under infinitesimal strains are introduced and the corresponding simplified expressions used in the context of infinitesimal deformations are obtained. The concept of compatibility equations for the infinitesimal strain tensor is also introduced. Contents: - Assumptions of the theory of infinitesimal deformations - Tensor displacement gradient - Infinitesimal strain tensor - Variation of infinitesimal volume - Polar decomposition - Stretching - Variation of angles - Matrix notation - Compatibility Equations Problems

#### Specific Objectives

The main objectives are to introduce the hypothesis of infinitesimal deformation theory and particularizing tensors introduced in the nonlinear case, the case of infinitesimal deformations.

#### Dedication

2h Large group + 2h Medium group + 1h Laboratory classes + 7h Self StudyThis topic describes the concepts of forces and stresses on a continuous medium. Contents: - Body forces and surface forces - Cauchy's theorems - Cauchy stress tensors and first Piola-Kirchhoff stress tensor - Piola transformation and Piola identity - Kirchhoff and second Piola-Kirchhoff stress tensors

#### Specific Objectives

The main objectives are to introduce the concepts of forces and stress tensors.

#### Dedication

2h Large group + 2h 48m Self StudyThis topic describes the basic principles of conservation / balance of a continuous medium are introduced, global / local shape and material / spatial. The ultimate goal is to obtain the governing equations of a problem of continuum mechanics. Contents: - Basic equations of conservation / balance - Conservation of Mass - Convective flux of a property - Lemma of Reynols - Reynolds Transport Theorem - Balance of momentum - Balance of moment of momentum - Thermodynamics - First law of thermodynamics. Energy Balance - Second Law of Thermodynamics - Thermodynamic Processes - Governing Equations - Uncoupled mechanical and thermal problems Problems

#### Specific Objectives

The main objectives are to introduce the basic principles of conservation / balance and get in space and equipment locally, the governing equations of a problem in continuum mechanics.

#### Dedication

2h Large group + 4h Medium group + 3h Laboratory classes + 12h 36m Self StudyThis topic describes the main concepts of linear elasticity are presented, leading to pose and solve the linear elastic problems. Contents: - Linear elastic model - Linear Elastic Problem - Solution of linear elastic problem - Orthogonal curvilinear coordinates Problems

#### Specific Objectives

The main objectives are to introduce the constitutive equations for linear elastic model and come to formulate and solve the linear elastic problem.

#### Dedication

2h Large group + 3h Medium group + 2h Laboratory classes + 9h 48m Self StudyThis chapter introduces additional concepts needed to formulate a generic problem in fluid mechanics. Contents: - Introduction - Constitutive Equations - Governing Equations

#### Specific Objectives

The main objective is to introduce the constitutive equations for a fluid mechanics problem.

#### Dedication

1h Large group + 1h 24m Self StudyIn this chapter, the constitutive equation for a Newtonian fluid is introduced and the formulation and solution of a fluid dynamics problem for Newtonian fluids is shown. content: - Constitutive equation - Governing Equations - Boundary Conditions - Orthogonal curvilinear coordinates Problem

#### Specific Objectives

The main objectives are to introduce the constitutive equations for a Newtonian fluid and get to pose and solve a problem in fluid mechanics for a Newtonian fluid.

#### Dedication

2h Large group + 1h 30m Medium group + 2h Laboratory classes + 7h 42m Self Study