Book contents
- Frontmatter
- Contents
- Preface
- A few words about notations
- PART I FUNDAMENTAL CONCEPTS IN CONTINUUM MECHANICS
- 1 Describing the motion of a system: geometry and kinematics
- 2 The fundamental law of dynamics
- 3 The Cauchy stress tensor and the Piola-Kirchhoff tensor. Applications
- 4 Real and virtual powers
- 5 Deformation tensor, deformation rate tensor, constitutive laws
- 6 Energy equations and shock equations
- PART II PHYSICS OF FLUIDS
- PART III SOLID MECHANICS
- PART IV INTRODUCTION TO WAVE PHENOMENA
- Appendix: The partial differential equations of mechanics
- Hints for the exercises
- References
- Index
1 - Describing the motion of a system: geometry and kinematics
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Preface
- A few words about notations
- PART I FUNDAMENTAL CONCEPTS IN CONTINUUM MECHANICS
- 1 Describing the motion of a system: geometry and kinematics
- 2 The fundamental law of dynamics
- 3 The Cauchy stress tensor and the Piola-Kirchhoff tensor. Applications
- 4 Real and virtual powers
- 5 Deformation tensor, deformation rate tensor, constitutive laws
- 6 Energy equations and shock equations
- PART II PHYSICS OF FLUIDS
- PART III SOLID MECHANICS
- PART IV INTRODUCTION TO WAVE PHENOMENA
- Appendix: The partial differential equations of mechanics
- Hints for the exercises
- References
- Index
Summary
Deformations
The purpose of mechanics is to study and describe the motion of material systems. The language of mechanics is very similar to that of set theory in mathematics: we are interested in material bodies or systems, which are made of material points or matter particles. A material system fills some part (a subset) of the ambient space (ℝ3), and the position of a material point is given by a point in ℝ3; a part of a material system is called a subsystem.
We will almost exclusively consider material bodies that fill a domain (i.e., a connected open set) of the space. We will not study the mechanically important cases of thin bodies that can be modeled as a surface (plates, shells) or as a line (beams, cables). The modeling of the motion of such systems necessitates hypotheses that are very similar to the ones we will present in this book, but we will not consider these cases here.
A material system fills a domain Ω0 in ℝ3 at a given time t0. After deformation (think, for example, of a fluid or a tennis ball), the system fills a domain Ω in ℝ3. A material point, whose initial position is given by the point a ∈ Ω0, will be, after transformation, at the point x ∈ Ω.
- Type
- Chapter
- Information
- Mathematical Modeling in Continuum Mechanics , pp. 3 - 23Publisher: Cambridge University PressPrint publication year: 2005