Mathematical Models of Solids and Fluids: a short introduction

BookMathematical Models of Solids and Fluids: a short introduction

Mathematical Models of Solids and Fluids: a short introduction


September 15th, 2021

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This textbook provides an introduction to continuum mechanics, which models the behaviour of elastic solids and viscous fluids. It assumes only a working knowledge of classical mechanics, linear algebra and multivariable calculus. Every chapter contains exercises, with detailed solutions. The book is aimed at undergraduate students from scientific disciplines. Mathematics students will find examples of applications involving techniques from different branches of mathematics, such as geometry and differential equations. Physics students will find a gentle introduction to the notions of stress and material laws. Engineering students will find examples of classic exactly-solvable problems. The emphasis is on the thorough derivation of exact solutions, but estimates of the relevant orders of magnitude are provided.

Author Information

Pascal Grange is Assistant Professor in the Department of Physics at Xi'an Jiaotong-Liverpool University.

Table of Contents

Table of Contents
Section TitlePagePrice
1 Description of solids and fluids11
1.1 Tensors and continuum mechanics: motivations and history11
1.1.1 Continuous media: solids and fluids11
1.1.2 Systems of coordinates12
1.2 Notations and conventions12
1.2.1 Notation: the summation rule over repeated indices13
1.2.2 Scalar product, Kronecker symbol14
1.3 Changes of orthonormal basis15
1.3.1 Transformation of the components of vectors15
1.3.2 Transformation of the matrix of an endomorphism under a change of orthonormal basis18
1.4 Generalisation: definition of tensors20
1.5 Exercises22
2 Kinematics of continuous media25
2.1 Lagrange description of flows25
2.2 Transformation of tangent vectors27
2.3 Application to deformations31
2.4 Exercises35
3 Dynamics of continuous media37
3.1 Body forces, surface forces, Cauchy postulate37
3.2 Stokes' theorem39
3.3 Surface forces in the normal vector40
3.3.1 Balance equation of a thin cylinder40
3.3.2 Balance equation of an infinitesimal tetrahedron41
3.4 Balance equations for a continuous medium44
3.4.1 Forces sum to zero at static equilibrium45
3.4.2 Moments of forces sum to zero at static equilibrium45
3.5 Exercises48
4 Boundary conditions50
4.1 The hydrostatic pressure50
4.2 Boundary conditions52
4.3 Statically admissible stress tensors54
4.4 Exercises56
5 Linear elasticity: material laws59
5.1 Hooke's law (for a cylinder)59
5.2 Small deformations61
5.3 Strain as a function of stress64
5.4 Stress as a function of strain66
5.5 Exercises68
6 Elasticity, elementary problems70
6.1 The Navier equations of linear elasticity70
6.2 Solution for a spherical shell71
6.2.1 Explicit form of the Navier equations with spherical symmetry71
6.2.2 Determination of the integration constants75
6.3 Exercises80
7 Viscous fluids83
7.1 Description of continuous media83
7.2 Description of fluids84
7.2.1 The Euler description of fluids84
7.2.2 Conservation of mass (the continuity equation)86
7.2.3 Acceleration of a particle of fluid89
7.3 Viscous fluids92
7.3.1 Thought experiment on friction92
7.3.2 Model: material law for Newtonian fluids92
7.3.3 Boundary conditions on the velocity field for Newtonian fluids93
7.4 Exercises95
8 Viscosimetric flows99
8.1 Navier–Stokes equations for Poiseuille flow99
8.1.1 Functional form of the velocity field102
8.1.2 Incompressibility102
8.1.3 Explicit form of each term in the Navier–Stokes equations102
8.1.4 Solution of the equations of motion103
Separation of variables104
Boundary conditions106
8.2 The Poiseuille law108
8.2.1 Derivation of the flow rate108
8.2.2 The Poiseuille flow is viscosimetric108
8.3 Exercises112
9 Cylindrical Couette flow114
9.1 Couette flow114
9.2 Solution of the Navier–Stokes equations115
9.2.1 Cylindrical coordinates115
9.2.2 Navier–Stokes equations in cylindrical coordinates117
9.2.3 Integration of the Navier–Stokes equations118
9.3 Application: Couette flow as a viscosimeter120
10 Solutions to the exercises124
Exercises in Chapter 1124
Exercises in Chapter 2130
Exercises in Chapter 3133
Exercises in Chapter 4136
Exercises in Chapter 5142
Exercises in Chapter 6144
Exercises in Chapter 7152
Exercises in Chapter 8157
Exercises in Chapter 9162
Glossary of terms167
Physical constants,orders of magnitude169