Mathematical Models of Solids and Fluids

171 Pages·2021·6.816 MB·English

by  Pascal Grange

Mathematical Models
Mathematical Models
of Solids and Fluids,
a short introduction
of Solids and Fluids
Pascal Grange
Department of Physics, Xi’an Jiaotong-Liverpool University
A Short Introduction
Pascal Grange

Mathematical Models
of Solids and Fluids,
a short introduction
Pascal Grange
Department of Physics, Xi’an Jiaotong-Liverpool University
XJTLU IMPRINT
First published 2021 by
XJTLU Imprint,
a collaboration between
Xi’an Jiaotong-Liverpool University
and Liverpool University Press.
Copyright © 2021 Pascal Grange
Pascal Grange has asserted the right to be identified as the author of this book in
accordance with the Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this book may be reproduced, stored in a
retrieval system, or transmitted, in any form or by any means, electronic,
mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher.
British Library Cataloguing-in-Publication data
A British Library CIP record is available
eISBN 978-1-80085-557-1
Typeset by Carnegie Book Production, Lancaster
Contents
1 Description of solids and fluids 4
1.1 Tensors and continuum mechanics: motivations and history 4
1.1.1 Continuous media: solids and fluids 4
1.1.2 Systems of coordinates 5
1.2 Notations and conventions 5
1.2.1 Notation: the summation rule over repeated indices 6
1.2.2 Scalar product, Kronecker symbol 7
1.3 Changes of orthonormal basis 8
1.3.1 Transformation of the components of vectors 8
1.3.2 Transformation of the matrix of an endomorphism
under a change of orthonormal basis 12
1.4 Generalisation: definition of tensors 13
1.5 Exercises 15
2 Kinematics of continuous media 18
2.1 Lagrange description of flows 18
2.2 Transformation of tangent vectors 21
2.3 Application to deformations 24
2.4 Exercises 28
3 Dynamics of continuous media 30
3.1 Body forces, surface forces, Cauchy postulate 30
3.2 Stokes’ theorem 32
3.3 Surface forces in the normal vector 33
3.3.1 Balance equation of a thin cylinder 33
3.3.2 Balance equation of an infinitesimal tetrahedron 34
3.4 Balance equations for a continuous medium 38
3.4.1 Forces sum to zero at static equilibrium 38
3.4.2 Moments of forces sum to zero at static equilibrium 38
3.5 Exercises 41
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4 Boundary conditions 43
4.1 The hydrostatic pressure 43
4.2 Boundary conditions 45
4.3 Statically admissible stress tensors 47
4.4 Exercises 49
5 Linear elasticity: material laws 52
5.1 Hooke’s law (for a cylinder) 52
5.2 Small deformations 55
5.3 Strain as a function of stress 57
5.4 Stress as a function of strain 59
5.5 Exercises 61
6 Elasticity, elementary problems 63
6.1 The Navier equations of linear elasticity 63
6.2 Solution for a spherical shell 64
6.2.1 Explicit form of the Navier equations
with spherical symmetry 64
6.2.2 Determination of the integration constants 68
6.3 Exercises 73
7 Viscous fluids 76
7.1 Description of continuous media 76
7.2 Description of fluids 77
7.2.1 The Euler description of fluids 78
7.2.2 Conservation of mass (the continuity equation) 79
7.2.3 Acceleration of a particle of fluid 82
7.3 Viscous fluids 85
7.3.1 Thought experiment on friction 85
7.3.2 Model: material law for Newtonian fluids 85
7.3.3 Boundary conditions on the velocity field for New-
tonian fluids 87
7.4 Exercises 88
8 Viscosimetric flows 92
8.1 Navier–Stokes equations for Poiseuille flow 92
8.1.1 Functional form of the velocity field 95
8.1.2 Incompressibility 95
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8.1.3 Explicit form of each term in the Navier–Stokes
equations 95
8.1.4 Solution of the equations of motion 97
Separation of variables 97
Boundary conditions 99
8.2 The Poiseuille law 101
8.2.1 Derivation of the flow rate 101
8.2.2 The Poiseuille flow is viscosimetric 101
8.3 Exercises 105
9 Cylindrical Couette flow 107
9.1 Couette flow 107
9.2 Solution of the Navier–Stokes equations 108
9.2.1 Cylindrical coordinates 108
9.2.2 Navier–Stokes equations in cylindrical coordinates 110
9.2.3 Integration of the Navier–Stokes equations 111
9.3 Application: Couette flow as a viscosimeter 113
9.4 Exercises 115
10 Solutions to the exercises 117
10.1 Exercises in Chapter 1 117
10.2 Exercises in Chapter 2 123
10.3 Exercises in Chapter 3 126
10.4 Exercises in Chapter 4 129
10.5 Exercises in Chapter 5 135
10.6 Exercises in Chapter 6 137
10.7 Exercises in Chapter 7 145
10.8 Exercises in Chapter 8 150
10.9 Exercises in Chapter 9 155
Introduction
Models of solids and fluids have been developed gradually, sometimes
with very close ties to mathematical research, since the 17th century.
The notation and vocabulary we use nowadays are often substantially
differerent from those used in the original works (for instance linear alge-
bra as we know it was formalised in the second half of the 19th century).
Thefirstfourchaptersofthepresentbookintroducegeneralconcepts
toformulatethelawsofclassicalmechanicsinacontinuousmedium. The
following historical miletsones are described in today’s terminology:
17th century. Hydrostatic pressure and atmospheric pressure are
understood and measured. Hydrostatic pressure is reviewed in Chapter
4 as an important example of the integration of the static-equilibrium
equations with boundary conditions.
Late 17th century, early 18th century. The deformation of a
cylinder under traction is understood to be proportional to the applied
force (“Ut tensio sic vis”, the first formulation of Hooke’s law). The no-
tion of elasticity modulus (Young’s modulus in today’s teminology, see
Chapter 6) was gradually formalised in the 18th century.
1755. The equation of motion for perfect fluids is written (the Euler
equation, reviewed in Chapter 7 before attempting a model of viscous
fluids).
1773. Franklin’s oil-drop experiment (see Exercises in Chapter 1)
paves the way for the description of matter at molecular scales, which
will develop gradually in the 19th and 20th centuries.
Circa 1820. The notion of viscosity is introduced, gradually leading
to the equation of motion for viscous fluids (the Navier–Stokes equation,
derived in Chapter 7).
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Circa 1840. The Poiseuille law, relating the delivery of a cylindrical
pipe to the fourth power of its diameter (see Chapter 8).
The guiding principle is to keep the mathematical apparatus mini-
mal and the presentation self-contained (assuming a working knowledge
of calculus, linear algebra and classical mechanics). For example, all
problems are solved in Cartesian coordinates (cylindrical coordinates are
introduced in a self-contained way in Chapter 9). Extremely important
notions and methods are left out, including dimensional analysis, virtual
work of forces, Mohr circles, matched asymptotic expansions, turbulence
andphasetransitions. Moreover,theeffectoftemperatureonthesystems
is disregarded. The bibliography is kept minimal, but an effort has been
made to give precise references to a few of the excellent and much more
extensive available books: they are gathered at the end of each chap-
ter, after the exercises. The very comprehensive introduction by Jean
Salen¸con (Salen¸con, 2012) has had a particularly strong influence on the
presentation of this book. Much emphasis is put on detailed derivations
of exact results. However, estimates of orders of magnitude based on
educated guesses are sometimes proposed, in the spirit of Franklin’s oil-
drop experiment (see Chapter 8 for estimates of the viscosity of water
and volcano lava). Indeed, a model can be defined in Feynman’s words
as our best guess. Chapters 38 to 41 of the Feynman Lectures on Physics
(Volume II of (Feynman, 1964)) provide a more intuitive presentation of
models of solids and fluids.
The present introduction, written by a non-specialist, was devel-
oped with the participation of several cohorts of students in the BSc
Applied Mathematics programme at Xi’an Jiaotong-Liverpool Univer-
sity (XJTLU). The content corresponds to a one-semester module in the
final year of the programme. It is usually taught over 14 weeks, with
two hours of lectures and two hours of tutorials per week. The emphasis
is put on working out in detail a few exactly solvable cases. On a less
formal note, a few examples of Fermi questions (estimates of orders of
magnitude) are given. The two approaches are shown to be complemen-
tary when we discuss the assumption of negligible gravity in problems
of linear elasticity, and when we estimate the viscosity of volcano lava.
Detailedsolutionstotheexercisesarepresentedattheendofthebookto
make it more suitable for self-study. However, there are very few figures
inordertoengagethereadersintoillustratingtheproblemsbydrawings.
Someoftheexercisesaredrawnfrompastexampapers,whichgreatly
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benefitted from the review of the moderators at XJTLU and the Uni-
versity of Liverpool: Stephen James Shaw, Tai-jun Chen and Natalia
Movchan. Moreover,substantialcontributionsbyXJTLUstudentsZuqian
Huang, Jin Yan, Ran Bi, Zhiwei Cheng, Qier Yu, Henger Li, Chang Liu
and Chenxia Gu are gratefully acknowledged.