Mathematical Modeling in Continuum Mechanics

by Roger Temam , Alain Miranville

• ISBN13: 9780521617239
• ISBN10: 0521617235
• Edition: 2nd
• Format: Paperback
• Copyright: 2005-06-20
• Publisher: Cambridge University Press

Summary

Temam and Miranville present core topics within the general themes of fluid and solid mechanics. The brisk style allows the text to cover a wide range of topics including viscous flow, magnetohydrodynamics, atmospheric flows, shock equations, turbulence, nonlinear solid mechanics, solitons, and the nonlinear Schr_dinger equation. This second edition will be a unique resource for those studying continuum mechanics at the advanced undergraduate and beginning graduate level whether in engineering, mathematics, physics or the applied sciences. Exercises and hints for solutions have been added to the majority of chapters, and the final part on solid mechanics has been substantially expanded. These additions have now made it appropriate for use as a textbook, but it also remains an ideal reference book for students and anyone interested in continuum mechanics.

Table of Contents

Preface	ix
A few words about notations	xi
PART I FUNDAMENTAL CONCEPTS IN CONTINUUM MECHANICS
1 Describing the motion of a system: geometry and kinematics	3	(21)
1.1 Deformations	3	(3)
1.2 Motion and its observation (kinematics)	6	(4)
1.3 Description of the motion of a system: Eulerian and Lagrangian derivatives	10	(3)
1.4 Velocity field of a rigid body: helicoidal vector fields	13	(5)
1.5 Differentiation of a volume integral depending on a parameter	18	(6)
2 The fundamental law of dynamics	24	(18)
2.1 The concept of mass	24	(6)
2.2 Forces	30	(2)
2.3 The fundamental law of dynamics and its first consequences	32	(2)
2.4 Application to systems of material points and to rigid bodies	34	(4)
2.5 Galilean frames: the fundamental law of dynamics expressed in a non-Galilean frame	38	(4)
3 The Cauchy stress tensor and the Piola-Kirchhoff tensor. Applications	42	(15)
3.1 Hypotheses on the cohesion forces	42	(3)
3.2 The Cauchy stress tensor	45	(3)
3.3 General equations of motion	48	(2)
3.4 Symmetry of the stress tensor	50	(2)
3.5 The Piola-Kirchhoff tensor	52	(5)
4 Real and virtual powers	57	(13)
4.1 Study of a system of material points	57	(4)
4.2 General material systems: rigidifying velocities	61	(2)
4.3 Virtual power of the cohesion forces: the general case	63	(4)
4.4 Real power: the kinetic energy theorem	67	(3)
5 Deformation tensor, deformation rate tensor, constitutive laws	70	(20)
5.1 Further properties of deformations	70	(5)
5.2 The deformation rate tensor	75	(2)
5.3 Introduction to rheology: the constitutive laws	77	(10)
5.4 Appendix. Change of variable in a surface integral	87	(3)
6 Energy equations and shock equations	90	(13)
6.1 Heat and energy	90	(5)
6.2 Shocks and the Rankine-Hugoniot relations	95	(8)
PART II PHYSICS OF FLUIDS
7 General properties of Newtonian fluids	103	(13)
7.1 General equations of fluid mechanics	103	(6)
7.2 Statics of fluids	109	(5)
7.3 Remark on the energy of a fluid	114	(2)
8 Flows of inviscid fluids	116	(21)
8.1 General theorems	116	(4)
8.2 Plane irrotational flows	120	(10)
8.3 Transsonic flows	130	(4)
8.4 Linear accoustics	134	(3)
9 Viscous fluids and thermohydraulics	137	(21)
9.1 Equations of viscous incompressible fluids	137	(1)
9.2 Simple flows of viscous incompressible fluids	138	(6)
9.3 Thermohydraulics	144	(2)
9.4 Equations in nondimensional form: similarities	146	(2)
9.5 Notions of stability and turbulence	148	(4)
9.6 Notion of boundary layer	152	(6)
10 Magnetohydrodynamics and inertial confinement of plasmas	158	(14)
10.1 The Maxwell equations and electromagnetism	159	(4)
10.2 Magnetohydrodynamics	163	(2)
10.3 The Tokamak machine	165	(7)
11 Combustion	172	(13)
11.1 Equations for mixtures of fluids	172	(2)
11.2 Equations of chemical kinetics	174	(2)
11.3 The equations of combustion	176	(2)
11.4 Stefan-Maxwell equations	178	(3)
11.5 A simplified problem: the two-species model	181	(4)
12 Equations of the atmosphere and of the ocean	185	(16)
12.1 Preliminaries	186	(2)
12.2 Primitive equations of the atmosphere	188	(4)
12.3 Primitive equations of the ocean	192	(1)
12.4 Chemistry of the atmosphere and the ocean	193	(2)
Appendix. The differential operators in spherical coordinates	195	(6)
PART III SOLID MECHANICS
13 The general equations of linear elasticity	201	(14)
13.1 Back to the stress-strain law of linear elasticity: the elasticity coefficients of a material	201	(2)
13.2 Boundary value problems in linear elasticity: the linearization principle	203	(5)
13.3 Other equations	208	(3)
13.4 The limit of elasticity criteria	211	(4)
14 Classical problems of elastostatics	215	(20)
14.1 Longitudinal traction-compression of a cylindrical bar	215	(3)
14.2 Uniform compression of an arbitrary body	218	(1)
14.3 Equilibrium of a spherical container subjected to external and internal pressures	219	(4)
14.4 Deformation of a vertical cylindrical body under the action of its weight	223	(2)
14.5 Simple bending of a cylindrical beam	225	(4)
14.6 Torsion of cylindrical shafts	229	(4)
14.7 The Saint-Venant principle	233	(2)
15 Energy theorems, duality, and variational formulations	235	(13)
15.1 Elastic energy of a material	235	(2)
15.2 Duality - generalization	237	(3)
15.3 The energy theorems	240	(3)
15.4 Variational formulations	243	(3)
15.5 Virtual power theorem and variational formulations	246	(2)
16 Introduction to nonlinear constitutive laws and to homogenization	248	(11)
16.1 Nonlinear constitutive laws (nonlinear elasticity)	249	(2)
16.2 Nonlinear elasticity with a threshold (Henky's elastoplastic model)	251	(2)
16.3 Nonconvex energy functions	253	(2)
16.4 Composite materials: the problem of homogenization	255	(4)
17 Nonlinear elasticity and an application to biomechanics	259	(12)
17.1 The equations of nonlinear elasticity	259	(3)
17.2 Boundary conditions - boundary value problems	262	(2)
17.3 Hyperelastic materials	264	(2)
17.4 Hyperelastic materials in biomechanics	266	(5)
PART IV INTRODUCTION TO WAVE PHENOMENA
18 Linear wave equations in mechanics	271	(18)
18.1 Returning to the equations of linear acoustics and of linear elasticity	271	(4)
18.2 Solution of the one-dimensional wave equation	275	(1)
18.3 Normal modes	276	(5)
18.4 Solution of the wave equation	281	(4)
18.5 Superposition of waves, beats, and packets of waves	285	(4)
19 The soliton equation: the Korteweg-de Vries equation	289	(14)
19.1 Water-wave equations	290	(2)
19.2 Simplified form of the water-wave equations	292	(3)
19.3 The Korteweg-de Vries equation	295	(4)
19.4 The soliton solutions of the KdV equation	299	(4)
20 The nonlinear Schrödinger equation	303	(16)
20.1 Maxwell equations for polarized media	304	(2)
20.2 Equations of the electric field: the linear case	306	(3)
20.3 General case	309	(4)
20.4 The nonlinear Schrödinger equation	313	(3)
20.5 Soliton solutions of the NLS equation	316	(3)
Appendix. The partial differential equations of mechanics	319	(2)
Hints for the exercises	321	(11)
References	332	(5)
Index	337