Marcelo Epstein

The Elements of Continuum Biomechanics (eBook, ePUB)

• Format: ePub

Produktbeschreibung

An appealing and engaging introduction to Continuum Mechanics in Biosciences This book presents the elements of Continuum Mechanics to people interested in applications to biological systems. It is divided into two parts, the first of which introduces the basic concepts within a strictly one-dimensional spatial context. This policy has been adopted so as to allow the newcomer to Continuum Mechanics to appreciate how the theory can be applied to important issues in Biomechanics from the very beginning. These include mechanical and thermodynamical balance, materials with fading memory and chemically reacting mixtures. In the second part of the book, the fully fledged three-dimensional theory is presented and applied to hyperelasticity of soft tissue, and to theories of remodeling, aging and growth. The book closes with a chapter devoted to Finite Element analysis. These and other topics are illustrated with case studies motivated by biomedical applications, such as vibration of air in the air canal, hyperthermia treatment of tumours, striated muscle memory, biphasic model of cartilage and adaptive elasticity of bone. The book offers a challenging and appealing introduction to Continuum Mechanics for students and researchers of biomechanics, and other engineering and scientific disciplines. Key features: * Explains continuum mechanics using examples from biomechanics for a uniquely accessible introduction to the topic * Moves from foundation topics, such as kinematics and balance laws, to more advanced areas such as theories of growth and the finite element method.. * Transition from a one-dimensional approach to the general theory gives the book broad coverage, providing a clear introduction for beginners new to the topic, as well as an excellent foundation for those considering moving to more advanced application

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Produktdetails

• Verlag: John Wiley & Sons
• Seitenzahl: 392
• Erscheinungstermin: 13. Juli 2012
• Englisch
• ISBN-13: 9781118361009
• Artikelnr.: 37337657

Autorenporträt

Marcelo Epstein is Professor of Mechanical and Manufacturing Engineering and Adjunct Professor of Kinesiology at the University of Calgary. He is a fellow of the American Academy of Mechanics, recipient of the CANCAM prize and University Professor of Rational Mechanics.

Inhaltsangabe

Dedication ix Preface xi Part One A one-dimensional context 1 1 Material
bodies and kinematics 3 1.1 Introduction 3 1.2 Continuous vs. discrete 6
1.3 Configurations and deformations 9 1.4 The deformation gradient 14 1.5
Change of reference configuration 15 1.6 Strain 16 1.7 Displacement 18 1.8
Motion 19 1.9 The Lagrangian and Eulerian representations of fields 22 1.10
The material derivative 24 1.11 The rate of deformation 26 1.12 The cross
section 27 2 Balance laws 29 2.1 Introduction 29 2.2 The generic Lagrangian
balance equation 30 2.2.1 Extensive properties 30 2.2.2 The balance
equation 31 2.3 The generic Eulerian balance equation 35 2.4 Case study:
Blood flow as a traffic problem 37 2.5 Case study: Diffusion of a pollutant
39 2.5.1 Derivation of the diffusion equation 39 2.5.2 A discrete diffusion
model 41 2.6 The thermo-mechanical balance laws 42 2.6.1 Conservation of
mass 42 2.6.2 Balance of (linear) momentum 43 2.6.3 The concept of stress
44 2.7 Case study: Vibration of air in the ear canal 45 2.8 Kinetic energy
50 2.9 The thermodynamical balance laws 55 2.9.1 Introduction 55 2.9.2
Balance of energy 56 2.9.3 The entropy inequality 58 2.10 Summary of
balance equations 59 2.11 Case study: Bioheat transfer and malignant
hyperthermia 61 3 Constitutive equations 69 3.1 Introduction 69 3.2 The
principle of determinism 70 3.3 The principle of equipresence 72 3.4 The
principle of material frame-indifference 72 3.5 The principle of
dissipation 75 3.6 Case study: Memory aspects of striated muscle 79 3.7
Case study: The thermo(visco)elastic effect in skeletal muscle 85 3.8 The
theory of materials with fading memory 90 3.8.1 Groundwork 90 3.8.2 Fading
memory 93 3.8.3 Stress relaxation 95 3.8.4 Finite linear viscoelasticity 96
4 Mixture theory 99 4.1 Introduction 99 4.2 The basic tenets of mixture
theory 99 4.3 Mass balance 101 4.4 Balance of linear momentum 102 4.4.1
Constituent balances 102 4.4.2 Mixture balance 103 4.5 Case study: Confined
compression of articular cartilage 106 4.5.1 Introduction 106 4.5.2
Empirical facts 107 4.5.3 Field equations 108 4.5.4 Nonlinear creep 112
4.5.5 Hysteresis 115 4.5.6 The linearized theory 115 4.6 Energy balance 121
4.6.1 Constituent balances 121 4.6.2 Mixture balance 123 4.7 The entropy
inequality 124 4.8 Chemical aspects 125 4.8.1 Stoichiometry 125 4.8.2
Thermodynamics of homogeneous systems 129 4.8.3 Enthalpy and heats of
reaction 131 4.8.4 The meaning of the Helmholtz free energy 134 4.8.5
Homogeneous mixtures 135 4.8.6 Equilibrium and stability 137 4.8.7 The
Gibbs free energy as a Legendre transformation 138 4.9 Ideal mixtures 140
4.9.1 The ideal gas paradigm 140 4.9.2 Mixtures of ideal gases 141 4.9.3
Other ideal mixtures 145 4.10 Case study: Bone as a chemically reacting
mixture 145 Part Two Toward three spatial dimensions 151 5 Geometry and
kinematics 153 5.1 Introduction 153 5.2 Vectors and tensors 153 5.2.1 Why
Linear Algebra? 153 5.2.2 Vector spaces 155 5.2.3 Linear independence and
dimension 156 5.2.4 Linear operators, tensors, matrices 158 5.2.5
Inner-product spaces 161 5.2.6 The reciprocal basis 162 5.3 Geometry of
classical space-time 164 5.3.1 A shortcut 164 5.3.2 R3 as a vector space
165 5.3.3 E3 as an affine space 166 5.3.4 Frames 166 5.3.5 Space-time and
observers 169 5.3.6 Fields and the divergence theorem 170 5.4 Eigenvalues
and eigenvectors 176 5.4.1 General concepts 176 5.4.2 More on principal
invariants 178 5.4.3 The symmetric case 180 5.4.4 Functions of symmetric
matrices 182 5.5 Kinematics 183 5.5.1 Material bodies 183 5.5.2
Configurations, deformations, motions 183 5.5.3 The deformation gradient
185 5.5.4 Local configurations 187 5.5.5 A word on notation 187 5.5.6
Decomposition of the deformation gradient 188 5.5.7 Measures of strain 193
5.5.8 The displacement field and its gradient 194 5.5.9 The geometrically
linearized theory 196 5.5.10 Volume and area 198 5.5.11 The material
derivative 201 5.5.12 Change of reference configuration 203 5.5.13 The
velocity gradient 204 6 Balance laws and constitutive equations 207 6.1
Preliminary notions 207 6.1.1 Extensive properties 207 6.1.2 Transport
theorem 208 6.2 Balance equations 210 6.2.1 The general balance equation
210 6.2.2 The balance equations of Continuum Mechanics 214 6.3 Constitutive
theory 223 6.3.1 Introduction and scope 223 6.3.2 The principle of material
frame-indifference and its applications 224 6.3.3 The principle of
thermodynamic consistency and its applications 228 6.4 Material symmetries
231 6.4.1 Symmetries and groups 231 6.4.2 The material symmetry group 232
6.5 Case study: The elasticity of soft tissue 236 6.5.1 Introduction 236
6.5.2 Elasticity and hyperelasticity 236 6.5.3 Incompressibility 238 6.5.4
Isotropy 241 6.5.5 Examples 242 6.6 Remarks on initial and boundary-value
problems 248 7 Remodelling, aging, growth 255 7.1 Introduction 255 7.2
Discrete and semi-discrete models 262 7.2.1 Challenges 262 7.2.2 Cellular
automata in tumour growth 264 7.2.3 A direct model of bone remodelling 266
7.3 The continuum approach 268 7.3.1 Introduction 268 7.3.2 The balance
equations of volumetric growth and remodelling 269 7.4 Case study: tumour
growth 273 7.5 Case study: Adaptive elasticity of bone 277 7.5.1 The
isothermal quasi-static case 281 7.6 Anelasticity 282 7.6.1 Introduction
282 7.6.2 The notion of material isomorphism 283 7.6.3 Non-uniqueness of
material isomorphisms 286 7.6.4 Uniformity and homogeneity 287 7.6.5
Anelastic response 289 7.6.6 Anelastic evolution 290 7.6.7 The Eshelby
stress 296 7.7 Case study: Exercise and growth 301 7.7.1 Introduction 301
7.7.2 Checking the proposed evolution law 301 7.7.3 A numerical example 303
7.8 Case study: Bone remodelling and Wolff's law 305 8 Principles of the
Finite Element Method 309 8.1 Introductory remarks 309 8.2 Discretization
procedures 310 8.2.1 Brief review of the method of finite differences 310
8.2.2 Non-traditional methods 313 8.3 The Calculus of Variations 313 8.3.1
Introduction 313 8.3.2 The simplest problem of the Calculus of Variations
315 8.3.3 The case of several unknown functions 321 8.3.4 Essential and
natural boundary conditions 323 8.3.5 The case of higher derivatives 326
8.3.6 Variational problems with more than one independent variable 329 8.4
Rayleigh, Ritz, Galerkin 330 8.4.1 Introduction 330 8.4.2 The method of
Rayleigh and Ritz 332 8.4.3 The methods of weighted residuals 334 8.4.4
Approximating differential equations by Galerkin's method 336 8.5 The
finite element idea 341 8.5.1 Introduction 341 8.5.2 A piecewise linear
basis 343 8.5.3 Automating the procedure 348 8.6 The FEM in Solid Mechanics
353 8.6.1 The Principle of Virtual Work 353 8.6.2 The principle of
stationary potential energy 358 8.7 Finite element implementation 359 8.7.1
General considerations 359 8.7.2 An ideal element 360 8.7.3 Meshing,
insertion maps and the isoparametric idea 362 8.7.4 The contractibility
condition and its consequences 363 8.7.5 The element IVW routine 366 8.7.6
The element EVW routine 368 8.7.7 Assembly and solution 369